PI and PID controller tuning rules for time delay processes: a summary
Technical Report AOD-00-01, Edition 1
A. O’Dwyer,
School of Control Systems and Electrical Engineering,
Dublin Institute of Technology, Kevin St., Dublin 8, Ireland.
15 May 2000
Phone: 353-1-4024875
Fax: 353-1-4024992
e-mail: aodwyer@dit.ie
Abstract: The ability of proportional integral (PI) and proportional integral derivative (PID) controllers to
compensate many practical industrial processes has led to their wide acceptance in industrial applications. The
requirement to choose either two or three controller parameters is perhaps most easily done using tuning rules. A
summary of tuning rules for the PI and PID control of single input, single output (SISO) processes with time
delay are provided in this report. Inevitably, this report is a work in progress and will be added to and extended
regularly.
Keywords: PI, PID, tuning rules, time delay.
1. Introduction
The ability of PI and PID controllers to compensate most practical industrial processes has led to their
wide acceptance in industrial applications. Koivo and Tanttu [1], for example, suggest that there are perhaps 510% of control loops that cannot be controlled by SISO PI or PID controllers; in particular, these controllers
perform well for processes with benign dynamics and modest performance requirements [2, 3]. It has been stated
that 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers [4] and that, in
process control applications, more than 95% of the controllers are of PID type [3]. The PI or PID controller
implementation has been recommended for the control of processes of low to medium order, with small time
delays, when parameter setting must be done using tuning rules and when controller synthesis is performed
either once or more often [5]. However, Ender [6] states that, in his testing of thousands of control loops in
hundreds of plants, it has been found that more than 30% of installed controllers are operating in manual mode
and 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. the
automatic controllers are poorly tuned); this is rather sobering, considering the wealth of information available in
the literature for determining controller parameters automatically. It is true that this information is scattered
throughout papers and books; the purpose of this paper is to bring together in summary form the tuning rules for
PI and PID controllers that have been developed to compensate SISO processes with time delay. Tuning rules for
the variations that have been proposed in the ‘ideal’ PI and PID controller structure are included. Considerable
variations in the ideal PID controller structure, in particular, are encountered; these variations are explored in
more detail in Section 2.
2. PID controller structures
The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domain
as follows:
U( s) = G c (s) E (s)
(1)
with
G c (s) = Kc (1 +
1
+ Tds)
Ts
i
(2)
and with Kc = proportional gain, Ti = integral time constant and Td = derivative time constant. If Ti = ∞
and Td = 0 (i.e. P control), then it is clear that the closed loop measured value, y, will always be less than the
desired value, r (for processes without an integrator term, as a positive error is necessary to keep the measured
value constant, and less than the desired value). The introduction of integral action facilitates the achievement of
equality between the measured value and the desired value, as a constant error produces an increasing controller
output. The introduction of derivative action means that changes in the desired value may be anticipated, and
thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PID
controller allows contributions from present controller inputs, past controller inputs and future controller inputs.
Many tuning rules have been defined for the ideal PI and PID structures. Tuning rules have also been
defined for other PI and PID structures, as detailed in Section 4.
3. Process modelling
Processes with time delay may be modelled in a variety of ways. The modelling strategy used will
influence the value of the model parameters, which will in turn affect the controller values determined from the
tuning rules. The modelling strategy used in association with each tuning rule, as described in the original
papers, is indicated in the tables. Of course, it is possible to use the tuning rules proposed by the authors with a
different modelling strategy than that proposed by the authors; applications where this occurs are not indicated
(to date). The modelling strategies are referenced as indicated. The full details of these modelling strategies are
provided in Appendix 2.
K e − sτ m
A. First order lag plus delay (FOLPD) model ( G m ( s) = m
):
1 + sTm
Method 1: Parameters obtained using the tangent and point method (Ziegler and Nichols [8], Hazebroek and
Van den Waerden [9]); Appendix 2.
Method 2: Km , τ m assumed known; Tm estimated from the open loop step response (Wolfe [12]); Appendix
2.
Method 3: Parameters obtained using an alternative tangent and point method (Murrill [13]); Appendix 2.
Method 4: Parameters obtained using the method of moments (Astrom and Hagglund [3]); Appendix 2.
Method 5: Parameters obtained from the closed loop transient response to a step input under proportional
control (Sain and Ozgen [94]); Appendix 2.
Method 6: Km , Tm , τ m assumed known.
Method 7: Parameters obtained using a least squares method in the time domain (Cheng and Hung [95]);
Appendix 2.
Method 8: Parameters obtained in the frequency domain from the ultimate gain, phase and frequency
determined using a relay in series with the closed loop system in a master feedback loop. The
model gain is obtained by the ratio of the integrals (over one period) of the process output to the
controller output. The delay and time constant are obtained from the frequency domain data
(Hwang [160]).
Method 9: Parameters obtained from the closed loop transient response to a step input under proportional
control (Hwang [2]); Appendix 2.
Method 10: Parameters obtained from two points estimated on process frequency response using a relay and
a relay in series with a delay (Tan et al. [39]); Appendix 2.
Method 11: Tm and τ m are determined from the ultimate gain and period estimated using a relay in series
with the process in closed loop; Km assumed known (Hang and Cao [112]); Appendix 2.
Method 12: Parameters are estimated using a tangent and point method (Davydov et al. [31]); Appendix 2.
Method 13: Parameters estimated from the open loop step response and its first time derivative (Tsang and
Rad [109]); Appendix 2.
Method 14: Tm and τ m estimated from Ku , Tu determined using Ziegler-Nichols ultimate cycle method;
Km estimated from the process step response (Hang et al. [35]); Appendix 2.
Method 15: Tm and τ m estimated from Ku , Tu determined using a relay autotuning method; Km estimated
from the process step response (Hang et al. [35]); Appendix 2.
Method 16: G p ( jω135 ) , ω135 and Km are determined from an experiment using a relay in series with the
0
0
process in closed loop; estimates for Tm and τ m are subsequently calculated. (Voda and
Landau [40]); Appendix 2.
Method 17: Parameter estimates back-calculated from discrete time identification method (Ferretti et al.
[161]); Appendix 2.
* Method 18: Parameter estimates calculated from process reaction curve using numerical integration
procedures (Nishikawa et al. [162]).
* Method 19: Parameter estimates determined graphically from a known higher order process (McMillan [58]
… also McMillan (1983), pp. 34-40.
* Method 20: Km estimated from the open loop step response. T90% and τ m estimated from the closed loop
step response under proportional control (Astrom and Hagglund [93]?)
Method 21: Parameters estimated from linear regression equations in the time domain (Bi et al. [46]);
Appendix 2.
Method 22: Tm and τ m estimated from relay autotuning method (Lee and Sung [163]); Km estimated
from the closed loop process step response under proportional control (Chun et al. [57]);
Appendix 2.
* Method 23: Parameters are estimated from a step response autotuning experiment – Honeywell UDC 6000
controller (Astrom et al. [30]).
Method 24: Parameters are estimated from the closed loop step response when process is in series with a
PID controller (Morilla et al. [104a]); Appendix 2.
Method 25: τm and Tm obtained from an open loop step test as follows: Tm = 1.4( t 67% − t 33% ) ,
τ m = t 67% − 1.1Tm . K m assumed known (Chen and Yang [23a]).
Method 26: τm and Tm obtained from an open loop step test as follows: Tm = 1.245 ( t 70% − t 33% ) ,
τ m = 1. 498 t 33% − 0. 498 t 70% . K m assumed known (Miluse et al. [27b]).
* Method 27: Data at the ultimate period is deduced from an open loop impulse response (Pi-Mira et al.
[97a]).
B. Non-model specific
Method 1: Parameters K u , K m , ωu are estimated from data obtained using a relay in series with the process
in closed loop and from the process step response (Kristiansson and Lennartsson [157]).
need to check how the other methods define these parameters –
C. Integral plus time delay (IPD) model ( G m ( s) =
K me− sτ m
s
)
Method 1: τ m assumed known; Km determined from the slope at start of the open loop step response
(Ziegler and Nichols [8]); Appendix 2.
Method 2: Km , τ m assumed known.
Method 3: Parameters estimated from the ultimate gain and frequency values determined from an experiment
using a relay in series with the process in closed loop (Tyreus and Luyben [75]); Appendix 2.
Method 4: Parameters are estimated from the servo or regulator closed loop transient response, under PI
control (Rotach [77]); Appendix 2.
Method 5: Parameters are estimated from the servo closed loop transient response under proportional
control (Srividya and Chidambaram [80]); Appendix 2.
Method 6: K u and Tu are estimated from estimates of the ultimate and crossover frequencies. The ultimate
frequency estimate is obtained by placing an amplitude dependent gain in series with the
process in closed loop; the crossover frequency estimate is obtained by also using an amplitude
dependent gain (Pecharroman and Pagola [165]); Appendix 2.
D. First order lag plus integral plus time delay (FOLIPD) model ( G m (s) =
Km e− sτ
)
s(1 + sTm )
m
* Method 1: Method of moments (Astrom and Hagglund [3]).
Method 2: Km , Tm , τ m assumed known.
Method 3: Parameters estimated from the open loop step response and its first and second time derivatives
(Tsang and Rad [109]); Appendix 2.
Method 4: K u and Tu are estimated from estimates of the ultimate and crossover frequencies (Pecharroman
and Pagola [165]) – as in Method 6, IPD model.
E.
Second order system plus time delay (SOSPD) model ( G m (s) =
Km e− sτ
)
s + 2ξ m Tm1s + 1 (1 + Tm1s)(1 + Tm2 s)
K m e− sτ
Tm1
2 2
m
m
,
Method 1: Km , Tm1 , Tm2 , τ m or Km , Tm1 , ξ m , τ m assumed known.
Method 2: Parameters estimated using a two-stage identification procedure involving (a) placing a relay in
series with the process in closed loop and (b) placing a proportional controller in series with the
process in closed loop (Sung et al. [139]); Appendix 2.
* Method 3: Parameters obtained in the frequency domain from the ultimate gain, phase and frequency
determined using a relay in series with the closed loop system in a master feedback loop. The
model gain is obtained by the ratio of the integrals (over one period) of the process output to the
controller output. The other parameters are obtained from the frequency domain data (Hwang
[160]).
Method 4: Tm and τ m estimated from Ku , Tu determined using a relay autotuning method; Km estimated
from the process step response (Hang et al. [35]); Appendix 2.
* Method 5: Parameter estimates back-calculated from discrete time identification method (Ferretti et al.
[161]).
Method 6: Parameteres estimated from the underdamped or overdamped transient response in open loop to a
step input (Jahanmiri and Fallahi [149]); Appendix 2.
* Method 7: Parameters estimated from a least squares time domain method (Lopez et al. [84]).
Method 8: Parameters estimated from data obtained when the process phase lag is − 900 and − 180 0 ,
respectively (Wang et al. [143]); Appendix 2.
* Method 9: Parameter estimates back-calculated from discrete time identification method (Wang and
Clements [147]).
Method 10: Km , Tm1 and τm are determined from the open loop time domain Ziegler-Nichols response
(Shinskey [16], page 151); Tm 2 assumed known.
Method 11: Parameters estimated from two points determined on process frequency response using a relay
and a relay in series with a delay (Tan et al. [39]); Appendix 2.
* Method 12: Parameter estimated back-calculated from discrete time identification method (Lopez et al. [84]).
* Method 13: Parameters estimated from a step response autotuning experiment – Honeywell UDC 6000
controller (Astrom et al. [30]).
Method 14: Tm1 = T m2 . τm and T m1 obtained from an open loop step test as follows:
Tm1 = 0. 794 ( t 70% − t 33% ) , τ m = 1.937 t 33% − 0.937 t 70% . K m assumed known (Miluse et al.
[27b]).
Method 15: K u and Tu are estimated from estimates of the ultimate and crossover frequencies (Pecharroman
and Pagola [165]) – as in Method 6, IPD model.
F. Integral squared plus time delay ( I 2PD ) model ( G m (s) =
K me − sτ m
)
s2
Method 1: Km , Tm , τ m assumed known.
G. Second order system (repeated pole) plus integral plus time delay (SOSIPD) model ( G m (s) =
Km e −s τ
m
s (1 + sTm )
2
)
Method 1: K u and Tu are estimated from estimates of the ultimate and crossover frequencies (Pecharroman and
Pagola [165]) – as in Method 6, IPD model.
Method 2: Km , Tm , τ m assumed known.
H. Third order system plus time delay (TOLPD) model ( G m (s) =
Km e − sτ
).
(1 + sTm1 )(1 + sTm2 )(1 + sTm 3 )
m
Method 1: Km , Tm1 , Tm2 , Tm3 , τ m known.
I. Unstable first order lag plus time delay model ( G m (s) =
K me − sτ
1 − sTm
m
)
Method 1: Km , Tm , τ m known.
Method 2: The model parameters are obtained by least squares fitting from the open loop frequency
response of the unstable process; this is done by determining the closed loop magnitude and
phase values of the (stable) closed loop system and using the Nichols chart to determine the
open loop response (Huang and Lin [154], Deshpande [164]).
J. Unstable second order system plus time delay model ( G m (s) =
K m e− s τ
)
(1 − sTm1 )(1 + sTm 2 )
m
Method 1: Km , Tm1 , Tm2 , τ m known.
Method 2: The model parameters are obtained by least squares fitting from the open loop frequency
response of the unstable process; this is done by determining the closed loop magnitude and
phase values of the (stable) closed loop system and using the Nichols chart to determine the
open loop response (Huang and Lin [154], Deshpande [164]).
K. Second order system plus time delay model with a positive zero ( G m (s) =
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
)
Method 1: Km , Tm1 , Tm2 , Tm3 , τ m known.
L. Second order system plus time delay model with a negative zero ( G m (s) =
K m (1 + sTm3 )e − sτ
m
(1 + sTm1 )(1 + sTm 2 )
)
Method 1: Km , Tm1 , Tm2 , Tm3 , τ m known.
M. Fifth order system plus delay model ( G m ( s) =
K m (1 + b1s + b2 s2 + b3s3 + b4 s4 + b ss5 ) e− s τ
(1 + a s + a s
2
1
2
Method 1: Km , b1 , b 2 , b 3 , b 4 , b5 , a1 , a2 , a3 , a4 , a5 , τ m known.
N. General model with a repeated pole ( G m ( s) =
+ a3s 3 + a 4 s4 + a 5s5
)
m
)
K m e − sτ
)
(1 + sTm ) n
m
* Method 1: Strejc’s method
O. General stable non-oscillating model with a time delay
P.
Delay model ( G m ( s) = e − sτ m )
Note: * means that the procedure has not been fully described to date.
4. Organisation of the report
The tuning rules are organised in tabular form, as is indicated in the list of tables below. Within each table, the
tuning rules are classified further; the main subdivisions made are as follows:
(i) Tuning rules based on a measured step response (also called process reaction curve methods).
(ii) Tuning rules based on minimising an appropriate performance criterion, either for optimum regulator or
optimum servo action.
(iii) Tuning rules that gives a specified closed loop response (direct synthesis tuning rules). Such rules may be
defined by specifying the desired poles of the closed loop response, for instance, though more generally,
the desired closed loop transfer function may be specified. The definition may be expanded to cover
techniques that allow the achievement of a specified gain margin and/or phase margin.
(iv) Robust tuning rules, with an explicit robust stability and robust performance criterion built in to the design
process.
(v) Tuning rules based on recording appropriate parameters at the ultimate frequency (also called ultimate
cycling methods).
(vi) Other tuning rules, such as tuning rules that depend on the proportional gain required to achieve a quarter
decay ratio or magnitude and frequency information at a particular phase lag.
Some tuning rules could be considered to belong to more than one subdivision, so the subdivisions cannot be
considered to be mutually exclusive; nevertheless, they provide a convenient way to classify the rules. Tuning
rules for the variations that have been proposed in the ‘ideal’ PI and PID controller structure are included in the
appropriate table. In all cases, one column in the tables summarise the conditions under which the tuning rules
are designed to operate, if appropriate ( Y ( s) = closed loop system output, R( s) = closed loop system input).
Tables 1-3: PI tuning rules – FOLPD model - G m ( s) =
K me − sτ
1 + sTm
m

1
Table 1: Ideal controller G c ( s) = Kc  1 +
 . Eighty-five such tuning rules are defined; the references are
Ts

i 
(a) Process reaction methods: Ziegler and Nichols [8], Hazebroek and Van der Waerden [9], Astrom
and Hagglund [3], Chien et al. [10], Cohen and Coon [11], Wolfe [12], Murrill [13] – page 356,
McMillan [14] – page 25, St. Clair [15] – page 22 and Shinskey [15a]. Twelve tuning rules are
defined.
(b) Performance index minimisation (regulator tuning): Minimum IAE - Murrill [13] – pages 358-363,
Shinskey [16] – page 123, ** Shinskey [17], Huang et al. [18], Yu [19]. Minimum ISE - Hazebroek
and Van der Waerden [9], Murrill [13]– pages 358-363, Zhuang and Atherton [20], Yu [19].
Minimum ITAE - Murrill [13] – pages 358-363, Yu [19]. Minimum ISTSE - Zhuang and Atherton
[20]. Minimum ISTES – Zhuang and Atherton [20]. Thirteen tuning rules are defined.
(c) Performance index minimisation (servo tuning): Minimum IAE – Rovira et al. [21], Huang et al. [18].
Minimum ISE - Zhuang and Atherton [20], han and Lehman [22]. Minimum ITAE - Rovira et al. [21].
Minimum ISTSE - Zhuang and Atherton [20]. Minimum ISTES – Zhuang and Atherton [20]. Seven
tuning rules are defined.
(d) Direct synthesis: Haalman [23], Chen and Yang [23a], Pemberton [24], Smith and Corripio [25], Smith
et al. [26], Hang et al. [27], Miluse et al. [27a], Gorecki et al. [28], Chiu et al. [29], Astrom et al. [30],
Davydov et al. [31], Schneider [32], McAnany [33], Leva et al. [34], Khan and Lehman [22], Hang et
al. [35, 36], Ho et al. [37], Ho et al [104], Tan et al. [39], Voda and Landau [40], Friman and Waller
[41], Smith [42], Cox et al. [43], Cluett and Wang [44], Abbas [45], Bi et al. [46], Wang and Shao
[47]. Thirty-one tuning rules are defined.
(e) Robust: Brambilla et al. [48], Rivera et al. [49], Chien [50], Thomasson [51], Fruehauf et al. [52],
Chen et al. [53], Ogawa [54], Lee et al. [55], Isaksson and Graebe [56], Chun et al. [57]. Ten tuning
rules are defined.
(f) Ultimate cycle: McMillan [58], Shinskey [59] – page 167, **Shinskey [17], Shinskey [16] – page
148, Hwang [60], Hang et al. [65], Zhuang and Atherton [20], Hwang and Fang [61]. Twelve tuning
rules are defined.

1
Table 2: Controller G c ( s) = Kc  b +
 Two direct synthesis tuning rules are defined by Astrom and Hagglund
T

i s
[3] - page 205-208.

1 
E (s ) − α Kc R( s) . One performance index
Table 3: Two degree of freedom controller: U(s) = K c 1 +
Tis 

minimisation tuning rule is defined by Taguchi and Araki [61a].
Tables 4-7: PI tuning rules - non-model specific

1
Table 4: Controller G c ( s) = Kc  1 +
 . Nineteen such tuning rules are defined; the references are:
Ts

i 
(a) Ultimate cycle: Ziegler and Nichols [8], Hwang and Chang [62], ** Hang et al. [36], McMillan [14] –
page 90, Pessen [63], Astrom and Hagglund [3] – page 142, Parr [64] – page 191, Yu [122] – page
11. Seven tuning rules are defined.
(b) Other tuning rules: Parr [64] – page 191, McMillan [14] – pages 42-43, Parr [64] – page 192,
Hagglund and Astrom [66], Leva [67], Astrom [68], Calcev and Gorez [69], Cox et al. [70]. Eight
tuning rules are defined.
(c) Direct synthesis: Vrancic et al. [71], Vrancic [72], Friman and Waller [41], Kristiansson and
Lennartson [158a]. Four tuning rules are defined.

1 
Table 5: Controller G c ( s) = K c  b +
 . One direct synthesis tuning rule is defined by Astrom and Hagglund
Ti s 

[3] – page 215.

1 
Table 6: Controller U( s) = Kc  1 +
 E( s) + Kc ( b − 1)R (s) . One direct synthesis tuning rule is defined by
T

is
Vrancic [72].
Table 7: Controller U( s) = Kc Y(s) −
Kc
E ( s) . One direct synthesis tuning rule is defined by Chien et al. [74].
Ti s
K me− sτ m
s
Tables 8-11: PI tuning rules – IPD model G m ( s) =

1
Table 8: Controller G c ( s) = Kc  1 +
 . Twenty such tuning rules are defined; the references are:
Ts

i 
(a) Process reaction methods: Ziegler and Nichols [8], Wolfe [12], Tyreus and Luyben [75], Astrom and
Hagglund [3] – page 138. Four tuning rules are defined.
(b) Regulator tuning – performance index minimisation: Minimum ISE – Hazebroek and Van der
Waerden [9]. Minimum IAE - Shinskey [59] – page 74. Minimum ITAE - Poulin and Pomerleau [82].
Four tuning rules are defined.
(c) Ultimate cycle: Tyreus and Luyben [75], ** Shinskey [17]. Two tuning rules are defined.
(d) Robust: Fruehauf et al. [52], Chien [50], Ogawa [54]. Three tuning rules are defined.
(e) Direct synthesis: Wang and Cluett [76], Cluett and Wang [44], Rotach [77], Poulin and Pomerleau
[78], Kookos et al. [38]. Five tuning rules are defined.
(f) Other methods: Penner [79], Srividya and Chidambaram [80]. Two tuning rules are defined.

1 1
Table 9: Controller G c ( s) = Kc  1 +
. One robust tuning rule is defined by Tan et al. [81].

Ts

i  1 + Tf s
Table 10: Controller U( s) = Kc Y(s) −
Kc
E ( s) . One direct synthesis tuning rule is defined by Chien et al. [74].
Ti s

1 
E (s ) − α Kc R( s) . Two performance index
Table 11: Two degree of freedom controller: U(s) = K c 1 +
Tis 

minimisation - servo/regulator tuning rules are defined by Taguchi and Araki [61a] and Pecharroman
and Pagola [134b].
Tables 12-14: PI tuning rules – FOLIPD model G m (s) =
Km e− sτ
s(1 + sTm )
m

1
Table 12: Ideal controller G c ( s) = Kc  1 +
 . Six such tuning rules are defined; the references are:
Ts

i 
(a) Ultimate cycle: McMillan [58]. One tuning rule is defined.
(b) Regulator tuning – minimum performance index: Minimum IAE – Shinskey [59] – page 75. Shinskey
[59] – page 158. Minimum ITAE - Poulin and Pomerleau [82]. Four tuning rules are defined.
(c) Direct synthesis - Poulin and Pomerleau [78]. One tuning rule is defined.

1 
Table 13: Controller G c ( s) = Kc  b +
 . One direct synthesis tuning rule is defined by Astrom and Hagglund
Ti s 

[3] – pages 210-212.

1 
E (s ) − α Kc R( s) . Two performance index
Table 14: Two degree of freedom controller: U(s) = K c 1 +
Tis 

minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134b].
Tables 15-16: PI tuning rules – SOSPD model
K m e− sτ
Tm1 s2 + 2ξ m Tm1s + 1
2
Km e− sτ
(1 + Tm1s)(1 + Tm2 s)
m
m
or

1
Table 15: Ideal controller G c ( s) = Kc  1 +
 . Ten tuning rules are defined; the references are:
Ts

i 
(a) Robust: Brambilla et al. [48]. One tuning rule is defined.
(b) Direct synthesis: Tan et al. [39]. One tuning rule is defined.
(c) Regulator tuning – minimum performance index: Minimum IAE - Shinskey [59] – page 158, **
Shinskey [17], Huang et al. [18], Minimum ISE – McAvoy and Johnson [83], Minimum ITAE –
Lopez et al. [84]. Five tuning rules are defined.
(d) Servo tuning – minimum performance index: Minimum IAE - Huang et al. [18]. One tuning rule is
defined.
(e) Ultimate cycle: Hwang [60], ** Shinskey [17]. Two tuning rules are defined.

1 
E (s ) − α Kc R( s) . Three performance index
Table 16: Two degree of freedom controller: U(s) = K c 1 +

T
is 

minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134a],
[134b].
Table 17: PI tuning rules – SOSIPD model (repeated pole) G m (s) =
Km e −s τ
m
s (1 + sTm )
2

1 
E (s ) − α Kc R( s) . Two performance index
Table 17: Two degree of freedom controller: U(s) = K c 1 +
Tis 

minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134b].
Km e − sτ
(1 + sTm1 )(1 + sTm2 )(1 + sTm 3 )
m
Tables 18-19: PI tuning rules – third order lag plus delay (TOLPD) model

1
Table 18: Ideal controller G c ( s) = Kc  1 +
 . One *** tuning rule is defined. The reference is Hougen [85].
Ts

i 

1 
E (s ) − α Kc R( s) . One performance index
Table 19: Two degree of freedom controller: U(s) = K c 1 +
Tis 

minimisation tuning rule is defined by Taguchi and Araki [61a].
K me − sτ
1 − sTm
m
Table 20: PI tuning rules - unstable FOLPD model

1
Table 20: Ideal controller G c ( s) = Kc  1 +
 . Six tuning rules are defined; the references are:
Ts

i 
(a) Direct synthesis: De Paor and O’Malley [86], Venkatashankar and Chidambaram [87], Chidambaram
[88], Ho and Xu [90]. Four tuning rules are defined.
(b) Robust: Rotstein and Lewin [89]. One tuning rule is defined.
(c) Ultimate cycle: Luyben [91]. One tuning rule is defined.
K m e− s τ
(1 − sTm1 )(1 + sTm 2 )
m
Table 21: PI tuning rules - unstable SOSPD model

1
Table 21: Ideal controller G c ( s) = Kc  1 +
 . Three tuning rules are defined; the references are:
Ts

i 
(a) Ultimate cycle: McMillan [58]. One tuning rule is defined.
(b) Minimum performance index – regulator tuning: Minimum ITAE – Poulin and Pomerleau [82]. Two
tuning rules are defined.
Table 22: PI tuning rules – delay model e −s τ m

1
Table 22: Ideal controller G c ( s) = Kc  1 +
 . Two tuning rules are defined; the references are:
Ts

i 
(a) Direct synthesis: Hansen [91a].
(b) Minimum performance index – regulator tuning: Shinskey [57].
K m e− sτ
1 + sTm
m
Tables 23-40: PID tuning rules - FOLPD model


1
Table 23: Ideal controller G c ( s) = Kc  1 +
+ Td s . Fifty-seven tuning rules are defined; the references are:
Ti s


(a) Process reaction: Ziegler and Nichols [8], Astrom and Hagglund [3] – page 139, Parr [64] – page
194, Chien et al. [10], Murrill [13]- page 356, Cohen and Coon [11], Astrom and Hagglund [93]pages 120-126, Sain and Ozgen [94]. Eight tuning rules are defined.
(b) Minimum performance index – regulator tuning: Minimum IAE – Murrill [13] – pages 358-363,
Cheng and Hung [95]. Minimum ISE - Murrill [13] – pages 358-363, Zhuang and Atherton [20].
Minimum ITAE - Murrill [13] – pages 358-363. Minimum ISTSE - Zhuang and Atherton [20].
Minimum ISTES - Zhuang and Atherton [20]. Minimum error - step load change - Gerry [96]. Eight
tuning rules are defined.
(c) Minimum performance index – servo tuning: Minimum IAE - Rovira et al. [21], Wang et al. [97].
Minimum ISE - Wang et al. [97], Zhuang and Atherton [20]. Minimum ITAE - Rovira et al. [21],
Cheng and Hung [95], Wang et al. [97]. Minimum ISTSE – Zhuang and Atherton [20]. Minimum
ISTES – Zhuang and Atherton [20]. Nine tuning rules are defined.
(d) Ultimate cycle: Pessen [63], Zhuang and Atherton [20], Pi-Mira et al. [97a], Hwang [60], Hwang and
Fang [61], McMillan [58], Astrom and Hagglund [98], Li et al. [99], Tan et al. [39], Friman and
Waller [41]. Fourteen tuning rules are defined.
(e) Direct synthesis: Gorecki et al. [28], Smith and Corripio [25], Suyama [100], Juang and Wang [101],
Cluett and Wang [44], Zhuang and Atherton [20], Abbas [45], Camacho et al.[102, Ho et al. [103],
Ho et al [104], Morilla et al. [104a]. Fourteen tuning rules are defined.
(f) Robust: Brambilla et al. [48], Rivera et al. [49], Fruehauf et al. [52], Lee et al. [55]. Four tuning rules.

 1
1
Table 24: Ideal controller with first order filter G c ( s) = Kc  1 +
+ Td s
. Three robust tuning rules are
Ti s

 Tf s + 1
defined by ** Morari and Zafiriou [105], Horn et al. [106] and Tan et al. [81].


1
1 + b1s
Table 25: Ideal controller with second order filter G c ( s) = Kc  1 +
+ Td s
. One robust tuning
2
Ti s

 1 + a 1s + a 2s
rule is defined by Horn et al. [106].


1
Table 26: Ideal controller with set-point weighting G c ( s) = Kc  b +
+ Td s . One direct synthesis tuning rule
Ti s


is defined by Astrom and Hagglund [3] – pages 208-210.
Table 27. Ideal controller with first order filter and set-point weighting:

 1 

1
1 + 0.4Trs
U(s) = K c 1 +
+ Td s 
− Y( s)  . One direct synthesis tuning rule is defined
R (s )
1 + sTr
 Tis
 Tf s + 1 

by Normey-Rico et al. [106a].

1  1 + sTd

Table 28: Classical controller G c ( s) = K c 1 +
. Twenty tuning rules are defined; the references are:
T
Ti s 

1+ s d
N
(a) Process reaction: Hang et al. [36] – page 76, Witt and Waggoner [107], St. Clair [15] – page 21,
Shinskey [15a]. Five tuning rules are defined.
(b) Minimum performance index – regulator tuning: Minimum IAE - Kaya and Scheib [108], Witt and
Waggoner [107]. Minimum ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108],
Witt and Waggoner [107]. Five tuning rules are defined.
(c) Minimum performance index – servo tuning: Minimum IAE - Kaya and Scheib [108], Witt and
Waggoner [107]. Minimum ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108],
Witt and Waggoner [107]. Five tuning rules are defined.
(d) Direct synthesis: Tsang and Rad [109], Tsang et al. [111]. Two tuning rules are defined.
(e) Robust: Chien [50]. One tuning rule is defined.
(f) Ultimate cycle: Shinskey [59] – page 167, Shinskey [16] – page 143. Two tuning rules are defined.





1 
Td s
 . Two tuning rules are defined; the
 E( s) −
Table 29: Non-interacting controller U (s) = K c 1 +
Y
(
s
)
Td s

Tis 

1+


N


references are:
(a) Minimum performance index – regulator tuning: Minimum IAE - Huang et al. [18].
(b) Minimum performance index – servo tuning: Minimum IAE - Huang et al. [18].

1
Td s
Table 30: Non-interacting controller U( s) = Kc  1 +
Y(s) . Five tuning rules are defined; the
 E ( s) −
sT
Ti s

1+ d
N
references are:
(a) Minimum performance index – servo tuning: Minimum ISE - Zhuang and Atherton [20], Minimum
ISTSE - Zhuang and Atherton [20]. Minimum ISTES - Zhuang and Atherton [20]. Three tuning
rules are defined.
(b) Ultimate cycle: Zhuang and Atherton [20], Shinskey [16] – page 148. Two tuning rules are defined.

Td s
1 
E (s) −
Table 31: Non-interacting controller U (s) =  K c +
Y ( s) . Six tuning rules are defined; the

sTd
T
s

i 
1+
N
references are:
(a) Minimum performance index – regulator tuning: Minimum IAE – Kaya and Scheib [108]. Minimum
ISE – Kaya and Scheib [108]. Minimum ITAE – Kaya and Scheib [108].
(b) Minimum performance index – servo tuning: Minimum IAE – Kaya and Scheib [108]. Minimum ISE –
Kaya and Scheib [108]. Minimum ITAE – Kaya and Scheib [108].
Table 32: Non-interacting controller with setpoint weighting:

1
Kc Td s
U( s) = Kc  b +
Y( s) + Kc ( b − 1)Y( s) . Three ultimate cycle tuning rules are
 E (s) −
Ti s 
1 + Td s N

defined by Hang and Astrom [111], Hang et al. [65] and Hang and Cao [112].




1 
1 + Td s
Table 33: Industrial controller U( s) = Kc 1 +
Y(s)  . Six tuning rules are defined: the
  R( s) −
Ts

Ti s  

1+ d



N
reference are:
(a) Minimum performance index – regulator tuning: Minimum IAE - Kaya and Scheib [108]. Minimum
ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108]. Three tuning rules are
defined.
(b) Minimum performance index – servo tuning: Minimum IAE - Kaya and Scheib [108]. Minimum ISE Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108]. Three tuning rules are defined.

1 
Table 34: Series controller Gc ( s) = Kc  1 +
 (1 + sTd ) . Three tuning rules are defined; the references are:
Ti s 

(a) ******: Astrom and Hagglund [3] – page 246.
(b) Ultimate cycle: Pessen [63].
(c) Direct synthesis: Tsang et al. [110].



1 
sTd 
 . One robust tuning rule is
Table 35: Series controller with filtered derivative Gc ( s) = Kc  1 +
 1 +
sT

Ti s 
1 + d 


N 
defined by Chien [50].



1
Td s 
 . Three tuning rules are defined; the
Table 36: Controller with filtered derivative G c ( s) = Kc  1 +
+
Ti s 1 + s Td 



N
references are:
(a) Robust: Chien [50], Gong et al. [113]. Two tuning rules are defined.
(b) Direct synthesis: Davydov et al. [31]. One tuning rule is defined.

1 
Table 37: Alternative non-interacting controller 1 - U( s) = Kc 1 +
 E( s) − Kc Td sY ( s) . Six ultimate cycle
Ti s 

tuning rules are defined; the references are: Shinskey [59] – page 167, ** Shinskey [17], Shinskey [16] –
page 143, VanDoren [114].
2

1   1 + 05
. τm s + 0.0833τ m s 2 
Table 38: Alternative filtered derivative controller - G c ( s) = Kc  1 +
. One direct

2

Ti s  

[1 + 01. τ ms]

synthesis tuning rule is defined by Tsang et al. [110].
Table 39: I-PD controller U(s) =
Kc
E (s) − K c (1 + Td s)Y(s) . Two direct synthesis tuning rules are defined by Chien
Ti s
et al. [74] and Argelaguet et al. [114a].








1
T
s
β
T
s
d
d
 E( s) − K c  α +
 R (s) . One
Table 40: Two degree of freedom controller: U (s) = K c 1 +
+
 Tis
Td 

Td 
1+
s
1+
s


N 
N 


performance index minimisation tuning rule is defined by Taguchi and Araki [61a].
Tables 41-48: PID tuning rules - non-model specific


1
Table 41: Ideal controller G c ( s) = Kc  1 +
+ Td s . Twenty five tuning rules are defined; the references are
Ti s


(a) Ultimate cycle: Ziegler and Nichols [8], Blickley [115], Parr [64] – pages 190-191, De Paor [116],
Corripio [117] – page 27, Mantz andTacconi [118], Astrom and Hagglund [3] – page 142, Astrom
and Hagglund [93], Atkinson and Davey [119], ** Perry and Chilton [120], Yu [122] – page 11, Luo
et al. [121], McMillan [14] – page 90, McAvoy and Johnson [83], Karaboga and Kalinli [123], Hang
and Astrom [124], Astrom et al. [30], St. Clair [15] - page 17, Shin et al. [125]. Nineteen tuning rules
are defined.
(b) Other tuning: Harriott [126], Parr [64] – pages 191, 193, McMillan [14] - page 43, Calcev and Gorez
[69], Zhang et al. [127], Garcia and Castelo [127a]. Six tuning rules are defined.



1
Td s 
 . Eight tuning rules are defined; the
Table 42: Controller with filtered derivative Gc ( s) = Kc 1 +
+
Ti s 1 + Td s 



N 
references are:
(a) Direct synthesis: Vrancic [72], Vrancic [73], Lennartson and Kristiansson [157], Kristiansson and
Lennartson [158], Kristiansson and Lennartson [158a]. Six tuning rules are defined.
(b) Other tuning: Leva [67], Astrom [68]. Two tuning rules are defined.
Table 43: Ideal controller with set-point weighting:
1
U( s) = Kc Fp R(s) − Y(s) +
( Fi R(s) − Y(s)) + Td s( Fd R(s) − Y(s)) . One ultimate cycle tuning rule is
Ts
i
(
)
defined by Mantz and Tacconi [118].


1
Table 42: Ideal controller with proportional weighting G c ( s) = Kc  b +
+ Td s . One direct synthesis tuning
Ti s


rule is defined by Astrom and Hagglund [3] – page 217.

1 
K Ts
Table 44: Non-interacting controller U( s) = Kc 1 +
 E( s) − c d Y( s) . One ultimate cycle tuning rule is
sT
Ti s 

1+ d
N
defined by Fu et al. [128].

1
Table 45: Series controller U( s) = Kc  1 +
 (1 + sTd ) . Three ultimate cycle tuning rules are defined by Pessen

Ti s 
[131], Pessen [129] and Grabbe et al. [130].



1 
sTd 
 . One ultimate cycle tuning
Table 46: Series controller with filtered derivative U( s) = Kc 1 +
1 +
sT

Ti s  
1 + d 


N 
rule is defined by Hang et al. [36] - page 58.

1  1 + sTd
Table 47: Classical controller U( s) = Kc  1 +
. One ultimate cycle tuning rule is defined by Corripio

T
Ti s 

1+ s d
N
[117].

1
Table 48: Non-interacting controller U( s) = Kc  1 +
 E ( s) − Kc Td sY (s) . One ultimate cycle tuning rule is
Ti s

defined by VanDoren [114].
K m e− sτ m
s


1
Table 49: Ideal controller Gc ( s) = K c  1 +
+ Td s . Five tuning rules are defined; the references are:
T
s


i
Tables 49-58: PID tuning rules - IPD model
(a) Process reaction: Ford [132], Astrom and Hagglund [3] – page 139. Two tuning rules are defined.
(b) Direct synthesis: Wang and Cluett [76], Cluett and Wang [44], Rotach [77]. Three tuning rules are
defined.
Table 50: Ideal controller with first order filter, set-point weighting and output feedback:

 1 

1
1 + 0.4Trs
U(s ) = K c 1 +
+ Tds 
− Y(s )  − K 0 Y(s) . One direct synthesis tuning rule
 R( s)
Ti s
1 + sTr

 Tf s + 1 

has been defined by Normey-Rico et al. [106a].



1
Td s 
 . One robust tuning rule has
Table 51: Ideal controller with filtered derivative G c ( s) = Kc  1 +
+
sTd 
Ts

i
1+


N 
been defined by Chien [50].



1 
Td s 
 . One robust tuning rule has
Table 52: Series controller with filtered derivative G c ( s) = Kc  1 +
 1 +
Ts

Ts
i 
 1 + d 

N 
been defined by Chien [50].



1   1 + Td s 
 . Five tuning rules have been defined; the references
Table 53: Classical controller G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
are:
(a) Ultimate cycle: Luyben [133], Belanger and Luyben [134]. Two tuning rules have been defined.
(b) Robust: Chien [50]. One tuning rule has been defined.
(c) Performance index minimisation – regulator tuning: ** Minimum IAE - Shinskey [17], Shinskey [59]
– page 74. Two tuning rules have been defined.

1 
Table 54: Alternative non-interacting controller 1: U( s) = Kc 1 +
 E( s) − Kc Td sY ( s) . Two performance index
T

is
minimisation rules – minimum IAE regulator tuning have been defined by Shinskey [59] – page 74 and
** Shinskey [17].
Table 55: I-PD controller U(s) =
Kc
E (s) − K c (1 + Td s)Y(s) . One direct synthesis tuning rule has been defined by
Ti s
Chien et al. [74].
1
) E( s) + Kc ( b − 1) R (s) − Kc TdsY( s) . One direct synthesis tuning rule has
Tis
been defined by Hansen [91a].








1
T
s
β
T
s
d
d
 E( s) − K c  α +
 R (s) . Two
Table 57: Two degree of freedom controller: U (s) = K c 1 +
+
 Tis
Td 

Td 
1
+
s
1
+
s




N 
N 


minimum performance index – servo/regulator tuning have been defined by Taguchi and Araki [61a] and
Pecharroman and Pagola [134a].
Table 56: Controller U(s) = Kc (1 +
Km e − sτ
s(1 + sTm )
m
Tables 58-67: PID tuning rules - FOLIPD model


1
Table 58: Ideal controller Gc ( s) = K c  1 +
+ Td s . One ultimate cycle tuning rule has been defined by Millan
Ti s


[58].

 1
1
Table 59: Ideal controller with filter G c ( s) = Kc  1 +
+ Td s
. Three robust tuning rules have been
T
s

 1 + Tf s
i
defined by Tan et al. [81], Zhang et al. [135] and Tan et al. [136].


1
Table 60: Ideal controller with set-point weighting G c ( s) = Kc  b +
+ Td s . One direct synthesis tuning rule
Ti s


has been defined by Astrom and Hagglund [3] - pages 212-213.



1   1 + Td s 
 . Five tuning rules have been defined; the references
Table 61: Classical controller G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
are as follows:
(a) Robust: Chien [50]. One tuning rule is defined.
(b) Minimum performance index – regulator tuning: Minimum IAE – Shinskey [59] – page 75, Shinskey
[59] – pages 158-159, Minimum ITAE - Poulin and Pomerleau [82], [92]. Four tuning rules are
defined.



1 
Td s 
 . One robust tuning rule has
Table 62: Series controller with derivative filtering G c ( s) = Kc  1 +
 1 +
Ts

Ts
i 
 1 + d 

N 
been defined by Chien [50].

1 
U(s) = Kc 1 +  E (s) − K c Td sY ( s) . Two minimum
 Ti s 
performance index (minimum IAE) – regulator tuning rules have been defined by Shinskey [59] – page
75, page 159.



1
Td s 
 . One robust tuning rule has
Table 64: Ideal controller with filtered derivative: G c ( s) = Kc  1 +
+
Ti s 1 + s Td 



N
been defined by Chien [50].
Table 63: Alternative non-interacting controller 1:
Table 65: Ideal controller with set-point weighting:
(
)
Kc
[ Fi R(s) − Y(s)] + Kc Td s[ Fd R (s) − Y(s)] . One ultimate cycle tuning rule
Ti s
has been defined by Oubrahim and Leonard [138].



 1+ T s   1 + T s 
i
d

 . One direct synthesis tuning rule has
Table 66: Alternative classical controller: G c ( s) = Kc 
 1 + Td s   1 + Td s 

N 
N 
been defined by Tsang and Rad [109].
U(s) = Kc Fp R ( s) − Y( s) +
Table 67: Two degree of freedom controller:








1
T
s
β
T
s
d
d
 E( s) − K c  α +
 R (s) . Two minimum performance index –
U (s) = K c 1 +
+
 Tis
Td 

Td 
1+
s
1+
s


N 
N 


servo/regulator tuning have been defined by Taguchi and Araki [61a] and Pecharroman and Pagola
[134a].
Km e− sτ
K m e− sτ
or
2 2
(1 + sTm1 )(1 + sTm 2 ) Tm1 s + 2ξ m Tm1s + 1
m
Tables 68-79: PID tuning rules - SOSPD model
m


1
Table 68: Ideal controller Gc ( s) = K c  1 +
+ Td s . Twenty seven tuning rules have been defined; the
Ti s


references are:
(a) Minimum performance index – servo tuning: Minimum ITAE – Sung et al. [139]. One tuning rule is
defined.
(b) Minimum performance index – regulator tuning: Minimum ITAE – Sung et al. [139], Lopez et al.
[84]. One tuning rule is defined.
(c) Ultimate cycle: Hwang [60], Shinskey [16] – page 151. Three tuning rules are defined.
(d) Direct synthesis: Hang et al. [35], Ho et al. [140], Ho et al. [141], Ho et al. [142], Wang et al. [143],
Leva et al. [34], Wang and Shao [144], Pemberton [145], Pemberton [24], Suyama [100], Smith et al.
[146], Chiu et al. [29], Wang and Clemens [147], Gorez and Klan [147a], Miluse et al. [27a], Miluse et
al. [27b], Seki et al. [147b], Landau and Voda [148]. Nineteen tuning rules are defined.
(e) Robust: Brambilla et al. [48], Chen et al. [53], Lee et al. [55]. Three tuning rules are defined.

 1
1
Table 69: Filtered controller G c ( s) = Kc  1 +
+ Td s
. One robust tuning rule has been defined by Hang
Ti s

 Tf s + 1
et al. [35].

 b s+1
1
Table 70: Filtered controller G c ( s) = Kc  1 +
+ Td s 1
. One robust tuning rule has been defined by
T
s

 a 1s + 1
i
Jahanmiri and Fallahi [149].



1   1 + Td s 
 . Seven tuning rules have been defined; the
Table 71: Classical controller G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
references are:
(a) Minimum performance index – regulator tuning: Minimum IAE - Shinskey [59] – page 159, **
Shinskey [59], ** Shinskey [17], ** Shinskey [17]. Minimum ISE – McAvoy and Johnson [83]. Five
tuning rules are defined.
(b) Direct synthesis: Astrom et al. [30], Smith et al. [26]. Two tuning rules are defined.

1   1 + NTd s 
Table 72: Alternative classical controller G c ( s) = Kc  1 +

 . One ***** tuning rule has been
Ti s   1 + Td s 

defined by Hougen [85].

1
Table 73: Alternative non-interacting controller 1: U( s) = K c  1 +
 E ( s) − Kc Td sY ( s) . Three minimum
Ti s

performance index (minimum IAE) – regulator tuning rules have been defined by Shinskey [59] – page
158, ** Shinskey [17], ** Shinskey [17].

1 
Table 74: Series controller G c ( s) = K c 1 +
 (1 + Td s) . One minimum performance index - regulator tuning rule
 Ti s 
has been defined by Haalman [23].





1 
Td s
 . Two tuning rules have been
 E( s) −
Table 75: Non-interacting controller U (s) = K c 1 +
Y
(
s
)
Td s

Tis 

1+


N


defined. The references are:
(a) Minimum performance index – regulator tuning: Minimum IAE - Huang et al. [18].
(b) Minimum performance index – servo tuning: Minimum IAE - Huang et al. [18].
Table 76: Ideal controller with set-point weighting:
K
U(s) = Kc Fp R ( s) − Y( s) + c [ Fi R( s) − Y(s) ] + Kc Td s[ Fd R ( s) − Y( s)] . One ultimate cycle tuning rule
Ti s
(
)
has been defined by Oubrahim and Leonard [138].

1 
Table 77: Non-interacting controller U( s) = Kc  b + [ R (s) − Y(s) ] − ( c + Tds)Y( s) . One direct synthesis tuning
Ts

i 
rule has been defined by Hansen [150].



1
Td s 
 . Two tuning rules are defined;
Table 78: Ideal controller with filtered derivative G c ( s) = Kc  1 +
+
sT
Ts

i
1 + d 

N 
the references are:
(a) Direct synthesis: Hang et al. [151].
(b) Robust: Hang et al. [151].








1
T
s
β
T
s
d
d
 E( s) − K c  α +
 R (s) . Three
Table 79: Two degree of freedom controller: U (s) = K c 1 +
+
 Tis
Td 

Td 
1
+
s
1
+
s




N 
N 


minimum performance index – servo/regulator tuning rules have been defined by Taguchi and Araki
[61a] and Pecharroman and Pagola [134a], [134b].
Table 80: PID tuning rules - I 2PD model G m (s) =
K me − sτ m
s2
1
) E( s) + Kc ( b − 1) R (s) − Kc TdsY( s) . One direct synthesis tuning rule has
Tis
been defined by Hansen [91a].
Table 80: Controller U(s) = Kc (1 +
Table 81: PID tuning rules – SOSIPD model (repeated pole) G m (s) =
Km e −s τ
m
s (1 + sTm )
2








1
T
s
β
T
s
d
d
 E( s) − K c  α +
 R (s) . Two
Table 81: Two degree of freedom controller: U (s) = K c 1 +
+
 Tis
Td 

Td 
1+
s
1+
s


N 
N 


minimum performance index – servo/regulator tuning rules have been defined by Taguchi and Araki
[61a] and Pecharroman and Pagola [134a].
Tables 82-84: PID tuning rules - SOSPD model with a positive zero
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )



1
Td s 
 . One robust tuning rule has been
Table 82: Controller with filtered derivative G c ( s) = Kc  1 +
+
Ti s 1 + Td s 



N 
defined by Chien [50].



1   1 + Td s 
 . One robust tuning rule has been defined by Chien
Table 83: Classical controller G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
[50].



1 
Td s 
 . One robust tuning rule has
Table 84: Series controller with filtered derivative G c ( s) = Kc  1 +
 1 +
Ts

Ts
i 
 1 + d 

N 
been defined by Chien [50].
Tables 85-88: PID tuning rules - SOSPD model with a negative zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )


1
Table 85: Ideal controller Gc ( s) = K c  1 +
+ Td s . One minimum performance index tuning rule has been
Ti s


defined by Wang et al. [97].



1
Td s 
 . One robust tuning rule has been
Table 86: Ideal controller with filtered derivative G c ( s) = Kc  1 +
+
Ti s 1 + Td s 



N 
defined by Chien [50].



1   1 + Td s 
 . One robust tuning rule has been defined by Chien
Table 87: Classical controller G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
[50].



1 
Td s 
 . One robust tuning rule has
Table 88: Series controller with filtered derivative G c ( s) = Kc  1 +
 1 +
Ts

Ts
i 
 1 + d 

N 
been defined by Chien [50].
Km e− sτ
(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 )
m
Table 89-90: PID tuning rules - TOLPD model


1
Table 89: Ideal controller Gc ( s) = K c  1 +
+ Td s . Two minimum performance index tuning rules have been
Ti s


defined by Polonyi [153].








1
T
s
β
T
s
d
d
 E( s) − K c  α +
 R (s) .One
Table 90: Two degree of freedom controller: U (s) = K c 1 +
+
 Tis
Td 

Td 
1+
s
1+
s


N 
N 


minimum performance index tuning rule has been defined by Taguchi and Araki [61a].
K m e− sτ
1 − sTm
m
Tables 91-93: PID tuning rules - unstable FOLPD model


1
Table 91: Ideal controller Gc ( s) = K c  1 +
+ Td s . Three direct synthesis tuning rules are defined by De Paor
Ti s


and O’Malley [86], Chidambaram [88] and Valentine and Chidambaram [154].

1
KTs
Table 92: Non-interacting controller U( s) = Kc  1 +
 E ( s) − c d Y(s) . Two tuning rules have been
sT
Ti s

1+ d
N
defined; the references are:
(a) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155]
(b) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155]



1   1 + Td s 
 . One performance index minimisation – regulator
Table 93: Classical controller G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
tuning rule has been defined by Shinskey [16]– page 381.
Tables 94-97: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ
(1 − sTm1 )(1 + sTm 2 )
m


1
Table 94: Ideal controller Gc ( s) = K c  1 +
+ Td s . Two tuning rules have been defined; the references are
Ti s


(a) Ultimate cycle: McMillan [58]
(b) Robust: Rotstein and Lewin [89].



1   1 + Tds 
 . Two minimum performance index tuning rules
Table 95: Classical controller G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
(regulator - minimum ITAE) have been defined by Poulin and Pomerleau [82], [92].

1
Table 96: Series controller Gc ( s) = Kc 1 +
 (1 + Td s) . One direct synthesis tuning rule has been defined by
Ti s

Ho and Xu [90].
Table 97: Non-interacting controller

1
KTs
U( s) = Kc  1 +  E ( s) − c d Y(s) . Two tuning rules have been
sT
Ti s

1+ d
N
defined; the references are
(a) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155]
(b) Minimum performance index – regulator tuning: Minimum IAE - Huang and Lin [155]
Table 98: PID tuning rules – general model with a repeated pole G m ( s) =
K m e − sτ
(1 + sTm ) n
m


1
Table 98: Ideal controller Gc ( s) = K c  1 +
+ Td s . One direct synthesis tuning rule has been defined by
Ti s


Skoczowski and Tarasiejski [156]
Table 99: PID tuning rules – general stable non-oscillating model with a time delay


1
Table 99: Ideal controller Gc ( s) = K c  1 +
+ Td s . One direct synthesis tuning rule has been defined by Gorez
Ti s


and Klan [147a].
Tables 100-101: PID tuning rules – fifth order model with delay
K (1 + b1s + b2 s2 + b3s3 + b4 s4 + b ss5 ) e− s τ m
G m ( s) = m
1 + a1s + a2 s2 + a3s 3 + a 4 s4 + a 5s5
(
)


1
Table 100: Ideal controller Gc ( s) = K c  1 +
+ Td s . One direct synthesis tuning rule is defined by Vrancic et
Ti s


al. [159].



1
Tds 
 . One direct synthesis tuning rule is
Table 101: Controller with filtered derivative G c ( s) = Kc  1 +
+
Td s 
Ts

i
1+


N 
defined by Vrancic et al. [159].
** some more information needed.
The number of tuning rules in each table is included in the data. Servo and regulator tuning rules are counted
separately; otherwise, rules in which different tuning parameters are provided for a number of variations in
process parameters or desired response parameters (such as desired gain margin, phase margin or closed loop
response time constant) are counted as one tuning rule. Tabular summaries are provided below.
Table A: Model structure and tuning rules – a summary for PI controllers
Model
Stable
FOLPD
Non-model
specific
IPD
FOLIPD
SOSPD
SOSIPD
TOLPD
Unstable
FOLPD
Unstable
SOSPD
Delay
model
TOTAL
Process
reaction
Direct
Synthesis
Ultimate
cycle
Robust
tuning
Other rules
Total
12
Minimise
Performanc
e index
21
33
10
12
0
88 (53%)
0
0
7
7
0
8
23 (14%)
4
0
0
0
0
0
6
6
9
2
1
0
6
2
1
0
0
4
2
1
2
0
0
1
4
0
1
0
0
1
2
0
0
0
1
0
24 (14%)
9 (5%)
13 (7%)
2 (1%)
2 (1%)
6 (4%)
0
2
0
1
0
0
3 (2%)
0
1
1
0
0
0
2 (1%)
16
48
54
24
18
11
171
Table B: Model structure and tuning rules – a summary for PID controllers
Model
Stable
FOLPD
Non-model
specific
IPD
FOLIPD
SOSPD
I 2PD
SOSIPD
SOSPD –
pos. zero
SOSPD –
neg. zero
TOLPD
Unstable
FOLPD
Unstable
SOSPD
Higher
order
TOTAL
Process
reaction
Direct
Synthesis
Ultimate
cycle
Robust
tuning
Other rules
Total
13
Minimise
Performanc
e index
45
24
28
12
1
123 (44%)
0
0
7
27
0
8
42 (15%)
2
0
0
0
6
8
16
0
6
2
23
1
2
2
4
0
3
6
6
0
0
0
1
0
19 (7%)
18 (6%)
50 (17%)
1 (0%)
0
0
2
0
0
0
0
0
0
3
0
0
2 (1%)
3 (1%)
0
1
0
0
3
0
4 (1%)
0
0
3
3
0
3
0
0
0
0
0
0
3 (1%)
6 (2%)
0
4
1
1
1
0
7 (2%)
0
0
4
0
0
0
4 (1%)
15
88
71
64
34
10
282
Table C: Model structure and tuning rules – a summary for PI/PID controllers
Process
reaction
Model
Stable
FOLPD
Non-model
specific
IPD
FOLIPD
SOSPD
I 2PD
SOSIPD
SOSPD –
pos. zero
SOSPD –
neg. zero
TOLPD
Unstable
FOLPD
Unstable
SOSPD
Delay
model
Higher
order
TOTAL
Direct
Synthesis
Ultimate
cycle
Robust
tuning
Other rules
Total
25
Minimise
Performanc
e index
66
57
38
24
1
211 (47%)
0
0
14
34
0
16
65 (15%)
6
0
0
0
12
14
25
0
12
4
24
1
4
3
6
0
7
6
7
0
2
0
1
0
43 (9%)
27 (6%)
63 (14%)
1 (0%)
0
0
4
0
0
0
0
0
0
3
0
0
4 (1%)
3 (1%)
0
1
0
0
3
0
4 (1%)
0
0
4
3
0
7
0
0
0
1
1
0
5 (1%)
12 (3%)
0
6
1
2
1
0
10 (2%)
0
1
1
0
0
0
2 (0%)
0
0
4
0
0
0
4 (1%)
31
136
125
88
52
21
453
Table D: PI controller structure and tuning rules – a summary
Stable
FOLPD
Nonmodel
specific
IPD

1
G c ( s) = Kc  1 + 
Ts

i 
85
19
20

1
G c ( s) = Kc  b + 
T

i s
2
1

1 
E (s ) − α Kc R( s)
U(s) = K c 1 +
Tis 

1
Controller structure
FOLIP
D
SOSPD
Other
Total
6
10
12
152
(92%)
0
1
2
0
6
(4%)
1
2
2
1
3
10
(4%)
0
1
1
0
0
0
2
(1%)

1 1
G c ( s) = Kc  1 + 
Ts

i  1 + Tf s
0
0
1
0
0
0
Total
88
21
23
8
12
15
1
(0%)
167
U( s) = Kc Y(s) −
Kc
E ( s)
Ti s
Table E: PID controller structure and tuning rules – a summary
Stable
FOLPD
Nonmodel
specific
IPD
57
25
5
3
8

 1
1
G c ( s) = Kc  1 +
+ Td s
Ti s

 Tf s + 1
3

 b s+1
1
G c ( s) = Kc  1 +
+ Td s 1
Ti s

 a 1s + 1
Controller structure
FOLIP
D
SOSPD
Other
Total
1
27
11
126
(45%)
1
1
2
3
0
0
3
1
0
7
(3%)
0
0
0
0
1
0
1
(0%)


1
1 + b1s
G c ( s) = Kc  1 +
+ Td s
T
s
1
+
a 1s + a 2s 2


i
3
0
0
0
0
0
3
(1%)


1
G c ( s) = Kc  b +
+ Td s
Ti s


1
1
0
1
0
0
3
(1%)
Subtotal
67
34
6
6
31
14
158
(56%)

1  1 + sTd

G c ( s) = K c 1 +
T
Ti s 

1+ s d
N
20
1
5
5
7
5
43
(15%)



 1+ T s   1 + T s 
i 
d 
G c ( s) = Kc 
 1 + Td s   1 + Td s 

N 
N 
0
0
0
1
0
0
1
(0%)

1   1 + NTd s 
G c ( s) = Kc  1 +


T

i s   1 + Td s 
0
0
0
0
1
0
1
(0%)

1 
Gc ( s) = Kc  1 +
 (1 + sTd )
Ti s 

3
3
0
0
1
1
8
(3%)
1
1
1
1
0
2
6
(2%)
2

1   1 + 05
. τ m s + 0.0833τ m s 2 
G c (s) = K c  1 +  
2


Ti s  

[1 + 01. τ ms]

1
0
0
0
0
0
1
(0%)
Subtotal
25
5
6
7
9
8
60
(22%)
5
0
0
0
0
0
5
(2%)
2
0
0
0
4
0
6
(2%)


1
G c ( s) = Kc  1 +
+ Td s
T
s


i


1
Td s
G c ( s) = Kc  1 +
+
Ti s 1 + s Td


N







1 
sTd
Gc ( s) = Kc  1 +   1 +
sT

Ti s 
1+ d


N






1
Td s
U( s) = Kc  1 +  E ( s) −
Y(s)
sTd
T
s

i 
1+
N





1 
Td s

 E( s) −
U (s) = K c 1 +
Y
(
s
)
Td s

Tis 

1+


N


18
(6%)
Stable
FOLPD
Nonmodel
specific
IPD

Td s
1 
E (s) −
U (s) =  K c +
Y ( s)

sTd
T
s

i 
1+
N
6
0
0

1 
K Ts
U( s) = Kc 1 +
 E( s) − c d Y( s)
sT
T
s

i 
1+ d
N
0
1
3
Controller structure
FOLIP
D
SOSPD
Other
Total
0
0
0
6
(2%)
0
0
1
0
6
(2%)
0
0
0
0
0
6
0
0
0
0
0
6
1
2
2
3
0
14
(5%)
0
0
0
0
1
0
1
(0%)
2
0
1
0
0
0
3
(1%)






 E(s ) − K  α + βTds  R (s )
c


T 
s
1+ d s

N 


1
0
1
1
3
3
9
(3%)
1
( Fi R(s) − Y(s)) + Td s( Fd R(s) − Y(s))
Ti s
0
1
0
1
1
0
1
) E (s ) + K c ( b − 1)R ( s) − K cTd sY(s )
Tis
0
0
1
0
0
1

 1 

1
1 + 0.4Trs
U( s) = K c 1 +
+ Td s
R (s)
− Y( s)

 T s +1
Ti s
1 + sTr

 f


1
0
0
0
0
0

 1 

1
1 + 0.4Tr s
U(s) = Kc 1 +
+ Td s 
− Y(s) − K0 Y (s )
R (s )
Ti s
1 + sTr

 Tf s + 1 

0
0
1
0
0
0
Subtotal
32
3
6
4
8
8
3
(1%)
2
(1%)
1
(0%)
1
(0%)
61
(22%)
Total
124
42
18
17
49
30

1
Kc Td s
U(s) = Kc  b +
Y(s) + Kc ( b − 1) Y(s)
 E( s) −
Ti s
1 + Td s N





1 
1 + Td s
U( s) = Kc 1 +
Y(s) 
  R( s) −
Ts

Ti s  

1+ d



N

1 
U( s) = Kc 1 +
 E( s) − Kc Td sY ( s)
Ti s 


1
U(s) = Kc  b + [R (s) − Y(s) ] − ( c + Td s) Y(s)
Ts

i 
U(s) =
Kc
E (s) − K c (1 + Td s)Y(s)
Ti s


1
Td s
U(s ) = Kc  1 +
+

T
Ti s
1+ d

N

(
)
U(s) = K c Fp R(s) − Y(s) +
U( s) = K c (1 +
3
(1%)
6
(2%)
280
3. Tuning rules for PI and PID controllers
Table 1: PI tuning rules - FOLPD model - G m ( s) =
K me − sτ

1
– ideal controller G c ( s) = Kc  1 +
 . 84 tuning
1 + sTm
Ts

i 
m
rules
Rule
Process reaction
Ziegler and Nichols [8]
Model: Method 1.
Hazebroek and Van der
Waerden [9]
Model: Method 1
τm Tm
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Kc
Ti
09
. Tm
K mτ m
3.33τm
α Tm
K mτ m
α
β
0.68
7.14
0.70
4.76
0.72
3.70
0.74
3.03
0.76
2.50
0.79
2.17
0.81
1.92
0.84
1.75
0.87
1.61
τ
α = 0.5 m + 01
.
Tm
τm Tm
α
0.90
1.49
0.93
1.41
0.96
1.32
0.99
1.25
1.02
1.19
1.06
1.14
1.09
1.10
1.13
1.06
1.17
1.03
τm
β=
16
. τ m − 1.2Tm
3.2τm
Chien et al. [10] - regulator
0.6Tm
K mτ m
4τ m
Chien et al. [10] - servo
Model: Method 1
Cohen and Coon [11]
process reaction
Model: Method 1.
07
. Tm
K mτ m
β
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
063
. Tm
Km τ m
Astrom and Hagglund [3] –
regulator – page 150
Model: Method 1
Quarter decay ratio.
τm
≤1
Tm
βτ m
Astrom and Hagglund [3] –
page 138
Model: Not relevant
Model: Method 1
Comment
2.33
τm
Km
τm Tm
α
β
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
1.20
1.28
1.36
1.45
1.53
1.62
1.71
1.81
1.00
0.95
0.91
0.88
0.85
0.83
0.81
0.80
τm
> 35
.
Tm
Ultimate cycle ZieglerNichols equivalent
0% overshoot τ
011
. < m < 10
.
Tm
20% overshoot τ
011
. < m < 10
.
Tm
07
. Tm
K mτ m
2.3τm
20% overshoot
035
. Tm
Km τ m
117
. Tm
0% overshoot τ
011
. < m < 10
.
Tm
0.6Tm
K mτ m
Tm
20% overshootτ
011
. < m < 10
.
Tm

1  Tm
+ 0083
. 
 0.9
K m  τm

2


 3.33 τ m + 0.31 τm  

Tm
 Tm  
Tm 

τ


1 + 2.22 m


Tm


Quarter decay ratio
Rule
Kc
Ti
τm Tm
α Tm
K mτ m
α
βτ m
Two constraints criterion Wolfe [12]
β
τm Tm
α
β
0.2
0.5
4.4
1.8
3.23
2.27
1.0
5.0
0.78
0.30
1.28
0.53
Model: Method 2.
Two constraints criterion Murrill [13] – page 356
 Tm 
0928
.
 
K m  τm 
0.946
Tm  τ m 
 
1078
.
 Tm 
Comment
Decay ratio = 0.4; minimum
error integral (regulator
mode).
0 .583
Quarter decay ratio;
minimum error integral
(servo mode).
τ
01
. ≤ m ≤ 10
.
Tm
Model: Method 3
McMillan [14] – page 25
Model: Method 3
St. Clair [15] – page 22
Model: Method 3
Shinskey [15a]
Model: Method 1
Regulator tuning
Minimum IAE - Murrill [13] –
pages 358-363
Model: Method 3
τm
Time delay dominant
processes
Tm
Tm
≤ 30
.
τm
3.78τ m
Tm
= 0.167
τm
Km
3
0333
. Tm
K mτ m
0. 667 Tm
K mτ m
Performance index minimisation
 Tm 
0984
.
 
Km  τm 
0.986
Tm  τ m 
 
0608
.
 Tm 
0. 707
01
. ≤
100
. Tm Km τ m
30
. τm
τm Tm = 0.2
Minimum IAE - Shinskey
[16] – page 123
Model: Method 6
104
. Tm K mτ m
2.25τ m
τm Tm = 0.5
111
. Tm Km τ m
145
. τm
τm Tm = 1
139
. Tm K mτ m
τm
τm Tm = 2
Minimum IAE - Shinskey
[17] – page 38
Model: Method 6
Minimum IAE – Huang et
al. [18]
Model: Method 6
0.95Tm K mτ m
34
. τm
τm Tm = 01
.
0.95Tm K mτ m
2.9τ m
τm Tm = 0.2
Minimum IAE – Yu [19]
K e− sτ L
(Load model = L
)
1 + sTL
Model: Method 6
Kc
0.685  TL 


K m  Tm 
( 1) 1
Ti
τm
0.214 − 0 .346
Tm
 τm 


 Tm 
− 1.256
Tm  TL 
 
0. 214  Tm 
01
. ≤
( 1)
τm
− 055
.
Tm
1977
.
 τm 
 
 Tm 
1123
.
TL
Tm
τm
Tm
 TL 
0874
.
 
Km  Tm 
− 0099
.
+ 0159
.
τm
Tm
 τm 
 
 Tm 
− 1041
.
Tm  TL 
 
0.415  Tm 
−4 .515
τm
+ 0. 067
Tm
 τm 
 
 Tm 
0 .876
2. 641
Kc
( 1)
1
=
Km
τm
≤1
Tm
≤ 2 .641
τm
Tm
τm
Tm
+ 016
. ;
≤ 0.35
TL
+ 0.16 ≤
τm
Tm
1
τm
≤ 10
.
Tm
Tm
≤1;
≤ 0.35
−0.9077
−0.063
0.5961
τ

τ 
 τm 
 τm 
τ
6.4884 + 4.6198 m + 0.8196 m 
− 52132
.
− 7.2712 
− 0.7241e T
 
Tm

 Tm 
 Tm 
 Tm 
m
m



2
3
4
5
6

 τm 
 τm 
 τm 
 τm 
 τm  
τ
Ti (1) = Tm 00064
.
+ 3.9574 m − 64789
.
+
9
.
4348
−
10
.
7619
+
7
.
5146
−
2
.
2236
 
 
 
 
  
Tm

 Tm 
 Tm 
 Tm 
 Tm 
 Tm  
Kc
Rule
Minimum IAE – Yu [19]
K e− sτ L
(Load model = L
)
1 + sTL
(continued)
Model: Method 6
Minimum ISE - Hazebroek
and Van der Waerden [9]
Ti
τ
− 0015
. + 0 .384 m
Tm
 TL 
0871
.
 
K m  Tm 
 TL 
0513
.
 
K m  Tm 
0 .218
τm
Tm
 τm 
 
 Tm 
 τm 
 
 Tm 
− 1055
.
− 1451
.
Tm  TL 
 
0. 444  Tm 
Comment
τ
− 0. 217 m − 0. 213
Tm
Tm  TL 
 
0. 670  Tm 
Tm 
τ 
. + 0.3 m 
 074
K mτ m 
Tm 
− 0. 003
τm
− 0. 084
Tm
 τm 
 
 Tm 
0867
.
 τm 
 
 Tm 
0 .56
τm Tm
α
β
τm Tm
0.2
0.3
0.5
0.80
0.83
0.89
7.14
5.00
3.23
0.7
1.0
1.5
0.96
1.07
1.26
2.44
1.85
1.41
2.0
3.0
5.0
0.959
Tm  τ m 
 
0492
.
 Tm 
0.739
0.945
Tm  τ m 
 
0535
.
 Tm 
0.586
Minimum ISE – Zhuang and
Atherton [20]
 Tm 
1279
.
 
K m  τm 
 Tm 
1346
.
 
Km  τm 
0 .675
Tm  τ m 
 
0552
.
 Tm 
0. 438
τm
Tm
0 .181− 0 .205
τm
Tm
− 0. 045 + 0. 344
 TL 
1289
.
 
K m  Tm 
Minimum ITAE - Murrill
[13] – pages 358-363
Model: Method 3
Minimum ITAE – Yu [19]
K e− sτ L
(Load model = L
)
1 + sTL
Model: Method 6
0. 04 + 0. 067
τm
Tm
τm
Tm
 Tm 
0859
.
 
Km  τm 
 TL 
0598
.
 
K m  Tm 
−1. 214
 τm 
 
 Tm 
Tm  TL 
 
0. 430  Tm 
τm
− 0 .49
Tm
0. 954
 τm 
 
 Tm 
0 .639
 τm 
 
 Tm 
−1. 014
Tm  TL 
 
0. 359  Tm 
τm
− 0 .292
Tm
− 2. 532
τm
≤ 10
.
Tm
11
. ≤
τm
≤ 2.0
Tm
TL
≤ 2.310
Tm
τm
 τm 
 
 Tm 
0. 899
− 1047
.
 τm 
 
 Tm 
− 0889
.
0. 977
τm
Tm
0. 272 − 0. 254
 τm 
 
 Tm 
 τm 
 
 Tm 
Tm  TL 
 
0.347  Tm 
τm
− 0. 094
Tm
− 1112
.
Tm  TL 


0596
.
 Tm 
τm
− 0. 44
Tm
0 .372
Tm  τ m 
 
0674
.
 Tm 
− 1341
.
Tm  TL 
 
0805
.
 Tm 
0 .304
 τm 


 Tm 
2 .310
τm
Tm
 τm 
 
 Tm 
 τm 
 
 Tm 
Tm
Tm
01
. ≤
0. 196
TL
Tm
τm
Tm
−0 .011− 1945
.
 τm 
 
 Tm 
−1. 055
Tm  TL 
 
0. 425  Tm 
−5. 809
τm
+ 0 .241
Tm
 τm 
 
 Tm 
2 .385
τm
Tm
τm
0 .084 + 0154
.
τm
Tm
 τm 
 
 Tm 
− 1. 042
Tm  TL 
 
0. 431  Tm 
− 0148
.
τm
− 0. 365
Tm
 τm 
 
 Tm 
0. 901
1<
TL
Tm
TL
Tm
≤1;
≤ 0.35
τm
Tm
≤ 0.35
> 035
.
τm
≤ 10
.
Tm
τm
Tm
+ 0.112 ;
≤ 0.35
+ 0112
.
≤
Tm
0.787  TL 
 
Km  Tm 
+ 0. 077 ;
≤ 0.35
≤ 2.385
τm
0 .901
Tm
≤3;
τm
0. 46
0.680
τm
− 0. 112
Tm
TL
1<
τm
+ 0.077 ≤
τm
0 .898
β
01
. ≤
Tm
0. 735  TL 
 
K m  Tm 
> 035
.
1.46
1.18
1.89
0.95
2.75
0.81
τ
01
. ≤ m ≤ 10
.
Tm
Tm
−0 .065+ 0 .234
≤ 0.35
α
Tm
107
.  TL 
 
K m  Tm 
Tm
βτ m
β
 TL 
1157
.
 
Km  Tm 
τm
τm Tm < 0.2
 Tm 
1305
.
 
K m  τm 
Model: Method 6
Tm
τm Tm
0. 921  TL 
 
K m  Tm 
≤3;
τm
143
. Tm
Minimum ISE - Murrill [13]
– pages 358-363
Model: Method 3
Minimum ISE – Yu [19]
K e− sτ L
(Load model = L
)
1 + sTL
Tm
α Tm
K mτ m
α
Model: Method 1
Model: Method 6
TL
1<
TL
Tm
≤1;
≤ 0.35
≤3;
τm
Tm
≤ 0.35
Kc
Rule
Minimum ITAE – Yu [19]
K e− sτ L
(Load model = L
)
1 + sTL
 TL 
0878
.
 
K m  Tm 
0172
.
− 0. 057
Ti
τm
Tm
 τm 


 Tm 
−0 .909
Tm  TL 
 
0. 794  Tm 
0 .228
Comment
τm
− 0257
.
Tm
 τm 
 
 Tm 
0. 489
τm
> 035
.
Tm
(continued)
Model: Method 6
Minimum ISTSE - Zhuang
and Atherton [20]
Model: Method 6
Minimum ISTES – Zhuang
and Atherton [20]
Model: Method 6
Servo tuning
Minimum IAE – Rovira et al.
[21]
Model: Method 3
Minimum IAE - Huang et al.
[18]
Model: Method 6
Minimum ISE - Zhuang and
Atherton [20]
Model: Method 6
Minimum ISE – Khan and
Lehman [22]
Model: Method 6
Minimum ITAE – Rovira et
al. [21]
Model: Method 3
 Tm 
1015
.
 
K m  τm 
0.957
Tm  τ m 
 
0667
.
 Tm 
0.552
 Tm 
1065
.
 
K m  τm 
0.673
Tm  τ m 
 
0687
.
 Tm 
0.427
 Tm 
1021
.
 
Km  τ m 
0.953
Tm  τ m 
 
0629
.
 Tm 
0.546
 Tm 
1076
.
 
Km  τm 
0 .648
Tm  τ m 
 
0650
.
 Tm 
0. 442
 Tm 
0758
.
 
K m  τm 
0.861
Kc( 2) 2
Performance index minimisation
Tm
τ
1.020 − 0.323 m
Tm
Ti (2 )
 Tm 
0980
.
 
Km  τm 
0.892
 Tm 
1072
.
 
K m  τm 
0.560
τm
≤ 2.0
Tm
01
. ≤
τm
≤ 10
.
Tm
11
. ≤
τm
≤ 2.0
Tm
01
. ≤
τm
≤ 10
.
Tm
01
. ≤
τm
≤1
Tm
τm
≤ 10
.
Tm
Tm
11
. ≤
τm
≤ 2.0
Tm
Kc Km
0.01 ≤
τm
≤ 0.2
Tm
0.2 ≤
τm
≤ 20
Tm
01
. ≤
τm
≤ 10
.
Tm
τ
0.648 − 0.114 m
Tm
0 .916
11
. ≤
01
. ≤
 0.5291 00003082

.

−

2
τm
 Tm τ m

Kc Km
 0.808 0.511 0.255  T

 m
+
−

 τm
Tm
Tm τ m  K m  0.095 + 0.846 − 0.381

 τ 2
τ m Tm τ m Tm τ m
 m
 Tm 
0586
.
 
Km  τ m 
τm
≤ 10
.
Tm
Tm
τ
0.690 − 0.155 m
Tm
 0.7388 03185
 Tm
.
+


Tm  Km
 τm
01
. ≤
Tm
τm
1.030 − 0165
.
Tm




−1.0169
3.5959
3.6843
τ

 τm 
 τm 
 τm 
1 
τm
T
− 130454
Kc =
.
− 9.0916
+ 0.3053 
+ 11075
.
− 2.2927 
+ 4.8259e 
 
Km 
Tm
 Tm 
 Tm 
 Tm 


2
3
4
5
6

 τm 
 τm 
 τm 
 τm 
 τm  
τm
(2 )

Ti = Tm 0.9771 − 0.2492
+ 3.4651  − 7.4538  + 8.2567   − 4.7536  + 11496
.
  
Tm

 Tm 
 Tm 
 Tm 
 Tm 
 Tm  


m
2
( 2)
m
Rule
Minimum ISTSE - Zhuang
and Atherton [20]
Model: Method 6
Minimum ISTES – Zhuang
and Atherton [20]
Model: Method 6
Direct synthesis
Haalman [23]
Model: Method 6
Chen and Yang [23a]
Model: Method 25
Minimum IAE – regulator Pemberton [24], Smith and
Corripio [25] – page 343346. Model: Method 6
Minimum IAE – servo Smith and Corripio [25] –
page 343-346. Model:
Method 6
5% overshoot – servo –
Smith et al. [26] – deduced
from graph.
Model: Method not stated
1% overshoot – servo –
Smith et al. [26] – deduced
from graph
Model: Method not stated
5% overshoot - servo Smith and Corripio [25] –
page 343-346. Model:
Method 6
5% overshoot - servo Hang et al. [27]
Model: Method 1
Miluse et al. [27a]
Model: Method not stated
Kc
Ti
 Tm 
0712
.
 
Km  τm 
0.921
 Tm 
0786
.
 
Km  τm 
0.559
 Tm 
0569
.
 
K m  τm 
0.951
 Tm 
0628
.
 
Km  τ m 
0.583
2Tm
3K m τ m
0. 7T m
Km τ m
Comment
Tm
01
. ≤
τm
≤ 10
.
Tm
Tm
11
. ≤
τm
≤ 2.0
Tm
Tm
01
. ≤
τm
≤ 10
.
Tm
Tm
11
. ≤
τm
≤ 2.0
Tm
τ
0.968 − 0.247 m
Tm
τ
0.883 − 0.158 m
Tm
τ
1.023 − 0.179 m
Tm
τ
1.007 − 0.167 m
Tm
Tm
Closed loop sensitivity
M s = 19
. . (Astrom and
Hagglund [3])
Ms = 1. 26 ; A m = 2. 24 ;
Tm
φ m = 500
01
. ≤
τm
≤ 0.5
Tm
Tm
01
. ≤
τm
≤ 05
.
Tm
052
. Tm
K mτ m
Tm
0.04 ≤
τm
≤ 14
.
Tm
044T
. m
K mτ m
Tm
0.04 ≤
τm
≤ 14
.
Tm
Tm
Km τ m
3Tm
5Km τ m
Tm
Tm
2K m τ m
Tm
13Tm
25K m τ m
Tm
0. 368Tm
K m τm
Tm
0. 514Tm
K m τm
Tm
Closed loop response
overshoot = 0%
(Model: Method 26 – Miluse
et al. [27b])
Closed loop response
overshoot = 5%
0. 581Tm
K m τm
Tm
Closed loop response
overshoot = 10%
Rule
Kc
Ti
Comment
Miluse et al. [27a] continued
0. 641Tm
K m τm
Tm
Closed loop response
overshoot = 15%
0. 696Tm
K m τm
Tm
Closed loop response
overshoot = 20%
0. 748Tm
K m τm
Tm
Closed loop response
overshoot = 25%
0. 801Tm
K m τm
Tm
Closed loop response
overshoot = 30%
0. 853Tm
K m τm
Tm
Closed loop response
overshoot = 35%
0. 906Tm
K m τm
Tm
Closed loop response
overshoot = 40%
0. 957 Tm
K mτ m
Tm
Closed loop response
overshoot = 45%
1.008 Tm
Km τ m
Tm
Closed loop response
overshoot = 50%
Model: Method not stated
Regulator – Gorecki et al.
[28]
(considered as 2 rules)
Model: Method 6
Chiu et al. [29]
Model: Method 6
Kc
Astrom et al. [30]
Kc
Ti
Ti ( 4)
Low freq. part of magnitude
Bode diagram is flat
λ Tm
K m (1 + λτm )
Tm
λ variable; suggested
values: 0.2, 0.4, 0.6, 1.0.
1
Model: Method 12


τ
K m  1.905 m + 0.826

Tm

2


 τ 
2 Tm 
=
2 +  m  − 1 e

K m τm 
 2Tm 


= τm
Tm
Honeywell UDC 6000
controller


τm
.
+ 0.362  Tm
 0153
Tm


Closed loop response
damping factor =
0.9; 02
. ≤ τ m Tm ≤ 1 .
2
 τm 
τm
2 +
 −2 −
2Tm
 2 Tm 
 τ 
1+  m 
 2Tm 
2
2
3
2
2

 τ  τ 
 τ 
 τ  
 τ 
3 +  m  +  m  +  m  − 2 +  m   2 +  m 
 2Tm   Tm 
 2Tm 

 2Tm  
 2Tm 
2
Kc ( 4) =
Pole is real and has max.
attainable multiplicity
( 3)
Kc( 4) 3
Davydov et al. [31]
( 3)
( 3)
Ti
3
 3τ m 
K m1 +


Tm 
Model: Method 22
3
( 3)
1
Km
T 
T 
Tm
+ 6 m  + 6 m 
τm
 τm 
 τm 
2

 Tm  
Tm

4 1+ 3
+ 3  
τm
 τm  


3
2
1+3
T 
T 
Tm
+ 6 m  + 6 m 
τm
 τm 
 τm 
2

 Tm  
Tm

3 1+ 2
+ 2  
τm
 τm  


1+ 3
, Ti ( 4) = τ m
3
Kc
Rule
Ti
Comment
Schneider [32]
0.368
Tm
K mτ m
Tm
Closed loop response
damping factor = 1
Model: Method 6
0.403
Tm
K mτ m
Tm
Closed loop response
damping factor = 0.6
McAnany [33]
(1.44Tm + 0.72Tm τ m − 0.43τ m − 2.14)
(
K m 12
. τ m + 0.36 τ m 2 + 2
Model: Method 5
Leva et al. [34]
ω cn Tm
Km
Model: Method 16
Khan and Lehman [22]
Tm
Km
Tm
Km
Model: Method 11
Or
Model: Method 14
Gain and phase margin – Ho
et al. [37]
)
Closed loop time constant =
167
. Tm .
π


tanφ m − + τ m ω cn + tan −1 (ω cn Tm ) 
2


ω cn
1 + ω cn 2 Tm 2
1 + ω cn 2 Ti 2
 0.404 0.256
01275
.

+
−
 τm
T
τ
m

m Tm




ω cn
 Tm  π

2.82 − φ m − tan −1 
 − φm 
 
 τ m  2
=
τm
KcKm
 0.4104 0.00024 0.525

−
−

τ m2
Tm 2 
 τ m Tm
KcKm
 0.719 0.0808
0.324

+
−
2
 τ m Tm
τ
τ
m

m τ mTm




0.01 ≤
τm
≤ 0.2
Tm
0.2 ≤
τm
≤ 20
Tm
1048
. Tm
Km τ m
Tm
Gain Margin = 1.5
Phase Margin = 30 0
07854
.
Tm
K mτ m
Tm
Gain Margin = 2
Phase Margin = 45 0
0524
. Tm
K mτ m
Tm
Gain Margin = 3
Phase Margin = 60 0
0.393Tm
K mτ m
Tm
Gain Margin = 4
Phase Margin = 67 .5 0
0314T
.
m
Km τ m
Tm
Gain Margin = 5
Phase Margin = 72 0
ω p Tm
1
K m Am
4ω p τ m
ωp =
2
2ω p −
(considered as 2 rules)
Model: Method 6
2
4Tm + 128
. τ m − 2.4
 0. 3852 0.723
τ 

+
− 0.404 m2 
Tm
 τm
Tm 
Model: Method 6
Hang et al. [35, 36]
)
(
Km 556
. + 2τ m + τ m
π
+
A m φ m + 0.5πA m (A m − 1)
(A
1
2
m
)
− 1 τm
Tm
Given A m , ISE is minimised when φ m = 688884
.
− 34.3534 A m + 91606
.
τm
for servo
Tm
tuning (Ho et al [104]).
Given A m , ISE is minimised when φ m = 45.9848A m 0.2677 ( τ m Tm )
0. 2755
for regulator
tuning (Ho et al [104]).
(
Tan et al. [39]
β Ti ω φ 1 + βTm ω φ
Model: Method 10
A m 1 + βTi ω φ
Symmetrical optimum
principle - Voda and Landau
[40]
(
)
)
2
[
ωφ < ω u
2
1
4.6
ω135
35
. G p ( jω135 )
0
1
Model: Method not relevant
1
βω φ tan − tan − 1 βT mω φ − βτ m ω φ − φ
2.828 G p ( jω135 )
1
115
. G p ( jω 135 0 ) + 0.75K m
0
4.6 G p ( jω135 ) − 0.6K m
0
[
β = 08
. ,
τm
< 05
. ;
Tm
β = 0.5,
τm
> 05
.
Tm
τm
≤ 01
.
Tm
0
4
ω135
]
01
. <
0
ω 135 0 2.3 G p ( jω 1350 ) − 0.3K m
]
τm
≤ 015
.
Tm
0.15 <
τm
≤1
Tm
Rule
Kc
Ti
Comment
Friman and Waller [41]
Model: Method 6
0.2333
1
τ m > 2Tm . Gain margin = 3;
G p ( jω135 )
ω135
Model: Method 6
Tm
2K mτ m
Tm
Smith [42]
Model: Method not
specified
0.35
Km
042
. τm
Kc( 5)
Ti (5)
0.5Tm
Km τ m
Tm
0
0
Voda and Landau [40]
Modulus optimum principle
- Cox et al. [43]4
Model: Method 17
Cluett and Wang [44]
Model: Method 6
Bi et al. [46]
Model: Method 20
4
Kc
( 5)
=
0.5
Km
6 dB gain margin - dominant
delay process
τm
≤1
Tm
τm
>1
Tm
0.099508τ m + 0.99956Tm
Closed loop time constant =
τm
4τm
0.99747τ m − 8742510
.
. − 5 Tm
0.05548 τ m + 0.33639Tm
K mτm
016440
.
τ m + 0.99558Tm
Closed loop time constant =
τm
2τm
0.98607τ m − 15032
.
.10− 4 Tm
0.092654τ m + 0.43620Tm
K mτm
0.20926τ m + 098518
.
Tm
Closed loop time constant =
τm
133
. τm
0.96515τ m + 4.255010
. −3 Tm
012786
.
τ m + 051235
.
Tm
Km τ m
0.24145τ m + 0.96751Tm
Closed loop time constant =
τ
0.93566 τ m + 2.298810
. −2 Tm m
τm
016051
.
τ m + 0.57109Tm
Km τ m
0.26502τ m + 0.94291Tm
Closed loop time constant =
τ
089868
.
τ m + 6.935510
. −2 Tm m
08
. τm
0.28242τ m + 0.91231Tm
τ
0.85491τ m + 015937
.
Tm m
−1.045
Model: Method 6
Phase margin = 60 0 ;
τ
0.25 ≤ m ≤ 1
Tm
0.019952τ m + 0.20042Tm
K mτm
019067
.
τ m + 0.61593Tm
Km τ m
Abbas [45]
Phase margin = 45 0
τ 
0148
.
+ 0186
.  m
 Tm 
Km (0.497 − 0.464 V0.590 )
05064
.
Tm
Km τ m
Tm + 0.5τ m
Closed loop time constant =
067
. τm
V = fractional overshoot,
0 ≤ V ≤ 0.2
τ
01
. ≤ m ≤ 5.0
Tm
Tm
 Tm 3 + Tm2 τ m + 0.5Tm τ m 2 + 0.167τ m 3 
 Tm3 + Tm 2τ m + 0.5Tmτ m 2 + 0.167τ m 3 
( 5)


,


T
=
i
2
2
3
2
2

Tm τm + Tm τ m + 0.667τ m


Tm + Tm τ m + 0.5τ m

Kc
Rule
Wang and Shao [47]
Kc
Ti
( 5a ) 5
Ti
Comment
λ = inverse of the maximum
of the absolute real part of
the open loop transfer
function; λ = [1.5, 2.5]
(5 a )
Model: Method 6
Robust
Tm + 0.5τ m
K m( λ + τ m )
Brambilla et al. [48] -
Closed loop response has less than 5% overshoot with no model uncertainty:
τ
τ
τ
λ = 1 , 01
. ≤ m ≤ 1 ; λ = 1 − 05
. log 10 m , 1 < m ≤ 10
Tm
Tm
Tm
Model: Method 6
Rivera et al. [49]
Tm
λKm
Tm
λ ≥ 17
. τ m , λ > 01
. Tm .
Model: Method 6
2Tm + τ m
2λK m
Tm + 0.5τ m
λ ≥ 17
. τ m , λ > 01
. Tm .
Tm
K m (τ m + λ)
Tm
τm
2Km ( τ m + λ )
05
. τm
Fruehauf et al. [52]
5Tm
9τ m Km
5τ m
τm
< 0.33
Tm
Model: Method 1
Tm
2τ m Km
Tm
τm
≥ 0.33
Tm
Chen et al. [53]
0. 50Tm
τm K m
Tm
A m = 3.14 , φ m = 61. 40 ,
Ms = 1. 00
Tm
A m = 2.58 , φ m = 55. 00 ,
Ms = 1. 10
0. 67Tm
τ mK m
Tm
A m = 2. 34 , φ m = 51.6 0 ,
Ms = 1. 20
0. 70Tm
τm K m
Tm
A m = 2. 24 , φ m = 50.0 0 ,
M s = 1. 26
0. 72Tm
τm K m
Tm
A m = 2.18 , φ m = 48. 70 ,
Ms = 1. 30
Chien [50]
Model: Method 6
Thomasson [51]
Model: Method not defined
0. 61Tm
τ m Km
Model: Method 6
5
Kc
( 5a )
f 1 (ω90
0
0
( 5a )
=
λ = Tm [50];
λ > Tm + τ m , τ m << Tm
(Thomasson [51])
τ m >> Tm ; λ = desired
closed loop time constant

1
1 
f 2 ( ω900 ) −
,
λf 1 (ω90 0 ) 
ω90 0 
Km
2
2
2
)=
( Tm + {1 + ω90 Tm }τ m ) sin(−ω90 τm − tan −1 ω90 Tm ) − ω90 Tm cos(− ω90 τ m − tan− 1 ω90 Tm )
2
2 1 .5
1 + ω90 Tm
=
(
)
0
f 2 ( ω90 ) = −
Ti
Tm + 0.5τ m
1
1 + ω 90 Tm
2
0
[
2
[
0
[(T
m
0
0
0
+ {1 + ω 90 Tm }τ m ) cot(−ω90 τm − tan −1 ω90 Tm ) + ω90 Tm
2
2
0
0
0
0
0
2
]
]
ω900 Tm + (1 + ω90 0 2Tm 2 ) τ m cos(− ω90 0 τm − tan− 1 ω90 0 Tm ) + (1 + 2ω 900 2 Tm 2 ) sin(− ω90 0 τm − tan −1 ω900 Tm )
− ω900 Tm cos(−ω90 0 τm − tan −1 ω90 0 Tm ) + ω90 0 [Tm + (1 + ω900 Tm ) τm ] sin(− ω900 τ m − tan− 1 ω90 0 Tm )
3
2
2
2
2
0
]
Rule
Kc
Chen et al. [53]
- continued
Model: Method 6
Ogawa [54] – deduced from
graphs
Model: Method 6
Tm
A m = 2. 07 , φ m = 46. 50 ,
Ms = 1. 40
0. 80Tm
τm K m
Tm
A m = 1.96 , φ m = 44.10 ,
Ms = 1. 50
βTm
β
τm Tm
α
β
0.5
1.0
0.5
1.0
0.5
1.0
0.5
1.0
0.9
0.6
0.7
0.47
0.47
0.36
0.4
0.33
1.3
1.6
1.3
1.7
1.3
1.8
1.3
1.8
2.0
10.0
2.0
10.0
2.0
10.0
2.0
10.0
0.45
0.4
0.4
0.35
0.32
0.3
0.3
0.29
2.0
7.0
2.2
7.5
2.4
8.5
2.4
9.0
Ti
K m ( λ + τm )
Tm +
Km (τ m + λ )
Model: Method 21
Ultimate cycle
τm
2( λ + τ m )
2
λ = 0.333 τm
Tm > τ m
Tm ( τ m + 2λ ) − λ2
Tm ( τ m + 2λ ) − λ2
Chun et al. [57]
20% uncertainty in the
process parameters
33% uncertainty in the
process parameters
50% uncertainty in the
process parameters
60% uncertainty in the
process parameters
Desired closed loop
e− τ m s
response =
,
( λ s + 1)
Tm + 025
. τm
Tm + 0.25τ m
Km λ
Isaksson and Graebe [56]
Model: Method 6
λ = 0.4Tm
τ m + Tm
2




0.65
  T
 

1881
.
Tm 
1
m
166
. τ m 1 + 
 

0.65 
Km τ m   T


  Tm + τ m  
m
1 +  T + τ 

m
  m

Regulator – minimum IAE –
Shinskey [59] – page 167.
Model: Method not
specified
Regulator – minimum IAE –
Shinskey [17] – page 121.
Model: Method not
specified
T
3.05 − 0.35 u
τm
µ1 =
(
Tuning rules developed
from Ku , Tu
2

T  
T
Tu  0.87 − 0.855 u + 0.172  u  

τm
 τ m  

Ku
Regulator – minimum IAE –
Shinskey [16] – page 148
Model: Method 6
Regulator – nearly minimum
IAE, ISE, ITAE – Hwang
[60]
Model: Method 8
0. 76Tm
τm K m
τm Tm
Model: Method 6
Model: Method 1 or
Method 18
Comment
α
Km
α
Lee et al. [55]
McMillan [58]
Ti
055
. Ku
0.78Tu
τ m Tm = 0.2
0. 48Ku
0.47 Tu
τ m Tm = 1
0.5848Ku
0.81Tu
τ m Tm = 0.2
0.5405K u
0.66Tu
τ m Tm = 0.5
0.4762K u
0.47Tu
τ m Tm = 1
0.4608 K u
0.37Tu
τ m Tm = 2
( 1 − µ1 ) K u ,
114
. 1 − 0.482ω u τ m + 0.068ω 2u τ 2m
Ku K m 1 + K u K m
Kc µ 2 Ku ω u ,
)
µ2 =
(
0.0694 −1 + 2.1ω u τ m −
0.367ω 2u τ m2
Ku Km 1 + Ku Km
)
Decay ratio = 0.15 τ
01
. ≤ m ≤ 2.0
Tm
Kc
Rule
Regulator – nearly minimum
IAE, ISE, ITAE – Hwang
[60]
(continued)
Model: Method 8
Servo – small IAE – Hwang
[60]
µ1 =
Model: Method not
specified
0.0724ω 2u τ 2m
K u K m 1 + K u Km
( 1 − µ1 ) K u ,
µ1 =
µ1 =
µ1 =
µ1 =
Hang et al. [65]
(
1.09 1 − 0.497ω u τ m +
Model: Method 8
r = parameter related to the
position of the dominant real
pole.
( 1 − µ1 ) K u ,
(
1.03 1 − 0.51ωu τm +
(
0.0759ω u2τ m2
Ku Km 1 + Ku Km
)
1.07 1 − 0.466ω uτ m + 0.0667ω u2τ m2
Ku Km 1 + Ku Km
(
(
−1 + 2.54ω u τm − 0.457ωu2 τm2
K u Km 1 + K u Km
)
( 1 − µ1 ) K u ,
0.0647ω 2u τ m2
(
0.0386 −1 + 3.26ω uτ m − 0.6ω 2u τ m2
K u Km 1 + K u Km
µ2 =
(
0.0328 −1 + 2.21ω u τm −
Decay ratio = 0.25 τ
01
. ≤ m ≤ 2.0
Tm
)
0.338ω 2u τ m2
K u Km 1 + K u Km
)
µ2 =
(
0.0477 − 1 + 2.07ω u τ m −
0.333ω 2u τ 2m
µ2 =
)
Ku K m 1 + K u K m
Kc µ 2 Ku ω u ,
)
Decay ratio = 0.1, r = 0.5 τ
01
. ≤ m ≤ 2.0
Tm
)
Kc µ 2 Ku ω u ,
0.0657ω 2u τ 2m
Ku Km 1 + Ku Km
Decay ratio = 0.2 τ
01
. ≤ m ≤ 2.0
Tm
0.054
µ2 =
µ2 =
)
Ku Km 1 + K u K m
114
. 1 − 0.466ωu τ m +
Kc µ 2 Ku ω u ,
Kc µ 2 Ku ω u ,
( 1 − µ1 ) K u ,
(
Comment
Kc µ 2 Ku ω u ,
( 1 − µ1 ) K u ,
111
. 1 − 0.467ω u τ m +
)
Ti
(
0.0609 −1 + 197
. ω uτ m − 0.323ωu2 τm2

 τm


 11 T + 13  

 12 + 2
m




τ
 37 m − 4 
4

 0.2


5
15
 Tm


Ku
6
 τm



 11 T + 13 
m
 15 + 14


 τm

37
− 4



 Tm

Ku Km 1 + Ku Km
)
Decay ratio = 0.1, r = 0.75 τ
01
. ≤ m ≤ 2.0
Tm
Decay ratio = 0.1, r = 1.0 τ
01
. ≤ m ≤ 2.0
Tm
 τm
 
τ
11 T + 13  
016
. ≤ m < 096
. ;
m

 + 1Tu
Tm

 37 τm − 4 
Servo response: 10%
 T
 
m

 
overshoot, 3% undershoot
Servo – minimum ISTSE –
Zhuang and Atherton [20]
0.361K u
0.083 (1.935 K m K u + 1)Tu
Model: Method not relevant
Regulator – minimum ISTSE  1892

. K m K u + 0244
.
 0.706Km K u − 0.227 
 Ku 
 Ku
– Zhuang and Atherton [20] 
3249
.
K
K
+
2
.
097
.


 0.7229Km K u + 12736

m u
Model: Method not relevant
2
( Kc / K uω u )
Servo – nearly minimum IAE 
τm
 τ m  
2
.
+ 0.0376
 Ku 
 τ m  
and ITAE – Hwang and  0.438 − 0110
τm
Tm
T
 m  
 0.0388 + 0108
.
− 0.0154 
 

Tm
 T m  
Fang [61]

Model: Method 9
2
( Kc / K uω u )
Regulator – nearly minimum 
 τ m  
τm
2
 τm  
IAE and ITAE – Hwang and  0.515 − 0.0521 Tm + 0.0254  Tm   Ku 
τm
0.0877 + 0.0918
− 0.0141  

Tm
 Tm  
Fang [61]

Model: Method 9
2
( Kc / K u ω u )

Simultaneous
  
 0.46 − 0.0835 τ m + 0.0305 τ m   K
2
u



  
Servo/regulator – Hwang 
Tm
 Tm  
 0.0644 + 0 .0759 τ m − 0.0111  τ m  

Tm
and Fang [61]
 Tm  

Model: Method 9
01
. ≤
τm
≤ 2.0
Tm
01
. ≤
τm
≤ 2.0
Tm
τm
≤ 2.0 ;
Tm
decay ratio = 0.03
01
. ≤
τm
≤ 2.0 ;
Tm
decay ratio = 0.12
01
. ≤
01
. ≤
τm
≤ 2.0
Tm
K me − sτ

1
– Controller G c ( s) = Kc  b +
 . 2 tuning rules
1 + sTm
Ti s

m
Table 2: PI tuning rules - FOLPD model - G m ( s) =
Kc
Rule
Ti
Direct synthesis
(Maximum sensitivity)
029
. e
Astrom and Hagglund [3] dominant pole design –
page 205-208
Comment
− 2.7 τ + 3.7τ 2
Tm
K mτ m
8.9τ m e −6.6 τ + 3.0 τ or
2
,
τ = τ m ( τ m + Tm )
0.79Tm e −1.4 τ + 2.4 τ
2
Model: Method 3 or 4
078
. e− 4.1τ + 5.7 τ Tm
Km τ m
8.9τ m e −6.6 τ + 3.0 τ or
Astrom and Hagglund [3] modified Ziegler-Nichols –
page 208
Model: Method 3 or 4
04
. Tm
K mτ m
0.7Tm
2
2
0.79Tm e −1.4 τ + 2.4 τ
2
b = 0.81e0.73 τ + 1.9 τ
τ
Ms = 1.4 , 014
. ≤ m ≤ 55
.
Tm
2
b = 0.44e0.78 τ − 0.45τ ;
τ
Ms = 2.0, 014
. ≤ m ≤ 55
.
Tm
2
b = 0.5; 01
. ≤
τm
≤2
Tm
Table 3: PI tuning rules - FOLPD model - G m ( s) =
K me − sτ
– Two degree of freedom controller:
1 + sTm
m

1 
E (s ) − α Kc R( s) . 1 tuning rule
U(s) = K c 1 +
Tis 

Rule
Minimum servo/regulator
Taguchi and Araki [61a]
Model: ideal process
Kc
Ti
Performance index minimisation




1 
0.7382

0
.
1098
+

τm
K m 
− 0.002434

Tm


Ti
( 5b ) 6
τ 
τ
α = 0. 6830 − 0.4242 m + 0.06568  m 
Tm
 Tm 
6
Ti
( 5b )
Comment
2
3

τ 
τ  
τ
= Tm  0.06216 + 3.171 m − 3.058  m  + 1. 205  m  


Tm
 Tm 
 Tm  

2
τm
≤ 1.0 .
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of tuning
rules of Chien et al. [10]

1
Table 4: PI tuning rules - non-model specific – controller G c ( s) = Kc  1 +
 . 20 tuning rules
Ts

i 
Kc
Ti
Comment
0.45Ku
0.83Tu
Quarter decay ratio
0.45Ku
1 
5.22 
. −
 522

p1 
T1 
p1 , T1 = decay rate, period
measured under
proportional control when
Kc = 05
. Ku
** Hang et al. [36]
0.25Ku
0.2546Tu
dominant time delay process
McMillan [14] – page 90
0.3571K u
Tu
Pessen [63]
0.25Ku
0.042 K uTu
dominant time delay process
0.4698 K u
0.4373Tu
01988
.
Ku
0.0882Tu
0.2015 Ku
01537
.
Tu
Parr [64] – page 191
05
. Ku
043
. Tu
Gain margin = 2, phase
margin = 20 0
Gain margin = 2.44, phase
margin = 61 0
Gain margin = 3.45, phase
margin = 46 0
Quarter decay ratio
Yu [122] – page 11
0.33Ku
2Tu
Other tuning rules
Parr [64] – page 191
0667
. K25%
T25%
Quarter decay ratio
042
. K25%
T25%
‘Fast’ tuning
0.33K25%
T25%
‘Slow’ tuning
0333
.
G p ( jω u )
2Tu
Bang-bang oscillation test
0.5
4
Alfa-Laval Automation
ECA400 controller
Rule
Ultimate cycle
Ziegler and Nichols [8]
Hwang and Chang [62]
Astrom and Hagglund [3] –
page 142
McMillan [14] – pages 4243
Parr [64] – page 192
Hagglund and Astrom [66]
G p ( jω 135 )
ω135
0.25
1.6
ω135
0
G p ( jω 135 )
0
0
0
Alfa-Laval Automation
ECA400 controller - process
has a long delay
Kc
Rule
(
)
tan φ m − φ p ω − 0 .5 π
Leva [67]
(
)
Ti
(
)
tan φ m − φ pω − 05
. π
G p ( jω ) 1 + tan φ m − φ p ω − 05
.π
ω
sin φ m
tanφ m
ω 90
2
Astrom [68]
G p ( jω 90 )
Comment
φ m > φ pω + 0.5π ,
φ pω = process phase at
frequency ω
0
0
1
ωu
φ m = 450 ,‘small’ τ m
016
. Tu tan φ m
V = relay amplitude, A = limit
cycle amplitude.
A3
A2
A m ≥ 2 , φ m ≥ 60 0
1
Calcev and Gorez [69]
2 2 G p ( jω u )
Cox et al. [70]
020
. VTu sin φ m
A
φ m = 15 0 , ‘large’ τ m
Direct synthesis
Vrancic et al. [71]
05
. A3
A1A2 − Km A 3
Vrancic [72]
05
. A3
A1A2 − Km A 3
1
333
. τm
3.7321
ω150
Gain margin = 2,
Phase margin = 30 0
1.18K u K m − 1.72
(0.33K u K m − 0.17) ωu
0.1 ≤ Ku Km ≤ 0.5
0. 50K u K m − 0. 36
(0.33K u K m − 0.17) ωu
K u K m > 0.5
0.4830
Friman and Waller [41]
G p ( jω150 )
0
0
1.18 K u K m − 1. 72
Kristiansson and
Lennartson [158a]
K uK m
2
0. 50K u K m − 0. 36
Ku Km
2
20K135 Km − 160
20K135 K m − 160
0
0
K135 K m
2
5.4K135 K m − 13. 6
0
K135 K m
2
0
1
(0.315 K135 Km − 0.175 )ω u
5. 4K135 K m − 13. 6
K u K m < 0. 1 ;
K 135 K m > 0 .1
0
(1.32K135 K m − 3.2)ω u
0
A1 = y1( ∞) , A 2 = y 2 ( ∞) , A 3 = y3 ( ∞)
Alternatively, if the process model is G m (s) = K m
(
1 + b1 s + b2 s2 + b 3s3 − s τ
e
, then
1 + a1s + a 2 s3 + a 3s3
m
)
A1 = Km ( a1 − b1 + τ m ) , A2 = K m b 2 − a2 + A1a1 − b1 τm + 05
. τ m2 ,
(
K u K m < 0. 1 ;
K 135 0 K m ≤ 0 .1
0
0
Modified Ziegler-Nichols
process reaction method
A3 = Km a 3 − b3 + A 2 a1 − A1a 2 + b2 τ m − 05
. b1τ m 2 + 0167
. τ m3
)
0

1 
Table 5: PI tuning rules - non-model specific – controller G c ( s) = K c  b +
 . 1 tuning rule
Ti s 

Rule
Kc
Ti
Direct synthesis
(Maximum sensitivity)
2.9 κ − 2.6κ 2
Astrom and Hagglund [3] Ms = 1.4 – page 215
Comment
0.053 Ku e
κ = 1 Km Ku
,
b = 11
. e −0.0061κ + 1.8 κ ;
0 < Km K u < ∞ .
maximum sensitivity Ms
=1.4
2
0.90Tu e− 4.4κ + 2. 7κ
2
b = 048
. e 0.40κ − 0.17 κ ;
0 < Km K u < ∞ .
Ms = 2.0
2
013
. K ue
1.9κ − 1.3κ 2
0.90Tu e
− 4.4κ + 2. 7κ 2

1 
Table 6: PI tuning rules - non-model specific – controller U( s) = Kc  1 +
 E( s) + Kc ( b − 1)R (s) . 1 tuning rule
Ti s 

Kc
Rule
Ti
Comment
Direct synthesis
b = [0.5,08
. ] - good servo
A1
Vrancic [72]
2
Kc
Kc
( 6)
( 6)
=
=
A 1A 2 − K m A 3 −
A 1A 2 − K m A 3 +
2
(1 − b )( K
0
2
2
2 K c (6 )
+
K c( 6) K m 2
2
(1 − b )
2
m
A 3 + A 1 − 2K mA 1A 2
3
)
( K m A3 − A 1A 2 ) 2 − (1 − b2 )A 3 ( Km 2 A3 + A13 − 2K m A1A2 )
(1 − b )( K
y( τ ) 

y1( t) =  Km −
 dτ , y2 ( t) =

∆u 
∫
Km +
1
and regulator response
( K mA 3 − A 1A 2 ) 2 − (1 − b2 )A 3 ( Km 2 A3 + A13 − 2K mA 1A2 )
2
2
t
Kc
( 6)
m
A 3 + A 1 − 2K mA 1A 2
3
t
∫ (A 1 − y1 (τ))dτ ,
0
)
t
y3 ( t) =
∫(A
0
2
− y2 ( τ) )dτ
, Km A 3 − A1A 2 < 0
, Km A 3 − A1A 2 > 0
Table 7: PI tuning rules - non-model specific – controller U( s) = Kc Y(s) −
Kc
E ( s) . 1 tuning rule
Ti s
Kc
Ti
Comment
− TCL + 1414
.
TC LTm + τ mTm
− T CL + 1414
. T C L Tm + Tm τ m
Underdamped system
response - ξ = 0.707 .
Rule
Direct synthesis
2
Chien et al. [74]
(
K m TC L2
+ 1414
. TCL τ m + τ m
2
2
)
Tm + τ m
τm > 0.2Tm
Table 8: PI tuning rules - IPD model G m ( s) =
K me− sτ m
s


- controller G c ( s) = Kc  1 + 1  . 21 tuning rules

Rule
Kc
Ti
Process reaction
Ziegler and Nichols [8]
Model: Method 1.
0.9
Km τ m
333
. τm
0.6
Km τ m
2.78τ m
Two constraints method –
Wolfe [12]
Model: Method 1
087
.
Km τ m
4.35τm
0.487
Km τ m
8.75τ m
Tyreus and Luyben [75]
Model: Method 2 or 3
Astrom and Hagglund [3] –
page 138
Model: Not relevant
Regulator tuning
Minimum ISE – Hazebroek
and Van der Waerden [9]
Model: Method 1
Shinskey [59] – minimum
IAE regulator – page 74.
Model: Method not
specified
Poulin and Pomerleau [82] –
minimum ITAE (process
output step load
disturbance)
Model: Method 2
Poulin and Pomerleau [82] –
minimum ITAE (process
input step load disturbance)
Model: Method 2
Ultimate cycle
Tyreus and Luyben [75]
0.63
Km τ m
Comment
Quarter decay ratio
Decay ratio = 0.4; minimum
error integral (regulator
mode).
Decay ratio is as small as
possible; minimum error
integral (regulator mode).
Maximum closed loop log
modulus = 2dB ; closed loop
time constant = τ m 10
Ultimate cycle ZieglerNichols equivalent
3.2τm
Performance index minimisation
15
.
K mτ m
556
. τm
09259
.
Km τ m
4τ m
05264
.
K mτ m
4.5804 τ m
05327
.
K mτ m
38853
.
τm
0.31Ku
2.2Tu
061
. Ku
Tu
05
.
Km τ m
5τ m
Model: Method 2 or 3.
Regulator – minimum IAE –
Shinskey [17] – page 121.
Model: method not
specified
Robust
Fruehauf et al. [52]
Model: Method 5
Ts
i 
Maximum closed loop log
modulus = 2dB ; closed loop
time constant = τ m 10
Rule
Kc
Chien [50]
1  2λ + τ m 
Km  [λ + τ m ]2 
Model: Method 2
λ
15
. τm
Ogawa [54] – deduced from
graph
Model: Method 5
Overshoot
58%
TS
6τ m
Ti
Comment
2λ + τm
 1

λ=
, τm  [50];
 Km

λ > τ m + Tm (Thomasson
[51])
PP
17
. Km
τm
RT
7Km
τm
2.5τ m
35%
11τ m
2.0K m
τm
16K m
τm
35
. τm
26%
16τ m
2.2K m
τm
23Km
τm
45
. τm
22%
20τ m
25
. Km
τm
30K m
τm
Zhang et al. [135]
λ = [15
. τ m ,4.5τ m ]
- values deduced from
graphs
11τ m
20% uncertainty in the
process parameters
Kmτm
12τ m
30% uncertainty in the
process parameters
0.34
K mτ m
13τ m
40% uncertainty in the
process parameters
0.30
Km τ m
14τ m
50% uncertainty in the
process parameters
0.27
K mτ m
15τ m
60% uncertainty in the
process parameters
0.45
K mτ m
0.39
Rule
Kc
Ti
α
K mτ m
βτ m
Comment
Direct synthesis
Wang and Cluett [76] –
deduced from graph
Model: Method 2
Cluett and Wang [44]
Model: Method 2
Rotach [77]
Model: Method 4
Poulin and Pomerleau [78]
Model: Method 2
Closed Damp. Gain Phase
loop Factor margin margin
time
ξ
Am
φm
const.
[deg.]
τm
0.707
1.3
11
Closed Damp. Gain Phase
loop Factor margin margin α
β
time
ξ
Am
φm
const.
[deg.]
0.9056 2.6096 τ m
1.0
1.3
14 0.8859 3.212
2τm
0.707
2.5
33
0.5501 4.0116
2τm
1.0
2.3
37
0.6109 5.2005
3τ m
0.707
3.6
42
0.3950 5.4136
3τ m
1.0
3.0
46
0.4662 7.1890
α
β
4τm
0.707
4.7
47
0.3081 6.8156
4τm
1.0
4.0
52
0.3770 9.1775
5τ m
0.707
5.9
50
0.2526 8.2176
5τ m
1.0
4.8
56
0.3164 11.166
6τ m
0.707
7.1
52
0.2140 9.6196
6τ m
1.0
5.6
59
0.2726 13.155
7τm
0.707
8.2
54
0.1856 11.022
7τm
1.0
6.3
61
0.2394 15.143
8τ m
0.707
9.2
55
0.1639 12.424
8τ m
1.0
7.2
62
0.2135 17.132
9τm
0.707
10.4
56
0.1467 13.826
9τm
1.0
8.0
64
0.1926 19.120
10τ m 0.707
11.5
57
0.1328 15.228 10τ m
1.0
8.7
65
0.1754 21.109
11τm 0.707
12.7
58
0.1213 16.630 11τm
1.0
9.6
66
0.1611 23.097
12τ m 0.707
13.8
59
0.1117 18.032 12τ m
1.0
10.4
67
0.1489 25.086
13τ m 0.707
14.9
59
0.1034 19.434 13τ m
1.0
11.2
67
0.1384 27.074
14τ m 0.707
16.0
60
0.0963 20.836 14τ m
1.0
12.0
68
0.1293 29.063
15τ m 0.707
17.0
60
0.0901 22.238 15τ m
1.0
12.7
68
0.1213 31.051
16τ m 0.707
18.2
60
0.0847 23.640 16τ m
1.0
13.6
69
0.1143 33.040
0.9588
Km τ m
30425
.
τm
Closed loop time constant =
τm
0.6232
Km τ m
52586
.
τm
Closed loop time constant =
2τ m
0.4668
Km τ m
7.2291τ m
Closed loop time constant =
3τ m
0.3752
Km τ m
91925
.
τm
Closed loop time constant =
4τ m
0.3144
Km τ m
111637
.
τm
Closed loop time constant =
5τ m
0.2709
Km τ m
131416
.
τm
Closed loop time constant =
6τ m
0.75
K mτ m
241
. τm
034
. K u or
2.13
Km Tu
104
. Tu
Damping factor for
oscillations to a disturbance
input = 0.75.
Maximum sensitivity
= 5 dB
Rule
Gain and phase margin –
Kookos et al. [38]
Model: Method 2
Representative results
Kc
Ti
ωp
1
ω p 05
. π − ωpτ m
A m Km
(
Comment
)
ωp =
A m φ m + 0.5πA m (A m − 1)
(A
2
m
)
− 1 τm
0.942
K mτ m
4510
. τm
Gain Margin = 1.5
Phase Margin = 22 .5 0
0.698
K mτ m
4.098τ m
Gain Margin = 2
Phase Margin = 30 0
0.491
K mτ m
6942
.
τm
Gain Margin = 3
Phase Margin = 45 0
0384
.
K mτ m
18.710τm
Gain Margin = 4
Phase Margin = 60 0
Other methods
Penner [79]
Model: Method 2
Srividya and Chidambaram
[80]
Model: Method 5
0.58
K mτ m
10τ m
Maximum closed loop gain =
1.26
0.8
K mτ m
59
. τm
Maximum closed loop gain =
2.0
0.67075
K mτ m
36547
.
τm
Table 9: PI tuning rules - IPD model G m ( s) =
Rule
K me− sτ m
s


- controller G c ( s) = Kc  1 + 1 

Kc
Ti
0463
. λ + 0.277
K mτ m
τm
0.238λ + 0.123
1
. 1 tuning rule
Ts
1
+
Tf s

i
Comment
Robust
Tan et al. [81]
Model: Method 2
Tf =
τm
,
5.750λ + 0.590
λ = 0.5
Table 10: PI tuning rules - IPD model G m ( s) =
Kc
Rule
Direct synthesis
Chien et al. [74]
Model: Method 1
K me− sτ m
s
(
1.414TCL + τ m
K m TCL + 1.414TCL τ m + τ m
2
- controller U( s) = Kc Y(s) −
Kc
E ( s) . 1 tuning rule
Ti s
Ti
Comment
1414
. TCL + τ m
Underdamped system
response - ξ = 0.707 .
τ m ≤ 0.2Tm
Table 11: PI tuning rules - IPD model G m ( s) =
K me− sτ m
s
- Two degree of freedom controller:

1 
E (s ) − α Kc R( s) . 1 tuning rule
U(s) = K c 1 +
Tis 

Rule
Kc
Servo/regulator tuning
Taguchi and Araki [61a]
Model: ideal process
Ti
Performance index minimisation
0.049 Ku
2. 826 Tu
τm
≤ 1.0 .
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of tuning
rules of Chien et al. [10]
α = 0.506 , φ c = −1640
0.066 Ku
2. 402 Tu
α = 0.512 , φ c = −1600
0.099 Ku
1.962 Tu
α = 0.522 , φ c = −1550
0.129 Ku
1.716 Tu
α = 0.532 , φ c = −1500
0.159 Ku
1.506 Tu
α = 0.544 , φ c = −1450
0.189 Ku
1.392 Tu
α = 0.555 , φ c = −1400
0.218 K u
1.279 Tu
α = 0.564 , φ c = −1350
0.250 Ku
1.216 Tu
α = 0.573 , φ c = −1300
0.286 Ku
1.127 Tu
α = 0.578 , φ c = −1250
0.330 Ku
1.114 Tu
α = 0.579 , φ c = −1200
0.351K u
1.093Tu
α = 0.577 , φ c = −1180
0. 7662
K m τm
4.091 τm
α = 0. 6810
Minimum ITAE Pecharroman and Pagola
[134b]
Km =1
Model: Method 6
Comment
Table 12: PI tuning rules – FOLIPD model G m (s) =
Km e− sτ


- controller G c ( s) = Kc  1 + 1  . 6 tuning rules
s(1 + sTm )
Ts

i 
m
Kc
Rule
Ti
Comment
Ultimate cycle
2




  T  0.65 
1477
.
Tm 
1

332
. τ m 1 +  m  
Tuning rules developed
  τ m  
Model: Method not relevant K m τ 2    0.65 
from Ku , Tu
m 
Tm

1
+




  τm  
Regulator tuning
Minimum performance index
Minimum IAE – Shinskey
[59] – page 75.
0.556
3.7( τ m + Tm )
Model: Method not
K m ( τ m + Tm )
specified
Minimum IAE – Shinskey
[59] – page 158
0.952
4(Tm + τ m )
Model: Open loop method
Km (Tm + τ m )
not specified
2
Minimum ITAE – Poulin and
b
Tm
Pomerleau [82] – deduced K m ( τ m + Tm ) a τ + T 2 + 1
a( τ m + Tm )
( m m)
from graph
McMillan [58]
Model: Method 2
τm Tm
a
b
τm Tm
a
b
τm Tm
a
b
Output step load
disturbance
0.2
0.4
0.6
0.8
5.0728
4.9688
4.8983
4.8218
0.5231
0.5237
0.5241
0.5245
1.0
1.2
1.4
1.6
4.7839
4.7565
4.7293
4.7107
0.5249
0.5250
0.5252
0.5254
1.8
2.0
4.6837
4.6669
0.5256
0.5257
0.2
0.4
0.6
0.8
3.9465
3.9981
4.0397
4.0397
0.5320
0.5315
0.5311
0.5311
1.0
1.2
1.4
1.6
4.0397
4.0337
4.0278
4.0278
0.5311
0.5312
0.5312
0.5312
1.8
2.0
4.0218
4.0099
0.5313
0.5314
(2 tuning rules)
Input step load disturbance
Direct synthesis
Poulin and Pomerleau [78]
Model: Method 2
034
. K u or
2.13
Km Tu
104
. Tu
Maximum sensitivity
= 5 dB
Table 13: PI tuning rules – FOLIPD model G m (s) =
Rule
Kc
Km e− sτ


- controller G c ( s) = Kc  b + 1  . 1 tuning rule
s(1 + sTm )
Ti s 

m
Ti
Comment
Direct synthesis
041
. e− 0.23τ + 0.019τ
,
Km ( Tm + τ m )
b = 0.33e2.5τ −1.9 τ .
τ
Ms = 1.4; 014
. ≤ m ≤ 55
.
Tm
2
Astrom and Hagglund [3] maximum sensitivity – pages
210-212
Model: Method 1.
2
57
. τ m e1.7 τ − 0.69τ
2
τ = τ m ( τ m + Tm )
081
. e −1.1τ + 0.76 τ
Km (Tm + τ m )
2
b = 078
. e−1.9 τ + 1. 2τ .
τ
Ms = 2.0; 014
. ≤ m ≤ 55
.
Tm
2
3.4τ me 0.28τ − 0.0089 τ
2
Table 14: PI tuning rules - FOLIPD model G m (s) =
Km e− sτ
- Two degree of freedom controller:
s(1 + sTm )
m

1 
E (s ) − α Kc R( s) . 1 tuning rule.
U(s) = K c 1 +
Tis 

Rule
Kc
Ti
Servo/regulator tuning
Taguchi and Araki [61a]
Model: ideal process
Minimum performance index
0.049 Ku
2. 826 Tu
τm
≤ 1.0 .
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of tuning
rules of Chien et al. [10]
α = 0.506 , φ c = −1640
0.066 Ku
2. 402 Tu
α = 0.512 , φ c = −1600
0.099 Ku
1.962 Tu
α = 0.522 , φ c = −1550
0.129 Ku
1.716 Tu
α = 0.532 , φ c = −1500
0.159 Ku
1.506 Tu
α = 0.544 , φ c = −1450
0.189 Ku
1.392 Tu
α = 0.555 , φ c = −1400
0.218 K u
1.279 Tu
α = 0.564 , φ c = −1350
0.250 Ku
1.216 Tu
α = 0.573 , φ c = −1300
0.286 Ku
1.127 Tu
α = 0.578 , φ c = −1250
0.330 Ku
1.114 Tu
α = 0.579 , φ c = −1200
0.351K u
1.093Tu
α = 0.577 , φ c = −1180




1 
0 .2839

0 .1787 +


τm
Km
+ 0. 001723

Tm


τ 
τ
4. 296 + 3. 794 m + 0. 2591 m 
Tm
 Tm 
τ
α = 0. 6551+ 0. 01877 m
Tm
Minimum ITAE Pecharroman and Pagola
[134b]
K m = 1 ; Tm = 1
Model: Method 4
Comment
2
Table 15: PI tuning rules - SOSPD model
K m e− sτ
Km e− sτ
(1 + Tm1s)(1 + Tm2 s)
m
m
Tm1 s2 + 2ξ m Tm1s + 1
2
or

1
- controller G c ( s) = Kc  1 +
 . 11 tuning rules.
Ts

i 
Kc
Ti
Comment
Tm1 + Tm 2 + 0.5τ m
K m τ m ( 2λ + 1)
Tm1 + Tm 2 + 05
. τm
λ varies graphically with
τm (Tm1 + Tm 2 ) -
Rule
Robust
Brambilla et al. [48]
τ m (Tm1 + Tm 2 )
Model: Method 1
0.1
0.2
0.5
Direct synthesis
Gain and phase margin - Tan
et al. [39] – repeated pole
Model: Method 11
Regulator tuning
Minimum IAE - Shinskey
[59] – page 158
Model: Open loop method
not specified
Model: Method 7
Minimum IAE – Shinskey
[17] – page 48. Model:
method not specified.
τ m (Tm1 + Tm 2 )
λ
3.0
1.8
1.0
(
β Ti ω φ 1 + βTm ω φ
(
A m 1 + βTi ω φ
)
)
2
τ m (Tm1 + Tm 2 )
λ
0.6
0.4
0.2
1.0
2.0
5.0
1
[
]
β = 0.8,
ωφ < ω u
2
λ
0.2
10.0
βω φ tan − 2 tan −1 βTm ω φ − βτ m ω φ − φ m
τm
< 0.33 ;
Tm
β = 0.5,
τm
> 0.33
Tm
Minimum performance index
100Tm1
K m (τ m
Minimum ISE – McAvoy
and Johnson [83] – deduced
ξm
from graph
1
1
Model: Method 1
1
Minimum ITAE – Lopez et
al. [84] – deduced from
graph
01
. ≤ τ m ( Tm1 + Tm 2 ) ≤ 10
T m1


−

+ Tm2 ) 50 + 551 − e τ m + T m2







3 Tm1



−
τ +T
τ m  0.5 + 35
. 1 − e ( m m2 ) 





α
Km
βτ m
τ m Tm 1
α
β
ξm
τ m Tm 1
α
β
ξm
τ m Tm 1
α
β
0.5
4.0
10.0
0.8
5.7
13.6
1.82
12.5
25.0
4
4
0.5
4.0
4.3
27.1
3.45
6.67
7
7
0.5
4.0
7.8
51.2
3.85
5.88
α
β
α
Km
ξm
0.5
0.5
0.5
βTm
α
τ m Tm 1
0.1
3.0
1.0
0.2
10.0
0.3
0.77 Tm1
K m τm
070
. Tm1
Km τ m
080
. Tm1
Km τ m
080
. Tm1
Km τ m
β
ξm
τ m Tm 1
α
β
ξm
τ m Tm 1
2.86
0.83
4.0
1
1
1
0.1
1.0
10.0
7.0
0.95
0.35
2.00
2.22
5.00
4
4
4
0.1
1.0
4.0
283
. ( τ m + Tm 2 )
2.65( τ m + Tm 2 )
2.29( τ m + Tm 2 )
167
. ( τ m + Tm2 )
τm
Tm1
40.0 0.83
6.0 3.33
0.75 10.0
T
= 02
. , m2 = 01
.
Tm 1
τm
T
= 02
. , m2 = 0.2
Tm1
Tm1
τm
T
= 02
. , m2 = 05
.
Tm1
Tm 1
τm
T
= 02
. , m2 = 10
.
Tm1
Tm 1
Rule
Kc
Minimum IAE - Huang et al.
[18]
K c ( 31)
Ti
1
Comment
0<
Ti ( 31)
Tm 2
τ
≤ 1 ; 01
. ≤ m ≤1
Tm1
Tm1
Model: Method 1
1
Kc (31) =
− 0.9077
−0.063

 τm 
 τm 
1 
τ
Tm2
τ mTm2
6.4884 + 4.6198 m − 3491

.
− 253143
.
+
0
.
8196
−
52132
.




2
Km 
Tm 1
Tm1
Tm1
 Tm1 
 Tm1 


0.5961
0.7204
1.0049
1.005
 τm 
 Tm2 
 Tm2 
 Tm 2 
1 
T
− 7.2712
+
− 180448
.
+ 5.3263
+ 139108
.
+ 0.4937 m2






Km 
τm
 Tm1 
 Tm1 
 Tm1 
 Tm1 

1
+
Km
0.8529
0.5613
0.557
1.1818


Tm 2  τ m 
Tm 2  τ m 
τm  Tm2 
τ m  Tm 2 
191783

.
+ 12.2494
+ 8.4355
− 17.6781








Tm1  Tm1 
Tm1  Tm1 
Tm1  Tm1 
Tm1  Tm1 


1
+
Km
Ti
( 31)



τ

− 0.7241eT


τ m Tm 2
Tm 2
m
m1
− 2.2525e
Tm 1
+ 54959
.
e
Tm1 2




2
2
3

 τm 
 Tm2 
 τm  
τm
Tm2
τ m Tm2

= Tm1 0.0064 + 3.9574
+ 4.4087
− 6.4789
.
− 15083
.
 − 128702

 + 9.4348 
 
2
Tm1
Tm1

 Tm1 
Tm1
 Tm1 
 Tm1  

T
+ Tm 1 17.0736 m2
Tm1

2
 τm 
τ

 + 15.9816 m
T
T
 m1 
m1
2
2
 Tm2 
T 
τ 
.  m 2  − 10.7619 m 

 − 3909
T
T
 m1 
 m1 
 Tm1 
3
2
2

 τ  T 
T τ 
τ
+ Tm 1 − 10.684 m2  m  − 22.3194 m   m 2  − 6.6602 m
Tm1  Tm1 
Tm1

 Tm1   Tm1 
3
4



 Tm2 
T 

 + 6.8122 m 2 
 Tm 1 
 Tm1 
4



5
4
2
3
2
3

 τm 
 Tm2   τ m 
 τ m   Tm 2  
Tm2  τ m 
+ Tm1  75146
.
.
.
.

 + 28724

 + 114666

 
 + 111207

 
 
Tm1  Tm1 

 Tm1 
 Tm1   Tm 1 
 Tm1   Tm1  
4
5
6
5
6

 τm   Tm 2 
 Tm 2 
 τm 
 Tm2  
Tm2  τm 

+ Tm 1 − 12174
.
.
.


 − 4.3675
 − 2.2236
 − 0112

 + 10308

 
Tm1  Tm1 

 Tm1   Tm1 
 Tm1 
 Tm1 
 Tm1  
4
2
3
3
2
4

 τ  T 
 τm   Tm2 
 τ m   Tm2 
τm
+ Tm1 − 1. 9136 m   m2  − 34994
.
−
15777
.
.

 


 
 + 11408
T
T
T
T
T
T
T

 m1   m1 
 m1   m1 
 m1   m1 
m1
 Tm 2 


 Tm1 
5



Rule
Kc
Minimum IAE - Huang et al.
[18]
K c ( 32)
Ti
Comment
0.4 ≤ ξ m ≤ 1 ;
2
Ti ( 32)
0.05 ≤
Model: Method 1
2
Kc
( 32)
1
=
Km
1.4439
0.1456


 τm 
 τm 
τm
τm
 − 10.4183 − 209497

.
− 55175
.
ξ m − 265149
.
ξm
+ 42.7745
+ 105069
.



Tm1
Tm1

 Tm1 
 Tm1 

+
0.3157
−0.0541

τ 
τ 
1 
15.4103 m 

+ 34.3236ξ m 3.7057 − 17.8860ξ m 4 .5359 − 54.0584 ξ m 1.9593 + 22.4263ξ m  m 
Km 
 Tm1 
 Tm1 


+
4. 7426
τ 
τ 
τm
T 
1 
2.7497ξ m  m 
+ 50.2197ξ m 1.8288  m  − 171968
.
ξ m 2.7227 + 10293
.
ξ m m1 
Km 
 Tm1 
 Tm1 
Tm1
τm 


+
τ
ξ
1 
− 167667
.
e T + 14.5737eξ − 7.3025 e
Km 

m
Ti
( 32)
τm
≤1
Tm1
m1
m
m
τm
Tm1



2
3

 τm 
 τm  
τm
τm
2

= Tm1 11447
.
+ 45128
.
− 75.2486ξ m − 110807
. 
+ 345.3228ξ m + 191.9539
 − 12.282ξ m
 
Tm1
Tm1

 Tm1 
 Tm1  
2
4

τ 
τ  
τ
+ Tm1 359 .3345ξ m  m  − 158.7611 m ξ m 2 − 770.2897ξ m2 − 153633
.  m 
Tm1

 Tm1 
 Tm1  
3
2


τ 
τ 
τm
+ Tm1 − 412 .5409 ξ m  m  − 414 .7786ξ m 2  m  + 4850976
.
ξ m 3 + 864.5195ξ m 4 
Tm1

 Tm1 
 Tm1 

5
4
3
2


τ 
τ 
τ 
 τm 
3
+ Tm 1 55.4366 m  + 222 .2865 ξ m  m  + 275166
. ξ m 2  m  + 2052493
.

 ξm 

 Tm1 
 Tm1 
 Tm1 
 Tm1 

6
5


 τm  4
 τm 
 τm 
5
6

+ Tm 1 − 479 .5627
.
ξ m − 6.547
.
ξ m
 ξ m − 4731346
 − 432822
 + 99.8717ξ m 

 Tm1 
 Tm1 
 Tm1 

4
3
2


 τm 
 τm 
 τm 
τ
2
3
4
5
+ Tm1 − 735666
.
ξ
−
56
.
4418
ξ
−
37
.
497

 m

 m

 ξ m + 160.7714 m ξ m 
Tm1

 Tm1 
 Tm1 
 Tm1 

Rule
Servo tuning
Minimum IAE - Huang et al.
[18]
Model: Method 1
3
Kc (33) =
Kc
Ti
Comment
Minimum performance index
K c ( 33)
3
0<
Ti ( 33)
Tm 2
τ
≤ 1 ; 01
. ≤ m ≤1
Tm1
Tm1
− 1.0169
3.5959

τ 
 τm 
1 
τ
T
τ T
− 130454

.
− 9.0916 m + 2.6647 m2 + 9.162 m m2 2 + 0.3053 m 
+ 11075
.


Km 
Tm1
Tm 1
Tm1 
Tm1 
T



m
1

+
3.6843
0.8476
2.6083
2.9049
τ 
 Tm2 
T 
T 
1 
T 
− 2.2927 m 
− 310306
.
− 13.0155 m2 
+ 9.6899 m 2 
− 0.6418 m 2 


Km 
τm 
 Tm1 
 Tm1 
 Tm1 
 Tm1 


−0.2016
1.3293
0.801

1 
Tm2  τ m 
Tm 2  τ m 
τ m  Tm 2 
189643

+
.
− 39.7340
+ 28155
.






Km 
Tm1  Tm1 
Tm1  Tm1 
Tm 1  Tm1 


Ti
( 33)
τ T
3.956
τ
T

1 
τ m  Tm2 
T
T
− 2.0067
+


+ 4.8259e + 2.1137e + 84511
.
eT 
Km 
Tm1  Tm1 



2
2
3

 τm 
 Tm 2 
 τm  
τm
Tm2
τ m Tm2
= Tm1 0.9771− 0.2492
+ 0.8753
+ 3.4651
 − 38516
.
+ 7.5106 
 − 7 .4538
 
2
Tm1
Tm1

 Tm1 
Tm1
 Tm1 
 Tm1  


2
2
3
4

Tm2  τ m 
τ m  Tm 2 
 Tm 2 
 τm  
+ Tm1 116768
.
.
.

 − 10.9909

 − 161461

 + 82567

 
Tm1  Tm1 
Tm1  Tm1 
 Tm1 
 Tm1  


m
m2
m1
m1
m
m2
2
m1
3
2
2
3
4

 τ  T 
 Tm 2  
Tm2  τ m 
τ m  Tm 2 
+ Tm 1 − 181011
.
.
.

 + 6.2208 m   m2  + 219893

 + 158538

 
Tm1  Tm1 
Tm 1  Tm1 

 Tm1   Tm1 
 Tm1  
5
4
2
3
2
3

τ 
T τ 
T   τ 
 τ  T  
+ Tm1  − 4.7536 m  + 14.5405 m 2  m  − 2.2691 m2   m  − 8.387  m   m 2  
 Tm 1 
Tm1  Tm1 
 Tm1   Tm1 
 Tm1   Tm1  


4
5
6
5
6

 τ m   Tm2 
 Tm 2 
 τm 
 Tm 2  
Tm2  τ m 

+ Tm 1 − 16.651
.
.
.

 − 71990

 + 11496

 − 4.728

 + 11395

 
Tm 1  Tm1 

 Tm1   Tm 1 
 Tm1 
 Tm1 
 Tm1  
4
2
3
3
2
4

 τ  T 
 τ m   Tm 2 
 τ m   Tm2 
τ
+ Tm 1 0.6385 m   m 2  + 10885
.
+
31615
.

 


 
 + 4.5398 m
T
T
T
T
T
T
T

 m1   m1 
 m1   m1 
 m1   m1 
m1
 Tm 2 


 Tm1 
5



Rule
Kc
Minimum IAE - Huang et al.
[18]
K c ( 34 )
Ti
Comment
0.4 ≤ ξ m ≤ 1 ;
4
Ti ( 34)
0.05 ≤
Model: Method 1
4
Kc
Ti
( 34)
( 34)
1
=
Km
τm
≤1
Tm1
3
2

 τm 
 τm  
τm
τm
− 10.95 − 18845
.
− 3.4123ξ m + 4.5954ξ m
− 17002
.
.
ξ m

 − 21324
 
Tm1
Tm1

 Tm1 
 Tm 1  
+
0.421
0.1984
1.8033

τ 
τ 
τ 
 τm 
1 
 − 14.4149ξ m 2  m  − 0.7683ξ m 3 + 7.5142 m 

+ 3.7291 m 
+ 53444
.


Km 
 Tm1 
 Tm1 
 Tm1 
 Tm1 


+
−0.6753
−0.1642

τ 
τ 
1 
 − 0.0819 ξ m 19 .5419 − 3603

. ξ m 1.0749 + 71163
.
ξ m 1.1006 + 3206
. ξm m 
− 7.8480ξ m  m 
Km 
 Tm1 
 Tm1 


+
τ
τ 
1 
113222
.
ξ m1.9948  m  + 2.4239e T
Km 
 Tm1 

m
m1
+ 34137
.
eξ + 10251
.
e
m
ξ m τm
Tm1
− 05593
.
ξm
Tm1 

τm 

2
3

 τm 
 τm  
τm
τm
2

= Tm1 2.4866 − 233234
.
+ 53662
.
ξ m + 656053
.
− 24.1648ξ m − 83.6796

 + 29.0062ξ m
 
Tm1
Tm1

 Tm1 
 Tm1  
2
4

τ 
τm
τ  
+ Tm1  − 1359699
.
ξ m  m  + 431477
.
ξ m 2 + 519749
.
ξ m 3 + 86.0228 m  
 Tm1 
Tm 1
 Tm1  


3
2


τ 
τ 
τm
+ Tm 1 704553
.
ξ m  m  + 1534877
.
ξ m 2  m  − 1250112
.
ξ m 3 − 685893
.
ξ m4
Tm1

 Tm1 
 Tm1 

5
4
3
2


τ 
τ 
τ 
τ 
+ Tm 1 − 62.7517 m  + 27.6178 ξ m  m  − 152 .7422 ξ m 2  m  + 20.8705 m  ξ m 3 

 Tm1 
 Tm1 
 Tm1 
 Tm 1 

6
5


 τm  4
 τm 
 τm 
5

+ Tm1 54.0012
.
.
ξm 
.
ξm 6 
 ξ m + 58.7376ξ m + 131193

 + 202645
 − 232064

 Tm1 
 Tm1 
 Tm1 

4
3
2


 τm 
 τm 
 τm 
τm
2
3
4
+ Tm 1 − 616742
.
ξ
+
136
.
2439
ξ
−
954092
.
.
ξm 5 

 m

 m

 ξ m + 204168
Tm1

 Tm1 
 Tm1 
 Tm1 

Kc
Rule
Ti
Comment
Ultimate cycle
Decay ratio = 0.15 - ε < 2.4 ,
Regulator - nearly minimum
IAE, ISE, ITAE – Hwang
[60]
Kc
( 7)
Ti
( 7)
0.2 ≤
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.15 -
Kc( 8)
Model: Method 3
Ti (8)
2.4 ≤ ε < 3 , 0.2 ≤
τm
Tm1
≤ 2.0 ,
0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.15 -
Kc ( 9 ) 5
Ti ( 9)
3 ≤ ε < 20 , 0.2 ≤
τm
Tm1
≤ 2.0 ,
0.6 ≤ ξ m ≤ 4.2
6Tm1 + 4ξ m Tm1 τm + K H Km τ m
9
, KH =
2
2
2 Tm1 τ mω H
2τ m K m
2
5
ε=
ωH =
Tm12
Kc
( 7)
Kc
(8 )
Kc
( 9)
2
1 + KH K m
2T τ ξ
K K τ 2
+ m1 m m + H m m
3
6
[
2

0.674 1 − 0.447ω H τ m + 0.0607 (ω H τ m )

= 1 −
K H K m (1 + K H K m )


[
2

0.778 1 − 0.467ω H τ m + 0.0609(ω H τ m )

= 1 −
K H K m (1 + K H K m )


[
]
2
 τm 2
ξ m Tm1τ m 
2
−
T
−

,
m1
18
 18

] K


H
, Ti ( 7) =


H
, Ti (8 ) =
] K
(
K c (1 + K H K m )
0.0607ω H K m 1 + 1.05ω H τ m − 0.233ω H τ m
(
2
2
K c (1 + K H K m )
0.0309ω H K m 1 + 2.84ω H τ m − 0.532 ω H τ m
2
2
ω τ

131
. ( 0519
. )
1 − 103
. ε + 0514
.
ε 2 
K c (1 + KH K m )
( 9)

= 1 −
K H , Ti =

KH Km 1 + KH K m
0.0603 1 + 0.929 ln[ ω H τ m ] 1 + 2.01 ε − 12
. ε2


H
m
(
)
(
)(
)
)
)
Kc
Rule
Kc
Ti
( 10 ) 6
Ti
Comment
Decay ratio = 0.15 - ε > 20 ,
( 10)
0.2 ≤
Regulator – nearly minimum
IAE, ISE, ITAE - Hwang [60]
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.2 - ε < 2.4 ,
Kc
Model: Method 3
( 11)
Ti
( 11)
Ti
( 12)
0.2 ≤
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.2 -
Kc
( 12 )
2.4 ≤ ε < 3 , 0.2 ≤
τm
≤ 2.0 ,
Tm1
0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.2 -
Kc(13 )
Ti (13)
3 ≤ ε < 20 , 0.2 ≤
τm
≤ 2.0 ,
Tm1
0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.2 - ε > 20 ,
Kc(14 )
Ti (14)
Kc(15)
Ti (15)
(16 )
( 16)
0.2 ≤
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.25 - ε < 2.4 ,
0.2 ≤
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.25 -
Kc
Ti
2.4 ≤ ε < 3 , 0.2 ≤
τm
≤ 2.0 ,
Tm1
0.6 ≤ ξ m ≤ 4.2
6
Kc
( 10)
Kc
( 11)
Kc
( 12 )
[
2

114
. 1 − 0.482ω H τ m + 0.068( ω H τ m )

= 1 −
K H K m (1 + K H K m )


] K
[
2

0.622 1 − 0.435ω Hτ m + 0.052(ω H τm )

= 1 −
K H K m (1 + K H K m )




H


H
] K
[
2

0.724 1 − 0.469ω H τ m + 0.0609 (ω H τ m )

= 1 −
K H K m (1 + K H K m )


[
, Ti (10) =
, Ti (11) =
] K


]
H
(
K c (1 + K H K m )
2
)
0.0697 ω H K m 1 + 0.752ω H τ m − 0.145ω H τ m
2
)
2
)
0.0694ω H K m − 1 + 2.1ω H τ m − 0.367ω H τm
(
, Ti (12 ) =
2
K c (1 + K H K m )
2
(
K c (1 + K H K m )
0.0405ω H K m 1 + 1.93ω H τ m − 0.363ω H τ m
2
ω τ

126
. (0.506)
1 − 107
. ε + 0.616 ε 2 
K c (1 + K H K m )
Kc (13) =  1 −
K H , Ti (13) =

K H Km 1 + KH K m
0.0661(1 + 0.824 ln[ω H τ m ])(1 + 171
. ε − 117
. ε2 )


H
Kc
( 14 )
Kc
( 15)
Kc
( 16)
[
m
(
)
2

109
. 1 − 0.497ω H τ m + 0.0724(ω Hτ m )

= 1 −
K H K m (1 + K H K m )


[
] K


2

0.584 1 − 0.439ω H τ m + 0.0514 ( ω H τ m )

= 1 −
K H K m (1 + K H K m )


[
2

0.675 1 − 0.472ω H τ m + 0.061( ω H τ m )

= 1 −
K H K m (1 + K H K m )


H
] K


] K


H
, Ti (14 ) =
H
(
0.054 ω H K m − 1 + 2.54ω H τ m − 0.457ω H τ m
, Ti (15) =
, Ti (16) =
K c (1 + K H K m )
2
(
K c (1 + K H K m )
0.0714ω H K m 1 + 0.685ω H τ m − 0.131ω H 2 τ m 2
(
)
K c (1 + K H K m )
0.0484ω H K m 1 + 1.43ω H τ m − 0.273ω H τ m
2
2
)
2
)
Rule
Kc
Ti
Regulator - nearly minimum
IAE, ISE, ITAE - Hwang [60]
Kc(17 ) 7
Ti (17)
Comment
Decay ratio = 0.25 τm
3 ≤ ε < 20 , 0.2 ≤
≤ 2.0 ,
Tm1
0.6 ≤ ξ m ≤ 4.2
Model: Method 3
Decay ratio = 0.25 - ε > 20 ,
Kc (18 )
Ti (18)
0.2 ≤
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
Decay ratio = 0.1 - ε < 2.4 ,
Servo - nearly minimum IAE,
ISE, ITAE - Hwang [60]
Kc
( 19 )
Ti
0.2 ≤
( 19)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ m ≤ 0.776 + 0.0568
Model: Method 3
τ 
τm
+ 018
.  m
Tm1
 T m1 
2
Decay ratio = 0.1 - 2.4 ≤ ε < 3 ,
Kc
( 20)
Ti
0.2 ≤
( 20)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ m ≤ 0.776 + 0.0568
τ 
τm
+ 018
.  m
Tm1
 T m1 
2
Decay ratio = 0.1 - 3 ≤ ε < 20 ,
Kc
( 21)
Ti
0.2 ≤
( 21)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ m ≤ 0.776 + 0.0568
τ 
τm
+ 018
.  m
Tm1
 T m1 
2
Decay ratio = 0.1 - ε > 20 ,
Kc
( 22 )
Ti
0.2 ≤
( 22 )
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ m ≤ 0.776 + 0.0568
Kc
( 17)
Kc
( 18)
Kc
( 19 )
Kc
( 20)
[
]
ω τ
2
 12
. ( 0495
. )
1 − 11
. ε + 0.698 ε 
Kc (1 + KH K m )

 K , T (17 ) =
= 1−
i
H


0.0702 1 + 0.734 ln[ ω H τ m ] 1 + 148
. ε − 11
. ε2
KH Km (1 + K H Km )


H
7
τ 
τm
+ 018
.  m
Tm1
 T m1 
m
[
2

1.03 1 − 0.51ω H τ m + 0.0759( ω H τ m )

= 1 −
K H K m (1 + K H K m )


[
] K


2

0.822 1 − 0.549ω H τ m + 0.112(ω Hτ m )

= 1 −
K H K m (1 + K H K m )


[
H
] K


2

0.786 1 − 0.441ω H τ m + 0.0569(ω H τ m )

= 1 −
K H K m (1 + K H K m )


[
)(
(
, Ti (18) =
H
] K
]


0.0386ω H K m − 1 + 3.26ω H τ m − 0.6ω H τm
, Ti (19 ) =
H
(
K c (1 + K H K m )
2
(
2
, Ti ( 20) =
)
K c (1 + K H K m )
0.0142ω H K m 1 + 6.96ω H τ m − 177
. ωH τm
2
(
2
)
K c (1 + K H K m )
0.0172 ω H K m 1 + 4.62ω H τ m − 0.823ω H τ m
2
ω τ
 128
. ( 0542
. )
1 − 0.986 ε + 0558
.
ε 2 
K c (1 + K H K m )
( 21)
Kc ( 21) = 1 −
 KH , Ti = 0.0476 1 + 0.996 ln ω τ 1 + 213
K
K
1
+
K
K
. ε2 )
(
[ H m ])( . ε − 113


H
m
H m
H
Kc
( 22)
[
m
(
)
2

114
. 1 − 0.466 ω H τ m + 0.0647 ( ω H τ m )

= 1 −
K H K m (1 + K H K m )


] K


H
, Ti (22) =
)
(
K c (1 + K H K m )
0.0609ω H K m − 1 + 1.97ω H τ m − 0.323ω H 2 τ m2
)
2
)
2
Kc
Rule
Ti
Comment
Decay ratio = 0.1 - ε < 2.4 ,
Servo - nearly minimum IAE,
ISE, ITAE - Hwang [60]
8
K c ( 23)
0.2 ≤
Ti ( 23)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ > 0889
.
+ 0.496
Model: Method 3
 τ 
τm
+ 0.26 m 
Tm1
 Tm1 
2
Decay ratio = 0.1 - 2.4 ≤ ε < 3 ,
K c ( 24 )
0.2 ≤
Ti ( 24)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ > 0889
.
+ 0.496
 τ 
τm
+ 0.26 m 
Tm1
 Tm1 
2
Decay ratio = 0.1 - 3 ≤ ε < 20 ,
Kc
( 25)
Ti
0.2 ≤
( 25)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ > 0889
.
+ 0.496
 τ 
τm
+ 0.26 m 
Tm1
 Tm1 
2
Decay ratio = 0.1 - ε > 20 ,
K c (26)
0.2 ≤
Ti (26)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
ξ > 0889
.
+ 0.496
 τ 
τm
+ 0.26 m 
Tm1
 Tm1 
2
Decay ratio = 0.1 - ε < 2.4 ,
K c ( 27 )
0.2 ≤
Ti ( 27)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
τ 
τm
+ 018
.  m 
Tm1
 T m1 
ξ m > 0.776 + 0.0568
ξ m ≤ 0.889 + 0.496
8
Kc
( 23)
Kc
( 24)
Kc
( 25)
[
2
 0.794 1 − 0541
. ω H τ m + 0.126( ω H τ m )

= 1 −
K H K m (1 + KH K m )


[
 0.738 1 − 0.415ω τ + 0.0575 ω τ 2
( H m)
H m

= 1 −
KH K m (1 + KH K m )


[
[
Kc


H
, Ti (23) =
]  K


H
, Ti (24) =
]
K c (1 + K H K m )
(
0.0078ω H K m 1 + 8.38ω H τ m − 197
. ω H2 τ m 2
)
K c (1 + K H K m )
(
0.0124ω H K m 1 + 4.05ω H τ m − 0.63ω H 2 τ m 2
)
ω τ
2
 115
. (0.564) H m 1 − 0.959 ε + 0.773 ε 
K c (1 + K H K m )
 K , T (25) =
= 1 −

 H i
K
K
(
1
+
K
K
)
0
.
0355
1
+
0
.
947
ln[ω H τ m ] 1 + 19
. ε − 107
. ε2
H m
H m


 107
. 1 − 0.466ω H τ m + 0.0667( ω H τ m )

Kc ( 26) =  1 −
K H Km (1 + K H Km )


( 27)
] K
[
 0.789 1 − 0.527ω τ + 0.11 ω τ 2
( H m)
H m

= 1 −
K H K m (1 + K H K m )


2
] K


]  K


H
)(
(
H
, Ti (26) =
, Ti (27) =
(
τ 
τm
+ 0.26 m 
Tm1
 Tm1 
)
K c (1 + K H K m )
0.0328ω H K m − 1 + 2.21ω H τ m − 0338
. ω H2 τ m 2
K c (1 + K H K m )
(
0.009ω H K m 1 + 9.7ω H τ m − 2.4ω H 2 τ m 2
)
)
2
2
Kc
Rule
Ti
Comment
Decay ratio = 0.1 - 2.4 ≤ ε < 3 ,
Servo - nearly minimum IAE,
ISE, ITAE - Hwang [60]
9
K c (28)
0.2 ≤
Ti (28)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
τ 
τm
+ 018
.  m 
Tm1
 T m1 
ξ m > 0.776 + 0.0568
Model: Method 3
ξ m ≤ 0.889 + 0.496
τ 
τm
+ 0.26 m 
Tm1
 Tm1 
2
2
Decay ratio = 0.1 - 3 ≤ ε < 20 ,
K c ( 29 )
0.2 ≤
Ti ( 29)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
τ 
τm
+ 018
.  m 
Tm1
 T m1 
ξ m > 0.776 + 0.0568
ξ m ≤ 0.889 + 0.496
τ 
τm
+ 0.26 m 
Tm1
 Tm1 
2
2
Decay ratio = 0.1 - ε > 20 ,
K c ( 30)
0.2 ≤
Ti (30)
τm
Tm1
≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2
τ 
τm
+ 018
.  m 
Tm1
 T m1 
ξ m > 0.776 + 0.0568
ξ m ≤ 0.889 + 0.496
Regulator - minimum IAE Shinskey [17] – page 121.
Model: method not
specified
9
Kc
( 28)
048
. Ku
[
[
τm
T
= 02
. , m2 = 0.2
Tm1
Tm1
083
. Tu
2
 0.76 1 − 0426
. ω H τ m + 0.0551( ω H τ m )

= 1 −
K H Km ( 1 + K H Km )


] K


H
, Ti (28) =
]
(
τ 
τm
+ 0.26 m 
Tm1
 Tm1 
K c (1 + K H K m )
0.0153ω H K m 1 + 4.37ω H τ m − 0.743ω H 2 τ m 2
)
Kc
( 29)
ω τ
2
 122
. (0.55) H m 1 − 0.978 ε + 0.659 ε 
K c (1 + K H K m )

 K , T (29) =
= 1−

 H i
K
K
(
1
+
K
K
)
0.0421 1 + 0.969 ln[ω H τ m ] 1 + 2.02 ε − 111
. ε2
H m
H m


Kc
( 30)
2
 111
. 1 − 0.467ω H τ m + 0.0657( ω H τ m )

= 1 −
KH K m ( 1 + K H Km )


[
] K


)(
(
H
, Ti (30) =
(
)
K c (1 + K H K m )
0.0477ω H K m − 1 + 2.07ω H τ m − 0.333ω H2 τ m 2
)
2
2
Table 16: PI tuning rules - SOSPD model -
K m e− sτ
m
Tm1 s2 + 2ξ m Tm1s + 1
2


- Two degree of freedom controller: U(s) = K c 1 + 1 E (s ) − α Kc R( s) . 3 tuning rules.

Kc
Rule
Ti
Servo/regulator tuning
Taguchi and Araki [61a]
Model: ideal process
Tis 
Minimum performance index




1 
0.5613

0
.
3717
+

τm
K m 
+ 0 .0003414

Tm


Ti
1
Km
τ 
τ
τm
+ 0. 3087 m  − 0. 1201 m
Tm
 Tm 
 Tm




0
.
05627


0
.
1000
+


τm
2

[
+ 0.06041] 

T

m

Ti
2
τm
≤ 1.0 ; ξ m = 1
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of tuning
rules of Chien et al. [10]
( 30a ) 10
2
α = 0. 6438− 0. 5056
τ 
τ
τ
α = 0. 6178− 0. 4439 m − 7. 575 m  + 9. 317 m
Tm
T
m


 Tm
Minimum ITAE Pecharroman and Pagola
[134a]
Comment




3
( 30b )
3

τ
 − 3. 182 m

T

 m
4




τm
≤ 1.0 ; ξ m = 0.5
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of tuning
rules of Chien et al. [10]
α = 0. 4002 ,
φ c = − 139.65 0
K m = 1 ; Tm = 1 ; ξ m = 1
0.1713 Ku
1.0059 Tu
0.147 K u
1.150 Tu
α = 0. 411 , φ c = −1460
0.170 K u
1.013Tu
α = 0. 401 , φ c = −1400
0.195 K u
0.880 Tu
α = 0.386 , φ c = −1330
0.210 K u
0.720 Tu
α = 0.342 , φ c = −1250
0.234 K u
0.672 Tu
α = 0.345 , φ c = −1150
0.249 K u
0.610 Tu
α = 0.323 , φ c = −1050
0.262 K u
0.568 Tu
α = 0.308 , φ c = −940
0.274 K u
0.545 Tu
α = 0. 291 , φ c = −840
0.280 K u
0.512 Tu
α = 0. 281 , φ c = −730
0.291K u
0.503 Tu
α = 0.270 , φ c = −630
0.297 K u
0.483 Tu
α = 0.260 , φ c = −520
0.303 K u
0.462 Tu
α = 0.246 , φ c = −410
0.307 K u
0.431Tu
α = 0.229 , φ c = −300
Model: Method 15
Minimum ITAE Pecharroman and Pagola
[134b]
K m = 1 ; Tm = 1 ; ξ m = 1
Model: Method 15
2
3

 τm 
 τ m  
τm

= Tm 2.069 − 0.3692
+ 1.081  − 0.5524  


Tm
 Tm 
 Tm  

2
3
4

τ
τ 
τ 
τ  
= Tm  4. 340 − 16. 39 m + 30. 04 m  − 25 .85 m  + 8.567  m  

Tm
 Tm 
 Tm 
 Tm  

( 30a )
10
Ti
Ti
( 30b )
Rule
Kc
Ti
Comment
Minimum ITAE Pecharroman and Pagola
[134b] - continued
0.317 K u
0.386 Tu
α = 0. 171 , φ c = −190
0.324 K u
0.302 Tu
α = 0.004 , φ c = −100
0.320 K u
0.223 Tu
α = −0.204 , φ c = − 60
Table 17: PI tuning rules - SOSIPD model (repeated pole) -
K m e −sτ
m
s (1 + Tm1s) 2


- Two degree of freedom controller: U(s) = K c 1 + 1 E (s ) − α Kc R( s) . 1 tuning rule.

Rule
Kc
Servo/regulator tuning
Taguchi and Araki [61a]
Model: ideal process
Ti
K m = 1 ; Tm = 1
Model: Method 1
Comment
Minimum performance index
τm
Tm
τm
≤ 1.0 ;
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of tuning
rules of Chien et al. [10]
0.049 Ku
2. 826 Tu
α = 0.506 , φ c = −1640
0.066 Ku
2. 402 Tu
α = 0.512 , φ c = −1600
0.099 Ku
1.962 Tu
α = 0.522 , φ c = −1550
0.129 Ku
1.716 Tu
α = 0.532 , φ c = −1500
0.159 Ku
1.506 Tu
α = 0.544 , φ c = −1450
0.189 Ku
1.392 Tu
α = 0.555 , φ c = −1400
0.218 K u
1.279 Tu
α = 0.564 , φ c = −1350
0.250 Ku
1.216 Tu
α = 0.573 , φ c = −1300
0.286 Ku
1.127 Tu
α = 0.578 , φ c = −1250
0.330 Ku
1.114 Tu
α = 0.579 , φ c = −1200
0.351K u
1.093Tu
α = 0.577 , φ c = −1180




1 
0 .3840 
0
.
07368
+

τm
K m 
+ 0.7640

Tm


τ
8.549 + 4.029 m
Tm
α = 0. 6691 + 0.006606
Minimum ITAE Pecharroman and Pagola
[134b]
Tis 
Km e − sτ


- controller G c ( s) = Kc  1 + 1  . 1
Ts
(1 + sTm1 )(1 + sTm2 )(1 + sTm 3 )

i 
tuning rule
m
Table 19: PI tuning rules - TOLPD model
Rule
Hougen [85]
Kc
0.7  Tm1 


Km  τm 
Ti
15
. τm
0.08
Tm1( Tm 2 + Tm3 )
15
. τm
0.08
Tm1( Tm 2 + Tm3 )
τm
≤ 0.04 ; Tm1 ≥ Tm 2 ≥ Tm3
Tm1
Model: Method 1
0. 333
1   Tm 1 
T + Tm2 + Tm 3
0.7 

+ 0.8 m1
0. 333
2K m   τ m 
( Tm 1Tm 2 Tm 3 )





Comment
τm
> 0.04 ; Tm1 ≥ Tm 2 ≥ Tm3
Tm1
0.333
Table 20: PI tuning rules - TOLPD model (repeated pole) -
K m e −sτ
m
(1 + Tm1s ) 3


- Two degree of freedom controller: U(s) = K c 1 + 1 E (s ) − α Kc R( s) . 1 tuning rule.

Rule
Kc
Servo/regulator tuning
Taguchi and Araki [61a]
Model: ideal process
Ti
Ti
( 30c )
Comment
Minimum performance index




1 
0.7399 
0
.
2713
+

τm
K m 
+ 0.5009

Tm


Ti
( 30 c ) 1
τ 
τ
α = 0. 4908 − 0. 2648 m + 0. 05159  m 
Tm
 Tm 
1
Tis 
2

τ  
τ
= Tm  2.759 − 0. 003899 m + 0.1354  m  


Tm
 Tm  

2
τm
≤ 1.0 ;
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of tuning
rules of Chien et al. [10]
K me − sτ
1 − sTm
m
Table 21: PI tuning rules - unstable FOLPD model
Kc
Rule


- controller G c ( s) = Kc  1 + 1  . 6 tuning rules

Ts
i 
Ti
Comment
Direct synthesis
1
De Paor and O’Malley [86]
Kc
( 35)
Model: Method 1
Venkatashankar and
Chidambaram [87]
Model: Method 1
Chidambaram [88]
Model: Method 1
 1 − Tm τm 
Tm 
 tan( 0.5φm )
 Tm τ m 
Kc( 36)
25( Tm − τ m )
1 
T 
 1 + 0.26 m 
Km 
τm 
25Tm − 27τ m
ω p Tm
Ho and Xu [90]
A m Km
Model: Method 1
τm
<1
Tm
gain margin = 2;
φ m = tan − 1
1 − Tm τ m
(
− T m τ m 1 − Tm τ m
Tm τ m
τm
< 0.67
Tm
τm
< 06
.
Tm
ωp =
1
1.57ω p − ω p τ m −
2
1
Tm
A m φ m + 157
. A m (A m − 1)
(A
2
m
)
−1 τm
,
τm
< 0.62
Tm
Robust
Rotstein and Lewin [89]
 λ

Tm λ
+ 2
 Tm

2
λ Km
 λ

λ
+ 2
 Tm

Km uncertainty = 50%
τm Tm = 0.2
λ = [0.6Tm ,1.9Tm ]
τm Tm = 0.2
λ = [ 0.5Tm , 45
. Tm ]
τm Tm = 0.4
λ = [15
. Tm ,4.5Tm ]
τm Tm = 0.6
λ = [ 39
. Tm ,41
. Tm ]
Model: Method 1
Km uncertainty = 30%
λ obtained graphically –
sample values below
Ultimate cycle
Luyben [91]
Model: Method not relevant
031
. Ku
Maximum closed loop log
modulus = 2 dB; closed
loop time constant =
316
. τm
2.2Tu
2
2
Kc
Kc
( 35)
( 36)
=
Tm 
1 − Tm τ m
sin
cos (1 − Tm τ m )Tm τ m +
K m 
Tm τ m
1
=
Km

2
 0.98 1 + 0.04Tm
2

( Tm − τ m )


(1 − Tm τ m )Tm τ m 

β 2 Tm2 (Tm − τ m )
)
2
1+
 
(Tm − τ m ) 2 τ 2m :
τm
  25  β( T − τ )
< 025
.
β = 1373
. ,
m
m
  τm 
625
2
Tm

1+β2
T
−
τ
(
)
m
m
2
τm
β = 0.953, 0.25 ≤
τm
< 0.67
Tm
K m e− s τ


- controller G c ( s) = Kc  1 + 1  . 3
Ts
(1 − sTm1 )(1 + sTm 2 )

i 
tuning rules.
m
Table 22: PI tuning rules - unstable SOSPD model
Rule
Kc
Ti
Comment
Ultimate cycle
McMillan [58]
Model: Method not relevant
Kc( 37) 3
Ti (37 )
Tuning rules developed
from Ku , Tu
Regulator tuning
Minimum performance index
Minimum ITAE – Poulin and
Pomerleau [82] – deduced
from graph
Model: Method 1
Output step load
disturbance
(considered as 2 tuning
rules)
Input step load disturbance
bTm1 1 +
(
4Tm1 ( τ m + Tm2 )
aTm2 2
4( τm + Tm 2 )
aTm1 − 4( τ m + Tm2 )
2
K m aTm1 − 4[ τ m + Tm2 ]
)
τm Tm
a
b
τm Tm
a
b
τm Tm
a
b
0.05
0.10
0.15
0.20
0.9479
1.0799
1.2013
1.3485
2.3546
2.4111
2.4646
2.5318
0.25
0.30
0.35
0.40
1.4905
1.6163
1.7650
1.9139
2.5992
2.6612
2.7368
2.8161
0.45
0.50
2.0658
2.2080
2.9004
2.9826
0.05
0.10
0.15
0.20
1.1075
1.2013
1.3132
1.4384
2.4230
2.4646
2.5154
2.5742
0.25
0.30
0.35
0.40
1.5698
1.6943
1.8161
1.9658
2.6381
2.7007
2.7637
2.8445
0.45
0.50
2.1022
2.2379
2.9210
3.0003
2
3
Kc ( 37) =
1477
.
Tm1Tm 2
K m τ m2
Ti A max
Km
(
)
Tm1 Tm2 ω max + Tm1 + Tm 2 ω max + ω max
2
Kc ( 38) =




0.65
  (T + T )T T


 
1

m1
m2
m1 m 2
( 37)
,
T
=
332
.
τ
1
+






i
m
0.65
  ( Tm1 + Tm 2 )Tm1Tm 2  
  (Tm1 − Tm 2 )(Tm1 − τ m )τ m  

 
1+ 
  (Tm1 − Tm2 )(Tm1 − τ m )τ m  
2
6
2
1 + Ti ω max
2
2
2
4
2

1
Table 23: PI tuning rules – delay model e − sτ m - controller G c ( s) = Kc  1 +
 . 2 tuning rules
Ts

i 
Rule
Direct synthesis
Hansen [91a]
Model: Method not
specified
Regulator tuning
Shinskey [57] – minimum
IAE – regulator - page 67.
Model: method not
specified
Kc
Ti
0.2
0.3τm
Minimum performance index
0.4
Km
0.5τ m
Comment
K m e− sτ


1
- ideal controller G c ( s) = Kc  1 +
+ Td s . 56.
1 + sTm
Ti s


m
Table 25: PID tuning rules - FOLPD model
Kc
Rule
Process reaction
Ziegler and Nichols
[8]
Model: Method 1
Astrom and
Hagglund [3] – page
139
Model: Method 6
Parr [64] – page 194
Model: Method 1
Chien et al. [10] –
regulator
Model: Method 1
Chien et al. [10] servo
Model: Method 1
Three constraints
method - Murrill [13]page 356
[
Ti
Td
Comment
2τ m
05
. τm
Quarter decay ratio
0.94Tm
K mτ m
2τ m
05
. τm
Ultimate cycle ZieglerNichols equivalent
125
. Tm
K mτ m
2.5τ m
0. 4τ m
095
. Tm
Km τ m
2.38τ m
0.42τ m
0% overshoot;
τ
011
. < m <1
Tm
12
. Tm
K mτ m
2τ m
0.42τ m
20% overshoot;
τ
011
. < m <1
Tm
0.6Tm
K mτ m
Tm
0.5τ m
0% overshoot;
τ
011
. < m <1
Tm
095
. Tm
Km τ m
1.36Tm
0.47τ m
20% overshoot;
τ
011
. < m <1
Tm
12
. Tm 2Tm
,
]
K m τ m K m τm
 Tm 
1370
.
 
Km  τm 
0.950
Tm  τm 
 
1351
.
 Tm 
0.738
τ 
0.365Tm  m 
 Tm 
0.950
Model: Method 3
Quarter decay ratio;
minimum integral error
(servo mode);
Kc K m Td
= 05
. ;
Tm
01
. ≤
Cohen and Coon [11]
Model: Method 1
Astrom and
Hagglund [93]pages 120-126
2

  
 2.5 τ m + 0.46 τ m  

1 
Tm
 Tm  
.
+ 025
.  T  Tm
 135

Km 
τm
 m
τm

1 + 0.61


Tm


3
Km
Model: Method 19
Sain and Ozgen [94] 1 

T
.
 0. 6939 m + 01814

Model: Method 5 K m 
τm

Regulator tuning
Minimum IAE –
Murrill [13] – pages
358-363
Model: Method 3
0.37 τ m
τ
1 + 0.2 m
Tm
τm
≤1
Tm
Quarter decay ratio
T90% = 90% closed
loop step response
time. Leeds and
Northrup Electromax V.
T90%
05
. τm
0.8647Tm + 0.226τ m
Tm
+ 0.8647
τm
0.0565Tm
T
0.8647 m + 0.226
τm
Minimum performance index
 Tm 
1435
.
 
Km  τ m 
0.921
Tm  τm 
 
0878
.
 Tm 
0.749
τ 
0.482 Tm  m 
 Tm 
1.137
01
. <
τm
≤1
Tm
Rule
Modified minimum
IAE - Cheng and
Hung [95]
Model: Method 7
Minimum ISE Murrill [13] – pages
358-363
Model: Method 3
Minimum ISE Zhuang and
Atherton [20]
Kc
3  Tm 
 
K m  τm 
Ti
0 .921
Td
Tm  τm 
 
0878
.
 Tm 
Comment
0.749
τ 
0.482 Tm  m 
 Tm 
1.137
 Tm 
1495
.
 
Km  τ m 
0.945
Tm  τm 
 
1101
.
 Tm 
0.771
τ 
0.56Tm  m 
 Tm 
1. 006
 Tm 
1473
.
 
Km  τ m 
0.970
Tm  τm 
 
1115
.
 Tm 
0.753
τ 
0.55Tm  m 
 Tm 
0.948
 Tm 
1524
.
 
Km  τm 
0.735
Tm  τ m 
 
1130
.
 Tm 
0.641
τ 
0.552Tm  m 
 Tm 
0.851
 Tm 
1357
.
 
Km  τm 
0 .947
Tm  τ m 
 
0842
.
 Tm 
0.738
τ 
0.381Tm  m 
 Tm 
0.995
 Tm 
1468
.
 
Km  τm 
0.970
Tm  τm 
 
0942
.
 Tm 
0.725
τ 
0.443Tm  m 
 Tm 
0.939
 Tm 
1515
.
 
Km  τ m 
0.730
Tm  τm 
 
0957
.
 Tm 
0.598
τ 
0.444 Tm  m 
 Tm 
0 .847
Model: Method 6
Minimum ISTES Zhuang and
Atherton [20]
 Tm 
1531
.
 
Km  τm 
0.960
Tm  τ m 
 
0971
.
 Tm 
0.746
τ 
0.413Tm  m 
 Tm 
0.933
 Tm 
1592
.
 
Km  τm 
0.705
Tm  τm 
 
0957
.
 Tm 
0.597
τ 
0.414 Tm  m 
 Tm 
0.850
Model: Method 6
Model: Method 6
Minimum ITAE Murrill [13] – pages
358-363
Model: Method 3
Minimum ISTSE Zhuang and
Atherton [20]
Minimum error - step
load change - Gerry
[96]
Model: Method 6
Servo tuning
Minimum IAE Rovira et al. [21]
Model: Method 3
03
.
Km
 Tm 
1086
.
 
Km  τm 
0.5τm
0.869

0.6032 
 0.7645 +
 (Tm + 05
. τm)
τ m Tm 

Tm + 0.5τ m
Model: Method 6
Minimum ISE - Wang
et al. [97]

0.7524 
 0.9155 +
 (Tm + 0 .5τ m )
τ m Tm 

Tm + 0.5τ m
K m ( Tm + τ m )
K m (Tm + τ m )
Model: Method 6
Minimum ISE Zhuang and
Atherton [20]
Model: Method 6
 Tm 
1048
.
 
Km  τ m 
0.897
 Tm 
1154
.
 
K m  τm 
0.567
0.5Tm τ m
Tm + 05
. τm
0.5Tm τ m
Tm + 05
. τm
Tm
τ 
0.489 Tm  m 
 Tm 
0.888
Tm
τ 
0.490 Tm  m 
 Tm 
0.708
τ
1195
.
− 0.368 m
Tm
τ
1.047 − 0.220 m
Tm
τm
≤1
Tm
01
. ≤
τm
≤ 10
.
Tm
11
. ≤
τm
≤ 2.0
Tm
01
. <
τm
≤1
Tm
01
. ≤
τm
≤ 10
.
Tm
11
. ≤
τm
≤ 2.0
Tm
01
. ≤
τm
≤ 10
.
Tm
11
. ≤
τm
≤ 2.0
Tm
τm
>5
Tm
Tm
Minimum performance index
0.914
Tm
 τm 
0.348 Tm  
τ
0.740 − 0.13 m
 Tm 
Tm
Minimum IAE Wang et al. [97]
01
. <
01
. <
τm
≤1
Tm
0.05 <
τm
<6
Tm
0.05 <
τm
<6
Tm
01
. ≤
τm
≤ 10
.
Tm
11
. ≤
τm
≤ 2.0
Tm
Rule
Kc
Ti
Minimum ITAE Rovira et al. [21]
Model: Method 3
 Tm 
0965
.
 
K m  τm 
Modified minimum
ITAE - Cheng and
Hung [95]
Model: Method 7
Minimum ITAE –
Wang et al. [97]
12
.  Tm 
 
Km  τm 
0.85
0.855
Td
Tm
τ 
0.308 Tm  m 
 Tm 
Tm
τ 
0.308 Tm  m 
 Tm 
0.929
τ
0.796 − 0.1465 m
Tm
τ
0.796 − 0.147 m
Tm


05307
.
 0.7303 +
 (Tm + 0.5τ m )
τ m Tm 

K m (Tm + τm )
Tm + 0.5τ m
0.5Tm τ m
Tm + 05
. τm
Model: Method 6
Minimum ISTSE –
Zhuang and
Atherton [20]
Model: Method 6
Minimum ISTES –
Zhuang and
Atherton [20]
Model: Method 6
 Tm 
1042
.
 
Km  τm 
0 .897
 Tm 
1142
.
 
Km  τ m 
0.579
 Tm 
0968
.
 
Km  τm 
0.904
 Tm 
1061
.
 
Km  τ m 
0 .583
Comment
0.929
Tm
τ 
0.385Tm  m 
 Tm 
0.906
Tm
τ 
0.384 Tm  m 
 Tm 
0 .839
Tm
τ 
0.316 Tm  m 
 Tm 
0.892
Tm
τ 
0.315Tm  m 
 Tm 
0.832
τ
0.987 − 0.238 m
Tm
0.919 − 0.172
τm
Tm
τ
0.977 − 0.253 m
Tm
τ
0.892 − 0.165 m
Tm
Ultimate cycle
Regulator – minimum
IAE – Pessen [63]
0.7K u
0.4Tu
Model: Method 6
Servo – minimum
0.051( 3. 302 K m Ku + 1) Tu
ISTSE – Zhuang and
0.509Ku
Atherton [20]
Model: Method not
relevant
Servo – minimum
ISTSE – Pi-Mira et
0.604 K u
0.04 (4. 972 K m K u + 1)T
al. [97a]
Model: Method 27
Regulator - minimum
ISTSE - Zhuang and 4.434 K m K u − 0.966
1751
. K m K u − 0.612
Ku
Tu
Atherton [20]
512
. K m K u + 1734
.
3776
. K m K u + 1388
.
Model: Method not
relevant
0149
. Tu
0125
. Tu
01
. ≤
τm
≤1
Tm
Damping factor of
closed loop system =
0.707.
0.05 <
τm
<6
Tm
01
. ≤
τm
≤ 10
.
Tm
11
. ≤
τm
≤ 2.0
Tm
01
. ≤
τm
≤ 10
.
Tm
11
. ≤
τm
≤ 2.0
Tm
01
. ≤
τm
≤ 10
.
Tm
01
. ≤
τm
≤ 2.0
Tm
01
. ≤
τm
≤ 2.0
Tm
0.130 Tu
0144
. Tu
Kc
Rule
Ti
Td
Ti (38 )
0471
. Ku
K mω u
Comment
1
Regulator - nearly
minimum IAE, ISE,
ITAE - Hwang [60]
Kc( 38)
Model: Method 8
Decay ratio = 0.15 0.1 ≤
τm
≤ 2.0 ; ε < 2.4
Tm
Decay ratio = 0.15 -
Model: Method 8
Kc
( 39)
Ti
( 39 )
0471
. Ku
K mω u
0.1 ≤
τm
≤ 2.0 ;
Tm
2.4 ≤ ε < 3
Decay ratio = 0.15 -
Kc( 40)
Ti (40 )
0471
. Ku
K mω u
0.1 ≤
τm
≤ 2.0 ;
Tm
3 ≤ ε < 20
Decay ratio = 0.15 -
Kc( 41)
Ti (41)
0471
. Ku
K mω u
( 42)
( 42 )
0471
. Ku
K mω u
Ti (43)
0471
. Ku
K mω u
0.1 ≤
τm
≤ 2.0 ; ε ≥ 20
Tm
Decay ratio = 0.2 -
Kc
Ti
0.1 ≤
τm
≤ 2.0 ; ε < 2.4
Tm
Decay ratio = 0.2 -
Kc( 43)
0.1 ≤
τm
≤ 2.0 ;
Tm
2.4 ≤ ε < 3
2
Kc
Rule
1
KH =
ωH =
Kc
( 38)
Kc
( 39)
2
Ti
 2
9
. Km Kc Ku τ m
 τm − τm Tm + 1884
+
2 
18
9ωu
2 K mτ m 18

Td
τ m4 49 τm 2Tm 2 7Tm τm 3 0.471Ku K c
+
+
−
324
324
162
ωu
1+ KH Km
τ m Tm K H K m τ m
0.942 K c K u τ m
+
−
3
6
3ω u
[
2
2

0.674 1 − 0.447ω H τ m + 0.0607ω H τ m

= KH −

K m (1 + K H K m )

[
2
2

0.778 1 − 0.467ω H τ m + 0.0609ω H τ m
=  KH −

K m (1 + KH K m )

[

10τ m3 16 Tm τ m2  1775
. Kc 2 Ku 2 τ m2 
+

−
2

81 
81ω u
 81

2Tm ω u + K H K m τ m ω u − 1884
. Ku Kc
0471
. K c K uω H τ m
,ε=
2
Comment
] , T
i
=
] , T
=




( 38)
( 39)
i
Kc
( 38)
(
(1 + KH K m )
ω H Km 0.0607 1 + 105
. ω Hτ m − 0233
. ωH 2τ m2
Kc
(
( 39)
(1 + K H Km )
ω H K m 0.0309 1 + 2.84ω H τ m − 0532
. ω H2 τ m 2
]
ω τ
( 40)

131
. ( 0519
. ) H m 1 − 103
. ε + 0.514 ε 2 
K c (1 + K H K m )
Kc ( 40) =  K H −
, Ti (40) =


K m (1 + K H Km )
ω H K m 00603
.
1 + 0.929 ln[ ω Hτ m ] 1 + 2.01 ε − 12
. ε2


[

114
. 1 − 0.482ω H τ m + 0.068ω H 2 τ m 2
Kc ( 41) =  KH −
K m (1 + K H Km )

[
] , T

0622
.
1 − 0.435ω H τ m + 0.052ω H τ m
Kc ( 42) =  K H −
K m (1 + K H K m )

[
2


2
)(
(
( 41)
i
]  , T

0.724 1 − 0469
. ω H τ m + 00609
.
ωH 2 τm2
Kc ( 43) =  KH −
K m (1 + K H K m )



( 42)
i
] , T


=
i
Kc
( 41)
(
(1 + KH K m )
ω H K m 0.0694 − 1 + 21
. ω H τ m − 0.367ω H 2 τ m2
=
( 43)
Kc
( 42)
(
)
(1 + KH K m )
ω H K m 00697
.
1 + 0752
. ω H τ m − 0.145ω H 2 τ m 2
=
Kc
(
( 43)
(1 + K H K m )
ω H K m 0.0405 1 + 193
. ω H τ m − 0.363ω H 2 τ m 2
)
)
)
)
)
3
Regulator – nearly
minimum IAE, ISE,
ITAE - Hwang [60]
(continued)
Kc
Model: Method 8
Decay ratio = 0.2 -
( 44)
Ti
( 44 )
Kc( 45)
Ti (45)
Kc
( 46)
Ti
( 46 )
Kc
( 47)
Ti
( 47 )
0471
. Ku
K mω u
0.1 ≤
τm
≤ 2.0 ;
Tm
3 ≤ ε < 20
Decay ratio = 0.2 -
0471
. Ku
K mω u
τm
≤ 2.0 ; ε ≥ 20
Tm
0.1 ≤
Decay ratio = 0.25 -
0471
. Ku
K mω u
0.1 ≤
τm
≤ 2.0 ; ε < 2.4
Tm
Decay ratio = 0.25 -
0471
. Ku
K mω u
0.1 ≤
τm
≤ 2.0 ;
Tm
2.4 ≤ ε < 3
Decay ratio = 0.25 -
Kc( 48)
Servo - nearly min.
IAE, ISE, ITAE Hwang [60]
Kc
( 49)
Kc
( 5 0)
Ti (48 )
0471
. Ku
K mω u
Ti
( 49 )
0471
. Ku
K mω u
Ti
( 50)
0471
. Ku
K mω u
0.1 ≤
τm
≤ 2.0 ;
Tm
3 ≤ ε < 20
Decay ratio = 0.25 τm
≤ 2.0 ; ε ≥ 20
Tm
0.1 ≤
Decay ratio = 0.1 01
. ≤
τm
≤ 2.0 ; ε < 2.4
Tm
Model: Method 8
3
[
]
ω τ
( 44)

126
. ( 0.506) H m 1 − 107
. ε + 0616
.
ε 2 
K c (1 + K H K m )
Kc ( 44) =  K H −
, Ti (44) =


K m (1 + K H K m )
ω H K m 00661
.
1 + 0.824 ln[ ω H τ m ] 1 + 171
. ε − 117
. ε2


Kc
( 45)
[

109
. 1 − 0.497ω H τ m + 0.0724ω H2 τ m 2

=  KH −
K m (1 + K H Km )

]  , T


[

0.584 1 − 0439
. ω H τ m + 00514
.
ωH 2 τm2
Kc ( 46) =  K H −
K m (1 + K H Km )

[

0675
.
1 − 0.472ω H τ m + 0061
. ω H 2 τm2
Kc ( 47) =  K H −
K m (1 + KH K m )

[
)(
(
( 45)
i
] , T


=
]  , T


( 47)
i
(
ω H K m 0.054 − 1 + 2.54ω H τ m − 0.457 ω H τ m
=
( 46)
i
=
]
K c (45) (1 + K H K m )
2
Kc
( 46)
(
(1 + K H K m )
ω H K m 0.0714 1 + 0.685ω H τ m − 0131
. ωH 2 τm2
Kc
( 47)
(
(1 + K H K m )
ω H K m 00484
.
1 + 143
. ω H τ m − 0.273ω H 2 τ m 2
)
)
ω τ
( 48)

12
. ( 0.495) H m 1 − 11
. ε + 0.698 ε 2 
K c (1 + K H K m )
Kc ( 48) =  K H −
, Ti (48) =


K m (1 + K H K m )
ω H K m 0.0702 1 + 0.734 ln[ ω H τ m ] 1 + 148
. ε − 11
. ε2


Kc
( 49)
[

103
. 1 − 051
. ω H τ m + 0.0759ω H 2 τ m 2

= KH −
K m (1 + KH K m )

[
)(
(
]  , T



0.822 1 − 0.549ω H τ m + 0112
. ω H2 τ m 2
Kc (50) =  KH −
K m (1 + K H K m )

( 49)
i
]  , T


i
=
( 50)
Kc
( 49)
(
(1 + K H K m )
ω H K m 0.0386 −1 + 3.26ω H τ m − 0.6ω H 2 τ m2
=
Kc
(
( 50)
)
(1 + K H K m )
ω H K m 0.0142 1 + 6.96ω H τ m − 177
. ω H 2 τm2
)
2
)
)
)
Kc
Rule
Ti
Td
Ti (51)
0471
. Ku
K mω u
Comment
4
Servo - nearly min.
IAE, ISE, ITAE Hwang [60]
(continued)
Decay ratio = 0.1 -
Kc(51)
2.4 ≤ ε < 3
Decay ratio = 0.1 -
Kc(5 2)
Model: Method 8
Ti (52)
Kc(53)
Servo – nearly
minimum IAE, ITAE
– Hwang and Fang
[61]
Model: Method 9
2

  
c1 + c2 τ m + c3  τ m   K u

Tm
 Tm  


c 1 = 0.537, c 2 = −0.0165
c 3 = 0.00173
0471
. Ku
K mω u
3 ≤ ε < 20
Decay ratio = 0.1 -
0471
. Ku
K mω u
Ti (53)
01
. ≤
2
Kc

  
c + c τ m + c 9  τ m   K u
2

 τ m    7 8 Tm
τm
Tm   ω u Kc


K uω u c 4 + c 5
+ c6 
  

Tm
 Tm  

 c = 0.350, c = − 00344
.
c 4 = 0.0503 , c 5 = 0.163
7
8
c 6 = −0.0389
Model: Method 1 or
Method 18
Ku
Am
arbitrary
Ku cosφ m
αTd
Astrom and
Hagglund [98]
Model: Method not
relevant
  T


m
τ m 1 + 

T
+
τ

m
m 

[

0786
.
1 − 0441
. ω H τ m + 00569
.
ω H 2 τm2
Kc (51) =  K H −
Km (1 + KH K m )

[
]  , T


( 51)
i
0. 65




  T


m
0 .25 τ m 1 + 

T
+
τ

m
m 

0 .65




τm
≤ 2.0 Tm
decay ratio = 0.03
01
. ≤
Tu
Kc
( 51)
(
τm
≤ 2.0 Tm
decay ratio = 0.12
01
. ≤
01
. ≤
Tuning rules
developed from Ku , Tu
4
+ tan 2 φ m
α
4π
Specify phase margin
φ m ; α = 4 (Astrom et
al. [30])
(1 + K H K m )
ω H K m 0.0172 1 + 4.62ω H τ m − 0.823ω H2 τ m 2
]
τm
≤ 2.0
Tm
Specify gain margin
Am
Tu
4π 2 Ti
tan φ m +
=
τm
≤ 2.0 ; ε ≥ 20
Tm
c 9 = 0.00644
2
2
Kc


Simultaneous
τ
τ  

 
c 1 + c 2 m + c 3  m   K u
c 7 + c 8 τ m + c 9  τ m   K u
2







τ
τ
Servo/regulator Tm

Tm
 Tm 
 Tm   ω u K c
K u ω u c 4 + c 5 m + c 6  m   



Tm
 Tm  
nearly minimum IAE,

c 7 = 0.371, c 8 = −00274
.
c 1 = 0.713, c 2 = −0176
.
ITAE - Hwang and
c 4 = 0.149, c 5 = 0.0556
c 9 = 0.00557
c 3 = 0.0513
Fang [61]
c 6 = −0.00566
Model: Method 9
McMillan [58]





1.415 Tm 
1

0 .65 
K m τm 
 Tm  
 
1+ 

 Tm + τ m  
τm
≤ 2.0 ;
Tm
01
. ≤
2
2
Kc

Regulator – nearly 
  
 τm  
τm
c + c τ m + c  τ m   K u
2
minimum IAE, ITAE c 1 + c 2 T + c 3  T   K K uω u c 4 + c 5 τ m + c 6  τ m    7 8 Tm 9  T m   ω u Kc
m
m




Tm
 Tm  

– Hwang and Fang 

c 7 = 0.421, c 8 = 0.00915
c
=
0802
.
,
c
=
−
0154
.
1
2
[61]
c 4 = 0.190 , c 5 = 0.0532
c 9 = −000152
.
c 3 = 0.0460
Model: Method 9
c 6 = −0.00509
4
τm
≤ 2.0 ;
Tm
0.1 ≤
)
( 52)

128
. ( 0.542)
1 − 0.986 ε + 0.558 ε 2 
K c (1 + K H K m )
Kc (52 ) =  K H −
, Ti (52) =


K m (1 + K H K m )
ω H K m 0.0476 1 + 0.996 ln[ ω H τ m ] 1 + 2.131 ε − 113
. ε2


ωH τm
[

114
. 1 − 0.466ω H τ m + 0.0647ω H 2 τ m 2
Kc (53) =  KH −
K m (1 + KH K m )

]  , T


)(
(
i
( 53)
=
Kc
(
( 53)
(1 + K H K m )
ω H K m 0.0609 − 1 + 197
. ω H τ m − 0.323ω H 2 τ m2
)
)
Rule
Kc
Ti
Li et al. [99]
Kc(5 4) 5
εTd
Model: Method not
relevant
φm
0
15
20 0
25 0
η
Am
2
1.67
1.67
4h
πAA m
2.8
3.2
3.5
Tan et al. [39]
Model: Method 10
Kφ
1.67
1.43
1.43
30
35 0
40 0
0
3.8
4.0
4.2
45
50 0
55 0
Tu
ω uωφ
2
2
2
)
K u − r 2 Kφ
2
4
+ tan 2 φ m
α
4π
ω φ Ti
2
2
01443
.
ωu
04830
.
3.7321
ω150
0.9330
ω 150
)
0
(
0
η
60
65 0
1.11
1.11
5.4
5.5
simplified algorithm
Am = 2 , φ m = 450 ;
α chosen arbitrarily
ωφ ;
r = 01
. + 09
. (K u K φ )
05774
.
ωu
(
Am
φm
Arbitrary A m , φ m at
1
0.25
G p ( jω u )
G p jω 150 0
4.6
4.9
5.2
00796
.
T
αTd
(
1.25
1.25
1.25
tan φ m +
rK φ ω u − ω φ
Am
Model: Method not
relevant
η
Am
0
Comment
T 
4
2
η + η + 
4π 
η 
η
φm
Am
03183
.
T
Ku
cosφ m
Am
Friman and Waller
[41]
φm
Td
0
)
τm
< 0.25 , Am = 2 ,
Tm
φ m = 60 0
0.25 ≤
τm
≤ 2.0
Tm
Am = 2 , φ m = 450
 tan φ − b A 2 − b2 
m


4h cos φm
Kc =
, η= 
with ±h = amplitude of relay, ±b =
2πb
 2

1 + b A2 − b 2 
2
πA m  A − b +
(Ti − Td )


T


deadband of relay, A, T = limit cycle amplitude and period, respectively.
5
( 54 )
(
)
Kc
Rule
Ti
Td
Comment
Direct synthesis
Regulator - Gorecki
et al. [28]
Kc
( 55)
Ti
( 55)
Td
Model: Method 6
Regulator - minimum
IAE - Smith and
Corripio [25] – page
343-346
Model: Method 6
Servo – minimum
IAE – Smith and
Corripio [25] – page
343-346
Model: Method 6
Servo – 5%
overshoot – Smith
and Corripio [25] –
page 343-346
Model: Method 6
6
Kc
( 55)
Ti (56)
Td (5 6)
Tm
K mτ m
Tm
0.5τ m
5Tm
6Km τ m
Tm
0.5τ m
Tm
2K m τ m
Tm
0.5τ m
Kc
(2 tuning rules)
( 56) 6
2
2

 τm 
 τm   τm  
2 Tm 
τm
=
6+
3+
 −9− 
 −
 e
Km τ m 
2Tm
 2Tm 
 2Tm   2Tm  


2
Ti
( 55)
Pole is real and has
maximum attainable
τ
multiplicity; m < 2
Tm
( 5 5)
Low frequency part of
magnitude Bode
diagram is flat.
2
 τ 
τ
3+  m  − 3− m
2 Tm
 2Tm 
2

 τm 
 τ 
τm 
τ
 −9− m − m 
6 +
 3+
2Tm 
2Tm  2Tm 
 2Tm 

= τm
,
2
2
2
3

 τm 
 τm   τ m 
τm  τm  
τm
21 + 3
+
− 6
  3+ 
 − 36 − 4.5
 −

Tm  2Tm  
Tm
 2Tm 
 2Tm   2Tm 


2
Td
( 55)
= 0.5τ m
 τ 
3+  m  − 1
 2Tm 
2

 τm 
 τ 
τm 
τ
 −9− m − m 
6 +
 3+
2Tm 
2Tm  2Tm 
 2Tm 

2
2
Kc ( 56) =
1
Km
1
, Ti (56 ) = τ m

τ m  Tm
2 (56) 
+ 1 − 2
Ti  τ m

2
1+ 7
Td
( 56 )
= τm
3
T 
T 
T 
Tm
+ 135 m  + 240  m  + 180 m 
τm
τ
τ
 m
 m
 τm 
2
 Tm

 Tm  
Tm
152
+ 1 1 + 3
+ 6  
τm
 τm  
 τm
 

3
Tm
T 
T 
T 
+ 27 m  + 60 m  + 60 m 
τm
 τm 
 τm 
 τm 
2
7 + 42
7 + 42
3
4
T 
T 
T 
Tm
+ 135 m  + 240  m  + 180 m 
τm
 τm 
 τm 
 τm 
4
4
01
. ≤
τm
≤ 15
.
Tm
01
. ≤
τm
≤ 15
.
Tm
Suyama [100]
Model: Method 6

1 
Tm
+ 0.2236
 0.7236
Km 
τm

Tm + 0.309 τ m
2.236Tm τ m
7.236Tm + 2.236τ m
Kc
Rule
Juang and Wang
[101]
Model: Method 6
Cluett and Wang
[44]
Model: Method 6
Gain and phase
margin - Zhuang and
Atherton [20]
Model: Method not
relevant
α+
Ti
τ 
τm
+ 05
.  m
Tm
 Tm 

τ 
Km  α + m 
Tm 

Model: Method 6
α+
Tm
2
Td
τ 
τm
+ 05
.  m
Tm
 Tm 
2
Comment
2
τ 

τ
τ 
0.5 m  Tm α + m − 0.5α m 
Tm
Tm 
 Tm 

2

 τ m  
τ m  
τm
 α +
 α +
+ 0.5  

Tm  
Tm

 Tm  

τ 
α + m 
Tm 

Closed loop time
constant = αTm ,
0<α <1
0.019952 τ m + 0. 20042Tm
Km τm
0.099508τ m + 0.99956Tm
τm
0.99747 τ m − 8 .7425.10 − 5 Tm
−0.0069905τ m + 0.029480Tm
τm
0.029773τ m + 0.29907 Tm
Closed loop time
constant = 4τm
0.055548 τ m + 0. 33639Tm
Km τm
0 .16440τ m + 0.99558 Tm
τm
0.98607τ m − 15032
.
.10 − 4 T m
−0.016651τ m + 0.093641Tm
τm
0.093905τm + 0.56867 Tm
Closed loop time
constant = 2τm
0.092654 τ m + 0.43620Tm
Km τm
0.20926 τ m + 0 .98518Tm
τm
0.96515τ m + 4255010
. − 3 Tm
−0.024442τ m + 0.17669Tm
τm
0.17150τm + 080740
.
Tm
Closed loop time
constant = 133
. τm
012786
.
τ m + 0.51235Tm
K mτ m
0.24145τ m + 0 .96751Tm
τm
0.93566 τ m + 2 .298810
. − 2Tm
−0.030407 τ m + 0 .27480Tm
τm
0.25285 τ m + 1.0132Tm
Closed loop time
constant = τ m
016051
.
τ m + 057109
.
Tm
K mτ m
0 .26502τ m + 0 .94291Tm
τm
0.89868τ m + 6.935510
. − 2 Tm
−0.035204τ m + 0.38823Tm
τm
0.33303 τm + 11849
.
Tm
Closed loop time
constant = 08
. τm
019067
.
τ m + 0.61593Tm
K mτ m
0.28242τ m + 0.91231T m
τm
0.85491τ m + 0.15937 Tm
−0.039589τ m + 0.51941Tm
τm
0.40950τ m + 13228
.
Tm
Closed loop time
constant = 067
. τm
mK u cos(φ m ),
m = 0 .614(1 − 0.233 e
0
(
−0 .347 K m K u
− 0. 45K m Ku
φ m = 338
. 1 − 0.97e
τ 
0177
.
+ 0.348 m 
 Tm 
Abbas [45]
2
)
) α
tan(φ m ) +
4
+ tan2 (φ m )
α
2ω u
,
tan(φm ) +
α = 0.413( 3.302K mK u + 1)
− 1.002
Tm + 05
. τm
1 Tm + τ m
Km Tm τm
0.0821
Servo – minimum ISE 18578
.
φm
- Ho et al. [103]
K
A 0.9087
m
m
 τm 
 
 Tm 
4Tm τ m
Tm + τ m
− 0.9471
Ti
( 57)
V = fractional
overshoot
Tm τ m
2Tm + τ m
K m ( 0531
.
− 0.359V 0.713 )
Camacho et al.[102]
Model: Method 6
Gain margin = 2, phase
margin = 60 0
τ
01
. ≤ m ≤ 2.0
Tm
4
+ tan2 ( φm )
α
2ω u
0 ≤ V ≤ 0.2
τ
01
. ≤ m ≤ 5.0
Tm
Tm τ m
Tm + τ m
0.4899Tm φ m 0.1457  τ m 
 
0.0845
 Tm 
Am
7
1.0264
Am ∈[ 2 ,5] ,
[
]
φ m ∈ 30 0 ,60 0 ,
τm
≤ 10
. .
Tm
01
. ≤
Model: Method 6
621189
.
403182
.
τ m 76.2833τ m
+
−
Am
Tm
A m Tm
(Ho et al [104])
Am ∈[ 2 ,5] ,
Given A m , ISE is minimised when φ m = 29.7985 +
− 0.908
0.3678
1.0317
Regulator - minimum 10722
.
φ m −0.116  τ m 
12497
.
Tm φ m 1.0082  τ m 
0.4763Tm φ m −0.328  τ m 






0.2099
0.0961
ISE – Ho et al. [103] Km A m 0.8432  Tm 
 Tm 
 Tm 
Am
Am
Model: Method 6
Given A m , ISE is minimised when φ m = 46.5489A m0.2035 ( τ m Tm )
Kc
Rule
7
Ti
( 57 )
=
[
Ti
0.0211Tm 1 + 0.3289 A m + 6.4572φ m + 251914
.
( τ m Tm )
1 + 0.0625A m − 0.8079φ m + 0.347( τm Tm )
Td
]
0.3693
[
]
01
. ≤
τm
≤ 10
.
Tm
φ m ∈ 30 0 ,60 0 ,
(Ho et al [104])
Comment
Morilla et al. [104a]
Model: Method 24
Kc
Robust
Robust - Brambilla et
al. [48]
Model: Method 6
τm
( 57a ) 8
α Ti
1 + 1 − 4α
Tm + 0.5τ m
1  Tm + 05
. τm 


Km 
λτ m

α = 0. 1; δ 0 = [ 0. 2, 0. 5]
( 57 a )
τm
≤ 10 and no
Tm
model uncertainty -
01
. ≤
Tm τ m
2Tm + τ m
λ ≈ 0.35
1  Tm + 05
. τm 


Km  λ + 05
. τm 
Tm + 0.5τ m
Tm τ m
2Tm + τ m
λ > 01
. Tm ,
λ ≥ 0.8τ m .
Fruehauf et al. [52]
5Tm
9τ m Km
5τ m
≤ 0.5τ m
τm
< 033
.
Tm
Model: Method 1
Tm
2τ m Km
Tm
≤ 0.5τ m
τm
≥ 0.33
Tm
Lee et al. [55]
Ti
K m ( λ + τm )
Rivera et al. [49]
Model: Method 6
Tm +
Model: Method 6
2

τm
τ 
1 − m 
2( λ + τ m ) 
3Ti 
τm
2( λ + τ m )
2
λ + ατ m − 05
. τm
2λ + τ m − α
2
τm
3
3
2
Two degrees of
τm
ατ
− m
freedom
controller;
2
Tmα − 6
α
=
T
−
m
2λ + τ m − α
τ
2

λ  −T ;
Ti
Tm 1 −
 e
 Tm 
2
λ2 + ατ m − 05
. τm
Desired response =
2λ + τ m − α
e− τ m s
Tm + α −
Ti
K m ( 2λ + τm − α )
λ=
2
m
m
1 + λs
8
Kc
( 57a )
=
with δ 0 =
Ti
( 57 a )
(
K m [ 2δ 0 + ω n 0 τ m − Ti
1

2π 

1 + 
 log e [b a ] 
,
2
( 57 a )
)] , ω
n0
=
(
− δ 0 Tm + δ 0 Tm + Tm τ m − Ti
2
ωn0
(
2
Tm τ m − Ti
( 57 a )
)− αT
b
= desired closed loop response decay ratio
a
i
( 57a )
)− αT
i
( 57 a ) 2
( 57a ) 2
K m e− sτ
- ideal controller with first order filter
1 + sTm
m
Table 26: PID tuning rules - FOLPD model

 1
1
G c ( s) = Kc  1 +
+ Td s
. 3 tuning rules.
Ti s

 Tf s + 1
Rule
Kc
Ti
Td
1  Tm + 05
. τm 


K m  λ + τm 
Tm + 0.5τ m
Tm τ m
2Tm + τ m
TF =
τm Tm
τ m + Tm
Tf =
Comment
Robust
** Morari and
Zafiriou [105]
Horn et al. [106]
Model: Method 6
H∞ optimal – Tan et
al. [81]
Model: Method 6
2Tm + τ m
2 (λ + τ m ) K m
Tm + 0.5τ m

0.265λ + 0.307  Tm

+ 05
. 
Km
 τm

Tm + 0.5τ m
τ m Tm
τ m + 2Tm
λτ m
,
2( Tm + τ m )
λ > 0.25τ m ,
λ > 0.2Tm .
λτ m
;
2( λ + τ m )
λ > τ m , λ < Tm .
τm
5.314 λ + 0.951
λ = 2 - ‘fast’
response
λ = 1 - ‘robust’ tuning
λ = 1.5 - recommended
Tf =
K m e− sτ
- ideal controller with second order filter
1 + sTm
m
Table 27: PID tuning rules - FOLPD model


1
1 + b1s
G c ( s) = Kc  1 +
+ Td s
. 1 tuning rule.
2
Ti s

 1 + a 1s + a 2s
Kc
Ti
Td
2Tm + τ m
,
2( 2λ + τm − b1) Km
Tm + 05
. τm
τ m Tm
τ m + 2Tm
Rule
Comment
Robust
Horn et al. [106]
[
λ τ m + 2 Tm τ m ( τ m − λ )
2
Model: Method 6
b1 =
+
[
Tm (τ m + 2 λ)
2 λ (2T m − λ)
( τ m + 2λ)
]
]
Filter
1 + b1 s
1+
2λτ m + 2 λ + b1 τ m
a2 =
2
2( 2λ + τ m − b 1 )
λ 2τ m
2( 2λ + τ m − b1 )
λ < Tm
.
s + a 2s 2
; λ > τm ,
K m e− sτ
- ideal controller with set-point weighting
1 + sTm
m
Table 28: PID tuning rules - FOLPD model


1
G c ( s) = Kc  b +
+ Td s . 1 tuning rule.
Ti s


Kc
Rule
Ti
Direct synthesis
Comment
(Maximum sensitivity)
38
. e
Astrom and
Hagglund [3] –
pages 208-210
Td
−8.4 τ + 7.3τ 2
Tm
K mτ m
,
τ = τ m ( τ m + Tm )
52
. τ me
− 2.5τ − 1.4 τ 2
0.46Tme 2.8τ − 2.1τ
or
2
0.89 τm e −0.37 τ − 4.1τ or
2
0.077 Tm e5.0τ − 4 .8 τ
2
b = 0.40e 0.18 τ + 2 .8 τ ;
M s = 1.4 ;
2
014
. ≤
Model: Method 3
τm
≤ 55
.
Tm
b = 0.22e0.65τ + 0.051τ ;
Ms = 2.0 ;
2
8.4e− 9.6 τ + 9.8τ Tm
Km τ m
2
32
. τ m e− 1.5τ − 0.93τ or
2
0.28Tm e3.8τ − 1.6 τ
2
0.86τ m e− 1.9 τ − 0.44 τ or
2
0.076 Tm e3.4 τ − 1.1τ
2
014
. ≤
τm
≤ 55
.
Tm
K m e− sτ
- ideal controller with first order filter and set-point
1 + sTm
m
Table 29: PID tuning rules - FOLPD model

 1 

1
1 + 0.4Trs
weighting U(s) = K c 1 +
+ Td s 
− Y( s)  . 1 tuning rule.
R (s )
1 + sTr
 Tis
 Tf s + 1 

Rule
Direct synthesis
Normey-Rico et al.
[106a]
Model: Method not
specified
Kc
Ti
Td
Comment
0. 375(τ m + 2Tm )
K m τm
Tm + 0. 5τ m
Tm τ m
2Tm + τ m
Tf = 0.13τ m
Tr = 0.5τ m



K m e− sτ
1   1 + Td s 
 . 20.
Table 30: PID tuning rules - FOLPD model
- classical controller G c ( s) = Kc  1 +

1 + sTm

Ti s   1 + Td s 



N 
m
Kc
Ti
Td
083
. Tm
Km τ m
15
. τm
0.25τ m
τm
τm
Rule
Process reaction
Hang et al. [36] –
page 76
Model: Method 1
Witt and Waggoner
[107]
Model: Method 1
[
0.6Tm
Tm
,
]
K mτ m K mτ m
Witt and Waggoner
[107]
Model: Method 2
Kc
Regulator tuning
Minimum IAE - Kaya
and Scheib [108]
Model: Method 3
Minimum IAE – Witt
and Waggoner [107]
Model: Method 1
1
Kc
( 58 )
=
Kc
Ti
( 59 )
‘aggressive’ tuning;
05
. Tm
K mτ m
5τ m
0.5τ m
‘conservative’ tuning;
0. 889Tm
K m τm
1.75τ m
0.70 τm
τm
= 0.167
Tm
( 59 )
=
Minimum performance index
 Tm 
098089
.
 
Km  τ m 
0.76167
Tm  τm 
 
091032
.
 Tm 
Kc(5 9) 2
Ti (59)
Td (59 )
τ 
τ
0.7425 + 0.0150 m + 0.0625 m 
Tm
 Tm 
τ 
Tm
τ
+ 0.25 m 0.7425 + 0.0150 m + 0.0625 m 
τm
Tm
 Tm 
2
Tm
τ 
Tm
τ
1350
.
+ 025
. ± 07425
.
+ 0.0150 m + 0.0625 m 
τm
Tm
 Tm 
−0.921
τ 
0.964 Tm  m 
 Tm 
2
− 1.886 

 τm 
1 ± 1 − 1693
,
.  


 Tm 


1.137
τ 
1 − 1 m 1693
.  m
 Tm 
1.886
, Td
τ 
0.59974 Tm  m 
 Tm 
1.05221
Tm
 τm 
0718
.
=
 
Km  Tm 
Equivalent to Cohen
and Coon [11];
N = [10,20]
0.5τ m
2 Km
1350
.
2
Td
( 5 8)
5τ m
T
T
1.350 m + 0.25 ± m
τm
τm
Ti (58 ) =
Td (58 ) =
Ti
( 58)
Foxboro EXACT
controller ‘pretune’;
N=10
Equivalent to Ziegler
and Nichols [8];
N = [10,20]
Tm
Km τ m
St. Clair [15] – page
21
Model: Method 1
Shinskey [15a]
Model: Method 1
( 5 8) 1
Comment
( 59 )
=
τ 
0.964 Tm  m 
 Tm 
1.137
τ 
1 + 1 ± 1.693 m 
 Tm 
1.886
2
0.89819
0<
0.1 <
τm
≤ 1 ; N=10
Tm
τm
< 0.258
Tm
; N = [10,20]
Kc
Rule
Minimum ISE - Kaya
and Scheib [108]
Model: Method 3
Minimum ITAE Kaya and Scheib
[108]
Model: Method 3
Minimum ITAE Witt and Waggoner
[107]
Model: Method 1
Servo tuning
Minimum IAE - Kaya
and Scheib [108]
Model: Method 3
Minimum IAE - Witt
and Waggoner [107]
Model: Method 1
Minimum ISE - Kaya
and Scheib [108]
Model: Method 3
Minimum ITAE Kaya and Scheib
[108]
Model: Method 3
3
Kc
Td
( 60 )
Kc
Ti
=
( 60)
( 61)
( 61)
=
=
 Tm 
111907
.
 
Km  τ m 
1.06401
 Tm 
077902
.
 
K m  τm 
065
.  Tm 
 
Km  τm 
Tm  τ m 
 
07987
.
 Tm 
τ 
0.54766 Tm  m 
 Tm 
Tm  τ m 
 
114311
.
 Tm 
0.70949
τ 
0.57137 Tm  m 
 Tm 
1.03826
Ti( 60) 3
1.04432
τ 
0.762Tm  m 
 Tm 
1.03092
Td ( 61)
0.80368
0.86411
τ 
0.54568 Tm  m 
 Tm 
Tm
τ 
0.42844Tm  m 
 Tm 
τ
0.99783 + 0.02860 m
Tm
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
1.0081
τm
≤ 1 ; N=10
Tm
0.995
0.914


 τm 

τm 
1 ± 1 − 1392
.  
. − 013
.
074
 ,

Tm  
 Tm 



τ 
1 m 1 − 1.392 m 
 Tm 
0.869

τm 
 0.74 − 0.13 
Tm 

, Td
( 61)
=
τ 
0.696Tm  m 
 Tm 
τ 
1 ± 1 − 1.392  m 
 Tm 
0.869
τm
≤ 1 ; N = [10,20]
Tm
0<
.
0.914
τm
≤ 1 ; N=10
Tm
τm
≤ 1 ; N=10
Tm
0.995
τ 
0.696 Tm  m 
 Tm 
; N = [10,20]
0<
−1.733 

1 ± 1 − 1.283 τ m 
 T ( 60) =
,
1.733

 i
 Tm 


τ


1 − 1 m 1283
.  m
 Tm 
1.733
τm
< 0.379
Tm
0 .1 ≤
Tm
τ
112666
.
− 0.18145 m
Tm
0<
0 .1 <
Minimum performance index
1.08433
Tm
 τm 
0
.
50814
T

τ
m
0.9895 + 0.09539 m
 Tm 
Tm
τ 
0.762Tm  m 
 Tm 
τ 
1 + 1 ± 1.283 m 
 Tm 
− 0.869
Td ( 60)
Ti (61)
 Tm 
071959
.
 
Km  τ m 
Comment
0.87798
Kc( 61)
0.947
Td
0.9548
Kc( 60)
 Tm 
112762
.
 
Km  τ m 
0.679  τ m 
 
K m  Tm 
 τm 
1086
.
=
 
Km  Tm 
Ti
0.89711
0.914

τm 
0.74 − 0.13 
Tm 

Kc
Ti
Td
Kc( 62) 4
Ti (62 )
Td ( 62)
0809
. Tm
Km τ m
Tm
05
. τm
Overshoot = 16%; N=5
Tm
025
. τm
N = 2.5
Rule
Minimum ITAE Witt and Waggoner
[107]
Model: Method 1
Direct synthesis
Tsang and Rad [109]
Model: Method 13
aTm
Km τ m
Tsang et al. [111]
ξ
a
Model: Method 6
1.681
8
1.382
9
ξ
a
0.0 1.161
0
0.1 0.991
6
a
0.2 0.859
4
0.3 0.754
2
ξ
ξ
a
0.4 0.669
3
0.5 0.600
0
ξ
a
0.6 0.542
9
0.7 0.495
7
Comment
a
0.8 0.456
9
0.9
0 .1 ≤
τm
≤ 1 ; N = [10,20]
Tm
ξ
1.0
Robust
Chien [50]
Model: Method 6

1 
Tm


K m  λ + 05
. τm 
Tm
05
. τm
λ = [ τ m , Tm ] ; N=10
1  05
. τm 


K m  λ + 05
. τm 
05
. τm
Tm
λ = [ τ m , Tm ] ; N=10
K mτ m
3τ m − 0.32 Tu


Tu
Tu  015
.
− 0.05
τm


014
. Tu
095
. Tm Km τ m
1.43τ m
052
. τm
τ m Tm = 0.2
095
. Tm Km τ m
117
. τm
048
. τm
τ m Tm = 0.5
114
. Tm Km τ m
1.03τ m
0.40τ m
τ m Tm = 1
139
. Tm Km τ m
077
. τm
0.35τ m
τ m Tm = 2
Ultimate cycle
Minimum IAE
regulator – Shinskey
[59] – page 167.
Model: Method not
specified
Minimum IAE
regulator - Shinskey
[16] – page 143.
Model: Method 6
4
Kc
Ti
( 62)
( 62)
=
0.965  τ m 
=
 
K m  Tm 
− 0.85
0.929


 τm 

τm 
1 ± 1 − 1232
.  
.
0.796 − 01465
 ,

Tm  
 Tm 



τ 
0.616 Tm  m 
 Tm 
τ 
1 m 1 − 1.232 m 
 Tm 
0.85
0.929

τm 
 0.796 − 0.1465 
Tm 

, Td
( 62)
=
τ 
0.616Tm  m 
 Tm 
τ 
1 ± 1 − 1.232 m 
 Tm 
0.85
0.929

τm 
0.796 − 0.1465 
Tm 

K m e− sτ
- non-interacting controller
1 + sTm
m
Table 31: PID tuning rules - FOLPD model
Rule
Regulator tuning
Minimum IAE –
Huang et al. [18]
Model: Method 6
Servo tuning
Minimum IAE Huang et al. [18]
Model: Method 6





1 
Td s
 . 2 tuning rules.
 E( s) −
U (s) = K c 1 +
Y
(
s
)
Td s

Tis 

1+


N


Kc
Ti
Td
Comment
Minimum performance index
Kc( 63) 5
Ti (63)
Td ( 63)
01
. ≤
τm
≤ 1 ; N =10
Tm1
01
. ≤
τm
≤ 1 ; N =10
Tm
Minimum performance index
Kc( 64) 6
Ti (64 )
Td ( 64)
−0.8058
0.6642
2.1482
τ
 τm 
 τm 
 τm 
1 
τ
T
01098
.
− 8.6290 m + 11863
.
+
231098
.
+
20
.
3519
−
191463
.
e
 
 
 
Km 
Tm
 Tm 
 Tm 
 Tm 

m
5
Kc ( 63) =
Ti
( 63)
Td
( 63)
m



2
3
4
5

τ 
τ 
τ 
τ  
τ
= Tm − 0.0145 + 2 .0555 m − 4.4805 m  + 7 .7916 m  − 7 .0263 m  + 2.4311 m  
Tm
 Tm 
 Tm 
 Tm 
 Tm  

2
3
4
5
6
 τm 
 τm 
 τm 
 τm 
 τm  
Tm 
τm
 −0.0206 + 0.9385
=
− 2 .3820  + 7.2774   − 111018
.
  + 8.0849   − 2.274  
( 63)
Tm
K c 
 Tm 
 Tm 
 Tm 
 Tm 
 Tm  
0.0865
−0.4062
2 .6405
τ
 τm 
τ 
 τm 
1 
τ
7.0636 + 66.6512 m + 261928
.
+ 7.3453 m 
+ 336578
.
− 57.937e T
 
 
Km 
Tm
 Tm 
 Tm 
 Tm 

m
6
Kc ( 64) =
Ti
( 64 )
Td
( 64 )
m



2
3
4
5

 τm 
τ 
 τm 
τ  
τ
= Tm  0.9923 + 0.2819 m − 14510
.
.
  + 2.504 m  − 18759
  + 0.5862 m  
Tm
 Tm 
 Tm 
 Tm 
 Tm  

2
3
4
5
6
 τm 
 τm 
 τm 
 τm 
 τm  
Tm 
τm

= ( 64) 0.0075 + 0.3449
− 0.0793  + 0.8089   − 1.0884  + 0.352  + 0.0471  
Tm
K c 
 Tm 
 Tm 
 Tm 
 Tm 
 Tm  
K m e− sτ
- non-interacting controller
1 + sTm
m
Table 32: PID tuning rules - FOLPD model
Rule

Tds
1 
E (s ) −
U (s) = K c 1 +
Y (s) . 5 tuning rules.

sT
Tis 

1+ d
N
Kc
Ti
Td
Servo tuning
Minimum ISE Zhuang and
Atherton [20]
Model: Method 6
Minimum ISTSE Zhuang and
Atherton [20]
Model: Method 6
Minimum ISTES Zhuang and
Atherton [20]
Model: Method 6
 Tm 
1260
.
 
Km  τm 
0.887
 Tm 
1295
.
 
Km  τ m 
0.619
 Tm 
1053
.
 
Km  τ m 
0.930
 Tm 
1120
.
 
K m  τm 
0.625
 Tm 
1001
.
 
Km  τ m 
Minimum performance index
0.886
Tm
 τm 
0
.
375
T


τ
m
0.701 − 0.147 m
 Tm 
Tm
τ 
0.378 Tm  m 
 Tm 
0.756
τ 
0.349 Tm  m 
 Tm 
0 .907
Tm
τ 
0.350 Tm  m 
 Tm 
0.811
Tm
τ 
0.308 Tm  m 
 Tm 
0.897
Tm
τ 
0.308 Tm  m 
 Tm 
0.813
Tm
τ
0.661 − 0.110 m
Tm
Tm
0.736 − 0.126
τm
Tm
τ
0.720 − 0.114 m
Tm
 Tm 
0942
.
 
Km  τm 
0.933
0 .624
Comment
τ
0.770 − 0.130 m
Tm
τ
0.754 − 0.116 m
Tm
Ultimate cycle
Servo – minimum
.
ISTSE – Zhuang and 4.437K m K u − 1587
0.037(589
. K m K u + 1)Tu
Ku
Atherton [20]
8.024K m Ku − 1435
.
Model: Method not
relevant
Regulator - minimum
0.5556Ku
039
. Tu
IAE - Shinskey [16] –
0.4926Ku
0.34Tu
page 148.
05051
.
Ku
0.33Tu
Model: Method 6
0.4608Ku
0.28Tu
0112
. Tu
01
. ≤
τm
≤ 10
. ; N=10
Tm
11
. ≤
τm
≤ 2.0 ; N=10
Tm
01
. ≤
τm
≤ 10
. ; N=10
Tm
11
. ≤
τm
≤ 2.0 ; N=10
Tm
01
. ≤
τm
≤ 10
. ; N=10
Tm
11
. ≤
τm
≤ 2.0 ; N=10
Tm
01
. ≤
τm
≤ 2.0 ; N=10
Tm
014
. Tu
τ m Tm = 0.2
014
. Tu
τ m Tm = 0.5
013
. Tu
τ m Tm = 1
013
. Tu
τ m Tm = 2
K m e− sτ
- non-interacting controller
1 + sTm
m
Table 33: PID tuning rules - FOLPD model
Rule

Td s
1 
E (s) −
U (s) =  K c +
Y ( s) . 6 tuning rules.

sT
Ti s 

1+ d
N
Kc
Ti
Td
Regulator tuning
Minimum IAE - Kaya
and Scheib [108]
Model: Method 3
Minimum ISE - Kaya
and Scheib [108]
Model: Method 3
Minimum ITAE Kaya and Scheib
[108]
Model: Method 3
Servo tuning
Minimum IAE - Kaya
and Scheib [108]
Model: Method 3
 Tm 
113031
.
 
Km  τ m 
Minimum ISE - Kaya
and Scheib [108]
Model: Method 3
 Tm 
126239
.
 
Km  τm 
0.8388
Minimum ITAE Kaya and Scheib
[108]
Model: Method 3
 Tm 
098384
.
 
Km  τ m 
0.49851
Comment
Minimum performance index
 Tm 
131509
.
 
Km  τm 
0.8826
1.3756
τ 
0.5655Tm  m 
 Tm 
1.25738
τ 
0.79715Tm  m 
 Tm 
0.41941
τ 
0.49547Tm  m 
 Tm 
0.41932
Tm  τm 
 
12587
.
 Tm 
 Tm 
13466
.
 
Km  τm 
0.9308
Tm  τ m 
 
16585
.
 Tm 
 Tm 
13176
.
 
Km  τm 
0.7937
Tm  τ m 
 
112499
.
 Tm 
0.81314
1.42603
0.4576
Minimum performance index
0.17707
Tm
 τm 
0.32175Tm  
τ
5.7527 − 5.7241 m
 Tm 
Tm
Tm
τ 
0.47617Tm  m 
 Tm 
0.24572
Tm
τ 
0.21443Tm  m 
 Tm 
0.16768
τ
6.0356 − 6.0191 m
Tm
τ
2.71348 − 2.29778 m
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
K m e− sτ
- non-interacting controller with set-point weighting
1 + sTm
m
Table 32: PID tuning rules - FOLPD model
Rule

1 
K Ts
U( s) = Kc  b +
 E( s) − c d Y( s) + Kc ( b − 1)Y(s) . 3 tuning rules.
sT
Ti s 

1+ d
N
Kc
Ti
Td
Comment
Ultimate cycle
Hang and Astrom
[111]
0.6Ku
0.5Tu
0125
. Tu
τm
< 0.3 .
Tm
b = 2( x − 01
. )+
N=10
Model: Method 1
0.6Ku
0.5Tu
0.6Ku

τm 
0.5 15
. −.083
.
 Tu
Tm 

0125
. Tu
0.6Ku

τm 
0.5 15
. −.083
.
 Tu
Tm 

0125
. Tu
0.6Ku
0.335Tu
0125
. Tu
0.5Tu
0125
. Tu
0.6Ku
Hang et al. [65]
Model: Method 1
36
15 − K
b=
,
,
'
27 + 5K '
15 + K
10% overshoot - servo 20% overshoot - servo
'
b=
0.6Ku
Hang and Cao [112]
Model: Method 11
K =
 11[τ m Tm ] + 13

2
 37[ τ m Tm ] − 4 
0.6Ku

τ 
. − 0.22 m  Tu
 053
Tm 

'
x = overshoot.
τ
0.3 ≤ m < 0.6 .
Tm
b = 2x +
0.6 ≤
K =
'
08
. ≤
0125
. Tu

τ T
. − 0.22 m  u
 053
Tm  4

τm
Tm
τm
< 10
. ; b=0.8
Tm
10
. <
016
. ≤
0.57 ≤
01
. ≤
τm
Tm
τm
< 0.8 .
Tm
b = 16
. −
 11[τ m Tm ] + 13

2
 37[ τ m Tm ] − 4 
0.222 K ' Tu
8 4 ' 
 K + 1 ,

17  9
20% overshoot,
10% undershoot;
servo
b=
0125
. Tu
166
. τm
,
Tm
τm
; b=0.8
Tm
τm
< 0.57 ;
Tm
N=10
τm
< 0.96 ;
Tm
N=10
τm
< 0.5 ; N=10
Tm
K m e− sτ
- industrial controller
1 + sTm
m
Table 33: PID tuning rules - FOLPD model




1 
1 + Td s
U( s) = Kc 1 +
Y(s)  . 6 tuning rules.
  R( s) −
Ts

Ti s  

1+ d



N
Rule
Kc
Ti
Regulator tuning
Minimum IAE - Kaya
and Scheib [108]
Model: Method 3
Minimum ISE - Kaya
and Scheib [108]
Model: Method 3
Minimum ITAE Kaya and Scheib
[108]
Model: Method 3
Servo tuning
Minimum IAE - Kaya
and Scheib [108]
Model: Method 3
 Tm 
081699
.
 
Km  τ m 
Minimum ISE - Kaya
and Scheib [108]
Model: Method 3
 Tm 
11427
.
 
Km  τm 
0.9365
Minimum ITAE Kaya and Scheib
[108]
Model: Method 3
 Tm 
08326
.
 
Km  τm 
0.7607
Td
Comment
Minimum performance index
091
.  Tm 
 
Km  τm 
0 .7938
Tm  τ m 
 
101495
.
 Tm 
 Tm 
11147
.
 
Km  τm 
0.8992
Tm  τ m 
 
09324
.
 Tm 
 Tm 
07058
.
 
Km  τm 
0.8872
Tm  τ m 
 
103326
.
 Tm 
1.004
1.00403
0.8753
0.99138
τ 
0.5414Tm  m 
 Tm 
τ 
0.56508 Tm  m 
 Tm 
0.7848
0.91107
τ 
0.60006Tm  m 
 Tm 
0.971
Minimum performance index
0.97186
Tm
 τm 
τ 0.44278Tm  
1.09112 − 0.22387 m
 Tm 
Tm
Tm
τ 
0.35308Tm  m 
 Tm 
0.78088
Tm
τ 
0.44243Tm  m 
 Tm 
1.11499
τ
0.99223 − 0.35269 m
Tm
τ
1.00268 + 0.00854 m
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
0<
τm
≤ 1 ; N=10
Tm
K m e− sτ

1 
- series controller Gc ( s) = Kc  1 +
 (1 + sTd ) . 3 tuning
1 + sTm
Ti s 

m
Table 34: PID tuning rules - FOLPD model
rules.
Kc
Ti
Td
Comment
5Tm
6Km τ m
15
. τm
0.25τ m
Foxboro EXACT
controller
035
. Ku
0.25
Tu
0.25
Tu
Tm
025
. τm
Rule
Autotuning **
Astrom and
Hagglund [3] – page
246
Model: Not specified
Ultimate cycle
Pessen [63]
Model: Method 6
01
. ≤
Direct synthesis
aTm
Km τ m
Tsang et al. [110]
Model: Method 13
a
1.819
4
1.503
9
ξ
a
0.0 1.269
0
0.1 1.089
4
ξ
a
0.2 0.949
2
0.3 0.837
8
ξ
a
0.4 0.748
2
0.5 0.675
6
ξ
a
0.6 0.617
0
0.7 0.570
9
ξ
a
0.8 0.541
3
0.9
ξ
1.0
τm
≤1
Tm
K m e− sτ
- series controller with filtered derivative
1 + sTm
m
Table 35: PID tuning rules – FOLPD model


1 
sTd
Gc ( s) = Kc  1 +   1 +
sT

Ti s 
1+ d


N


 . 1 tuning rule.


Kc
Ti
Td
Comment
Chien [50]
Tm
K m (λ + 0.5τm )
Tm
05
. τm
λ = [ τ m , Tm ] ; N=10
Model: Method 6
0.5τ m
K m ( λ + 0.5τm )
05
. τm
Tm
λ = [ τ m , Tm ] ; N=10
Rule
Robust
K m e− sτ
- controller with filtered derivative
1 + sTm
m
Table 36: PID tuning rules – FOLPD model


1
Td s
G c ( s) = Kc  1 +
+
Ti s 1 + s Td


N
Rule
Robust
Chien [50]
Model: Method 6
Gong et al. [113]


 . 3 tuning rules.


Kc
Ti
Td
Comment
Tm + 0.5τ m
K m ( λ + 0.5τm )
Tm + 05
. τm
Tm τ m
2Tm + τ m
λ = [ τ m , Tm ] ; N=10
Tm + 0.3866τ m
K m (λ + 1.0009τm )
Tm + 0.3866τ m
03866
.
Tm τm
Tm + 0.3866τ m
N = [3,10]
Model: Method 6
λ=
( 0.1388 + 0.1247N )Tm + 0.0482 Nτ m τ
0.3866 ( N − 1)Tm + 0.1495 Nτ m
m
Direct synthesis
Davydov et al. [31]
Model: Method 12
1


τm
Km  1552
.
+ 0078
. 

Tm



τm
 0186
.
+ 0532
.  Tm
T


m
1


τm
K m  1209
.
+ 0103
. 

Tm



τ
 0.382 m + 0.338 Tm
Tm


Closed loop response


τm
0.25 0186
.
+ 0532
.  Tm
damping factor =
T


m
0.9; 02
. ≤ τ m Tm ≤ 1 ;
N = Km
Closed loop response


τ
0.4 0.382 m + 0.338 Tm
damping factor =
Tm


0.9; 02
. ≤ τ m Tm ≤ 1 ;
N = Km
K m e− sτ
- alternative non-interacting controller 1 1 + sTm
m
Table 37: PID tuning rules – FOLPD model

1 
U( s) = Kc 1 +
 E( s) − Kc Td sY ( s) . 6 tuning rules.
Ti s 

Rule
Kc
Ti
Td
Comment
012
. Tu
Tu
< 2 .7
τm
012
. Tu
Tu
≥ 2.7
τm
Ultimate cycle
2
Ku
Regulator - minimum
IAE - Shinskey [59] –
page 167.
Model: method not
specified
Minimum IAE Shinskey [17] – page
117. Model: Method
6
Minimum IAE Shinskey [16] – page
143.
Model: Method 6
Regulator - minimum
IAE - Shinskey [16] –
page 148.
Model: Method 6
Regulator – minimum
IAE - Shinskey [17] –
page 121. Model:
method not specified
Process reaction –
VanDoren [114]
Model: Method 1
3.73 − 0.69
Tu
0125
.
Tu
τm
0125
.
Tu
τm
τm
Ku
T
2.62 − 0.35 u
τm
2
132
. Tm K m τ m
180
. τm
0.44τm
τ m Tm = 01
.
132
. Tm K m τ m
1.77τm
0.41τ m
τ m Tm = 0.2
135
. Tm Km τ m
1.43τ m
0.41τ m
τ m Tm = 0.5
149
. Tm K m τ m
117
. τm
0.37τm
τ m Tm = 1
182
. Tm Km τ m
0.92τ m
0.32τ m
τ m Tm = 2
0.7692 Ku
0.48Tu
011
. Tu
τ m Tm = 0.2
0.6993Ku
0.42Tu
012
. Tu
τ m Tm = 0.5
0.6623Ku
0.38Tu
012T
. u
τ m Tm = 1
0.6024K u
0.34Tu
012
. Tu
τ m Tm = 2
0.7576 Ku
0.48Tu
011
. Tu
τ m Tm = 0.2
15
. Tm
Km τ m
25
. τm
04
. τm
K m e− sτ
- Alternative filtered derivative controller 1 + sTm
m
Table 38: PID tuning rules – FOLPD model
2

1   1 + 05
. τm s + 0.0833τ m s 2 
G c ( s) = Kc  1 +
. 1 tuning rule.

2

Ti s  

[1 + 01. τ ms]

Kc
Ti
Td
aTm
Km τ m
Tm
025
. Tm
Rule
Comment
Direct synthesis
Tsang et al. [110]
Model: Method 13
a
1.851
2
1.552
0
ξ
a
0.0 1.329
3
0.1 1.159
5
ξ
a
0.2 1.028
0
0.3 0.924
6
ξ
a
0.4 0.841
1
0.5 0.768
0
ξ
a
0.6 0.695
3
0.7 0.621
9
ξ
a
0.8 0.552
7
0.9
ξ
1.0
K m e− sτ
K
- I-PD controller U(s) = c E (s) − K c (1 + Td s)Y(s) . 2 tuning
1 + sTm
Ti s
m
Table 39: PID tuning rules – FOLPD model
rules.
Kc
Rule
Ti
Td
Comment
Ti ( 64a )
Td ( 64 a )
Underdamped system
response - ξ = 0.707 .
Direct synthesis
Chien et al. [74]
K c ( 64a )
1
τ m ≤ 0.2Tm
Model: Method 6
Minimum ISE –
Argelaguet et al.
[114a]. Model:
Method not defined
1
Kc (64a ) =
Td (64 a ) =
Tm + 0. 5τ m
2Tm + τ m
2K m τ m
1.414TCLTm + τ m Tm + 0.25τ m 2 − TCL2
(
Km TCL2
+ 0.707TCL τ m + 0.25τ m
2
)
, Ti (64 a ) =
0707
. TmTC Lτ m + 0.25Tm τ m2 − 05
. τ m TCL2
2
Tm τ m + 025
. τ m + 1414
. TCLTm − TC L2
Tm τ m
2Tm + τ m
1414
. TCL Tm + τ mTm + 0.25τ m2 − TCL2
Tm + 05
. τm
First order Pade
approximation for τm
Table 40: PID tuning rules - FOLPD model - G m ( s) =
K me − sτ
– Two degree of freedom controller:
1 + sTm
m


1
Td s
U (s) = K c 1 +
+
 Tis
T
1+ d

N

Kc
Rule
Ti
Servo/regulator
tuning
Kc
( 64b ) 2
Ti
Model: ideal process
Kc
Td
( 64 b)
( 64 b)
Td
Comment
Minimum performance index
Taguchi and Araki
[61a]
2






β
T
s
d
 E( s) − K c  α +
 R (s) . 1 tuning rule.


Td 
s
1+
s

N 




1 
1. 224
=
0.1415 +
τm
Kc 

− 0.001582
Tm

( 64 b)
Td
( 64b )
τm
≤ 1.0 ;
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of
tuning rules of Chien
et al. [10]


2
3



( 64 b)
 0.01353 + 2.200 τm − 1. 452  τm  + 0. 4824  τm  
,
T
=
T
 
  
m
 i
Tm
 Tm 
 Tm  



2
2

 τm  
τm
τm 
τm

= Tm 0.0002783 + 0. 4119
− 0. 04943   , α = 0. 6656 − 0. 2786
+ 0. 03966   ,


Tm
Tm
 Tm 
 Tm  

β = 0. 6816 − 0.2054
τm
τ 
+ 0.03936  m 
Tm
Tm 
2


1
Table 41: PID tuning rules - non-model specific – ideal controller G c ( s) = Kc  1 +
+ Td s . 25 tuning rules.
Ti s


Rule
Kc
Ti
Td
Comment
Ultimate cycle
Ziegler and Nichols
[8]
[ 0.6Ku , Ku ]
0.5Tu
0125
. Tu
Quarter decay ratio
Blickley [115]
0.5K u
Tu
. Tu , 0167
. Tu ]
[ 0125
Quarter decay ratio
0.5K u
Tu
0.2Tu
Parr [64] – pages
190-191
05
. Ku
034
. Tu
008
. Tu
Overshoot to servo
response ≈ 20%
Quarter decay ratio
De Paor [116]
0.866Ku
0.5Tu
0125
. Tu
phase margin = 30 0
Corripio [117] – page
27
Mantz andTacconi
[118]
0. 75Ku
0. 63Tu
01
. Tu
Quarter decay ratio
0.906K u
0.5Tu
0125
. Tu
phase margin = 25 0
0.4698 K u
0.4546Tu
01136
.
Tu
01988
.
Ku
12308
.
Tu
0.3077Tu
Gain margin = 2, phase
margin = 20 0
Gain margin = 2.44,
phase margin = 61 0
0.2015K u
0.7878Tu
01970
.
Tu
Gain margin = 3.45,
phase margin = 46 0
0.35Ku
0.77Tu
0.19Tu
Atkinson and Davey
[119]
** Perry and Chilton
[120]
0. 25Ku
0.75Tu
0. 25Tu
033
. Ku
05
. Tu
033
. Tu
Gain margin ≥ 2 , phase
margin ≥ 450
20% overshoot - servo
response
‘Some’ overshoot
02
. Ku
05
. Tu
033
. Tu
‘No’ overshoot
Yu [122] – page 11
033
. Ku
05
. Tu
0125
. Tu
‘Some’ overshoot
02
. Ku
05
. Tu
0125
. Tu
‘No’ overshoot
Luo et al. [121]
0.48Ku
0.5Tu
0125
. Tu
McMillan [14] –
page 90
McAvoy and
Johnson [83]
Karaboga and Kalinli
[123]
0.5K u
0.5Tu
0125
. Tu
054
. Ku
Tu
0.2Tu
Astrom and
Hagglund [3] – page
142
Astrom and
Hagglund [93]
. Ku ,0.6Ku ]
[ 032
 04267
.
K u 15
. Ku 
,


Tu
Tu 

Tu (1 − cos φ m )
π sin φm
. Ku Tu ,015
. Ku Tu ]
[ 008
Tu (1 − cos φ m )
4π sin φ m
φ m = phase margin
Tu 

2
 tan φ m + 1 + tan φ m 

4π 
Tu 

2
 tan φ m + 1 + tan φ m 

16π 
‘small’ time delay
05
. Ku
12
. Tu
0125
. Tu
‘aggressive’ tuning
0.25Ku
12
. Tu
0125
. Tu
‘conservative’ tuning
Hang and Astrom
[124]
Ku sinφ m
Astrom et al. [30]
Ku cos φ m
St. Clair [15] page 17
Kc
Rule
Pole placement - Shin
et al. [125]
Kc
Other rules
Harriott [126] –
pages 179-180
Ti
Td
αTi
( 65)
Comment
Typical α : 0.1
Typical ξ : [0.3,0.7]
( 65)
K25%
0167
. T25%
0.667T25%
Quarter decay ratio
Quarter decay ratio
K25%
0.67T25%
017
. T25%
Parr [64] – pages
191, 193
05
.
G p ( jω u )
Tu
0.25Tu
McMillan [14] Page 43
083
. K25%
0.5T25%
01
. T25%
‘Fast’ tuning
0.67K25%
0.5T25%
01
. T25%
‘Slow’ tuning
1
Ti
4
φ m = 450 ,‘small’ τ m
2 2 G p ( jω u )
1
ωu
Kc( 66) 2
Ti (66 )
Td ( 66)
Calcev and Gorez
[69]
Zhang et al. [127]
1
Ti
( 65) 1
K c ( 65) =
Ti ( 65) =
ρ


1 
1 
( 65)
( 65)
a 1  αω uTi −
 − a 2  αω 1Ti −
 − b2 − ρ a 2
( 65)
( 65)

ω u Ti 

ω 1Ti 


φ m = 15 0 , ‘large’ τ m
,

1
( a2 − a1 ) + (a 1 − a 2 ) 2 + 4(αρa1ω u + αb2 ω 1 ) ρωa1 + ωb 2 
2(αρa1ω u + αb2 ω 1 )
u
1
( ω − ω1 ) 1 − ξ2 , b = 1 sin ∠G ( jω ) , a = − 1
cos ∠G p ( jω 1 ) , ρ = u
2
p
1
1
K1
ωu
ξ
K1
Ku
K1 , ω1 = modified ultimate gain and corresponding angular frequency
(
a2 =
2
)
(
1
Kc ( 66) =
(
1 + tan φm − φ p
(
π 2 A2 − ε 2
)
) + sin
)
, d, ε = relay amp. and deadband, A = limit cycle amp.
φm
16d 2
Crossing point of the Nyquist curve and relay with hysteresis is outside the unit circle:
2
2
2
Ti ( 66) =
 ωu 
  −1
 ωc 


ω
ω u  β − u tan φ m − φ p 

ωc

(
(
)
, 10
. +
ωu
ω
tan φ m − φ p ≤ β ≤ 1.2 + u tan φ m − φ p
ωc
ωc
(
)
(
)
)
βω u − ω c tan φm − φ p
66
Td ( ) =
, ω c = frequency when the open loop gain equals unity.
2
2
ωu − ωc
Crossing point of the Nyquist curve and relay with hysteresis is within the unit circle:
2
Ti ( 66) =
 ωu 
  −1
 ωc 


ω
ω u  β + u tan φ p − φ m 

ωc

(
(
)
)
,
ωu − ωr
ω − ωr
− 0.4 ≤ β ≤ u
+ 0.4
ωr
ωr
βω u − ω c tan φm − φ p
66
Td ( ) =
, ω r = oscillation frequency when a pure relay is switched into closed loop.
2
2
ωu − ωc
Kc
Rule
Garcia and Castelo
[127a]
3
Kc
Td
( 66a )
( 66a )
=
=
Kc
( 66a ) 3
(
cos 180 0 + φm − ∠G p ( jω1 )
(
G p ( jω1 )
Td
( 66a )
( 66a )
d
Ti
), T
( 66a )
i
)
sin 180 0 + φm − ∠G p ( jω1 ) + 1
(
Ti
2ω1 cos 180 0 + φm − ∠G p ( jω1 )
=
(
T
)
2[sin 180 0 + φ m − ∠G p ( jω1 ) + 1]
(
Comment
ω1 cos 180 + φ m − ∠G p ( jω1 )
0
)
,
) , ω = oscillation frequency when a sine function is placed in series
with the process in closed loop; ω1 < ωu .
1
Table 42: PID tuning rules - non-model specific – controller with filtered derivative



1
Td s 
 . 8 tuning rules.
Gc ( s) = Kc 1 +
+
Ti s 1 + Td s 



N 
Kc
Ti
Ti
2( A 1 − Ti )
A3
Rule
Td
Comment
Td ( 67) 4
N < 10
A3A 4 − A2 A 5
2
A 3 − A1A 5
N ≥ 10
Direct synthesis
A 2 − Td
( 67)
Td
A1 −
N
Vrancic [72]
A3
(
2 A 1A 2 − A 3K m − Td A
2
1
A3
A 2 − Td A1
)
05
. Ti
A1 − Km Ti
( 67 ) 2
A2 − A 2 2 − 4χ A1A 3
2χ A1
χTi
N = 10
χ = [ 02
. , 0.25]
Td (67 a)
8 ≤ N ≤ 20
A3
Vrancic [73]
Kc
− A3 − A 5A1 +
2
Td ( 67) =
4
(A
2
3
− A 5A1
A 2 − Td (67 a) A 1 −
)
2
−
Td (67 a)
Km
N
4
( A 3A 2 − A5 )( A5 A2 − A4 A 3 )
N
2
( A 3A2 − A5 )
N
t
t
y( τ ) 

y1( t) =  Km −
 dτ , y2 ( t) =

∆u 
∫
0
t
y5 ( t) =
2
( 67a )
∫ [A
4
∫ (A 1 − y1 (τ))dτ ,
t
y3 ( t) =
0
∫ ( A 2 − y2 (τ))dτ ,
t
y 4 (t ) =
0
∫ [A
3
]
− y 3 ( τ ) dτ ,
0
]
− y 4 ( τ) dτ , A1 = y1( ∞) , A 2 = y 2 ( ∞) , A 3 = y3 ( ∞) , A4 = y 4 ( ∞) , A 5 = y5 ( ∞)
0
Alternatively, if the process model is G m ( s) = K m
(
1 + b1s + b2 s 2 + b3 s3 + b4 s 4 + b5 s5 − sτ
e
, then
1+ a 1s + a 2 s2 + a 3s 3 + a 4 s4 + a 5s 5
m
)
A1 = Km ( a1 − b1 + τ m ) , A2 = K m b 2 − a2 + A1a1 − b1 τm + 05
. τ m2 ,
(
(b − a
(a − b
)
A3 = Km a 3 − b3 + A 2 a1 − A1a 2 + b2 τ m − 05
. b1τ m 2 + 0167
. τ m3 ,
)
A4 = Km
4
4
+ A 3a 1 − A 2a 2 + A 1a 3 − b3 τ m + 05
. b2 τ m 2 + 0167
. b1τ m 3 + 0.042 τ m 4 ,
A5 = Km
5
5
+ A 4 a1 − A 3a 2 + A2 a 3 − A1a 4 + b 4 τm − 05
. b3 τ m2 + 0167
. b2 τ m 3 − 0.042b1τ m 4 + 0.008τm 5
K c ( 67a ) =
A3


[ T (67 a ) ]2
( 67 a )
2
2 A 1A 2 − A 0A 3 − Td
A1 − d
A 0A 1 
N


Td (67a ) is obtained from a solution of the following equation:
[
A0 A3
( 67a )
Td
N3
]
4
+
[
A1A 3
( 67 a )
Td
N2
]
3
−
[
A 0A 5 − A 2A 3
( 67 a )
Td
N
] + (A
2
2
3
)[
− A 1A5 Td
( 67a )
] + (A A
2
5
− A 3A4 ) = 0
)
Rule
Kc
Ti
Td
Comment
Direct synthesis
Lennartson and
Kristiansson [157]
Model: Method 1
Kc(111) 5
Ti (111)
0.4Ti (111)
Ku Km ≤ 0.6
Kc(111a )
Ti (111a)
0.4Ti (111a)
Ku Km > 10
ω u K u Km ≤ 0.4
Kc(111b )
Ti (111b )
0.4Ti (111b)
Ku K m > 10
ω u Ku K m ≥ 0.4
Kc(111c) 6
Ti (111c)
0.4Ti (111c)
Ku Km > 10
ω u K u Km ≤ 0.4
Kc(111d )
Ti (111d)
0.4Ti (111d )
Ku Km > 10
ω u Ku K m ≥ 0.4
( 111e)
( 111e)
( 111e)
Kristiansson and
Lennartson [157]
Model: Method 1
Kc
5
12Ku K m − 35K u Km + 30
2
Kc (111) = Ku K m
N=
0.4Ti
2
[
Ku K m + 25
. 12K u Km − 35Ku K m + 30
2
2
]
, Ti
( 111)
K c (111)
=
−0.053ω u + 0.47ω u − 014
. ω u + 0.11
3
2
,
2.5 
35
30 
+
12 −

Km K u 
K m K u K m 2 Ku 2 
Kc (111a ) = Ku Km
N=
Ti
Ku K m > 167
.
ω u Ku K m > 0.45
N = 2.5
2
2
K c (111a )
12 Ku Km − 35Ku Km + 30
(111a )
,
T
=
,
i
−0.525ω u3 + 0.473ω u2 − 0.143ω u + 0.113
Ku Km + 3 12Ku 2K m2 − 35KuK m + 30
[
]
3 
35
30 
+
12 −
2
2
Km Ku 
K mK u Km Ku 
2
2
Kc
12Ku Km − 35Ku Km + 30
( 111b)
, Ti
=
,
3
2
−0185
. ω u + 1052
. ω u − 0854
. ω u + 0.309
K uKm + 3 12Ku2 Km 2 − 35Ku Km + 30
( 111b)
Kc (111b ) = Ku Km
N=
6
3
Km Ku
[

35
30 
+
12 −
2
2
K mK u Km Ku 

Kc (111c ) = Ku K m
N=
]
( 111c )
12Ku K m − 35K u K m + 30
2
2
[
Ku K m + 25
. 12K u Km − 35Ku Km + 30
2
2
]
, Ti
( 111c )
=
−0525
. ωu
3
Kc
,
2
+ 0.473ω u − 0143
. ω u + 0113
.
2.5 
35
30 
+
12 −

Km K u 
K m K u K m 2 Ku 2 
Kc (111d ) = Ku K m
12Ku K m − 35K u Km + 30
2
[
( 111d)
2
Ku K m + 25
. 12K u Km − 35Ku K m + 30
2
2
]
, Ti
( 111d )
=
−0185
. ωu
3
Kc
,
2
+ 1052
. ω u − 0.854ω u + 0309
.
2.5 
35
30 
+
12 −

Km K u 
K m K u K m 2 Ku 2 
7.71
914
.
K c (111e )
( 111e )
(111e )
Kc
=
+ 314
. , Ti
=
2
2 −
3
K m Ku
−0.63ω u + 0.39ω u 2 + 0.15ω u + 0.0082
Km K u
N=
Rule
Kc
Ti
Td
Kristiansson and
Lennartson [158]
Model: Method 1
Kc(111f) 7
Ti (111f )
0.4Ti (111f )
Comment
(111g)
Kristiansson and
Lennartson [158a]
Kc
Kc
Model: Method not
specified
7
Kc (111f ) =
N=
2.5
Km K u
Ti
(111g )
(111g )
Td
(111g )
Tf
(111g)
3
2
Kc
Ti
Td
(111h )
Tf
(111h )
Td
(111g )
Td
(111h )
( 111f )
]
Tf
given below;
0.1 ≤ K u K m ≤ 0.5
Tf
[
3
2
]
 1. 6( −20 + 13 K m K u )(1+ 0.37 K m K u ) 
− 1

2
2
2
K m K u ( −20 + 13K m K u )(1 + 0.37 K m K u ) 2  1.1K m K u − 2.3K m K u + 1. 6

2
2
K K (1. 1K u K m − 2. 3K u K m + 1.6) 1.6( −20 + 13K m K u )(1 + 0. 37K m K u ) 
= m u
− 1

ω u ( −20 + 13K m K u )(1 + 0. 37K m K u ) 2  1. 1K m 2 K u 2 − 2. 3K m K u + 1.6

2
2
 ( −20 + 13K m K u ) 2 (1 + 0.37 K m K u ) 2



2
2
2
2
2
K K (1. 1K u K m − 2.3K u K m + 1. 6) 
(1.1K m K u − 2.3K m K u + 1.6)

= m u
−
1

ω u ( −20 + 13K m K u )(1 + 0.37 K m K u ) 2  1.6( −20 + 13K m K u )(1 + 0. 37K m K u )
−
1


2
2
 1.1K m K u − 2. 3K m K u + 1.6

=
K m K u (1.1K u K m − 2.3K u K m + 1.6)
=
2
ω u ( − 20 + 13 K m K u )(1 + 0.37 K m K u ) 2
=
2
2
2

(1.1K u K m − 2.3K u K m + 1. 6)  4. 8K m K u (1 + 0.37 K m K u )
− 1

2
2
2
2
2
3ω u K m K u (1+ 0.37 K m K u )  1. 1K m K u − 2. 3K m K u + 1.6 
2
2

(1.1K u K m − 2.3K u K m + 1.6)  4.8K m K u (1+ 0.37 K m K u )
− 1

2
2
2
3ω u (1 + 0.37 K m K u )
1.1K m K u − 2.3K m K u + 1.6 
 3K m 2 K u 2 (1+ 0.37 K m K u ) 2



2
2
2
2
2
(1.1K u K m − 2.3K u K m + 1.6)  (1.1K m K u − 2.3K m K u + 1. 6)

=
− 1
2

4
.
8
K
K
(
1
+
0
.
37
K
K
)
3ω u (1 + 0. 37K m K u )
m
u
m
u

−1 
 1. 1K m 2 K u 2 − 2.3K m K u + 1.6



1.1K u K m − 2. 3K u K m + 1.6
2
=
2
3ω u (1 + 0.37 K m K u ) 2
(111h )
given below;
K u K m > 0.5

35
30 
+
12 −

K m K u K m 2 Ku 2 

(1. 1K u K m − 2. 3K u K m + 1.6)
=
(111h )
(111h )
Ti
(111h )
2
[
2
8
(111h ) 8
Ti
(111g )
12 Ku Km − 35Ku Km + 30Ku
Kc
Km Ku
, Ti (111f ) =
,
3
3
2
2
2
2
Km Ku + 25
. 12K u Km − 35Ku Km + 30
ω u 095
. Km Ku − 2Km Ku + 14
.
2
Kc
(111g )
Kc
Rule
Kristiansson and
Lennartson [158a]
(continued)
Model: Method not
specified
Other
Kc
Ti
(111i) 9
Ti
Leva [67]
(1 − ω
2
G p ( jω )
(1 − ω
)
2
+ ω Ti 2
2
Ti Td
)
2
+ ω Ti
2
ω 1+
2π
β
)
(
tan φ m − φ ω − 0.5π
)
αTd (68) 10
Comment
Tf
(111i )
[
2 tan φ m + 1 + tan 2 φ m
ωu
]
(111i )
given below;
K u K m < 0.1
2π
βω
β = 10
φω > φm − π
Td ( 68)
6 ≤ α ≤ 10
φω < φm − π
Ti
4
Parameters determined
at φ m = 300 ,450 ,600
2
Ku cosφ m
Astrom [68]
Td
tan φm − φ ω − 0.5π
Ti Td
ωTi
2
(111i )
(
ωTi
G p ( jω )
Td
 20 .8(1.8 + 0.3K m K135 0 ) 
− 1

13 K m (1.8 + 0.3K m K 1350 )  ( −6 + 3. 7K m K 1350 )

( −6 + 3.7K 1350 K m ) K 1350 K m  20. 8(1.8 + 0. 3K m K 1350 ) 
(111i )
Ti
=
− 1

13ω1350 (1. 8 + 0.3K m K135 0 ) 2  (− 6 + 3.7K m K 1350 )

2
 169 (1. 8 + 0.3K m K135 )



( −6 + 3. 7K m K135 )K m K135
( −6 + 3.7K m K 135 ) K m K 135  ( −6 + 3. 7K m K 135 ) 2

(111i )
( 111i)
Td
=
− 1 , Tf
=
2  20 .8(1. 8 + 0 .3K K
13ω 135 (1.8 + 0.3K m K 135 ) 2
13ω135 (1. 8 + 0.3K m K135 ) 
m 135 )
−1 
 ( −6 + 3.7K m K135 )



9
Kc
(111i )
=
( −6 + 3.7K 1350 K m ) 2
2
0
0
0
0
0
0
0
10
Td ( 68) =
− αω + α 2ω 2 + 4αω 2 tan 2 ( φ m − φ ω − 05
. π)
2αω 2 tan(φ m − φω − 0.5π )
0
0
0
0
0
Table 43: PID tuning rules - non-model specific – ideal controller with set-point weighting
1
U( s) = Kc Fp R(s) − Y(s) +
( Fi R(s) − Y(s)) + Td s( Fd R(s) − Y(s)) . 1 tuning rule.
Ts
i
(
)
Rule
Kc
Ti
Td
Comment
Ultimate cycle
Mantz and Tacconi
[118]
0.6K u
0.5Tu
Fi = 1
0125
. Tu
Fd = 0.654
Quarter decay ratio
Fp = 017
.
Table 44: PID tuning rules - non-model specific – ideal controller with proportional weighting


1
G c ( s) = Kc  b +
+ Td s . 1 tuning rule.
Ti s


Kc
Rule
Direct synthesis
Astrom and
Hagglund [3] – page
217
(0.33e
− 0.31κ − κ
Ti
2
)K
u
κ = 1 Km Ku
,
Td
Comment
)T
b = 058
. e −1.3κ + 3.5κ
0 < Km Ku < ∞
maximum Ms = 1.4
)T
b = 025
. e 0.56 κ − 0.12 κ
0 < Km Ku < ∞
maximum Ms = 2.0
2
(0.76e
−1.6κ − 0.36κ
(0.59e
−1.3κ + 0.38 κ
2
)T
. e
(017
−0.46κ − 2.1κ
)T
. e
(015
−1.4κ + 0.56κ
u
2
u
2
(0.72e
− 1.6κ + 1.2 κ
2
)K
u
2
u
2
u
Rule
Ultimate cycle
Fu et al. [128]
Table 45: PID tuning rules - non-model specific – non-interacting controller

1 
K Ts
U( s) = Kc 1 +
 E( s) − c d Y( s) . 1 tuning rule.
sT
Ti s 

1+ d
N
Kc
Ti
Td
0.5K u
0.34Tu
0.08Tu
Comment

1
Table 46: PID tuning rules - non-model specific – series controller U( s) = Kc  1 +
 (1 + sTd ) . 3 tuning rules.

Ti s 
Rule
Kc
Ti
Td
Comment
025
. Ku
033
. Tu
05
.
Tu
‘optimum’ servo
response
0.2K u
0. 25Tu
05
.
Tu
0.2K u
0.5Tu
0. 33Tu
0. 33Ku
0. 33Tu
0.5Tu
‘optimum’ regulator
response - step
changes
No overshoot; close to
optimum regulator
‘Some’ overshoot
025
. Ku
033
. Tu
0.5Tu
Ultimate cycle
Pessen [131]
Pessen [129]
Grabbe et al. [130]
Table 47: PID tuning rules - non-model specific – series controller with filtered derivative



1 
sTd 
 . 1 tuning rule.
U( s) = Kc 1 +
1 +
sT

Ti s  
1 + d 


N 
Rule
Kc
Ti
Td
Ultimate cycle
Hang et al. [36]
- page 58
0.35Ku
113
. Tu
0. 20Tu
Comment

1  1 + sTd
Table 48: PID tuning rules - non-model specific – classical controller U( s) = Kc  1 +
. 1 tuning rule.

T
Ti s 

1+ s d
N
Rule
Kc
Ti
Td
Comment
Ultimate cycle
Corripio [117] – page
27
0.6K u
05
. Tu
0125
. Tu
10 ≤ N ≤ 20
Table 49: PID tuning rules - non-model specific – non-interacting controller

1
U( s) = Kc  1 +  E ( s) − Kc Td sY (s) . 1 tuning rule.
Ti s

Rule
Kc
Ti
Td
0.75K u
0.625Tu
01
. Tu
Ultimate cycle
VanDoren [114]
Comment
Table 50: PID tuning rules - IPD model
K m e− sτ m
s
Kc
Rule


- ideal controller Gc ( s) = K c 1 + 1 + Td s . 5 tuning rules.

Ti s
Ti
Td
βτ m
χτ m

Comment
Direct synthesis
α
Wang and Cluett [76]
K mτ m
– deduced from
Closed loop Damping
graph
time
Factor, ξ
constant
Model: Method 2
τm
0.707
Gain margin Phase margin
Am
φ m [degrees]
α
β
χ
2.0
32
0.9056
2.6096
0.3209
2τm
0.707
3.1
40
0.5501
4.0116
0.2205
3τ m
0.707
4.4
46
0.3950
5.4136
0.1681
4τm
0.707
5.5
49
0.3081
6.8156
0.1357
5τ m
0.707
6.7
52
0.2526
8.2176
0.1139
6τ m
0.707
7.8
54
0.2140
9.6196
0.0980
7τm
0.707
8.9
55
0.1856
11.0216
0.0861
8τ m
0.707
10.0
56
0.1639
12.4236
0.0767
9τm
0.707
11.2
57
0.1467
13.8256
0.0692
10τ m
0.707
12.2
58
0.1328
15.2276
0.0630
11τm
0.707
13.4
59
0.1213
16.6296
0.0579
12τ m
0.707
14.5
59
0.1117
18.0316
0.0535
13τ m
0.707
15.6
59
0.1034
19.4336
0.0497
14τ m
0.707
16.7
60
0.0963
20.8356
0.0464
15τ m
0.707
17.8
60
0.0901
22.2376
0.0436
16τ m
0.707
19.0
60
0.0847
23.6396
0.0410
τm
1.0
2.0
37
0.8859
3.2120
0.3541
2τm
1.0
2.9
46
0.6109
5.2005
0.2612
3τ m
1.0
3.8
52
0.4662
7.1890
0.2069
4τm
1.0
4.6
56
0.3770
9.1775
0.1713
5τ m
1.0
5.5
58
0.3164
11.1660
0.1462
6τ m
1.0
6.4
61
0.2726
13.1545
0.1275
7τm
1.0
7.1
62
0.2394
15.1430
0.1311
8τ m
1.0
8.0
64
0.2135
17.1315
0.1015
9τm
1.0
8.7
65
0.1926
19.1200
0.0921
10τ m
1.0
9.5
66
0.1754
21.1085
0.0843
11τm
1.0
10.4
67
0.1611
23.0970
0.0777
12τ m
1.0
11.1
67
0.1489
25.0855
0.0721
13τ m
1.0
12.0
68
0.1384
27.0740
0.0672
14τ m
1.0
12.8
68
0.1293
29.0625
0.0630
15τ m
1.0
13.4
69
0.1213
31.0510
0.0592
16τ m
1.0
14.4
69
0.1143
33.0395
0.0559
Rule
Kc
Ti
Td
Comment
Cluett and Wang [44]
0.9588
Km τ m
30425
.
τm
0.3912 τ m
Closed loop time
constant = τ m
Model: Method 2
0.6232
Km τ m
52586
.
τm
0.2632 τ m
Closed loop time
constant = 2τm
0.4668
Km τ m
7.2291τ m
0.2058τ m
Closed loop time
constant = 3τ m
0.3752
Km τ m
91925
.
τm
01702
.
τm
Closed loop time
constant = 4τ m
0.3144
Km τ m
111637
.
τm
01453
.
τm
Closed loop time
constant = 5τ m
0.2709
Km τ m
131416
.
τm
01269
.
τm
Closed loop time
constant = 6τ m
160
. τm
0.48τ m
1.48
K mτm
2τ m
0.37τ m
Decay ratio: 2.7:1
094
.
Km τ m
2τ m
0.5τ m
Ultimate cycle ZieglerNichols equivalent
Rotach [77]
Model: Method 4
Process reaction
Ford [132]
Model: Method 2
Astrom and
Hagglund [3] – page
139
Model: Method not
relevant
121
.
K mτ m
Damping factor for
oscillations to a
disturbance input =
0.75.
Table 51: PID tuning rules - IPD model
K m e− sτ m
s
- Ideal controller with first order filter, set-point weighting and

 1 

1
1 + 0.4Trs
output feedback U(s ) = K c 1 +
+ Tds 
− Y(s )  − K 0 Y(s) . 1 tuning rule.
 R( s)
Ti s
1 + sTr

 Tf s + 1 

Rule
Direct synthesis
Normey-Rico et al.
[106a]
Model: Method not
specified
Kc
Ti
Td
Comment
Tf = 0. 13τm
0.563
K m τm
1.5τm
0.667 τm
1
2Km τ m
Tr = 0.75τ m
K0 =
Table 52: PID tuning rules - IPD model
K m e− sτ m
s
- ideal controller with filtered derivative


1
Td s
G c ( s) = Kc  1 +
+
sT
Ts

i
1+ d

N
Rule
Robust
Chien [50]
Model: Method 2


 . 1 tuning rule.


Kc
Ti
Td
2
K m ( λ + 0.5τm )
2λ + τm
τ m ( λ + 025
. τm )
2 λ + τm
Comment
λ=
1
; N=10
Km
K m e− sτ m
- series controller with filtered derivative
s



1 
Td s 
 . 1 tuning rule.
G c ( s) = Kc  1 +  1 +
Ts

Ts
i 
 1 + d 

N 
Table 53: PID tuning rules - IPD model
Rule
Kc
Ti
Td
Comment
1  2λ + 0.5τ m 
2
K m  [ λ + 05
. τ m ] 
2λ + 05
. τm
05
. τm
 1

λ=
, τ m  ; N=10
K m


1 
05
. τm

2
K m  [ λ + 05
. τ m ] 
05τ m
2λ + 05
. τm
 1

λ=
, τ m  ; N=10
K m

Robust
Chien [50]
Model: Method 2



1   1 + Td s 
.
- classical controller G c ( s) = Kc  1 +  

Ti s   1 + Td s 



N 
5 tuning rules.
K e− sτ m
Table 54: PID tuning rules - IPD model m
s
Rule
Kc
Ti
Td
Comment
Ultimate cycle
Luyben [133]
Model: Method 2
0.46Ku
2.2Tu
016
. Tu
maximum closed loop
log modulus of +2dB ;
N=10
311
. Ku
2.2Tu
364T
. u
N=10
0.5τm
2λ + 05
. τm
 1

λ=
, τ m  ; N=10
K
 m

2λ + 0.5τ m
0.5τm
 1

λ=
, τ m  ; N=10
K m

Belanger and
Luyben [134]
Model: Method 2
Robust
Chien [50]
Model: Method 2
Regulator tuning
Minimum IAE Shinskey [17] – page
121. Model: Method
not specified
Minimum IAE Shinskey [17] – page
117. Model: Method
1
Minimum IAE –
Shinskey [59] – page
74
Model: Method not
specified
1
Km
 05
. τ m 

 λ + 0.5τ 2 
[
m] 
1  2λ + 0.5τ m 
2
K m  [ λ + 05
. τ m ] 
Minimum performance index
056
. Ku
0.39Tu
015
. Tu
0.93
Km τ m
157
. τm
056
. τm
09259
.
Km τ m
160
. τm
0.58τ m
N=10
09259
.
Km τ m
1.48τ m
0.63τm
N=20
Table 55: PID tuning rules - IPD model
K m e− sτ m
s
- Alternative non-interacting controller 1 -

1 
U( s) = Kc 1 +
 E( s) − Kc Td sY ( s) . 2 tuning rules.
Ti s 

Rule
Regulator tuning
Minimum IAE Shinskey [59] – page
74.
Model: Method not
specified
Minimum IAE –
Shinskey [17] – page
121. Model: Method
not specified
Minimum IAE –
Shinskey [17] – page
117. Model: Method
1
Kc
Ti
Td
Minimum performance index
12821
.
K m τm
19
. τm
046
. τm
0. 77K u
0. 48Tu
015
. τm
128
.
K mτ m
1.90τ m
0.48τ m
Comment
Table 56: PID tuning rules - IPD model
Kc
Rule
Direct synthesis
Chien et al. [74]
Model: Method 1
1
Kc
( 68a )
(
Kc
( 68a ) 1
1.414 TCL + τ m
2
CL
Km T
+ 0.707TCL τ m + 0.25τ m
2
)
K m e− sτ m
- I-PD controller U(s) = Kc E (s) − K c(1 + Td s)Y(s) .
s
Ti s
1 tuning rule.
Ti
Td
Comment
1414
. TCL + τ m
0.25τ m2 + 0.707TC Lτ m
1.414TCL + τ m
Underdamped system
response - ξ = 0.707 .
τ m ≤ 0.2Tm
Table 57: PID tuning rules - IPD model G m ( s) =
U(s) = Kc (1 +
Rule
Direct synthesis
Hansen [91a]
Model: Method not
specified
K me− sτ m
s
- controller
1
) E( s) + Kc ( b − 1) R (s) − Kc TdsY( s) . 1 tuning rule.
Tis
Kc
Ti
Td
Comment
0.938 K m τm
2. 7τm
0.313 τm
b = 0.167
Table 58: PID tuning rules - IPD model
K m e− sτ m
s
- Two degree of freedom controller:


1
Td s
U (s) = K c 1 +
+
 Tis
T
1+ d

N

Rule
Kc
Servo/regulator
tuning
1  1.253 


K m  τ m 
Model: ideal process
α = 0. 6642
φ c = phase
corresponding to the
crossover frequency;
Km =1
Td
Comment
Minimum performance index
Taguchi and Araki
[61a]
Minimum ITAE Pecharroman and
Pagola [134a]
Ti






β
T
s
d
 E( s) − K c  α +
 R (s) . 2 tuning rules.


Td 
s
1+
s

N 


2.388 τ m
0.4137 τm
β = 0.6797
1.672 K u
0.366 Tu
0.136 Tu
1.236 K u
0.427 Tu
0.149 Tu
0.994 Ku
0.486 Tu
0.155 Tu
0.842 Ku
0.538 Tu
0.154 Tu
0.752 Ku
0.567 Tu
0.157 Tu
0.679 Ku
0.610 Tu
0.149 Tu
0.635 K u
0.637 Tu
0.142 Tu
0.590 Ku
0.669 Tu
0.133 Tu
0.551K u
0.690 Tu
0.114 Tu
0.520 K u
0.776 Tu
0.087 Tu
0.509 Ku
0.810 Tu
0.068 Tu
τm
≤ 1.0 ;
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of
tuning rules of Chien
et al. [10]
α = 0. 601 , β = 1 ,
N = 10, φ c = −1640
α = 0.607 , β = 1 ,
N = 10, φ c = −1600
α = 0.610 , β = 1 ,
N = 10, φ c = −1550
α = 0.616 , β = 1 ,
Model: Method 6
N = 10, φ c = −1500
α = 0.605 , β = 1 ,
N = 10, φ c = −1450
α = 0.610 , β = 1 ,
N = 10, φ c = −1400
α = 0.612 , β = 1 ,
N = 10, φ c = −1350
α = 0.610 , β = 1 ,
N = 10, φ c = −1300
α = 0.616 , β = 1 ,
N = 10, φ c = −1250
α = 0.609 , β = 1 ,
N = 10, φ c = −1200
α = 0. 611 , β = 1 ,
N = 10, φ c = −1180
Km e − sτ


- ideal controller Gc ( s) = K c 1 + 1 + Td s .
s(1 + sTm )
Ti s


m
Table 59: PID tuning rules - FOLIPD model
1 tuning rule.
Rule
Kc
Ti
Td
Comment
Ultimate cycle
McMillan [58]
Model: Method not
relevant
Kc( 69) 1
Ti ( 69 )
Td ( 69)
Tuning rules
developed from Ku , Tu
2
1
Kc ( 69)




  T  0.65 
  T  0.65 
1111
.
Tm 
1

( 69 )
( 69 )
m
=
. τ m 1 +  m  

 , Ti = 2τ m 1 +    , Td = 05
K m τ m 2   T  0.65 
 τm  

  τ m  
m


1 +  τ  
m


Km e − sτ
- ideal controller with filter
s(1 + sTm )
m
Table 60: PID tuning rules - FOLIPD model

 1
1
G c ( s) = Kc  1 +
+ Td s
. 3 tuning rules.
Ti s

 1 + Tf s
Kc
Ti
Td
0.463λ + 0277
.
[
]
2
Km τ m
τm
Tm +
0.238λ + 0123
.
Tm τ m
0238
.
λ
+
0123
. ) Tm + τ m
(
Rule
Comment
Robust
Tan et al. [81]
Model: Method 2
Zhang et al. [135]
Model: Method 2
Tan et al. [136]
Model: Method 2
([ 0238
. λ + 0123
. ]Tm + τ m )
(
3λ + Tm + τ m
K m 3λ
2
+ 3λτ m + τ m2
)
3λ + Tm + τm
( 3λ + τ m ) Tm
3λ + τ m + Tm
λ = 15
. τ m ….Overshoot = 58%, Settling time = 6τ m
λ = 25
. τ m ….Overshoot = 35%, Settling time = 11τ m
λ = 35
. τ m ….Overshoot = 26%, Settling time = 16τm
λ = 45
. τm ….Overshoot = 22%, Settling time = 20τm
τm
5.750λ + 0.590
λ = 0.5 - performance
λ = 0.1 - robustness
λ = 0.25 - acceptable
Tf =
Tf =
λ3
3λ 2 + 3λτ m + τ m 2
15
. τ m ≤ λ ≤ 4.5τ m
Obtained from graph
Tmτ m

0. 0337Tm 
τm
1 +

2
01225
.
T
Km τ m 
m 
Tm + 81633
.
τm
01225
.
Tm + τ m

0. 0754Tm 
τm
1+

2 
0
.
1863
T
Km τ m 
m 
Tm + 53677
.
τm
01863
.
Tm + τ m

01344
.
Tm 
τm
 1+

0.2523Tm 
K mτ m2 
Tm + 3.9635τ m
0.2523Tm + τ m
Tf = 05549
.
τm
Tm τ m
Tf = 0. 4482τ m
Tm τ m
Tf = 0. 2863τm
Km e − sτ
- ideal controller with set-point weighting
s(1 + sTm )
m
Table 61: PID tuning rules - FOLIPD model


1
G c ( s) = Kc  b +
+ Td s . 1 tuning rule.
Ti s


Rule
Kc
Ti
Td
Comment
Direct synthesis
Astrom and
Hagglund [3] - pages
212-213
56
. e −8 .8 τ + 6.8τ
,
K m ( Tm + τ m )
11
. τm e 6.7 τ − 4.4 τ
τ = τ m ( τ m + Tm )
Model: Method 1 or
Method 2.
86
. e −7.1τ + 5.4 τ
K m ( Tm + τ m )
10
. τ m e3.3τ − 2 .33τ
2
2
17
. τm e −6.4 τ + 2.0 τ
2
Maximum Ms = 1.4
2
b = 012
. e 6.9 τ − 6.6 τ
2
0.38τ me 0.056τ − 0.60 τ
2
2
Maximum Ms = 2.0
b = 056
. e −2 .2τ + 1.2 τ
2



Km e − sτ
1   1 + Td s 
.
Table 62: PID tuning rules - FOLIPD model
- classical controller G c ( s) = Kc  1 +  
s(1 + sTm )

Ti s   1 + Td s 



N 
5 tuning rules.
m
Kc
Rule
Ti
Td
Comment
Tm
2λ + τm
λ = [ τ m , Tm ] ; N=10
2λ + τm
Tm
λ = [ τ m , Tm ] ; N=10
Robust
Chien [50]
Model: Method 2
Regulator tuning
Minimum IAE –
Shinskey [59] – page
75.
Model: Method not
specified
1
Km
 T

m


 λ+τ 2
[
]
m 
1  2λ + τ m 
K m  [ λ + τ m ] 2 
Minimum performance index
1.38( τ m + Tm )
0. 78
K m ( τ m + Tm )
100
(
Minimum IAE 108K m τ m 122
. − 0.03 τ
m
Shinskey [59] –
pages 158-159
100
Model: Open loop
T
108K m τ m 1 + 0. 4 τ m
m
method not specified
Tm
(
Minimum ITAE Poulin and
Pomerleau [82], [92] –
deduced from graph
Model: Method 2
Output step load
disturbance
Input step load
disturbance
)
b
K m (τ m + Tm )
)
Tm 2
a (τ m + T m )
2
0.66(τ m + Tm )
τ m = Tm ; N=10
T


− m

157
. τ m  1 + 12
. 1 − e τ m






 

0.56τ m + 0.75Tm
Tm
> 05
.
τm
T


− m

157
. τ m 1 + 12
. 1 − e τ m







0.56τ m + 075
. Tm
Tm
≤ 05
.
τm
a(τ m + Tm )
+1
0≤
Tm
τm
≤2;
( Td N)
01
. Tm ≤
Td
≤ 033
. Tm
N
τm ( Td N)
a
b
τm ( Td N)
a
b
0.2
0.4
0.6
0.8
1.0
5.0728
4.9688
4.8983
4.8218
4.7839
0.5231
0.5237
0.5241
0.5245
0.5249
1.2
1.4
1.6
1.8
2.0
4.7565
4.7293
4.7107
4.6837
4.6669
0.5250
0.5252
0.5254
0.5256
0.5257
0.2
0.4
0.6
0.8
1.0
3.9465
3.9981
4.0397
4.0397
4.0397
0.5320
0.5315
0.5311
0.5311
0.5311
1.2
1.4
1.6
1.8
2.0
4.0397
4.0278
4.0278
4.0218
4.0099
0.5312
0.5312
0.5312
0.5313
0.5314
Km e − sτ
- series controller with derivative filtering
s(1 + sTm )
m
Table 63: PID tuning rules - FOLIPD model



1 
Td s 
 . 1 tuning rule.
G c ( s) = Kc  1 +  1 +
Ts

Ts
i 
 1 + d 

N 
Kc
Rule
Ti
Td
2λ + τm
Tm
Tm
2λ + τm
Robust
Chien [50]
Model: Method 2
1
Km
 2λ + τ 
m 

 λ + τ 2
[
m] 

1 
Tm

2
K m  [ λ + τ m ] 
Comment
λ = [ τ m , Tm ] ; N=10
(Chien and Fruehauf
[137])
λ = [ τ m , Tm ] ; N=10
Km e − sτ
- alternative non-interacting controller 1
s(1 + sTm )
m
Table 64: PID tuning rules - FOLIPD model

1 
U(s) = Kc 1 +  E (s) − K c Td sY ( s) . 2 tuning rules.
 Ti s 
Rule
Regulator tuning
Minimum IAE Shinskey [59] – page
75.
Model: Method not
specified
Minimum IAE Shinskey [59] – page
159.
Model: Open loop
method not defined
Kc
Ti
Td
Minimum performance index
118
.
K m (τ m + Tm )
1.28
K m τ m (1 + 0.24
Tm
τm
2
T 
− 014
.  m )
 τ m 
1.38( τ m + Tm )
0.55( τ m + Tm )
Tm 

−
19
. τ m 1 + 0.75[1 − e τ m ]




0.48τ m + 0.7Tm
Comment
Km e − sτ
- ideal controller with filtered derivative
s(1 + sTm )
m
Table 65: PID tuning rules - FOLIPD model



1
Td s 
 . 1 tuning rule.
G c ( s) = Kc  1 +
+
Ti s 1 + s Td 



N
Rule
Kc
Robust
Chien [50]
2λ + τ m + Tm
Model: Method 2
K m( λ + τ m)
Ti
2
2λ + Tm + τ m
Td
Tm ( 2λ + τ m )
2λ + Tm + τ m
Comment
λ = Tm ; N = 10
Km e − sτ
- ideal controller with set-point weighting
s(1 + sTm )
m
Table 66: PID tuning rules - FOLIPD model
(
)
U(s) = Kc Fp R ( s) − Y( s) +
Kc
[ Fi R(s) − Y(s)] + Kc Td s[ Fd R (s) − Y(s)] . 1 tuning rule.
Ti s
Rule
Kc
Ti
Td
Ultimate cycle
Oubrahim and
Leonard [138]
Model: Method not
relevant
06
. Ku
Fp = 01
.
05
. Tu
Fi = 1
0125
. Tu
Fd = 0.01
Comment
τm
< 0.8 ;
Tm
20% overshoot
0.05 <
Km e − sτ
- Alternative classical controller
s(1 + sTm )
m
Table 67: PID tuning rules - FOLIPD model



 1+ T s   1 + T s 
i 
d 
G c ( s) = Kc 
. 1 tuning rule.
 1 + Td s   1 + Td s 

N 
N 
Rule
Direct synthesis
Tsang and Rad [109]
Model: Method 3
Kc
Ti
Td
Comment
0.809
K mτ m
Tm
0.5τ m
Maximum overshoot =
16%; N = 8.33
Km e − sτ
- Two degree of freedom controller:
s(1 + sTm )
m
Table 68: PID tuning rules – FOLIPD model


1
Td s
U (s) = K c 1 +
+
 Tis
T
1+ d

N

Rule
Kc
Servo/regulator
tuning
Taguchi and Araki
[61a]
Ti






β
T
s
d
 E( s) − K c  α +
 R (s) . 2 tuning rules.


Td 
s
1+
s

N 


Td
Minimum performance index
Kc
( 69a ) 2
Ti
( 69a )
Td
( 69a )
Model: ideal process
Minimum ITAE Pecharroman and
Pagola [134a]
φ c = phase
corresponding to the
crossover frequency;
Km = 1 ; Tm = 1 ;
0.05 < τm < 0. 8 .
Comment
1.672 K u
0.366 Tu
0.136 Tu
1.236 K u
0.427 Tu
0.149 Tu
0.994 Ku
0.486 Tu
0.155 Tu
0.842 Ku
0.538 Tu
0.154 Tu
0.752 Ku
0.567 Tu
0.157 Tu
0.679 Ku
0.610 Tu
0.149 Tu
0.635 K u
0.637 Tu
0.142 Tu
0.590 Ku
0.669 Tu
0.133 Tu
0.551K u
0.690 Tu
0.114 Tu
τm
≤ 1.0 ;
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of
tuning rules of Chien
et al. [10]
α = 0. 601 , β = 1 ,
N = 10, φ c = −1640
α = 0.607 , β = 1 ,
N = 10, φ c = −1600
α = 0.610 , β = 1 ,
N = 10, φ c = −1550
α = 0.616 , β = 1 ,
N = 10, φ c = −1500
α = 0.605 , β = 1 ,
Model: Method 4
N = 10, φ c = −1450
α = 0.610 , β = 1 ,
N = 10, φ c = −1400
α = 0.612 , β = 1 ,
N = 10, φ c = −1350
α = 0.610 , β = 1 ,
N = 10, φ c = −1300
α = 0.616 , β = 1 ,
2
Kc
Td
( 69a )
( 69 a )


1 
0.5184
=
0.7608 +
τm
Kc 

[
− 0.01308 ] 2
Tm



2



( 69 a )
 0.03330 + 3.997 τ m − 0.5517  τm  
,
T
=
T
  
m
 i
Tm
 Tm  



2
3

 τm 
 τm  
τm

= Tm 0.03432 + 2. 058
− 1. 774   + 0.6878   ,


Tm
 Tm 
 Tm  

α = 0. 6647 , β = 0. 8653 − 0. 1277
N = 10, φ c = −1250
τm
τ 
+ 0.03330  m 
Tm
 Tm 
2
Rule
Minimum ITAE Pecharroman and
Pagola [134a] continued
Kc
Ti
Td
0.520 K u
0.776 Tu
0.087 Tu
0.509 Ku
0.810 Tu
0.068 Tu
Comment
α = 0.609 , β = 1 ,
N = 10, φ c = −1200
α = 0. 611 , β = 1 ,
N = 10, φ c = −1180
Km e− sτ
K m e− sτ
or
- ideal controller
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 69: PID tuning rules - SOSPD model
m


1
Gc ( s) = K c  1 +
+ Td s . 27 tuning rules.
Ti s


Kc
Rule
Servo tuning
Minimum ITAE –
Sung et al. [139]
Model: Method 2
Regulator tuning
Minimum ITAE –
Sung et al. [139]
Model: Method 2
Ti
Td
Comment
Minimum performance index
Kc( 70)
Ti (70 )
Td
( 70)
0.05 <
τm
≤2
Tm1
0.05 <
τm
≤2
Tm1
Minimum performance index
Kc( 71)
Ti (71)
Td
( 71)
1
1
Kc
Kc
( 70)
( 70)
=
− 0.983

 
τ 
1 
−0.04 + 0333
ξ m  , ξ m ≤ 0.9 or
.
+ 0.949  m 
Km 
 Tm1 

 


−0.832

 τm 
1 
τm
=
−0.544 + 0308
.
+ 1408
. 
ξ m  , ξ m > 0.9 .

Km 
Tm1
 Tm1 



τ

τ
τ 
τ 
Ti ( 70) = Tm1 2.055 + 0.072 m  ξ m , m ≤ 1 or Ti ( 70) = Tm1 1768
.
+ 0.329 m ξ m , m > 1
T
T
T
T

m1 
m1

m1 
m1
Td ( 70) =
τ 



ξ
T 

−
0.870
1 − e



1.060
m
m1
Kc
( 71)
m
Tm1

1.090

T  
 0.55 + 1.683 m1  
 τm  
 



−2.001
− 0.766
 τ
 τm 
 τm 
1 
=
 −067
. + 0297
. 
+ 2189
. 
ξ m  , m < 0.9 or


Km 
 Tm1 
 Tm1 
 Tm1

Kc ( 71) =
2
−0.766
 τ
τ

τ 
1 
− 0365
.
+ 0.260  m − 14
.  + 2189
.  m
ξ m  , m ≥ 0.9
Km 
 Tm1

 Tm1 
 Tm1


τ 
Ti ( 71) = Tm1  2.212 m 
 Tm1 

Ti
( 71)
0.520
 τ
− 03
.  , m < 0.4 or
 Tm1
ξm

−
2
τ
0.15+0.33 m
 τm


Tm1
= Tm1{−0.975 + 0.910
− 1.845 + 1 − e
 Tm1



Td ( 71) =

2

 τm
 
τ

} , m ≥ 0.4
5
.
25
−
0
.
88

−
2
.
8


 Tm1
  Tm1



Tm1
ξm

−

−1.121
 τ 
1.171
−0.530


− 0.15+ 0.939 m 
T  
τ 
 Tm 1 
1 − e
145
. + 0.969 m1   − 19
. + 1576
.  m


 τm  
 Tm1 





Kc
Rule
Minimum ITAE –
Lopez et al. [84]
- taken from plots
Ti
Td
Comment
ξm
τm Tm1
Kc
Ti
Td
0.5
0.1
25
Km
05
. Tm1
0.25Tm1
0.5
1.0
07
.
Km
1.3Tm1
12
. Tm1
0.5
10.0
0.35
Km
5Tm1
1.0Tm1
1.0
0.1
25
Km
0.5Tm1
0.2Tm1
1.0
1.0
18
.
Km
1.7Tm1
0.7Tm1
4.0
1.0
9.0
Km
2Tm1
0.45Tm1
Model: Method 12
Representative results
Ultimate cycle
Regulator – nearly
minimum IAE, ISE,
ITAE – Hwang [60]
Kc
( 72) 2
Ti
( 72 )
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

Decay ratio = 0.15 τ
0.2 ≤ m ≤ 2.0 and
Tm1
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
0.6 ≤ ξ m ≤ 4.2 ;
Model: Method 3
Kc
( 73)
Ti
(73)
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
ε < 2.4
Decay ratio = 0.15 τ
0.2 ≤ m ≤ 2.0 and
Tm1
0.6 ≤ ξ m ≤ 4.2 ;
2.4 ≤ ε < 3
2
KH =
 τm 2
ξ τ T
4 K mTd Kc τ m 
9
2
− Tm1 − m m m +
+
2 
9
9ω u
2 K m τm  18


4
2
2
2
2
2
3
3
3
2
2 2
2
9
 T 4 + τm + 49 τm ξ m Tm − τm Tm1 + 7Tmξ m τ m + 10 τm1 Tm1 ξ m − K T K  10 τm + 4 Tm1 τm (ξ m τ m + Tm1 )  − 8Kc Km Td τ m
m1
c d m
2 
2
324
81
9
81
9
81
9
2 Km τm
81ω u



ωH =
Tm12
ε=
Kc
1 + K HK m
6T 2 + 4Tm1ξ mτ m + K uK mτ m 2
, ε 0 = m1
2
2τ m Tm12ω u
2ξ τ T
K K τ
2 Kc KmTd τ m
+ m m m+ H m m −
3
6
3ω u
6Tm12 + 4 Tm1ξm τ m + KH Km τ m 2 − 4 τ mK mK cTd
( Km Kc Td τ m 2 + 2 τ mTm12 )ω H
( 72)
[
]  , T
[
]  , T

0674
.
1 − 0.447ω H τ m + 0.0607ω H2 τ m 2

= KH −
K m (1 + KH K m )


0.778 1 − 0.467ω H τ m + 0.0609ω H 2 τ m 2
Kc ( 73) =  KH −
K m (1 + K H K m )



i


i
( 72)
( 73)
=
=
Kc
( 72)
(
(1 + K H K m )
ω H K m 00607
.
1 + 105
. ω H τ m − 0.233ω H 2 τ m 2
Kc
(
( 73)
)
(1 + K H K m )
ω H K m 0.0309 1 + 2.84ω H τ m − 0.532ω H 2 τ m 2
)




Rule
Kc
Ti
Td
Comment
Regulator – nearly
minimum IAE, ISE,
ITAE – Hwang [60] –
continued
Kc( 74) 3
Ti (74 )
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

Decay ratio = 0.15 τ
0.2 ≤ m ≤ 2.0 and
Tm1
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
0.6 ≤ ξ m ≤ 4.2 ;
3 ≤ ε < 20
Model: Method 3
Kc( 75)
Decay ratio = 0.15 τ
0.2 ≤ m ≤ 2.0 and
Tm1
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

Ti (75)
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
0.6 ≤ ξ m ≤ 4.2 ; ε ≥ 20
Kc( 76)
Decay ratio = 0.2 τ
0.2 ≤ m ≤ 2.0 and
Tm1
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

Ti (76 )
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
0.6 ≤ ξ m ≤ 4.2 ; ε < 2.4
Kc
( 77)
Ti
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

( 77 )
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.2 τ
0.2 ≤ m ≤ 2.0 and
Tm1
0.6 ≤ ξ m ≤ 4.2 ;
2.4 ≤ ε < 3
Kc( 78)
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

Ti (78)
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.2 τ
0.2 ≤ m ≤ 2.0 and
Tm1
0.6 ≤ ξ m ≤ 4.2 ;
3 ≤ ε < 20
Kc( 79)
Decay ratio = 0.2 τ
0.2 ≤ m ≤ 2.0 and
Tm1
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

Ti (79 )
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
0.6 ≤ ξ m ≤ 4.2 ; ε ≥ 20
3
[
]
( 74)

131
. ( 0519
.
. ε + 0514
.
ε 2 
) ω H τm 1 − 103
K c (1 + K H K m )
Kc ( 74) =  K H −
, Ti (74) =


K m (1 + KH K m )
ω H K m 0.0603 1 + 0.929 ln[ ω H τ m ] 1 + 2.01 ε − 12
. ε2


[

114
. 1 − 0.482ω H τ m + 0.068ω H 2 τ m 2
Kc ( 75) =  KH −
K m (1 + K H K m )

Kc
( 76)
Kc
( 77)
Kc
( 78)
[
] , T

0.622 1 − 0.435ω H τ m + 0.052ω H τ m

= KH −

K m (1 + K H K m )

2


2
=
( 75)
i
] , T


[

0.724 1 − 0.469ω H τ m + 0.0609ω H 2 τ m 2

= KH −

K m (1 + K H K m )

[
)(
(
( 76)
i
]  , T


Kc
( 75)
(
(1 + K H K m )
ω H K m 0.0694 − 1 + 21
. ω H τ m − 0.367ω H2 τ m 2
=
( 77)
i
]
)
K c (76) (1 + K H Km )
(
ω H Km 0.0697 1 + 0.752ω H τ m − 0145
. ω H 2 τm2
=
Kc
( 77 )
(
(1 + KH K m )
ω H K m 00405
.
1 + 193
. ω H τ m − 0363
. ωH 2 τm2
)
)
ω τ
( 78)

126
. ( 0.506) H m 1 − 1.07 ε + 0.616 ε 2 
K c (1 + K H K m )

= KH −
, Ti (78) =


K m (1 + KH K m )
ω H K m 0.0661 1 + 0.824 ln[ ω H τ m ] 1 + 171
. ε − 117
. ε2


[

109
. 1 − 0497
. ω H τ m + 00724
.
ω H 2 τm2
Kc ( 79) =  K H −
Km (1 + KH K m )

)(
(
]  , T


i
( 79)
=
(
Kc
( 79 )
(1 + KH K m )
ω H K m 0.054 − 1 + 2.54ω H τ m − 0457
. ω H 2 τm2
)
)
)
Rule
Kc
Ti
Td
Regulator – nearly
minimum IAE, ISE,
ITAE – Hwang [60] –
continued
Kc(8 0) 4
Ti (80)
Comment
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.25 τ
0.2 ≤ m ≤ 2.0 and
Tm1
0.6 ≤ ξ m ≤ 4.2 ; ε < 2.4
Kc
Model: Method 3
( 81)
Ti
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

( 81)
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.25 τ
0.2 ≤ m ≤ 2.0 and
Tm1
0.6 ≤ ξ m ≤ 4.2 ;
2.4 ≤ ε < 3
Kc(8 2)
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

Ti (82)
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
Decay ratio = 0.25 τ
0.2 ≤ m ≤ 2.0 and
Tm1
0.6 ≤ ξ m ≤ 4.2 ;
3 ≤ ε < 20
Kc(83)
1.45(1 + Ku K m ) 
116
. 
1 −

2
ε0 
Km ω u

Ti (83)
Decay ratio = 0.25 τ
0.2 ≤ m ≤ 2.0 and
Tm1
. ω u2τ m2 )
( 1 − 0.612ω u τ m + 0103
0.6 ≤ ξ m ≤ 4.2 ; ε ≥ 20
Servo– nearly
minimum IAE, ISE,
ITAE – Hwang [60].
In all, decay ratio =
0.1 with
τ
0.2 ≤ m ≤ 2.0 and
Tm1
Kc(8 4)
Ti (84 )
0471
. Ku
K mω u
Kc(85)
Ti (85)
0471
. Ku
K mω u
τ
τm
+ 0117
.  m
Tm1
 Tm1



2
ξ ≤ 0613
.
+ 0613
.
τ 
τm
+ 0117
.  m 
Tm1
 Tm1 
2
ξ ≤ 0613
.
+ 0613
.
0.6 ≤ ξ m ≤ 4.2
Model: Method 3
4
Kc
( 80)
[

0.584 1 − 0.439ω H τ m + 0.0514ω H 2 τ m 2

=  KH −
K m (1 + K H K m )

[




( 80)
i
]  , T

0675
.
1 − 0.472ω H τ m + 0061
. ω H 2 τm2
Kc (81) =  K H −
K m (1 + KH K m )

[
]  , T
( 81)
i
=
Kc
(
(1 + KH K m )
ω H K m 0.0714 1 + 0.685ω H τ m − 0.131ω H 2 τ m 2
Kc
=
( 80)
( 81)
(
(1 + K H K m )
ω H K m 00484
.
1 + 143
. ω H τ m − 0.273ω H 2 τ m 2
]
)
ω τ
( 82)

12
. ( 0.495) H m 1 − 11
. ε + 0.698 ε 2 
K c (1 + K H K m )
Kc (82 ) =  K H −
, Ti (82) =


K m (1 + K H K m )
ω H K m 0.0702 1 + 0.734 ln[ ω H τ m ] 1 + 148
. ε − 11
. ε2


[

103
. 1 − 0.51ω H τ m + 0.0759ω H 2 τ m 2
Kc (83) =  KH −
K m (1 + K H K m )

[
] , T

0.822 1 − 0.549ω H τ m + 0112
. ωH τm
Kc (84 ) =  K H −
K m (1 + K H K m )

[
2
)(
(


2
( 83)
i
]  , T

0.786 1 − 0.441ω H τ m + 0.0569ω H 2 τ m 2
Kc (85) =  KH −
K m (1 + K H K m )



( 84)
i
]  , T


i
Kc
=
( 83)
(
(1 + KH K m )
ω H K m 0.0386 −1 + 3.26ω H τ m − 0.6ω H 2 τ m 2
=
( 85)
Kc
( 84 )
(
)
(1 + K H K m )
ω H K m 0.0142 1 + 6.96ω H τ m − 177
. ω H 2 τm2
=
Kc
(
( 85)
)
)
(1 + K H Km )
ω H K m 0.0172 1 + 4.62ω H τ m − 0.823ω H 2 τ m 2
)
)
Kc
Rule
Servo– nearly
minimum IAE, ISE,
ITAE – Hwang [60]
(continued)
Kc
Servo – nearly
minimum IAE, ISE,
ITAE – Hwang [60]
Ti
( 8 6) 5
Ti
Td
Ti (87 )
( 8 8)
( 88)
Kc
Ti
0471
. Ku
K mω u
( 86)
Kc(8 7)
0471
. Ku
K mω u
0471
. Ku
K mω u
Ti (89)
0471
. Ku
K mω u
Kc( 90)
Ti (90)
0471
. Ku
K mω u
Kc
( 91)
Ti
( 91)
0471
. Ku
K mω u
Kc
( 92)
Ti
( 92 )
0471
. Ku
K mω u
Kc(8 9)
Comment
τ
τm
+ 0117
.  m
Tm1
 Tm1



2
ξ ≤ 0613
.
+ 0613
.
τ 
τm
+ 0117
.  m 
Tm1
 Tm1 
2
ξ ≤ 0613
.
+ 0613
.
ξ > 0649
.
+ 058
.
τm
Tm1
ξ > 0649
.
+ 058
.
6
Model: Method 3
Kc( 93)
5
ξ ≤ 0649
.
+ 058
.
]
τm
Tm1
τm
Tm1
τm
Tm1
2
 τm 

− 0005
. 
 Tm1 
2
 τm 

− 0005
. 
 Tm1 
2
 τ 2
τm
− 0005
.  m 
Tm1
 Tm1 
ξ > 0613
.
+ 0613
.
ξ ≤ 0649
.
+ 058
.
 τ 2
τm
− 0005
.  m 
Tm1
 Tm1 
ξ > 0613
.
+ 0613
.
[
Kc
( 88)
]  , T


[

0.794 1 − 0.541ω H τ m + 0126
. ω H 2 τ m2

= KH −

K m (1 + K H K m )

[

0.738 1 − 0415
. ω H τ m + 0.0575ω H 2 τ m 2
Kc (89 ) =  K H −
K m (1 + K H K m )

[
( 87)
i
]  , T


=


(
=
( 89)
i
( 87)
(1 + K H K m )
ω H K m 0.0609 − 1 + 197
. ω H τ m − 0.323ω H 2 τ m 2
=
( 88)
i
]  , T
Kc
Kc
( 88)
(
(1 + KH K m )
ω H K m 0.0078 1 + 8.38ω H τ m − 197
. ω H 2 τm2
Kc
( 89 )
(
(1 + K H K m )
ω H K m 0.0124 1 + 4.05ω H τ m − 0.63ω H 2 τ m 2
]
 τ 2
τm
+ 0117
.  m 
Tm1
 Tm1 
)
)
)
ω τ
( 90)

115
. ( 0.564) H m 1 − 0.959 ε + 0.773 ε 2 
K c (1 + K H K m )
Kc ( 90) =  K H −
, Ti (90) =


K m (1 + K H K m )
ω H K m 0.0335 1 + 0947
.
ln[ ω H τ m ] 1 + 19
. ε − 1.07 ε 2


[

107
. 1 − 0.466ω H τ m + 0.0667ω H 2 τ m 2
Kc ( 91) =  KH −
K m (1 + K H K m )

[

0.789 1 − 0.527ω H τ m + 011
. ω H τm
Kc ( 92) =  K H −
K m (1 + KH K m )

[
2
2
] , T



0.76 1 − 0.426ω H τ m + 0.0551ω H 2 τ m 2
Kc ( 93) =  KH −
K m (1 + K H K m )

( 92)
i
] , T


( 91)
i
]  , T


)(
(
i
=
=
( 93)
Kc
( 91)
(
(1 + KH K m )
ω H K m 0.0328 −1 + 2.21ω H τ m − 0.338ω H 2 τ m 2
Kc
( 92)
(
(1 + K H K m )
ω H K m 0009
.
1 + 9.7ω H τ m − 2.4ω H2 τ m 2
=
Kc
(
( 93)
)
(1 + KH K m )
ω H K m 0.0153 1 + 4.37ω H τ m − 0.743ω H 2 τ m 2
)
)
,
 τ 2
τm
+ 0117
.  m 
Tm1
 Tm1 
)(
(
2
 τm 

− 0005
. 
 Tm1 
ω τ
( 86)

128
. ( 0.542) H m 1 − 0986
.
ε + 0558
.
ε 2 
K c (1 + K H K m )
Kc (86) =  KH −
, Ti (86) =


K m (1 + K H Km )
ω H K m 0.0476 1 + 0.996 ln[ ω H τ m ] 1 + 2.13 ε − 113
. ε2



114
. 1 − 0466
. ω H τ m + 0.0647ω H 2 τ m 2
Kc (87 ) =  K H −
K m (1 + K H K m )

6
ξ > 0649
.
+ 058
.
0471
. Ku
K mω u
Ti (93)
[
ξ > 0649
.
+ 058
.
 τm 

− 0005
. 
 Tm1 
)
)
,
Kc
Rule
Servo – nearly
minimum IAE, ISE,
ITAE – Hwang [60]
(continued)
Kc
( 94) 7
Kc
Minimum IAE
regulator – ultimate
cycle - Shinskey [16]
– page 151.
Model: Method 10.
Direct synthesis
Gain and phase
margin – Hang et al.
[35]
Model: Method 4
Gain and phase
margin – Ho et al.
[140]
Ti
Ti
( 95)
Ti
Gain and phase
margin – Ho et al.
[142]
Model: Method 1
7
Comment
0471
. Ku
K mω u
( 94 )
ξ ≤ 0649
.
+ 058
.
ξ > 0613
.
+ 0613
.
0471
. Ku
K mω u
(95)
 τ 2
τm
− 0005
.  m 
Tm1
 Tm1 
ξ ≤ 0649
.
+ 058
.
τ 
τm
+ 0117
.  m 
Tm1
 Tm1 
 τ 2
τm
− 0005
.  m 
Tm1
 Tm1 
ξ > 0613
.
+ 0613
.
0.38Tu
015
. Tu
Tm 2 Tm2 + τ m = 0.25
0.6766Ku
0.33Tu
019T
. u
Tm 2 Tm2 + τ m = 0.5
0.7874K u
0. 26Tu
0.21Tu
Tm 2 Tm2 + τ m = 0.75
πTm1
,
Am Km τ m
2Tm1 ,
0.5Tm1
Sample A m ,φ m
A m = 1.5, φm = 30 o
A m = 3.0 , φm = 60 o
A m = 2 .0, φm = 45o
A m = 4.0 , φ m = 67 .5o
ω p Tm1
1
A m Km
4ω p τ m
2
2ω p −
A m = 2 .0, φm = 45
π
+
A m = 5.0, φ m = 72 o
provided. Model has a
repeated pole ( Tm1 )
Tm 2
Tm1 > Tm 2 . Sample
A m ,φ m provided
A m φ m + 05
. πA m (A m − 1)
1
Tm1
A m = 3.0 , φm = 60 o
A m = 4.0, φ m = 70 o
Kc( 96)
Ti (96 )
Td ( 96)
A m = 2 .0, φm = 45o
A m = 3.0 , φm = 60 o
A m = 4.0, φ m = 70 o
Kc (97)
Ti (97 )
Td ( 97)
A m = 2 .0, φm = 45o
A m = 3.0 , φm = 60 o
A m = 4.0, φ m = 60 o
[
]
[

111
. 1 − 0467
. ω H τ m + 0.0657ω H 2 τ m 2
Kc ( 95) =  K H −
K m (1 + KH K m )

ωp =
(A
2
m
)
−1 τm
τm τm
,
≤1
Tm Tm
Sample A m ,φ m
provided *
ω nτ m < 2ξ m ; Sample
A m ,φ m provided
2ξ m ≥
] , T


( 97)
Td ( 97)
)(
(
( 95)
i
=
Kc
(
( 95)
(1 + KH K m )
ω H K m 0.0477 − 1 + 2.07ω H τ m − 0.333ω H 2 τ m 2
 πξ m
 ( 97) 2

2  πξ m

+ π − 2ω p τm  , Ti
= Tm1 
+ π − 2ω p τ m  ,
π
 ω p Tm1

 Tm1ω p

πTm1
=
 πξ

2 m + π Tm1 − 2Tm1ω p τm 
 ωp

=
)
2

2
τ 
π Tm1
( 96)
( 96)
 πξ m + π − 2 m  , Ti = Tm1( πξ m + π − 2τ m ) , Td =
π
πA m K m 
Tm1 
2( πξ mTm1 + π Tm1 − 2τ m )
2
2ω pTm1
π Am
2
Kc
,
 τ 2
τm
+ 0117
.  m 
Tm1
 Tm1 
ω τ
( 94)

122
. ( 0.55) H m 1 − 0.978 ε + 0.659 ε 2 
K c (1 + K H K m )
Kc ( 94) =  K H −
, Ti (94) =


K m (1 + K H K m )
ω H K m 0.0421 1 + 0969
.
ln[ ω H τ m ] 1 + 2.02 ε − 111
. ε2


Kc ( 96) =
,
2
0.6173Ku
o
Model: Method 1
Gain and phase
margin – Ho et al.
[141]
Model: Method 1
Td

8πτm ξ m 
 π + π2 +
 A m 2 − 1 − 2πA m ( A m − 1)
T


m1
* φm <
4A m
(
)
)
Kc
Rule
Ti
Td
Comment
ξ m > 0.7071 or
ξ m Tm1
Km τ m
Wang et al. [143]
0.05 <
05
. Tm1
ξm
2ξ m Tm1
07071
.
τm
Tm1 2ξ m 2 − 1
0.7071τm
Tm1 2 ξ m 2 − 1
Model: Method 8
or 0.05 <
ξmτ m
Tm1
Minimum of
2
2 Tm1
e
Km
−
Gain and phase
margin – Leva et al.
[34]
Tm τ m
ξm
Kc
,
07358
.
ξ m Tm1
Km τ m
( 98) 8
Ti
( 98 )
0.25Ti
> 1, ξ m ≥ 1
ξ mτ m
Tm1
< 015
. ,
> 1, ξ m < 1
ξ m ≤ 0.7071 and
05
. Tm1
ξm
2ξ m Tm1
< 015
.
0.15 ≤
ξmτ m
Tm1
≤1
ξ m ≤ 1 or
ξ m > 1 with
05
. π − φm >
3τ m 
2
ξ m + ξ m − 1

Tm1 
( 98)
Model: Method 5
ξ m > 1 with
Kc( 99)
Ti (99 )
3τ m 
2
ξ m − ξ m −1 

Tm1 
< 0.5π − φm ≤
3τ m 
2
ξ m + ξ m − 1 ,

Tm1 
Td ( 99)
φ m = 70 0 at least
8
Kc
( 98)
Ti( 98)
z=
=
ω cn Ti
Km
( 1 + ω T ) + 4ξ ω
(1 − T T ω ) + T ω
2
cn
2
2
2
m1
m
2
i d
cn
2
cn
2
2
i
Tm1
2
2
, ω cn =
cn
 2ξ τ T ( 05

1 
m m m1 . π − φm ) 
4.07 − φ m + tan −1 

2
2
2 
τm 
 ( 0.5π − φ m ) Tm1 − τ m 

 
2
 2ξ mω cnTm1  
2
1
  , Kc ( 99) = ω cnTm(99
=
tan 05
. ω cn τ m + φ m − 05
. π − tan− 1
2
2
ω cn
 
 ω cn Tm1 − 1 
K mTd )
ω cn





ω cn Tm1
−1 
tan φ m − 05
. π + ω cnτ m + tan 


  ξ m + ξm 2 − 1  
  


, Ti (99 ) =
Tm1z +  ξ m + ξ m 2 − 1




z ξ m + ξ m 2 − 1


ω cn 2 +
1 
 2ξ m ξ m 2 − 1 − 1

Tm12 
,
2
2
ω cn + z
, Td (99) =
Tm1


Tm1z +  ξ m + ξ m2 − 1 


Rule
Gain and phase
margin – Wang and
Shao [144]
Kc
other tuning rules
obtained using
authors method
Td
ξ m Tm1
Km τ m
2ξ m Tm1
05
.
Tm1
ξm
Am = 3 , φ m = 60 0
2.094
ξ mTm1
K mτ m
2ξ m Tm1
05
.
Tm1
ξm
A m = 15
. , φ m = 30 0
1571
.
ξ mTm1
K mτ m
2ξ m Tm1
05
.
Tm1
ξm
A m = 2 , φ m = 450
0.785
ξ m Tm1
K mτ m
2ξ m Tm1
05
.
Tm1
ξm
A m = 4 , φ m = 67.50
0.628
ξ mTm1
K mτ m
2ξ m Tm1
05
.
Tm1
ξm
A m = 5 , φ m = 72 0
Pemberton [145]
Model: Method 1
2( Tm1 + Tm2 )
3Km τ m
Tm1 + Tm2
Tm1Tm2
Tm1 + Tm 2
Pemberton [24]
(Tm1 + Tm2 )
K mτ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
2( Tm1 + Tm 2 )
3Km τ m
Tm1 + Tm2
Model: Method 1
Pemberton [145]
Model: Method 1
Comment
1047
.
Model: Method 8
Authors quote
tuning rule for
Am = 3 , φ m = 60 0 ;
Ti
Tm1 + Tm 2
4
01
. ≤
τm
≤ 10
. ;
Tm1
0.2 ≤
τm
≤ 10
.
Tm 2
01
. ≤
τm
≤ 10
.
Tm1
0.2 ≤
τm
≤ 10
.
Tm 2
Suyama [100]
Model: Method 1
Tm1 + Tm 2
2K m τ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Smith et al. [146]
Model: Method not
stated
Chiu et al. [29]
Tm1 + Tm 2
K m ( λ + τm )
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
λTm1 + Tm2
K m (1 + λτm )
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
λTm 1
K m (1 + λτ m )
λ variable; suggested
values are 0.2, 0.4, 0.6
and 1.0.
Tm1 > Tm2 . λ = pole of
Tm1
Tm2
2λξ m Tm1
K m (1 + λτ m )
2ξ m Tm1
Tm1
2ξ m
specified closed loop
overdamped response.
Underdamped
response
Gorez and Klan
[147a]
Model: Ideal
Miluse et al. [27a]
2ξ m Tm1
K m (2ξ m Tm1 + τ m )
2ξ m Tm1
Tm1
2ξ m
Non-dominant time
delay
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 0%
Model: Method not
specified
0. 368 (Tm1 + Tm 2 )
K mτ m
0. 514 (Tm1 + Tm 2 )
K mτ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 5%
0. 581( Tm1 + Tm 2 )
Km τ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 10%
0. 641( Tm1 + Tm 2 )
Km τ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 15%
Model: Method 6
Wang and Clemens
[147]
Model: Method 9
Overdamped
process;
Tm1 > Tm 2
Rule
Kc
Ti
Td
Comment
Miluse et al. [27a]
(continued)
0. 696 (Tm1 + Tm 2 )
K mτ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 20%
0. 748 (Tm1 + Tm 2 )
K m τm
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 25%
0. 801( Tm1 + Tm 2 )
Km τ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 30%
0. 853( Tm1 + Tm2 )
K m τm
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 35%
0. 906 (Tm1 + Tm 2 )
K mτ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 40%
0. 957 (Tm1 + Tm 2 )
K mτ m
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 45%
1.008 ( Tm1 + Tm 2 )
K m τm
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Closed loop overshoot
= 50%
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 0%
Model: Method not
specified
0. 736ξ mTm1
Km τ m
1.028 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 5%
Underdamped
process;
0.5 < ξ m ≤ 1
1.162 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 10%
1.282 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 15%
1.392 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 20%
1.496 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 25%
1.602 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 30%
1.706 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 35%
1.812 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 40%
1.914 ξ m Tm1
K m τm
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 45%
2.016 ξ m Tm1
K mτ m
2ξ m Tm1
Tm1
2ξ m
Closed loop overshoot
= 50%
0. 736 Tm1
K m τm
2Tm1
0.5Tm1
Tm1 = Tm 2
Miluse et al. [27a]
Miluse et al. [27b]
Model: Method 14
‘Re-tuning’ rule. ω 0 =
2
Seki et al. [147b]
ωc
tan θ c + 2 + tan 2 θ c
cos
θ
Model: Method not G c ( jω 0 )
c
2
ω0
ωc
specified
0.5Ti
freq. when controlled
system goes unstable
(crossover freq.);
ω c = new crossover
freq. ; θ c = phase lag
Rule
Autotuning –
Landau and Voda
[148]
Model: Method not
relevant
Kc
Ti
Td
Comment
4+β
β
4 2 2 G p ( jω )
135
4+β 1
β ω135
4
1
4 + β ω135
1≤ β ≤ 2 ;
τm < 0.25Tm1
3
3.2
ωu
08
.
ωu
ω uτ m ≤ 016
.
4
ωu
1
ωu
016
. < ω u τ m ≤ 0.2
Tm 1 + Tm2 + 05
. τm
Tm1 Tm2 + 0.5( Tm1 + Tm2 ) τ m
λ varies graphically
with τm (Tm1 + Tm 2 )
0
G p ( jω u )
19
.
G p ( jω u )
0
0
Robust
Brambilla et al. [48] –
values deduced from
graph
Model: Method 1
Chen et al. [53]
Tm1 + Tm 2 + 0.5τ m
K m τ m ( 2λ + 1)
τ m (Tm1 + Tm 2 )
λ
0.1
0.50
0.2
0.47
0.5
0.39
1.00 ξ m Tm
τ mK m
Tm1 + Tm 2 + 05
. τm
0.1 ≤ τ m ( Tm1 + Tm 2 ) ≤ 10
τ m (Tm1 + Tm 2 )
λ
τ m (Tm1 + Tm 2 )
λ
1.0
2.0
5.0
0.29
0.22
0.16
10.0
0.14
A m = 3.14 ,
2ξ m τm
τm
2ξ m
1.22 ξ m Tm
τ mK m
2ξ m τm
τm
2ξ m
A m = 2.58 ,
1.34 ξ m Tm
τ mK m
2ξ m τm
τm
2ξ m
A m = 2. 34 ,
1.40 ξ m Tm
τ mK m
2ξ m τm
τm
2ξ m
A m = 2. 24 ,
1.44 ξ m Tm
τ mK m
2ξ m τm
τm
2ξ m
A m = 2.18 ,
1.52 ξ m Tm
τ mK m
2ξ m τm
τm
2ξ m
1.60 ξ m Tm
τ mK m
2ξ m τm
τm
2ξ m
φ m = 61. 40 ,
Ms = 1. 00
Model: Method 1
φ m = 55. 00 ,
Ms = 1. 10
φ m = 51.6 0 ,
Ms = 1. 20
φ m = 50.0 0 ,
M s = 1. 26
φ m = 48. 70 ,
Ms = 1. 30
A m = 2. 07 ,
φ m = 46. 50 ,
Ms = 1. 40
A m = 1.96 ,
φ m = 44.10 ,
Ms = 1. 50
2
Lee et al. [55]
Model: Method 1
Ti
K m ( 2λ + τ m )
2ξ mTm1 −
2 λ − τm
2( 2λ + τ m )
2
2
Ti − 2ξ mTm1 +
Tm1
−
Ti
τm
6Ti ( 2λ + τm )
3
Desired closed loop
e− τ m s
response =
( λs + 1) 2
Rule
Kc
Lee et al. [55]
(continued)
Ti
K m ( 2λ + τ m )
Ti
K m( λ + τ m )
Ti
K m( λ + τ m )
Ti
2 λ2 − τ m 2
−
2( 2λ + τ m )
Tm1 + Tm2
2ξ mTm1
Td
τm
+
2( λ + τ m )
Tm1 + Tm2
2
τm
+
2 ( λ + τm )
2
Comment
Ti − Tm1 − Tm2 +
Tm1Tm2
Ti
Desired closed loop
e− τ m s
response =
( λs + 1) 2
τm
6Ti ( 2λ + τ m )
3
−


τ m3
 Tm1 2 −


6(λ + τ m ) 

Ti − 2ξ m Tm1 


Ti




Desired closed loop
e− τ m s
response =
( λ s + 1)

τ m3
 Tm1Tm 2 −

6 λ + τm
Ti − ( Tm1 + Tm2 ) 
Ti



Desired closed loop
e− τ m s
response =
( λ s + 1)
(


)




Km e− sτ
K m e− sτ
or
- filtered controller
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 70: PID tuning rules - SOSPD model
m

 1
1
G c ( s) = Kc  1 +
+ Td s
. 1 tuning rule.
Ti s

 Tf s + 1
Rule
Kc
Ti
2Tm1 + τ m
2( λ + τ m ) K m
Tm1 + 05
. τm
Td
Comment
Robust
Hang et al. [35]
Model: Method 4
Tm1τm
2Tm1 + τm
Model has a repeated
pole (Tm1 ) .
λ > 025
. τm .
Tf =
λτ m
2( λ + τ m )
Km e− sτ
K m e− sτ
or
- filtered controller
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 71: PID tuning rules - SOSPD model
m

 b s+1
1
G c ( s) = Kc  1 +
+ Td s 1
. 1 tuning rule.
Ti s

 a 1s + 1
Rule
Robust
Jahanmiri and Fallahi
[149]
Model: Method 6
Kc
Ti
Td
2ξ m Tm1
K m (τ m + λ)
λ = 0.25τ m + 01
. ξ m Tm1
2ξ m Tm1
Tm1
2ξ m
Comment
b1 = 0.5τ m
a1 =
λτ m
2( τ m + λ )
Km e− sτ
K m e− sτ
or
- classical controller
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 72: PID tuning rules - SOSPD model
m



1   1 + Td s 
 . 7 tuning rules.
G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
Kc
Rule
Regulator tuning
Ti
Td
T

−
1.5τ

τm 15
. −e


m1
Minimum IAE Shinskey [59] – page
159
1
Kc
( 100 )
0800
. Tm1
Km τ m
0770
. Tm1
K mτ m
0833
. Tm1
Km τ m
** Minimum IAE
Shinskey [17]
059
. Ku
085
. Tm1
K mτ m
087
. Tm1
Km τm
100
. Tm1
K mτ m
125
. Tm1
K mτ m
1
Kc (100) =


 48 + 57 1 − e







m2
25
. Tm1
K mτ m
** Minimum IAE Shinskey [17]
m
T


−
 1 + 0.9 1 − e τ



Model: Open loop
method not specified
** Minimum IAE Shinskey [59]
Comment
Minimum performance index
1.2 Tm1
−
τm



m



1.2 T

−

0.56τ m 1 − e τ


m1
m
Tm2
≤3
τm
0.6Tm2
τm + 0.2Tm2
τm + 0.2Tm2
15
. ( Tm2 + τ m )
060
. ( Tm2 + τ m )
12
. ( Tm2 + τ m )
0.70( Tm 2 + τ m )
075
. ( Tm2 + τ m )
060
. ( Tm2 + τ m )
0. 36Tu
0. 26Tu
198
. τm
086
. τm
2.30τ m
165
. τm
2.50τm
200
. τm
2.75τm
2.75τ m
100
2
 Tm 2  
Km τ m 
Tm2
1 + 0.34
− 0.2
 
Tm1 
τm
 τ m  

+


Tm2
>3
τm
Tm 2
= 025
.
Tm 2 + τm
Tm 2
= 0.5
Tm2 + τ m
Tm 2
= 0.75
Tm2 + τ m
τm
T
= 0.2 , m 2 = 0.2
Tm1
Tm1
τm
T
= 0.2 , m 2 = 01
.
Tm1
Tm1
Tm 2
= 0.2
Tm1
Tm 2
= 0.5
Tm1
Tm 2
= 10
.
Tm1
Rule
Kc
Ti
Td
Minimum ISE –
McAvoy and
Johnson [83] –
deduced from graph
α
Km
βτ m
χτ m
ξm
Model: Method 1
α
τm
Tm1
β
χ
ξm
τm
Tm1
α
β
χ
Comment
ξm
τm
Tm1
α
β
χ
N = 20
1
1
1
0.5 0.7 0.97 0.75
4.0 7.6 3.33 2.03
10.0 34.3 5.00 2.7
4
4
0.5
4.0
3.0 1.16 0.85
22.7 1.89 1.28
7
7
0.5
4.0
5.4 1.19 0.85
40.0 1.64 1.14
N = 10
1
1
1
0.5 0.9 1.10 0.64
4.0 8.0 4.00 1.83
10.0 33.5 6.25 2.43
4
4
0.5
4.0
3.2 1.33 0.78
23.9 2.17 1.17
7
7
0.5
4.0
5.9 1.39 1.04
42.9 1.89 1.37
Direct synthesis
3
Astrom et al. [30] –
Method 13
Smith et al. [26] –
deduced from graph
[ ρ = τ m ( Tm1 + Tm2 ) ]
Model: Method not
stated

3τ m 
K m 1 +


Tm1 + Tm 2 
Tm1 + Tm2
Tm 1Tm2
Tm1 + Tm 2
Tm1
Tm2
N = 8 - Honeywell
UDC6000 controller
ξm
αTm1
K mτ m
ρ
α
ξm
ρ
α
ξm
ρ
α
6
1.75
1.75
1.5
1.0
≥ 0.02
0.27
0.07
0.11
0.25
0.51
0.46
0.36
0.33
0.46
3
1.75
1.5
1.5
1.0
≥ 004
.
0.13
0.33
0.08
0.17
0.50
0.42
0.44
0.28
0.48
2
1.75
1.5
1.0
1.0
≥ 006
.
0.07
0.17
0.50
0.13
0.45
0.39
0.38
0.40
0.49
N = 10
Km e− sτ
K m e− sτ
or
- alternative classical
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 73: PID tuning rules - SOSPD model
m

1   1 + NTd s 
controller G c ( s) = Kc  1 +

 . 1 tuning rule.
Ti s   1 + Td s 

Rule
Kc
Ti
Td
Comment
******
0.7
Hougen [85]
Model: Method 1
08
. Tm1 Tm2
τm
0.8
0. 3
084
. Tm1 Tm 2
τm
05
. Tm1 + Tm 2
3
τm Tm1 Tm 2
N=10
0.2
0.53Tm1 + 1.3Tm2
0.08( τ m Tm1Tm2 )
0.28
N=30
Km e− sτ
K m e− sτ
or
- alternative non(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 74: PID tuning rules - SOSPD model
m

1
interacting controller 1: U( s) = K c  1 +
 E ( s) − Kc Td sY ( s) . 3 tuning rules.
Ti s

Kc
Rule
Regulator tuning
Minimum IAE Shinskey [59] – page
159
Model: open loop
method not specified
Minimum IAE Shinskey [17] – page
119. Model: method
not specified.
Kc (101) =


 38 + 401 − e



Td
Comment
Minimum performance index
1
Kc
( 101)
T

− m1 
τ m  0.5 + 1. 4[ 1 − e 1.5 τ m ]




T


− m2
 1 + 0.48 1 − e τ m








1.2 Tm1


−
0.42τ m  1 − e τm  +




0.6Tm2
Tm2
≤3
τm
τ m + 0.2Tm2
τ m + 0.2Tm2
118
. Tm1
Km τ m
2.20τ m
0.72τ m
τm
T
= 0.2 , m 2 = 01
.
Tm1
Tm1
125
. Tm1
K m τm
2.20τm
110
. τm
Tm 2
= 0.2
Tm1
167
. Tm1
K mτ m
2.40τ m
165
. τm
Tm 2
= 0.5
Tm1
25
. Tm1
K mτ m
2.15τ m
2.15τ m
Tm 2
= 10
.
Tm1
085
. Ku
035
. Tu
017
. Tu
τm
T
= 0.2 , m 2 = 0.2
Tm1
Tm1
333
. Tm 1
Km τm
Minimum IAE Shinskey [17] – page
121. Model: method
not specified.
1
Ti
1.5Tm1
−
τm



100
2
 Tm 2  
Km τ m 
Tm2
1 + 0.34
− 0.2
 
Tm1 
τm
 τ m  
Tm 2
>3
τm
Km e− sτ
K m e− sτ
or
- series controller
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 75: PID tuning rules - SOSPD model
m

1 
G c ( s) = K c 1 +
 (1 + Td s) . 1 tuning rule.
 Ti s 
Rule
Kc
Regulator tuning
Least mean square
error - Haalman [23]
Model: Method 1
Ti
Td
Comment
Minimum performance index
2Tm1
3τ m K m
Tm1
Tm2
Tm1 > Tm2 . Maximum
sensitivity = 1.9, Gain
margin = 2.36, Phase
margin = 50 0
Km e− sτ
K m e− sτ
or
- non-interacting
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 76: PID tuning rules - SOSPD model





1 
Td s
 . 2 tuning rules.
 E( s) −
controller U (s) = K c 1 +
Y
(
s
)

T
s

T
s
d

i 
1+


N


Kc
Ti
Td
Rule
Servo – Min. IAE Huang et al. [18]
Model: Method 1
2
Kc (102 ) =
m
Kc
( 102 ) 2
Ti
( 102)
Td
Comment
0<
( 102 )
0.1 ≤
Tm 2
≤1;
Tm1
τm
Tm1
≤ 1 ; N =10
0.0865

 τm 
1 
τ
T
τ T
7.0636 + 66.6512 m − 137 .8937 m2 − 122 .7832 m m2 2 + 261928
.



Km 
Tm1
Tm1
Tm1
 Tm1 


2.6405
1.0309
2.345
1.0570
 τm 
 Tm 2 
T 
T 
T
1 
336578
.
+ 30098
.
− 10.9347 m 2 
+ 141511
.  m2 
+ 29.4068 m2




Km 
 Tm1 
 Tm1 
 Tm1 
 Tm1 
τm

+



− 0.9450
−0.9282
0.8866

Tm2  τ m 
Tm 2  τ m 
τ m  Tm2 
1 
34.3156

+
− 701035
.
+ 152.6392






Km 
Tm 1  Tm1 
Tm1  Tm1 
Tm1  Tm1 


1
+
Km
Ti
( 102)
0 .8148
τ
T

τ T 
T
− 47.9791 m  m 2 
− 57.9370e + 10.4002e T
Tm1  Tm1 


m
m2
τ m Tm 2
m1
m1
Tm1 2
+ 6.7646e
τ 
+ 7.3453 m 
 Tm 
− 0.4062




2
2
3

 τm 
 Tm2 
 τm  
τm
Tm2
τ m Tm2
= Tm1 0.9923 + 0.2819
− 0.2679
− 1.4510
+ 0.6424 
 − 0.6712
 + 2 .504 
 
2
Tm1
Tm1
 Tm1 
Tm1
 Tm1 
 Tm1  

2
2
3
4
4

T τ 
τ T 
T 
 τm 
τ m  Tm2  
+ Tm1 2.5324 m 2  m  + 2.3641 m  m 2  + 2.0500  m 2  − 18759
.
+
08519
.



 
Tm1  Tm1 
Tm1  Tm1 
 Tm1 
 Tm1 
Tm1  Tm1  


3
2
2
3
4

 τ m   Tm2 
 Tm 2  
Tm2  τm 
τ T 
+ Tm 1 − 13496
.
.
.

 − 34972

 
 − 2.4216 m  m 2  − 31142

 
Tm1  Tm1 
Tm1  Tm1 

 Tm1   Tm1 
 Tm1  
5
4
2
3
2
3
5

 τm 
 Tm2   τ m 
 τm   Tm 2 
 Tm 2  
Tm2  τ m 
+ Tm 1 05862
.
.
.

 + 0.0797

 + 0.985
 
 + 12892

 
 + 12108

 
Tm1  Tm1 

 Tm1 
 Tm1   Tm1 
 Tm1   Tm1 
 Tm1  
Td
( 102)
2
2
3

 τm 
 Tm2 
 τm  
τm
Tm 2
τm Tm 2

= Tm1 0.0075 + 0.3449
+ 0.3924
− 0.0793
+ 0.6485
 + 2.7495
 + 0.8089 
 
2
Tm1
Tm1
 Tm1 
Tm1
 Tm1 
 Tm1  

2

T τ 
τ
+ Tm1 − 9.7483 m 2  m  + 3.4679 m
Tm1  Tm1 
Tm1

2
3
4
 Tm2 
 Tm2 
 τm 
.
.

 − 58194

 − 10884


 Tm1 
 Tm1 
 Tm1 



3
2
2
3
4

 τm   Tm 2 
 Tm 2  
T τ 
τ T 
+ Tm 1 12.0049 m 2  m  − 14056
.
.

 
 − 3.7055 m  m 2  + 100045

 
Tm1  Tm1 
Tm1  Tm1 

 Tm1   Tm1 
 Tm1  
5
4
2
3
2
3

 τm 
 Tm2   τ m 
 τ m   Tm 2  
Tm 2  τ m 

+ Tm 1 0.3520
.
 − 6.3603

 − 32980

 
 + 7.0404
 
 
Tm1  Tm1 

 Tm1 
 Tm1   Tm1 
 Tm1   Tm1  
4
5
6

 τ m   Tm 2 
 Tm 2 
 τm 
Tm 2
+ Tm1 14294
.
−
69064
.
+
0
.
0471
.






 + 11839
T
T
T
T
Tm1

 m1   m1 
 m1 
 m1 
5
 τm 
 Tm 2 
.

 + 17087


T
 m1 
 Tm1 
6



Kc
Rule
Servo-Min. IAE Huang et al. [18]
Kc
( 103) 3
Ti
(N=10)
Ti
Td
( 103)
( 103)
Td
Comment
0 .4 ≤ ξ m ≤ 1 ; 0 .05 ≤
4
2
3
3
2
4
5

 τ m   Tm 2 
 τ m   Tm2 
 τ m   Tm 2 
τ m  Tm 2  

+ Tm1 1. 7444
.
.
.
 
 − 12817

 
 − 21281

 
 + 15121

 
Tm1  Tm1  

 Tm1   Tm1 
 Tm1   Tm1 
 Tm1   Tm1 

3
Kc (103) =
−1.9009

τ 
τ 
1 
τ
−81727
.
− 32.9042 m + 319179
.
ξ m + 38.3405ξ m  m  + 0.2079 m 

Km 
Tm1
 Tm1 
 Tm1 


+
0.1571
1.2234

τ 
 τm 
1 
29.3215 m 
+ 359456
.
− 214045
.
ξ m0 .1311 + 51159
.
ξ m1.9814 − 219381ξ m1.737 


Km 
 Tm1 
 Tm 1 


−0.1303
1.2008

 τm 
 τm 
1 
τ
− 17.7448ξ m 
+
+ 268655
.
ξ m
− 52.9156 m ξ m1.1207 


Km 
Tm1
 Tm1 
 Tm1 


1
+
Km
Ti
( 103)
τ

τ
T
− 22.4297 m ξ m 0.3626 − 33331
.
e
Tm1

m
m1
+ 85175
.
e
ξm
− 15312
.
e
τ mξm
Tm1
+ 08906
.
ξ m Tm1 

τm 

2
2


τ 
τ 
τ
τ
2
= Tm1 11731
.
+ 6.3082 m − 0.6937ξ m + 8.5271 m  − 24 .7291ξ m m − 6.7123ξ m  m  + 7 .9559ξ m 
Tm1
Tm1
 Tm1 
 Tm1 


3
4
3

 τm 
 τm 
 τm 
 τ 
+ Tm1  − 32.3937 
.
.
ξ m2  m  
 − 271372

 + 166.9272ξ m 
 + 363954
 Tm1 
 Tm1 
 Tm1 
 Tm1  


2


τm
2  τm 
3

+ Tm1 − 94.8879ξ m 
.
ξ m 3 + 29.9159ξ m 4 
 − 22.6065 ξ m − 16084
Tm1

 Tm1 

5
4
3
2


τ 
τ 
τ 
 τm 
3
+ Tm 1 49.6314 m  − 84.3776ξ m  m  − 938912
.
ξ m 2  m  + 1101706
.

 ξm 

 Tm1 
 Tm1 
 Tm1 
 Tm1 

6
5


 τm  4
τ 
τ 
5
+ Tm 1 − 251896
.
.
ξ m  m  + 55268
.
ξ m6

 ξ m − 19.7569 ξ m − 12.4348  m  − 117589

 Tm1 
 Tm1 
 Tm1 

4
3
2


 τm 
 τm 
 τm 
τm
2
3
4
+ Tm 1 68.3097
.
.
ξm 5 
 ξ m − 17.8663
 ξ m − 225926

 ξ m + 95061
Tm1

 Tm1 
 Tm1 
 Tm1 

Td
( 104)
2
3

 τm 
 τm  
τm
τm
2

= Tm1 0.0904 + 0.8637
− 0.1301ξ m + 4 .9601
+ 0.7170ξ m − 12.5311
 + 14 .3899ξ m
 
Tm1
Tm1
 Tm1 
 Tm1  

2
4

τ 
τ 
τ  
+ Tm1 − 42.5012ξ m  m  − 214907
.
ξ m2  m  − 69555
.
ξ m 3 − 12.3016  m  

 Tm1 
 Tm1 
 Tm1  
3
2


τ 
τ 
τ
+ Tm1 102.9447ξ m  m  + 7.5855ξ m 2  m  + 19.1257 m ξ m 3 + 17.0952 ξ m 4 
 Tm1 
 Tm1 
Tm1


5
4
3
2


 τm 
 τm 
 τm 
2  τm 
3

+ Tm1 108688
.
.
ξm 

 − 17.2130ξ m 
 − 1100342
 + 50.6455
 ξm 

 Tm1 
 Tm1 
 Tm1 
 Tm1 

6
5


τ 
τ 
τ 
+ Tm 1 − 16.7073 m  ξ m 4 − 16.2013ξ m5 − 0. 0979 m  − 109260
.
ξ m  m  + 54409
.
ξ m6

 Tm1 
 Tm1 
 Tm1 

τm
≤1
Tm1
Rule
Kc
Ti
Td
Regulator – Min. IAE
- Huang et al. [18]
Model: Method 1
Kc(104 ) 4
Ti (104)
Td (104 )
Comment
Tm 2
≤1;
Tm1
0<
τm
0.1 ≤
Tm1
≤ 1 ; N =10
4
3
2

 τm 
 τm 
 τm 
τm 5 
2
3
4
+ Tm 1 29.4445
.
.
.
ξm 
 ξ m + 216061

 ξ m − 241917

 ξ m + 62798
Tm1

 Tm1 
 Tm1 
 Tm1 

4
Kc (104) =
− 0.8058

 τm 
1 
τm
Tm2
τ m Tm2
01098
.
− 86290
.
+ 766760
.
− 33397
.
+
11863
.



2
Km 
Tm1
Tm1
Tm1
 Tm1 


+
0.6642
2 .1482
0.8405
2.1123

 τm 
 τm 
T 
 Tm2 
1 
231098

.
+ 203519
.
− 52.0778 m 2 
− 121033
.






Km 
Tm1 
Tm1 
Tm1 
Tm1 






+
0.5306
−1.0781
−0.4500

1 
τ T 
T τ 
T τ 
9.4709 m  m2 

+ 13.6581 m 2  m 
− 19.4944 m2  m 
Km 
Tm1  Tm1 
Tm1  Tm1 
Tm1  Tm1 


1.1427
τ
T
1 
τ T 
− 28.2766 m  m2 
+
− 191463
.
e T + 8.8420e T
Km 
Tm1  Tm1 

Ti
( 104)
m
m2
m1
m1
τ m Tm 2
Tm1 2
+ 7.4298e
− 114753
.
Tm 2 

τm 

2
2
3

τ 
T 
τ  
τ
T
τ T
= Tm1 − 0.0145 + 2 .0555 m + 0.7435 m2 − 4.4805 m  + 1.2069 m m2 2 + 0.2584 m 2  + 7.7916  m  
Tm1
Tm1
 Tm1 
Tm1
 Tm1 
 Tm1  

2
2
3
4
3

 Tm 2 
 τm 
Tm 2  τ m 
τ m  Tm2 
Tm 2  τ m  

+ Tm1 − 6.0330
.

 + 3.9585

 − 30626

 − 7.0263
 + 7.0004

 
Tm1  Tm1 
Tm1  Tm1 
Tm1  Tm1  

 Tm1 
 Tm1 

2
2
3
4
5

T   τ 
 τm   Tm 2 
 Tm 2 
 τm  
+ Tm1 − 2.7755 m 2   m  − 15769
.
+
31663
.
+
2
.
4311






 

 T m1   Tm1 
 Tm1   Tm1 
 Tm1 
 Tm1  

T 
T   τ 
 τ m   Tm2 
T τ 
τ
+ Tm 1 − 0.9439 m 2  − 2.4506 m 2  m  − 0.2227 m2   m  + 19228
.

 
 − 0.5494 m
Tm1  Tm 1 
Tm1

 Tm1 
 Tm1   Tm1 
 Tm1   Tm1 
5
Td
( 104)
4
2
3
2
3
2
2
3

 τm 
 Tm2 
 τm  
τm
Tm 2
τ mTm 2
= Tm1 −0.0206 + 0.9385
+ 0.7759
− 2.3820
− 3.2336
 + 2 .9230
 + 7.2774 
 
2
Tm1
Tm1
 Tm1 
Tm1
 Tm1 
 Tm1  


T
+ Tm 1 − 9.9017 m 2
Tm1

2
 τm 
τ

 + 2.7095 m
T
T
 m1 
m1
2
3
 Tm 2 
 Tm2 
 τm 
.
.

 + 61539

 − 111018


T
T
 m1 
 m1 
 Tm1 
3
2
2

 τ m   Tm2 
T τ 
τ
+ Tm 1 10.6303 m2  m  + 57105
.

 
 − 7.9490 m
Tm1  Tm1 
Tm1

 Tm1   Tm1 
3
4



 Tm 2 
T 

 − 6.6597 m 2 
 Tm1 
 Tm1 
4



5
4
2
3
2
3

 τm 
T   τ 
 τ m   Tm2  
T τ 
+ Tm 1 80849
.
.

 − 4.4897 m 2  m  − 7.6469 m2   m  + 21155

 
 
Tm1  Tm1 

 Tm1 
 Tm1   Tm1 
 Tm1   Tm 1  
4
5
6
5
6

 τ m   Tm 2 
 Tm 2 
 τm 
 Tm 2  
Tm 2  τ m 

+ Tm 1 50694
.
.
.


 + 4.1225
 − 2.274 
 + 0519

 − 11295

 
Tm1  Tm1 

 Tm1   Tm1 
 Tm1 
 Tm1 
 Tm1  
4
2
3
3
2
4

 τ  T 
 τ  T 
 τm   Tm 2 
τm
+ Tm 1 2.2875 m   m2  + 0.9524 m   m 2  − 16307
.
.

 
 − 09321
T
T
T
T
T
T
T

 m1   m1 
 m1   m1 
 m1   m1 
m1
 Tm2 


 Tm1 
5



 Tm2 


 Tm1 
5



Rule
Regulator - Min. IAE Huang et al. [18]
Model: Method 1
5
Kc
( 105)
Kc
Kc
( 105) 5
Ti
Ti
Td
( 105)
Td
( 105)
Comment
[N=10]
0 .4 ≤ ξ m ≤ 1 ; 0 .05 ≤
τm
≤1
Tm1
3
0.086

 τm 
 τm 
 τm 
1 
τm
=
−357307
.
− 1419
.
+ 14023
.
ξ m + 6.8618ξ m 
.
ξ m 

 − 0.9773
 + 555898
Km 
Tm1
 Tm1 
 Tm1 
 Tm 


+
2
−1. 6624
−0.6951

τ 
 τm 
τ 
1 
τm
− 33093

.
ξ m  m  + 538651
.
ξ m 2 + 114911
.
ξ m3 + 08778
.
− 29.8822 m 


Km 
Tm 1 
Tm1
Tm1 
Tm1 





+
− 0.4762
−2.1208

τ 
τ 
1 
53535

.  m
− 16.9807 ξ m1.1197 − 254293
.
ξ m1.4622 − 01671
.
ξ m 58981 + 0.0034ξ m  m 
Km 
 Tm1 
 Tm 


τ
τ ξ
1 
τm
τ m 1.2103
ξ m Tm1 
3.0836
T
ξ
− 250355

+
.
ξm
− 54.9617
ξm
− 01398
.
e
− 82721
.
e + 63542
.
e T + 10479
.
Km 
Tm1
Tm1
τm 


m
m1
Ti
( 105)
m
m
m
m1
2
3

 τm 
 τm  
τm
τm
2

= Tm1 0.2563 + 11.8737
− 1.6547ξ m − 16.1913
+ 3.5927 ξ m + 19.5201
 − 9 .7061ξ m
 
Tm1
Tm1
 Tm1 
 Tm1  

2
4

τ 
τ 
τ  
+ Tm 1 − 14.5581ξ m  m  + 2.939ξ m 2  m  − 0.4592ξ m 3 − 34.6273 m  

 Tm1 
 Tm1 
 Tm1  
3
2


 τm 
τ
2  τm 
3
4
+ Tm 1 50.5163ξ m 
 + 8.9259ξ m 
 + 8.6966 m ξ m − 6.9436ξ m 
Tm1

 Tm1 
 Tm 1 

5
4
3
2


 τm 
 τm 
 τm 
2 τm 
3

+ Tm 1 27.2386
.
 − 20.0697ξ m 
 − 42.2833ξ m 
 + 85019

 ξm 

 Tm1 
 Tm1 
 Tm1 
 Tm1 

6
5


τ 
τ 
τ 
+ Tm 1 − 12.2957 m  ξ m 4 + 8.0694 ξ m 5 − 7.7887 m  + 2.3012 ξ m  m  − 2.7691ξ m 6 

 Tm1 
 Tm1 
 Tm1 

4
3
2


 τm 
 τm 
 τm 
τ
2
3
4
5
+ Tm 1 88984
.
.
.

 ξ m + 102494

 ξ m − 54906

 ξ m + 4.6594 m ξ m 
Tm1

 Tm1 
 Tm1 
 Tm1 

Td
( 105)
2
3

 τm 
 τm  
τm
τm
2
= Tm1 −0.021 + 3.3385
+ 0.185ξ m − 0.5164
− 0.8815ξ m + 0.584 
 − 0.9643ξ m
 
Tm1
Tm1
 Tm1 
 Tm1  

2
4

τ 
 τm  
2 τ 
3
+ Tm1  − 12513
. ξ m  m  + 13468
.
ξ m  m  + 2.3181ξ m − 52368
.

 

 Tm1 
 Tm1 
 Tm1  


3
2


τ 
τ
2 τ 
3
4
+ Tm1 153014
.
ξ m  m  + 119607
.
ξ m  m  − 2.0411 m ξ m − 31988
.
ξm 

Tm1

 Tm1 
 Tm1 


5
4
3
2


 τm 
 τm 
 τm 
2 τm 
3

+ Tm1 34675
.

 − 0.8219ξ m 
 − 15.0718ξm 
 − 18859
.

 ξm 


 Tm1 
 Tm1 
 Tm1 
 Tm1 


6
5


τ  4
τ 
τ 
5
6
+ Tm1  0.4841 m  ξ m + 2.2821ξ m − 0.9315 m  + 0529
. ξ m  m  − 06772
.
ξm 


 Tm1 
 Tm1 
 Tm1 


4
3
2


 τm 
 τm 
τ 
τ
2
3
4
5
+ Tm1  −14212
.

 ξ m + 71176
.

 ξ m − 2.3636 m  ξ m + 0.5497 m ξ m 

Tm1

 Tm1 
 Tm1 
 Tm1 


Km e− sτ
K m e− sτ
or
- ideal controller with
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 77: PID tuning rules - SOSPD model
(
)
set-point weighting U(s) = Kc Fp R ( s) − Y( s) +
Kc
Rule
m
Kc
[ Fi R(s) − Y(s)] + Kc Td s[ Fd R (s) − Y(s)] . 1 tuning rule.
Ti s
Ti
Td
Comment
05
. Tu
Fi = 1
0125
. Tu
Repeated pole
τ
01
. < m < 3;
Tm
Ultimate cycle
06
. Ku
Oubrahim and
Leonard [138]
Fp = 13
.
Model: Method not
specified
16 − Km Ku
17 + Km Ku
Fd = Fp
2
06
. Ku
Fp =
38
29 + 35Km K u
05
. Tu
Fi = 1
0125
. Tu
Fd = Fp
2
10% overshoot
Repeated pole
τ
01
. < m < 3;
Tm
20% overshoot
Km e− sτ
K m e− sτ
or
- non-interacting
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 78: PID tuning rules - SOSPD model
m

1 
controller U( s) = Kc  b + [ R (s) − Y(s) ] − ( c + Tds)Y( s) . 1 tuning rule.
Ts

i 
Rule
Kc
Ti
Td
Comment
Kc(105a) 6
154
. Ti ( 105a )
Td (105a)
c = K c ( 105a ) − 1 K m ;
Direct synthesis
b = 0.198
Hansen [150]
zero overshoot
b = 0.289
Model: Method 1
127
. Ti (105a )
Kc(105a)
Td (105a)
c = K c ( 105a ) − 1 K m ;
minimum IAE
b = 0.143
175
. Ti (105a )
Kc(105a)
Td (105a)
c = K c ( 105a ) − 1 K m ;
conservative tuning
6
Kc
Td
(105a )
( 105a )
=
=
[
2 Tm1Tm 2 + (Tm1 + Tm2 )τ m + 0.5τ m2
[
2[ T
]
3
9 Km Tm1Tm2 τ m + 0.5(Tm1 + Tm2 )τ m + 0167
. τm
[
m1Tm2
2
+ (Tm1 + Tm 2 )τ m + 05
. τ m2
]
[
3 Tm1Tm2 τ m + 05
. ( Tm1 + Tm 2 )τ m 2 + 0167
. τm3
2
3K m Tm1Tm 2 τ m + 0.5(Tm1 + Tm2 )τ m + 0167
. τm
2
]
3 2
, Ti (105a ) =
3
]
−
Tm1 + Tm2 − τ m
Km
[T
m1Tm 2
+ (Tm1 + Tm2 )τ m + 05
. τm
2
]
],
Km e− sτ
K m e− sτ
or
- ideal controller with
(1 + sTm1 )(1 + sTm 2 ) Tm12 s2 + 2ξ m Tm1s + 1
m
Table 79: PID tuning rules - SOSPD model
m


1
Td s
filtered derivative G c ( s) = Kc  1 +
+
sT
Ts

i
1+ d

N
Rule
Kc
Ti
ω p Tm 1
1
A m Km
4ω p 2 τ m


 . 2 tuning rules.


Td
Comment
Tm2
provided. N = 20.
Tm1 > Tm2
Direct synthesis
Hang et al. [151]
2ω p −
Model: Method 1
A m = 2 .0 , φ m = 45o
A m = 3.0 ,φ m = 60
Robust
Hang et al. [151]
Model: Method 1
Tm1
K m( λ + τ m)
π
Sample A m , φ m
1
+
Tm1
A m = 4.0 , φ m = 67.5 o
A m = 5.0, φ m = 72 o
Tm1
Tm2
o
ωp =
A m φ m + 0.5πA m (A m − 1)
(A
2
m
)
−1 τm
N = 20. Tm1 > Tm2
Table 80: PID tuning rules – SOSPD model
K m e− sτ
m
Tm1 s2 + 2ξ m Tm1s + 1
2


1
Td s
U (s) = K c 1 +
+
 Tis
T
1+ d

N

Kc
Rule
- Two degree of freedom controller:






β
T
s
d
 E( s) − K c  α +
 R (s) . 3 tuning rules.


Td 
s
1+
s

N 


Ti
Servo/regulator
tuning
Td
Minimum performance index
Taguchi and Araki
[61a]
Kc
(105 b) 7
Ti
(105 b )
Td
(105b )
(105 c )
Ti
(105 c )
Td
(105 c )
Model: ideal process
Kc
Minimum ITAE Pecharroman and
Pagola [134a]
0.7236 K u
0.5247 Tu
0.1650 Tu
Model: Method 15
7
Kc
Td
(105 b)
(105 b)


1 
0. 6978
=
1.389 +
τm
Kc 

[
− 0.02295 ] 2
Tm

2
Td
(105c )
(105 c )
τm
≤ 1.0 ; ξ m = 1. 0
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of
tuning rules of Chien
et al. [10]
τm
≤ 1.0 ; ξ m = 0.5
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of
tuning rules of Chien
et al. [10]
α = 0. 5840 , β = 1 ,
N = 10,
φ c = − 139.65 0
K m = 1 ; Tm = 1 ;
ξm =1


2
3



(105 b)
 0.02453 + 4.104 τ m − 3. 434  τm  + 1.231 τ m  
,
T
=
T




i
m



Tm
 Tm 
 Tm  



2
3
4

τm
 τm 
 τm 
 τ m  

= Tm 0.03459 + 1. 852
− 2.741  + 2.359   − 0. 7962   ,

Tm
 Tm 
 Tm 
 T m  

α = 0. 6726 − 0.1285
Kc
Comment
3
τm
τ 
τ 
τ
τ 
− 0.1371  m  + 0.07345  m  , β = 0. 8665 − 0.2679 m + 0.02724  m 
Tm
T
T
T
 m
 m
m
 Tm 


1 
0.5013
=
0.3363 +
τm
Kc 

[
− 0.01147 ] 2
Tm



2
3


 (105 c )
 − 0.02337 + 4.858 τm − 5.522  τm  + 2.054  τ m  
,
T
=
T
 
  
m
 i
Tm
 Tm 
 Tm  



2
3

τm
 τm 
 τ m  

= Tm 0.03392 + 2. 023
− 1.161  + 0.2826   ,

Tm
 Tm 
 Tm  

2
α = 0. 6678 − 0. 05413
2
3
τm
τ 
τ 
τ
τ 
− 0. 5680  m  + 0. 1699  m  , β = 0. 8646 − 0.1205 m − 0.1212  m 
Tm
Tm
Tm 
 Tm 
 Tm 
2
Rule
Kc
Ti
Td
Minimum ITAE Pecharroman and
Pagola [134b]
0.803 K u
0.509 Tu
0.167 Tu
φ c = phase
corresponding to the
crossover frequency;
K m = 1 ; Tm = 1 ;
0.1 < τm < 10
0.727 K u
0.524 Tu
0.165 Tu
0.672 Ku
0.532 Tu
0.161Tu
0.669 Ku
0.486 Tu
0.170 Tu
Model: Method 15
0.600 Ku
0.498 Tu
0.157 Tu
0.578 K u
0.481Tu
0.154 Tu
0.557 K u
0.467 Tu
0.149 Tu
0.544 Ku
0.466 Tu
0.141Tu
0.537 K u
0.444 Tu
0.144 Tu
0.527 K u
0.450 Tu
0.131Tu
0.521K u
0.440 Tu
0.129 Tu
0.515 K u
0.429 Tu
0.126 Tu
0.509 Ku
0.399 Tu
0.132 Tu
0.496 Ku
0.374 Tu
0.123 Tu
0.480 Ku
0.315 Tu
0.112 Tu
0.430 Ku
0.242 Tu
0.084 Tu
Comment
α = 0.585 , β = 1 ,
N = 10, φ c = −1460
α = 0.584 , β = 1 ,
N = 10, φ c = −1400
α = 0.577 , β = 1 ,
N = 10, φ c = −1340
α = 0.550 , β = 1 ,
N = 10, φ c = −1250
α = 0.543 , β = 1 ,
N = 10, φ c = −1150
α = 0.528 , β = 1 ,
N = 10, φ c = −1050
α = 0.504 , β = 1 ,
N = 10, φ c = −930
α = 0.495 , β = 1 ,
N = 10, φ c = −840
α = 0.484 , β = 1 ,
N = 10, φ c = −730
α = 0.477 , β = 1 ,
N = 10, φ c = −630
α = 0.454 , β = 1 ,
N = 10, φ c = −520
α = 0.445 , β = 1 ,
N = 10, φ c = −410
α = 0.433 , β = 1 ,
N = 10, φ c = −300
α = 0.385 , β = 1 ,
N = 10, φ c = −190
α = 0.286 , β = 1 ,
N = 10, φ c = −100
α = 0.158 , β = 1 ,
N = 10, φ c = −6 0
Table 81: PID tuning rules - I 2PD model G m ( s) =
U(s) = Kc (1 +
K m e− sτ m
- controller
s2
1
) E( s) + Kc ( b − 1) R (s) − Kc TdsY( s) . 1 tuning rule.
Tis
Rule
Kc
Direct synthesis
Hansen [91a]
Model: Method 1
3.75K m τ m
2
Ti
Td
Comment
5.4τ m
2. 5τ m
b = 0.167
Table 82: PID tuning rules – SOSIPD model (repeated pole)
K m e −s τ
s (1 + Tm1s) 2


1
Td s
U (s) = K c 1 +
+
 Tis
T
1+ d

N

Rule
Kc
- Two degree of freedom controller:






β
T
s
d
 E( s) − K c  α +
 R (s) . 2 tuning rules.


Td 
s
1+
s

N 


Ti
Servo/regulator
tuning
Taguchi and Araki
[61a]
m
Td
Comment
Minimum performance index
Kc
(105 d) 8
Ti
(105 d )
Td
τm
≤ 1.0 ;
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of
tuning rules of Chien
et al. [10]
α = 0. 601 , β = 1 ,
(105d )
Model: Method 2
Minimum ITAE Pecharroman and
Pagola [134a]
1.672 K u
0.366 Tu
0.136 Tu
φ c = phase
corresponding to the
crossover frequency;
K m = 1 ; Tm = 1 ;
0.1 < τm < 10
1.236 K u
0.427 Tu
0.149 Tu
0.994 Ku
0.486 Tu
0.155 Tu
0.842 Ku
0.538 Tu
0.154 Tu
Model: Method 1
0.752 Ku
0.567 Tu
0.157 Tu
0.679 Ku
0.610 Tu
0.149 Tu
0.635 K u
0.637 Tu
0.142 Tu
0.590 Ku
0.669 Tu
0.133 Tu
0.551K u
0.690 Tu
0.114 Tu
N = 10, φ c = −1640
α = 0.607 , β = 1 ,
N = 10, φ c = −1600
α = 0.610 , β = 1 ,
N = 10, φ c = −1550
α = 0.616 , β = 1 ,
N = 10, φ c = −1500
α = 0.605 , β = 1 ,
N = 10, φ c = −1450
α = 0.610 , β = 1 ,
N = 10, φ c = −1400
α = 0.612 , β = 1 ,
N = 10, φ c = −1350
α = 0.610 , β = 1 ,
N = 10, φ c = −1300
α = 0.616 , β = 1 ,
8
Kc
Ti
(105d )
(105 d)
Td
(105 d)


1 
0. 5667
=
0.1778 +
τm
Kc 

+ 0. 002325
Tm

N = 10, φ c = −1250



,


2
3
4

 τm 
 τm 
 τm  
τm

= Tm 0.2011 + 11.16
− 14. 98  + 13. 70  − 4.835  


Tm
 Tm 
 Tm 
 Tm  


τ 
τ
τ 
= Tm  1. 262 + 0.3620 m  , α = 0. 6666 , β = 0. 8206 − 0.09750 m + 0.03845  m 
Tm 
Tm

 Tm 
2
Rule
Minimum ITAE Pecharroman and
Pagola [134a] –
continued
Kc
Ti
Td
0.520 K u
0.776 Tu
0.087 Tu
0.509 Ku
0.810 Tu
0.068 Tu
Comment
α = 0.609 , β = 1 ,
N = 10, φ c = −1200
α = 0. 611 , β = 1 ,
N = 10, φ c = −1180
Table 83: PID tuning rules - SOSPD model with a negative zero
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
- controller with filtered



1
Td s 
 . 1 tuning rule.
derivative G c ( s) = Kc  1 +
+
Ti s 1 + Td s 



N 
Rule
Kc
Ti
Td
Tm1 + Tm2 − Tm 3
K m( λ + τ m )
Tm1 + Tm2 − Tm 3
Tm1 Tm2 − (Tm1 + T m2 − Tm3 )Tm 3
Tm1 + T m2 − Tm3
N=10; λ = [ Tm1 , τm ]
2ξTm 2 − Tm3
2ξTm1 − Tm 3
Tm2 1 − (2 ξTm1 − Tm3 )Tm3
N=10; λ = [ Tm1 , τm ]
Comment
Robust
Chien [50]
Model: Method 1
Km (λ + τ m )
2ξ Tm1 − Tm3
Tm1 > Tm 2 > Tm3
Table 84: PID tuning rules - SOSPD model with a negative zero
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
- classical controller



1   1 + Td s 
 . 1 tuning rule.
G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
Rule
Kc
Ti
Td
Comment
Tm 2
K m( λ + τ m )
Tm2
Tm1
Tm1 > Tm 2
Tm1
K m( λ + τ m )
Tm1
Tm2
Robust
Chien [50]
Model: Method 1
N=10; λ = [ Tm1 , τm ]
Tm1 > Tm 2
N=10; λ = [ Tm1 , τm ]
Table 85: PID tuning rules - SOSPD model with a negative zero
K m (1 − sTm3 ) e− s τ
m
(1 + sTm1 )(1 + sTm2 )
- series controller with



1 
Td s 
 . 1 tuning rule.
filtered derivative G c ( s) = Kc  1 +
 1 +
Ts

Ts
i 
 1 + d 

N 
Rule
Kc
Ti
Td
Comment
Tm 2
K m( λ + τ m )
Tm2
Tm1 − Tm3
Tm1 > Tm 2 > Tm3
Tm1
K m( λ + τ m )
Tm1
Tm 2 − Tm3
Robust
Chien [50]
Model: Method 1
N=10; λ = [ Tm1 , τm ]
Tm1 > Tm 2 > Tm3
N=10; λ = [ Tm1 , τm ]
Table 86: PID tuning rules - SOSPD model with a positive zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
- ideal controller


1
Gc ( s) = K c  1 +
+ Td s . 1 tuning rule.
T
s


i
Kc
Rule
ρ
Ti
( p0 q1 − p1 )
,
Minimum IAE, ISE
2
p0
and ITAE – Wang et
p 0 = Tm1 + Tm2 + Tm3 − τ m
al. [97]
q 1 = Tm1 + Tm 2 + 05
. τm
Model: Method 1
Td
Comment
Minimum performance index
p
p0 q 2 − p 2 p1
q1 − 1 ,
−
,
p0
p 0 q1 − p1 p 0
p2 = 05
. Tm1Tm 2 τ m
1
01
. ≤
τm
≤1
Tm1
p1 = Tm1Tm2 +
01
. ≤
Tm 2
≤1
Tm 1
q 2 = Tm1Tm2
01
. ≤
Tm 3
≤1
Tm1
0.5τ m (Tm1 + Tm2 ) − 0.5Tm3 τ m
+0.5τ m ( Tm1 + Tm 2 )
1
ρ = 35550
.
− 3.6167
+
Tm 2
Tm 1

τm
T
T 
− 0.4918 m 2 − 0.3318 m 3 
− 0.4685
Tm 1
Tm1
Tm 1 

ρ = 39395
.
− 3.2164
+
+
Tm3
Tm1

τm
T
T 
.
− 0.3318 m 2 + 2.5356 m 3  , minimum IAE
14746
T
T
Tm1 

m1
m1
τm
Tm 2
Tm 3 τ m 
τm
Tm2
Tm3 
+ 16185
.
− 58240
.
+
.
− 02383
.
+ 13508
.
10933

Tm1
Tm1
Tm1 Tm1 
Tm1
Tm1
Tm1 
Tm 2 
τm
T
T 
− 0.6679 m 2 − 0.0564 m3 
− 0.2383
Tm 1 
Tm1
Tm1
Tm1 
ρ = 32950
.
− 34779
.
+
τm
Tm 2
Tm 3
τ 
τm
Tm2
Tm3 
+ 21781
.
− 55203
.
+ m 14704
.
− 04685
.
+ 14746
.

Tm1
Tm1
Tm1 Tm1 
Tm1
Tm1
Tm1 
+
Tm3 
τm
T
T 
.
− 0.0564 m2 + 2.5648 m3  , minimum ISE
13508
Tm1 
Tm1
Tm1
Tm1 
τm
Tm2
Tm3
τ 
τm
T
Tm3 
+ 25336
.
− 55929
.
+ m 14407
.
− 0.5712 m2 + 15340
.

Tm1
Tm 1
Tm1 Tm 1 
Tm1
Tm1
Tm1 
Tm 2 
τm
T
T 
− 0.3268 m2 − 0.5790 m 3 
− 0.5712
Tm 1 
Tm1
Tm1
Tm 1 
+
Tm3
Tm1

τm
Tm2
T 
.
− 05790
.
+ 2.7129 m 3  , minimum ITAE
15340
Tm1
Tm1
Tm1 

Table 87: PID tuning rules - SOSPD model with a positive zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
- ideal controller with filtered



1
Td s 
 . 1 tuning rule.
derivative G c ( s) = Kc  1 +
+
Ti s 1 + Td s 



N 
Kc
Rule
Ti
Td
Comment
Robust
Tm1 + Tm2 +
Chien [50]
T m3 τ m
λ + T m3 + τ m
K m ( λ + Tm 3 + τ m )
Tm3 τ m
Tm1 + Tm2 +
Tm3 τ m
λ + Tm 3 + τ m
λ + Tm 3 + τ m
+
Tm1
Model: Method 1
2ξTm1 +
Tm3 τ m
λ + Tm3 + τ m
K m ( λ + Tm 3 + τ m )
2 ξTm1 +
Tm1 Tm2
Tm3 τ m
+ Tm2 +
λ + Tm 3 + τ m
Tm3 τ m
Tm3 τ m
λ + Tm3 + τ m
λ + Tm 3 + τ m
+
Tm2 3
Tm 3 τ m
2ξ Tm1 +
λ + Tm3 + τ m
Tm1 > Tm 2 > Tm3
N=10; λ = [ Tm1 , τm ]
Tm1 > Tm 2 > Tm3
N=10; λ = [ Tm1 , τm ]
Table 88: PID tuning rules - SOSPD model with a positive zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
- classical controller



1   1 + Td s 
 . 1 tuning rule.
G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
Rule
Kc
Ti
Td
Comment
Tm 2
K m ( λ + Tm3 + τm )
Tm2
Tm1
Tm1 > Tm 2 > Tm3
Tm1
K m ( λ + Tm3 + τm )
Tm1
Tm2
Robust
Chien [50]
Model: Method 1
N=10; λ = [ Tm1 , τm ]
Tm1 > Tm 2 > Tm3
N=10; λ = [ Tm1 , τm ]
Table 89: PID tuning rules - SOSPD model with a positive zero
K m (1 + sTm 3 ) e− sτ
m
(1 + sTm1 )(1 + sTm 2 )
- series controller with



1 
Td s 
 . 1 tuning rule.
filtered derivative G c ( s) = Kc  1 +
 1 +
Ts

Ts
i 
 1 + d 

N 
Rule
Kc
Ti
Tm 2
K m ( λ + Tm3 + τm )
Tm2
Tm1
K m ( λ + Tm3 + τm )
Tm1
Td
Robust
Chien [50]
Model: Method 1
Tm1 +
Tm 3τ m
λ + Tm 3 + τ m
Tm 2 +
Tm 3τ m
λ + Tm3 + τ m
Comment
N=10; λ = [ T, τ m ] , T =
dominant time
constant
N=10; λ = [ T, τ m ] , T =
dominant time
constant
Km e− sτ
- ideal controller
(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 )
m
Table 90: PID tuning rules - TOLPD model


1
Gc ( s) = K c  1 +
+ Td s . 1 tuning rule.
Ti s


Rule
Kc
Ti
Td
Comment
Minimum performance index
Standard form
optimisation binomial - Polonyi
[153]
Model: Method 1
Standard form
optimisation –
minimum ITAE Polonyi [153]
Model: Method 1


 1 − 4 Tm 2 + Tm2  Tm1


6
τ
T
m
m3  τ m

4 6Tm2 τ m + τ m
τm
Tm1 > 10( Tm2 + τm )

Tm2
T  T
 1 − 21
.
+ m2  m1

3
.
4
τ
Tm3  τ m
m

2.7 3.4Tm2 τ m + τ m
τm
Tm1 > 10( Tm2 + τm )
Table 91: PID tuning rules - TOLPD model - G m (s ) =
K m e− sτ
m
(1 + sTm ) 3
– Two degree of freedom controller:


1
Td s
U (s) = K c 1 +
+
 Tis
T
1+ d

N

Kc
Rule
Ti
Servo/regulator
tuning
Kc
(105 e ) 2
Ti
Model: ideal process
Kc
(105e )
Td
Comment
Minimum performance index
Taguchi and Araki
[61a]
2






β
T
s
d
 E( s) − K c  α +
 R (s) . 1 tuning rule.


Td 
s
1+
s

N 




1 
1.275
=
0.4020 +
τm
Kc 

− 0.003273
Tm

(105 e )
Td
(105 e )
τm
≤ 1.0 ;
Tm
Overshoot (servo step)
≤ 20% ; settling time
≤ settling time of
tuning rules of Chien
et al. [10]



,


2
3
4

 τm 
 τm 
 τm  
τm

Ti
= Tm 0. 3572 + 7. 647
− 12. 86  + 11 .77   − 4. 146  


Tm
 Tm 
 Tm 
 Tm  

2
2

 τm  
τm
 τm 
τm
(105 e )

Td
= Tm 0.8335 + 0.2910
− 0. 04000   , α = 0. 6661 − 0.2509
+ 0. 04773   ,


Tm
Tm
 Tm 
 Tm  

(105 e )
β = 0. 8131 − 0.2303
τm
τ 
+ 0.03621  m 
Tm
Tm 
2
K m e− sτ


1
- ideal controller Gc ( s) = K c  1 +
+ Td s .
1 − sTm
Ti s


m
Table 92: PID tuning rules - unstable FOLPD model
3 tuning rules.
Kc
Rule
Ti
Td
Comment
Direct synthesis
De Paor and
O’Malley [86]
Tm 
cos
K m 
(1 − T τ )T

Tm  1 − Tm τ m
sin
Km 
Tm τ m

m
m
m τm



(1 − T τ )T
m
m
m
τm
Model: Method 1
Chidambaram [88]
Model: Method 1
Valentine and
Chidambaram [154] dominant pole
placement
Model: Method 1
1 
T 
. + 0.3 m 
13
Km 
τm 
 Tm 
1165
.
 
Km  τ m 
0.245
1
+




 1− T τ
m m
Tm 

Tm τ m

φ m = tan − 1
0.245
 Tm 
1165
.
 
Km  τ m 
0.245
(
,
)
1  Tm τ m
2 
Tm  1 − Tm τ m
1 − Tm τ m
− Tm τm 1 − Tm τ m
Tm τm
(

τ 
Tm  25 − 27 m 
Tm 


τm 
 0176
T
.
+ 0.36
Tm  m

0.179 − 0.324
 Tm 
1165
.
 
Km  τ m 

 tan 0.75φ
m


τm
Tm
 τm 

+ 0.161
 Tm 
2
 1

 Ti
)
046
. τm
τm
< 0.6
Tm

τ 
.
+ 0.36 m  Tm
 0176
Tm 

τm
< 06
.
Tm


τ 
τ 
.
+ 0.36 m  25Tm  0176
.
+ 0.36 m  Tm
 0176
Tm 
Tm 


0.176Tm + 0.36τ m
τ
0.12 − 0.1 m
Tm
τm
<1
Tm

τ 
.
+ 0.36 m  Tm
 0176
Tm 

0.6 ≤
08
. <
τm
Tm
τm
Tm
≤ 08
.
≤1
K m e− sτ
- non-interacting controller
1 − sTm
m
Table 93: PID tuning rules - unstable FOLPD model

1
KTs
U( s) = Kc  1 +  E ( s) − c d Y(s) . 2 tuning rules.
sT
Ti s

1+ d
N
Kc
Rule
Servo tuning
Huang and Lin [155]
- minimum IAE
Model: Method 2
Regulator tuning
Huang and Lin [155]
- minimum IAE
Model: Method 2
Ti
Td
Comment
Minimum performance index
−
−1 .0251


τm
τ m 
1 

− 0 .433 + 0 .2056
+ 0 .3135


Km
Tm
 T m 


6.6423
7.6983τ m




τ 
τm
 T  − 0. 0312+ 1. 6333 τm + 00399
Tm  − 00018
.
+ 08193
.
+ 7. 7853 m 
.
e T m 

 m
Tm
Tm

 Tm 



0.01 ≤
τm
≤ 0.8 ; N=10
Tm
0.01 ≤
τm
≤ 0.8 ; N=10
Tm
Minimum performance index
−
−1.004

1 
τm
τ 

0.2675 + 01226
.
+ 0.8781 m 

Km 
Tm
 Tm 

2. 9123

τ m 
τ

Tm  00005
.
+ 2.4631 m + 95795
.
 


T
T
m

 m


τ 
Tm  0.0011 + 0.4759 m 
Tm 

K m e− sτ
- classical controller
1 − sTm
m
Table 94: PID tuning rules - unstable FOLPD model



1   1 + Td s 
 . 1 tuning rule.
G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
Rule
Kc
Regulator tuning
Shinskey [16] minimum IAE – page
381.
Method: Model 1
Ti
Td
Comment
Minimum performance index
170
. τm
0.60τm
τ m Tm = 01
.
Tm
K mτ m
1.90τ m
0.60τm
τ m Tm = 0.2
08929
.
Tm
Km τ m
2.00τ m
0.80τ m
τ m Tm = 0.5
08621
.
Tm
K m τm
2.25τm
0.90τm
τ m Tm = 0.67
08333
.
Tm
K m τm
2.40τ m
100
. τm
τ m Tm = 0.8
0.9091Tm
Km τ m
Table 95: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ
- ideal controller
(1 − sTm1 )(1 + sTm 2 )
m


1
Gc ( s) = K c  1 +
+ Td s . 2 tuning rules.
Ti s


Kc
Ti
Td
Comment
Kc(106 ) 3
Ti (106)
Td (106 )
Tuning rules
developed from Ku , Tu
  λ


Tm1  λ 
+ 2 + Tm 2 

  Tm1

2
λ
 λ

λ
+ 2 + Tm 2
 Tm1

Km uncertainty = 50%
τm Tm = 02
.
λ = [0.5Tm ,19
. Tm ]
τm Tm = 0.4
λ = [1.3Tm ,19
. Tm ]
τm Tm = 02
.
λ = [0.4Tm ,4. 3Tm ]
τm Tm = 0.4
λ = [11
. Tm ,4.3Tm ]
τm Tm = 0.6
λ = [ 2.2Tm , 4.3Tm ]
Rule
Ultimate cycle
McMillan [58]
Model: Method not
relevant
Robust
Rotstein and Lewin
[89]
Model: Method 1
Km uncertainty = 30%
 λ

λ
+ 2 Tm 2
 Tm1

 λ

λ
+ 2 + Tm 2
 Tm1

λ determined
graphically – sample
values provided
2
3
Kc
Ti
( 106)
( 106)
=
1111
.
Tm1Tm2
Km τ m 2






1

0.65  ,
 
  ( Tm1 + Tm 2 )Tm1Tm2
1 +  T − T T − τ τ  
m2 )( m1
m) m 
 
  ( m1
0.65
0.65
  (T + T ) T T
  (T + T )T T
 
 


m1
m2
m1 m2
( 106)
m1
m2
m1 m 2
= 2τ m 1 + 
= 05
. τ m 1 + 
  , Td
 
  ( Tm1 − Tm2 )( Tm1 − τ m ) τ m  
  (Tm1 − Tm 2 )( Tm1 − τ m ) τ m  
Table 96: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ
- classical controller
(1 − sTm1 )(1 + sTm 2 )
m



1   1 + Tds 
 . 2 tuning rules.
G c ( s) = Kc  1 +


Ti s   1 + Td s 



N 
Kc
Rule
Ti
Regulator tuning
bTm1 1 +
aTm2 2
Minimum ITAE 2
4( τm + Tm 2 )
Poulin and
Pomerleau [82], [92] – K m ( aTm1 − 4[ τ m + Tm2 ])
deduced from graph
Model: Method 1
(τ m + Td N) Tm1
Output step load
disturbance
Input step load
disturbance
Td
Comment
Minimum performance index
4Tm1 ( τ m + Tm2 )
aTm1 − 4( τ m + Tm2 )
0≤
Tm2
τm
≤2;
T
( d N)
01
. Tm ≤
Td
≤ 033
. Tm
N
a
0.9479
1.0799
1.2013
1.3485
1.4905
b
2.3546
2.4111
2.4646
2.5318
2.5992
(τ m + Td N) Tm1
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
a
1.6163
1.7650
1.9139
2.0658
2.2080
b
2.6612
2.7368
2.8161
2.9004
2.9826
0.05
0.10
0.15
0.20
0.25
1.1075
1.2013
1.3132
1.4384
1.5698
2.4230
2.4646
2.5154
2.5742
2.6381
0.30
0.35
0.40
0.45
0.50
1.6943
1.8161
1.9658
2.1022
2.2379
2.7007
2.7637
2.8445
2.9210
3.0003
Table 97: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ
- series controller
(1 − sTm1 )(1 + sTm 2 )
m

1
Gc ( s) = Kc 1 +
 (1 + Td s) . 1 tuning rule.
Ti s

Rule
Kc
Direct synthesis
Ho and Xu [90]
ω p Tm1
Model: Method 1
A m Km
Ti
Td
157
. ω p − ω p τm −
2
1
Tm1
Tm2
Comment
ωp =
A m φ m + 157
. A m (A m − 1)
(A
2
m
)
− 1 τm
Table 98: PID tuning rules - unstable SOSPD model G m ( s) =
Km e − sτ
- non-interacting controller
(1 − sTm1 )(1 + sTm 2 )
m

1
KTs
U( s) = Kc  1 +  E ( s) − c d Y(s) . 2 tuning rules.
sT
Ti s

1+ d
N
Rule
Servo tuning
Huang and Lin [155]
- minimum IAE
Kc
Ti
Td
Comment
Minimum performance index
Tm2 ≤ Tm1 ;
Kc(107 ) 4
Ti (107)
Td (107 )
0.05 ≤
Model: Method 2
4
Kc
( 107 )
τm
≤ 0.4
Tm1
−1.344
0.995
 τm 
 Tm 2 
1 
τm
Tm2
T τ 
=−
10.741 − 13363
.
+ 0099
. 
+ 727.914
− 708.481
+ 9.915 m 2 2m 


Km 
Tm1
 Tm1 
Tm1
 Tm1 
Tm1 


−
1.031
0.997
τ
T
T
T 
 Tm 2 
Tm 2
1 
T
T
T
84.273 m1 
−
90
.
959
+
9
.
034
e
−
2
.
386
e
−
16
.
304
e


Km 
 τm 
 Tm1 
τm

m
m2
m1
m1
m 2 τm
2
m1



2.12
0.985

τ 
T 
τm
T
T τ 
Ti (107 ) = Tm1  −149 .685 − 141418
.
− 88.717 m  − 17.29 m 2 + 20518
.  m2 
− 12.82 m 2 2m 
Tm1
 Tm1 
Tm1
 Tm1 
Tm1 


0.286
1.988
τ
T
T

τ 
 Tm 2 
T
+ Tm1 3.611 m 
+ 0.000805 m2 + 141.702e T − 2.032e T + 10.006e T


 Tm1 
 Tm1 
τm

Td
( 107 )
m
m2
m1
m1
m 2τ m
2
m1



2
2

 τm 
 Tm2 
τm
Tm2
T τ 
= Tm1  −0.4144 + 15805
.
− 142.327 
+ 01123
.
 + 0.7287

 − 18.317 m 2 2m 
Tm1
 Tm1 
Tm1
 Tm1 
Tm1 


3
3

 τ 2T 
τ T 2
τ 
T 
+ Tm1 48695
.  m  − 10542
.  m2  + 204.009 m 3m 2  + 47.26 m m32
 Tm1 
 Tm1 

 Tm1 
 Tm1

τ 
 + 396349
.  m
 Tm1 

4




 τ 3T 
τ T 3
 τ 2T 2 
T 
τ 
+ Tm1  − 138.038 m 4m 2  + 52155
.  m m42  − 646.848 m m4 2  + 19.302 m 2  − 4731.72 m 
 Tm1 
 Tm1 

 Tm1 
 Tm1 
 Tm1 
4

 τ 4T 
τ T 4
 τ 3T 2
+ Tm1 425789
.  m 5m 2  − 289 .746 m m52  − 841807
.  m m5 2

 Tm1 
 Tm1 
 Tm1
5




 τ 2T 3 
 Tm 2 
 + 1313.72 m m5 2  − 37688
.


 Tm1 

 Tm1 
6

 τ 5T 
 τ T 5
 τ 4T 2  
 τm 
+ Tm1 6264 .79
.  m 6m2  + 204.689 m m62  + 25706
.  m m6 2  
 − 161469
 Tm1 

 Tm1 
 Tm1 
 Tm1  
3
6

 τ 2T 4 
τ T 
T  
+ Tm1  − 791857
.  m m6 2  + 648.217 m m2 2  − 5.71 m2  
 Tm1  

 Tm1 
 Tm1 


5



Rule
Kc
Ti
Td
Huang and Lin [155]
- minimum IAE continued
Model: Method 2
Kc(108 ) 5
Ti (108)
Td (108 )
5
Kc (108) = −
Ti
Tm 1 < Tm 2 ≤ 10Tm1 ;
0.05 ≤
τm
≤ 025
.
Tm1
− 0.3055
0.5174
τ 
T 
1 
τm
T
T τ 
−1302
. + 85914
.
+ 34.82 m 
+ 10.442 m 2 − 22.547  m2 
− 14.698 m 2 2m 
Km 
Tm1
 Tm1 
Tm1
 Tm1 
Tm2 


1
−
Km
( 108)
Comment
1.0077
0.9879
τ
T

T 
 Tm 2 
 Tm 2 
T
52.408  m1 


− 5147
. 
 + 53.378e − 0.000001e T
τ
T
τ

 m
 m1 
 m

m
m2
m1
m1
Tm2 τ m
+ 0.286e
Tm1 2




2
2

 Tm 2 
 τm 
 Tm2 
τm
Tm2 τ m 
= Tm1 72.806 − 268.746
− 4.9221
. 
.

 + 246819
 + 0.6724
 + 151351
Tm1
 Tm1 
 Tm1 
 Tm1 
Tm12 


3
3
4

 τ 2T 
τ T 2
τ 
 Tm 2 
τ  
+ Tm1 − 6914.46 m  − 00092
.
.  m 3m 2  − 14.27 m m32  + 558017
.  m 

 − 795465

 Tm1 
 Tm1 
 Tm1 
 Tm1 
 Tm1  

 τ 3T 
 τ T 3
 τ 2T 2 
 Tm2 
+ Tm 1 1417 .65 m 4m 2  + 0.4057 m m42  + 55536
.  m m4 2  − 0001119
.



 Tm1 
 Tm1 
 Tm1 
 Tm1 
T
τ
T

T
T
+ Tm 1 − 44.903e + 0.000034e − 15694
. eT


m
m2
m1
m1
m2 τ m
2
m1
τ 
+ 678778 m 
 Tm1 
19.056
 Tm2 


 Tm1 
4
7.3464







1.1798
0.1064

τ 
T 
τ
T
T τ 
Td (108) = Tm1 175515
.
− 86.2 m + 348.727 m 
− 0.008207 m 2 − 55619
.  m2 
+ 0.0418 m 2 2m 
Tm1
 Tm1 
Tm1
 Tm1 
Tm1 


0.0355
0.0827
τ

τ 
 Tm2 
T 
+ Tm 1 78959
.  m


+ 0.005048 m 2  − 187 .01e T

 Tm 1 
 Tm1 
 τm 

Tm 2 τ m
Tm 2
m
m1
+ 0000001
.
e
Tm1
− 00149
.
e
Tm1 2




Kc
Rule
Regulator tuning
Huang and Lin [155]
– minimum IAE
Ti
Td
Comment
Minimum performance index
Tm2 ≤ Tm1 ;
Kc
( 109 ) 6
Ti
( 109)
Td
( 109 )
0.05 ≤
Model: Method 2
6
Kc
( 109 )
−1.164
2.54
 τm 
 Tm2 
1 
τm
Tm 2
T τ 
=−
− 174167
.
− 31364
.
+ 0.4642
− 103069
.
− 83916
. 

 − 66.962 m2 2m 
Km 
Tm1
 Tm1 
Tm1
 Tm1 
Tm1 


1
−
Km
Ti
( 109)
τm
≤ 0.4
Tm1
1.065
1.014
τ

T  T 
Tm 2
T
59.496  m1   m 2 
− 7079
.
+ 23121
. e
τm

 τm   Tm1 

Tm 2 τ m
Tm 2
m
m1
+ 126.924e
Tm 1
+ 26.944e
Tm 1 2




2
2
3

 τm 
 Tm2 
 τm  
τm
Tm 2
Tm2 τ m
= Tm1 0.008 + 2.0718
+ 6.431
+ 0.7503
− 18686
. 
 + 0.4556
 + 2.4484
 
Tm1
 Tm1 
Tm1
 Tm1 
Tm12
 Tm1  


3
4

T τ 2
τ T 2
 T τ 3
T 
τ 
+ Tm 1 − 2.9978  m 2  − 21135
.  m2 m3  + 12.822  m m32  + 39.001 m  + 22848
.  m 2 4m  

 Tm1 
 Tm1 
 Tm1 
 Tm1 
 Tm1  

 T 3τ 
 T 2τ 2 
T 
+ Tm1 − 4.754 m 2 4 m  − 0.527  m2 4m  + 164
.  m2 

 Tm1 
 Tm1 
 Tm1 
4



2
2
3

τ 
 Tm2 
τ  
τm
Tm 2
T τ
Td (109) = Tm1  −0.0301 + 11766
.
− 4.4623 m  + 05284
.
− 14281
.
.  m 

 + 4.6 m 2 2m + 11176
Tm1
 Tm1 
Tm1
 Tm1 
Tm1
 Tm1  


3
4

 τ m2 Tm 2 
 τ m Tm2 2 
 Tm 2 τm 3  
 Tm 2 
 τm 

+ Tm 1 10886
.

−
05039
.


−
98564
.
−
7528
.



 − 5.0229


3
3
4

 Tm1 
 Tm1 
 Tm1 
 Tm1 
 Tm1  
4

τ T 3
 τ 2T 2 
 Tm 2  
+ Tm1 − 2.3542  m m42  + 9.3804  m m4 2  − 01457
.

 

 Tm1 
 Tm1 
 Tm1  
Rule
Kc
Ti
Td
Huang and Lin [155]
- minimum IAE –
continued
Model: Method 2
Kc(110 ) 7
Ti (110)
Td (110 )
7
Kc
( 110)
Comment
Tm 1 < Tm 2 ≤ 10Tm1 ;
0.05 ≤
τm
≤ 025
.
Tm1
2.1984
0.791
 τm 
 Tm 2 
1 
τm
Tm 2
T τ 
=−
1750.08 + 1637.76
+ 1533.91
− 7.917
+ 6187
. 
− 6.451 m2 2m 


Km 
Tm1
 Tm1 
Tm1
 Tm1 
Tm1 


3.2927
1.0757
τ
T
 Tm1 
 Tm 2 
1 
Tm 2
T
0002452
−
.




+ 13729
.
− 1739.77e − 0.000296e T
Km 
τm
 τm 
 Tm1 

m
m2
m1
m1
Tm 2 τ m
+ 2.311e
Tm1 2




2
2
3

τ 
 Tm 2 
τ  
τ
Tm 2
T τ
Ti (110) = Tm1 51678
.
− 57.043 m + 1337.29 m  + 01742
.
− 01524
.

 + 7.7266 m 2 2m − 6011.57 m  
Tm1
 Tm1 
Tm1
 Tm1 
Tm1
 Tm1  


3
4

 Tm2 τ m 2 
 τ m Tm2 2 
 Tm2 τ m 3  
 Tm2 
 τm 

+ Tm 1 0.0213

+
0
.
0645


+
913552
.
+
274
.
851


 − 65.283


3
3
4

 Tm1 
 Tm1 
 Tm1 
 Tm1 
 Tm1  
4
τ
T

 T 3τ 
 T 2τ 2 
 Tm 2 
T
T
+ Tm 1 0.003926 m 2 4 m  − 2.0997 m 2 4m  − 0001077
.
−
49
.
007
e
+
0
.
000026
e


T
T
T

 m1 
 m1 
 m1 
T τ

+ Tm1 0.2977e T

m2
m1
m
m
m2
m1
m1






2
2
3

τ 
T 
τ  
τ
T
T τ
Td (110) = Tm1  −0.0605 + 4.6998 m − 29.478 m  + 0.0117 m 2 − 0.0129 m 2  + 0.6874 m2 2m + 140.135 m  
Tm1
 Tm1 
Tm1
 Tm1 
Tm1
 Tm1  


3
4

 τ m 2 Tm 2 
 τ mTm 2 2 
 Tm 2 τm 3  
 Tm 2 
 τm 

+ Tm 1 0.002455
.


−
01289
.


−
238
.
864
+
06884
.


 − 14712


3
3
4

 Tm1 
 Tm1 
 Tm1 
 Tm1 
 Tm1  
4

 τ T 3
 τ m 2Tm 2 2 
 Tm 2  
+ Tm 1 0.007725 m m42  − 01222
.


−
0
.
000135

 
4

 Tm1 
 Tm1 
 Tm1  
Table 99: PID tuning rules – general model with a repeated pole G m ( s) =
K m e − sτ
- ideal controller
(1 + sTm ) n
m


1
Gc ( s) = K c  1 +
+ Td s . 1 tuning rule.
Ti s


Kc
Rule
Direct synthesis
Skoczowski and
Tarasiejski [156]
Model: Method 1
1
≤
ω g Tm
Km
(1 + ω
2
g
Tm 2
)
Ti
Td
Tm
n−1
Tm , n ≥ 2
n+2
Comment
n− 1
1 + ω g 2 Td 2
1

2n + 4 τ m 4n + 2 
− Tm n 2 − 2n − 2 +
+
φm  ± b
π Tm
π


ωg =
 2
4 n + 2 τ m 2n − 2 
2
2Tm n − 4n + 3 +
+
φm 
π Tm
π


with
2
b = Tm
 2
2n + 4 τ m 4 n + 2 2 
2 
4 n + 2 τm 2 n − 2 

+
φ m  + 4( n + 2) 1 − φ m   n2 − 4n + 3 +
+
φm 
n − 2n − 2 +

π Tm
π
π 
π Tm
π



Table 100: PID tuning rules – general stable non-oscillating model with a time delay - ideal controller


1
Gc ( s) = K c  1 +
+ Td s . 1 tuning rule.
Ti s


Kc
Rule
Ti
Td
Comment
Direct synthesis
2
Kc
(110 a )
=
Ti
Ti
(110 a ) 2
Ti
(110 a )
Kc
(110 a )
Ti
(110 a )
Kc
(110 a )
Ti
(110 a )
Kc
Gorez and Klan
[147a]
Model: not specified
( 110a )
( 110a )
+ τm
, Ti
(110 a )
τ
1+ 1 + 2 m
 Tar
= Tar
2
Td
Ti
(110 a )
(110a )
τm
Tar
0.25Ti
 τm
1 −
 T
ar





(110 a )
2

τ
 −2 m

Tar

,
2 


T
τ
2τ m  
 Ti (110a ) + Tcr cr + Ti (110a ) − Taa − m 1 − K c (110a ) 1 +
 
(110 a )
Tar
2Tar 

 3Ti (110 a )  
Ti


Tar = average residence time of the process (which equals Tm + τ m for a FOLPD process, for example); T aa =
Td
(110 a )
=
Tar
(
)


(110 a )
1 + τ m
additional apparent time constant; Tcr = 1 − K c
 2T (110a )

i


 τ m


Table 101: PID tuning rules – fifth order model with delay
K m (1 + b1s + b2 s2 + b3s3 + b4 s4 + b ss5 ) e− s τ


1
G m ( s) =
– ideal controller Gc ( s) = K c  1 +
+ Td s .
2
3
4
5
Ti s
1 + a 1s + a 2 s + a 3s + a 4 s + a 5s


m
(
)
1 tuning rule.
Kc
Rule
Direct synthesis
Magnitude optimum
- Vrancic et al. [159]
Model: Method 1
Ti
Td
Ti (112)
Td (112 )
Comment
3
Kc
( 112 )
3
Kc ( 112 ) =
Td
( 112)
)
[
2 Km − a1 b 1 + a1a2 + a1b1 − a3 − b1b 2 + b 3 + ( a1 − b 1 ) τ m + ( a1 − b 1 ) τ m2 + 0.333τ m 3 − ( a1 − b 1 + τ m ) Td
2
2
2
(
2
)
a1 − a1 b 1 + a 1b 2 − 2 a1a 2 + a2 b 1 + a 3 − b 3 + τ m a 1 − a1b 1 − a 2 + b 2 + 0.5( a1 − b1 ) τm + 0.167τ m
3
Ti (112 ) =
(
a13 − a1 2b 1 + a1 b 2 − 2 a1a2 + a2 b 1 + a 3 − b 3 + τ m a1 2 − a1b1 − a 2 + b 2 + 0.5( a1 − b1 ) τ m 2 + 0167
. τm 3
2
[a
2
1
2
− a 1b 1 − a 2 + b 2 + ( a1 − b 1 ) τ m + 0.5τ m − ( a1 − b1 + τ m ) Td
… see attached sheet with δ = 0.
2
2
]
3
]
Table 102: PID tuning rules – fifth order model with delay
K m (1 + b1s + b2 s2 + b3s3 + b4 s4 + b ss5 ) e− s τ
G m ( s) =
– controller with filtered derivative
1 + a1s + a2 s2 + a3s 3 + a 4 s4 + a 5s5
m
(
)



1
Tds 
 . 1 tuning rule.
G c ( s) = Kc  1 +
+
Ts
Ts

i
1 + d 

N 
Rule
Direct synthesis
Magnitude optimum
- Vrancic et al. [73]
Model: Method 1
Kc
Ti
Td
Comment
Kc(113) 4
Ti (113)
Td (113)
8 ≤ N ≤ 20
4
Kc ( 113) =
Ti( 113 ) =
(
)
a13 − a12 b1 + a1b2 − 2a1a2 + a 2b 1 + a 3 − b 3 + τ m a12 − a1b 1 − a2 + b 2 + 0.5( a1 − b1 ) τ m2 + 0.167τ m3
 2

T2
2
2
2
2
3
2 K m −a1 b 1 + a1a2 + a1b 1 − a3 − b1b 2 + b 3 + ( a1 − b1) τ m + ( a1 − b1 ) τ m + 0.333τ m − ( a1 − b1 + τ m ) Td − d ( a1 − b1 + τ m ) 
N


(
)
a13 − a12 b1 + a1b 2 − 2a1a2 + a2 b1 + a 3 − b 3 + τ m a12 − a1b 1 − a 2 + b 2 + 0.5( a1 − b1 ) τ m2 + 0.167τ m3
 2
Td 2 
2
a1 − a1b 1 − a 2 + b 2 + ( a1 − b 1) τ m + 0.5τ m − ( a1 − b1 + τ m ) Td −

N

Td (113) = see attached sheet
4. Conclusions
The report has presented a comprehensive summary of the tuning rules for PI and PID controllers that
have been developed to compensate SISO processes with time delay. Further work will concentrate on evaluating
the applicability of these tuning rules to the compensation of processes with time delays, as the value of the time
delay varies compared with the other dynamic variables.
Year
1942
1950
1951
1952
1953
1954
1961
1964
1965
1967
1968
1969
1972
1973
1975
1979
1980
1982
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
1941-1950
Table 103: Tuning rules published by year and medium
Journal articles
Conference
Books/ Ph.D.
papers/
thesis
correspondence
1
1
1
1
3
1
1
1
1
1
1
1
2
2
1
1
2
1
2
1
2
1
1
1
1
1
3
4
2
4
3
4
1
5
1
2
5
4
1
5
2
2
14
2
1
8
6
2
11
4
1
6
7
12
1
1
1
16
2
0
0
Total
1
1
1
1
3
1
1
1
1
2
1
2
2
2
2
1
2
1
3
2
1
1
9
7
5
5
3
10
9
17
16
16
13
14
17
2
1951-1960
1961-1970
1971-1980
1981-1990
1991-2000
TOTAL
6
5
7
15
68
103
0
0
0
7
44
51
0
3
2
7
8
20
6
8
9
29
120
174
List of journals in which tuning rules were published and number of tuning rules published
Advances in Modelling and Analysis C, ASME Press
1
AIChE Journal
4
Automatica
12
British Chemical Engineering
1
Chemical Engineering Communications
5
Chemical Engineering Progress
1
Chemical Engineering Science
1
Control
1
Control and Computers
1
Control Engineering
7
Control Engineering Practice
4
European Journal of Control
1
Hydrocarbon Processing
2
Hungarian Journal of Industrial Chemistry
1
IEE Proceedings, Part D
9
(including IEE Proceedings - Control Theory and Applications, Proceedings of the IEE, Part 2)
IEEE Control Systems Magazine
1
IEEE Transactions on Control Systems Technology
4
IEEE Transactions on Industrial Electronics and Control Instrumentation
1
IEEE Transactions on Industrial Electronics
1
IEEE Transactions on Industry Applications
1
Industrial and Engineering Chemistry Process Design and Development 2
Industrial Engineering Chemistry Research
12
International Journal of Adaptive Control and Signal Processing
1
International Journal of Control
3
International Journal of Electrical Engineering Education
1
International Journal of Systems Science
1
Instrumentation
1
Instrumentation Technology
2
Instruments and Control Systems
2
ISA Transactions
2
Journal of Chemical Engineering of Japan
2
Journal of the Chinese Institute of Chemical Engineers
3
Pulp and Paper Canada
1
Process Control and Quality
1
Thermal Engineering (Russia)
2
Transactions of the ASME
7
(including Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control)
Transactions of the Institute of Chemical Engineers
1
Classification of journals in which tuning rules were published and number of tuning rules
published
Chemical Engineering Journals
32
(AIChE Journal, British Chemical Engineering, Chemical Engineering Communications, Chemical Engineering
Progress, Chemical Engineering Science, Transactions of the Institute of Chemical Engineers, Hungarian Journal
of Industrial Chemistry, Industrial and Engineering Chemistry Process Design and Development, Industrial
Engineering Chemistry Research, Journal of Chemical Engineering of Japan, Journal of the Chinese Institute of
Chemical Engineers)
Control Engineering Journals
53
(Automatica, Control, Control and Computers, Control Engineering, Control Engineering Practice, European
Journal of Control, IEE Proceedings, Part D, IEE Proceedings - Control Theory and Applications, Proceedings of
the IEE, Part 2, IEEE Control Systems Magazine, IEEE Transactions on Control Systems Technology,
International Journal of Adaptive Control and Signal Processing, International Journal of Control, International
Journal of Systems Science, Instrumentation, Instrumentation Technology, Instruments and Control Systems,
ISA Transactions, Process Control and Quality)
Mechanical Engineering Journals
8
(Advances in Modelling and Analysis C, ASME Press, Transactions of the ASME, Transactions of the ASME.
Journal of Dynamic Systems, Measurement and Control)
Electrical/Electronic Engineering Journals
4
(EE Transactions on Industrial Electronics and Control Instrumentation, IEEE Transactions on Industrial
Electronics, IEEE Transactions on Industry Applications, International Journal of Electrical Engineering
Education)
Trade Journals
5
(Hydrocarbon Processing, Pulp and Paper Canada, Thermal Engineering (Russia))
5. References
1.
Koivo, H.N. and Tanttu, J.T., Tuning of PID Controllers: Survey of SISO and MIMO techniques.
Proceedings of the IFAC Intelligent Tuning and Adaptive Control Symposium, Singapore, 1991, 75-80.
2.
Hwang, S.-H., Adaptive dominant pole design of PID controllers based on a single closed-loop test.
Chemical Engineering Communications, 1993, 124, 131-152.
3.
Astrom, K.J. and Hagglund, T., PID Controllers: Theory, Design and Tuning. Instrument Society of America,
Research Triangle Park, North Carolina, 2nd Edition, 1995.
4.
Bialkowski, W.L., Control of the pulp and paper making process. The Control Handbook. Editor: W.S.
Levine, CRC/IEEE Press, Boca Raton, Florida, 1996, 1219-1242.
5.
Isermann, R., Digital Control Systems Volume 1. Fundamentals, Deterministic Control. 2nd Revised Edition,
Springer-Verlag, 1989.
6.
Ender, D.B., Process control performance: not as good as you think. Control Engineering, 1993, September,
180-190.
7.
Astrom, K.J. and Wittenmark, B., Computer controlled systems: theory and design. Prentice-Hall
International Inc., 1984.
8.
Ziegler, J.G. and Nichols, N.B., Optimum settings for automatic controllers. Transactions of the ASME, 1942,
November, 759-768.
9.
Hazebroek, P. and Van der Waerden, B.L., The optimum adjustment of regulators. Transactions of the
ASME, 1950, April, 317-322.
10. Chien, K.-L., Hrones, J.A. and Reswick, J.B., On the automatic control of generalised passive systems.
Transactions of the ASME, 1952, February, 175-185.
11. Cohen, G.H. and Coon, G.A., Theoritical considerations of retarded control. Transactions of the ASME, 1953,
May, 827-834.
12. Wolfe, W.A., Controller settings for optimum control. Transactions of the ASME, 1951, May, 413-418.
13. Murrill, P.W., Automatic control of processes. International Textbook Co., 1967.
14. McMillan, G.K., Tuning and control loop performance - a practitioner’s guide. Instrument Society of
America, Research Triangle Park, North Carolina, 3rd Edition, 1994.
15. St. Clair, D.W., Controller tuning and control loop performance, Straight Line Control Co., Inc., 2nd Edition,
1997.
15a. Shinskey, F.G. (2000). PID-deadtime control of distributed processes, Preprints of the Proceedings of PID
’00: IFAC Workshop on digital control (Past, present and future of PID control), Terrassa, Spain, pp. 1418.
16. Shinskey, F.G., Process Control Systems - Application, Design and Tuning. McGraw-Hill Inc., New York, 3rd
Edition, 1988.
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Appendix 1: List of symbols used (more than once) in the paper.
a 1, a2 = PID filter parameters
A m = gain margin
b = setpoint weighting factor
b1 = PID filter parameter
c = derivative term weighting factor
E(s) = Desired variable, R(s), minus controlled variable, Y(s)
FOLPD model = First Order Lag Plus time Delay model
FOLIPD model = First Order Lag plus Integral Plus time Delay model
G c ( s) = PID controller transfer function
G p ( jω ) = process transfer function at frequency
G p ( jω ) = magnitude of G p ( jω ) ,
ω
∠Gp ( jω) = phase of G p ( jω )
IAE = integral of absolute error
IMC = internal model controller
IPD model = Integral Plus time Delay model
ISE = integral of squared error
ISTES = integral of squared time multiplied by error, all to be squared
ISTSE = integral of squared time multiplied by squared error
ITAE = integral of time multiplied by absolute error
ki = integral gain of the parallel PID controller
k p = proportional gain of the parallel PID controller
kd = derivative gain of the parallel PID controller
Kc = Proportional gain of the controller
Km = Gain of the process model
Ku = Ultimate gain
K25% = proportional gain required to achieve a quarter decay ratio
m = multiplication parameter in the lead controller
Ms = closed loop sensitivity
N = determination of the amount of filtering on the derivative term
PP = perturbance peak = peak of system output when unit step disturbance is added
R(s) = Desired variable
RT= recovery time = time for perturbed system output (when a unit step disturbance is added) to come to its final
value
SOSPD model = Second Order System Plus time Delay model
Td = Derivative time of the controller
TCL = desired closed loop system time constant
Tf = Time constant of the filter in series with the PID controller
Ti = Integral time of the controller
Tm = Time constant of the FOLPD process model
Tm1, Tm2 ,Tm3 = Time constants of the higher order process models
Tu = Ultimate time constant
T25% = period of the quarter decay ratio waveform
TOLPD model = Third Order Lag Plus time Delay model
TS = settling time
U(s) = manipulated variable
Y(s) = controlled variable
λ = Parameter that determines robustness of compensated system.
ξ m = damping factor of an underdamped process model, ξ = damping factor of the compensated system
κ = 1 K m Ku
φ = phase lag,
φ m = phase margin,
φ ω = phase lag at an angular frequency of
ω
φ c = phase corresponding to the crossover frequency
τ m = time delay of the process model,
ω = angular frequency,
τ = τ m ( τ m + Tm )
ω u = ultimate frequency,
ω φ = angular frequency at a phase lag of
φ