International Journal of Computational Cognition (http://www.YangSky.com/yangijcc.htm)
Volume 2, Number 3, Pages 63–100, September 2004
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Article electronically published on June 16, 2003 at http://www.YangSky.com/ijcc23.htm.
Please
cite this paper as:
hChing-Hung Lee, “A Survey of PID Controller Design Based on
Gain and Phase Margins(Invited Paper) ”, International Journal of Computational Cognition
(http://www.YangSky.com/yangijcc.htm), Volume 2, Number 3, Pages 63–100, September 2004i.
A SURVEY OF PID CONTROLLER DESIGN BASED ON
GAIN AND PHASE MARGINS(INVITED PAPER)
CHING-HUNG LEE
Abstract. This paper focuses on the PID controller design based on
the gain and phase margin (GPM) specifications for a given signalinput-signal-output plant. The basic ideas, techniques, and results
are presented in language and notation familiar. Since the definitions
of GPM equations are highly nonlinear, an analytical tuning method
is not yet available. Approximation and fuzzy neural network (FNN)
approaches are adopted to treat this problem. Then, they are used
to automatically tune the PID controller parameters for GPM specifications so that neither numerical methods nor graphical methods
need be used. Applications of usual used process- first order with
time delay plant are extended to general plant by the FNN approach.
In addition, the robust PID controller is definition and presented.
Finally, a novel PI-PD controller scheme based on GPM is firstly proposed to obtain the better performance. Simulation results are also
c
presented to show the effectiveness. °2003
Yang’s Scientific Research
Institute, LLC. All rights reserved.
1. Introduction
Even though control theory has been developed significantly, the proportional-integral-derivative (PID) controllers are used for a wide range of
process control, motor drives, magnetic and optic memories, automotive,
flight control, instrumentation, etc. In industrial applications, more than
90% of all control loops are PID type [45]. Integral, proportional and derivative feedback is based on the past (I), present (P) and future (D) controller.
Received by the editors June 12, 2003 / final version received June 15, 2003.
Key words and phrases. PID control, frequency response, gain margin, phase margin,
Kharitonov theorem.
The author would like to thank the Prof. C. C. Teng for the constructive comments
and suggestions. This work was supported by the National Science Council, Taiwan,
R.O.C., under Contracts NSC-91-2213-E-155-012.
c
°2003-2004
Yang’s Scientific Research Institute, LLC. All rights reserved.
63
64
CHING-HUNG LEE
Feedback control systems form one such area as witness in part by recent
special issues on the subject [1,25,26].
Over the past 50 years, several methods for determining PID controller
parameters have been developed for stable processes that are suitable for
auto-tuning and adaptive control [1-5,9,12,13,17-26,32-34,41-43,45-49]. Some
employ information about open-loop step response, for example, the CoonCohen reaction curve method; other methods use knowledge of the Nyquist
curve, for example, the Ziegler-Nichols frequency-response method [3,14,48].
However, these tuning methods use only a small amount of information
about the dynamic behavior of the system, and often do not provide good
tuning.
It is known that gain and phase margins (GPM) have served as important measures of robustness [3,15,29,36,40]. The phase margin is related
to the damping of the system from classical control theories, and therefore
also serves as a performance measurement. Their solutions are normally
obtained numerically or graphically by trial-and-error use of Bode plots.
Despite the fact that many PID tuning methods are available for achieving
the specified GPM, they can be divided into two categories. Firstly, approximation of tan−1 function is adopt to simplify the problem, but they
are only applicable to the simple models [4,9,18-26,45-49]. Secondly, inverse
function mapping was done by fuzzy neural network (FNN). The FNN identify the relationship between GPM and PID controllers, which is available
for general linear system [13,30,32-34]. For solving the problem, the first
obstacle is the difficulty in finding the stabilizing region of PID controllers.
The solution is a necessary first step to any rational design of PID controllers
based on GPM.
This survey focuses on the PID controller design based on the GPM
specification for a given SISO plant. We attempt to present the basic ideas,
techniques, and results are presented in language and notation familiar.
Because the equations of GPM are highly nonlinear, an analytical tuning
method is not yet available. Approximation simplified approach has been
adopted to treat this problem. To extend the applications to general plant it
is important to introduce the FNN for identifying the relationship between
PID controllers and GPM. It is used to automatically tune the PID controller parameters for GPM specifications so that neither numerical methods
nor graphical methods need be used. In addition, the robust PID controller
is definition and presented by the FNN approach. Finally, a novel PI-PD
controller scheme based on GPM is firstly proposed to obtain the better
performance. Simulation results are also presented to show the effectiveness.
A SURVEY OF PID CONTROLLER DESIGN
65
This paper is organized as follows. In Section 2, we show the problem
formulation. In Section 3, we present the methods for estimating stabilizing
region of PID controllers. Section 4 introduces the approximation simplified and the FNN approach for designing the PID controllers to achieve
the GPM specifications. In addition, for obtaining a better time-response
performance, a novel PI-PD controller to meet the GPM specification is proposed. Also, the so-called robust PID controller and tuning method based
on FNN is introduced. Finally, Section 5 contains conclusion and future
research.
2. Problem Formulation
A block diagram of a simple control loop system is shown in Fig. 1. The
system composes a process and a controller, where ddenotes the external
disturbance. The process has one input and one output, denotes u and
y, respectively. The desired value yr is called the set point (SP) or the
reference one. The purpose of the system is to keep the process output y
close to the desired one yr in spite of disturbances.
Figure 1. Block diagram of a simple feedback loop.
In general, the process is usually modeled by an nth-order one with time
delay L
(1)
Gp (s) =
bm sm + bm−1 sm−1 + · · · + b1 s + b0 −Ls
e
.
sn + an−1 sn−1 + · · · + a1 s + a0
66
CHING-HUNG LEE
Here, we assume n > m. For system identification, process coefficients ai
and bj can be determined from the step response of process. For simplifying
the controller design, system (1) is usually reduced as a lower order system
as typical industrial processes [2,3,40]. In general, the usually used process
model is described as a first-order with time-delay
(2)
KP e−sL
.
(1 + τ s)
GP (s) =
For time-delay element e−Ls , the Pade approximation can be usually used
to replace it, for example, the second-order one is,
Ls (Ls)2
+
a(s)
2
12 .
(3)
e−Ls =
=
b(s)
Ls (Ls)2
1+
+
2
12
The PID controller is given by
1−
(4)
GC (s) = KC (1 +
1
ki
+ TD s) = k +
+ kd s
sTI
s
and the implementation is
Z
Z
1
de(t)
de(t)
u(t) = KC (e(t) +
e(t)dt + TD
) = ke(t) + ki e(t)dt + kd
TI
dt
dt
where e = yr −y and u is the control signal. Thus, the loop transfer function
for system (2) and controller (4) is
(5)
GC (s)GP (s) =
KC KP (1 + sTI ) −sL
e
.
sTI (1 + τ s)
There is not surprising as over 90% of the industrial controllers are the
PID type [45]. It is well-known that Ziegler and Nichols presented the PID
tuning method for system (2) firstly [48]. The method is based on the system’s open-loop step response and is derived for model (2). Obviously, it
derived based on time domain performance. On the other hand, the frequency response tuning method was introduced for the closed-loop system
later [2,3]. A modification to the Ziegler-Nichols method can give a substantially improved system performance. Consider a first-order lag plus a
time-delay model
GP (s) =
α −sL
e
.
sL
A SURVEY OF PID CONTROLLER DESIGN
67
Utilize the Nyquist diagram, and give a point A, GP (jw) = rP ej(π+φP )
and then try to find a regulator to move this point to B = rS ej(π+φS ) . An
amplitude margin corresponds to φS = 0, r = 1/Am and a gain margin
corresponds to φS = φm , rS = 1 can be obtained, where Am and φm are
the gain and phase margins, respectively. This modification has a better
system response [2]. Thus, the PID controller can be designed roughly for
a given GPM specifications. Due from it, GPM are typical specifications
associate with the frequency response [29]. They not only provide the important indicators of systems robustness but also reflect on the performance
and stability of the system and thus are widely used for controller design
[2,3,29,36,40].
In this paper, the performance specifications for the PID controller design are GPM. It is also known from classical control that phase margin
is related the damping of the system and therefore could also be served as
a performance measure. In 1984, PID controller designed to satisfy GPM
criteria are presented firstly [4]. They typically have a gain margin of three
and a phase margin of 60 degrees.
Denote the process and controller transfer functions GP (s) and GC (s)
as Fig. 1. The gain margin is given by the solution of the following set of
equations:
(6)
∠GC (jwp )GP (jwP ) = −π
(7)
Am =
1
.
|GC (jwp )GP (jwP )|
Correspondingly, the phase margin can be obtained by
(8)
|GC (jwg )GP (jwg )| = 1
(9)
φm = ∠GC (jwg )GP (jwg ) + π
where the frequency wp at which the Nyquist curve has a phase of -π is
known in classical terminology as phase crossover frequency, and the frequency wg at which the Nyquist curve has amplitude of one as the gain
crossover frequency. Note that if at least one of pole is negative then the
process is an unstable plant. Figures 2(a) and 1(b) show the Bode and
Nyquist diagrams of an unstable first-order plus delay process with PI control. Obviously, unstable plant has more than one GPM. As the definitions
of the GPM [22,27], Am1 and Am2 are called gain margin (or upward gain
68
CHING-HUNG LEE
margin) and gain reduction margin (or downward margin). In addition,
applying the Nyquist criterion for stability, the Nyquist diagram should encircle the point (-1,0) [in the G(jw ) plane] exactly once in the anti-clockwise
direction. Base on the stability criterion, the gain margin Am1 is chosen in
this literature.
Bode Diagrams
Nyquist Diagram
Gm=2.361 dB (at 6.531 rad/sec), Pm=13.011 deg. (at 4.96 rad/sec)
2
200
150
1.5
100
KC=5,TI=1.3
50
1/A m2
1
-50
1/Am1
-100
0.5
-150
-200
Im
Phase (deg); Magnitude (dB)
0
-100
0
φm
-150
-0.5
-200
-250
Gain reduction margin:-12.29dB
Gain reduction margin: -12.2900
-300
-1
-350
Am1: Gain Margin
Am2: Gain Reduction Margin
-400
-1.5
-450
-500
-5
10
0
5
10
10
10
10
-2
-6
-5
-4
-3
-2
-1
0
1
Frequency (rad/sec)
Re
(a)
(b)
Figure 2. Bode and Nyquist diagrams of unstable firstorder plus delay process with PI control.
3. Stabilizing and Available GPM Regions
As previous discussion, instability is the disadvantage of feedback control
system. When using feedback, there is always a risk that the closed-loop
system will become unstable. Therefore, stability analysis is primary requirement for feedback system. And the major obstacle is the difficulty in
finding the stabilizing region of PID controllers. The solution is a necessary
first step to any rational design of PID controllers based on GPM. Thus, the
estimation of stabilizing region and available region are introduced here. As
description of literature [5], the stabilizing region of systems with monotone
transfer function is a convex set, but other transfer functions the stabilizing
region may contain several disjoint sets. In this paper, transfer functions of
considered plants are of monotone.
3.1. Estimation of the Stabilizing Region. The stability of closed-loop
system depends on the PID parameters we chosen. Thus, the method for
finding the parameters ranges ensures the overall closed-loop system stable.
Here, we introduce the Routh Hurwitz criterion [8,29,36] to present a simple
method to estimate the stabilizing region of PID controller. This method is
A SURVEY OF PID CONTROLLER DESIGN
69
simple and straightforward that avoids the complicate theoretical analysis
[5,17].
(s)
Let GP (s) = N
D(s) be the plant to be controlled. For a PID controller,
GC (s) is of the form (4)
ki + ks + kd s2
ki
+ kd s =
.
s
s
Thus, the closed-loop characteristic polynomial is
(10)
(11)
GC (s) = k +
δ(s, k, ki , kd ) = sD(s) + (ki + ks + kd s2 )N (s).
Now, the Routh Hurwitz criterion is used to propose a method to find
the PID controller’s stabilizing region, i.e., available region of parameters
ki , k, and kd . The procedure can be described directly as follows and Fig.
3 summaries the concept.
Procedure for estimating the stabilizing region.
Step 1: Roughly estimate the stabilizing region Ω for two parameters
of PID controller, e.g., k and ki .
Step 2: Uniformly partition this estimation region of k, ki into n × n
subdivision, the intersected points are denoted Ωlm = (k l , kim ), l =
0, . . . , n and m = 0, . . . , n.
Step 3: Substitute Ωlm = (k l , kim ) into Equation (12). Then for
each point Ωlm = (k l , kim ), the corresponding stabilizing region
lm
(k lm
d , k̄d ) of parameter kd can be computed by using Routh-Hurwitz
stability criterion.
Step 4: Sweeping over all points in the estimated region Ω, and finding the corresponding stabilizing region.
Step 5: Unite the estimated region and then we can obtain the stabilizing region.
In general, a large estimated region (in first step) and small subdivision
are usually chosen for accuracy. Other approaches for obtaining the stabilizing region can be found on literature [5,22,30,33] which are based on
Nyquist stability theorem, Hermite-Bieler theorem, and trial/error.
3.2. Estimation of Available Region. In addition, note that the performance specification Am and φm for given plant may not be achieved.
Therefore, the available region for Am and φm by PID controller should be
found. Firstly, there are some restrictions on the choice of Am and φm pairs.
One usual requirement is k, ki , kd > 0. Here, a method using approximation
70
CHING-HUNG LEE
Figure 3. Concept of estimating the stabilizing region method.
approach is introduced to find the available region of GPM. For example,
consider unstable process model with PI controller
(12)
GP (s) =
KP −sL
e
.
sτ − 1
Using approximation of tan−1 function [22], this requirement for unstable
plant (12) is given as
(13)
φm ≤
A2m − 1
π
h − (Am − 1)
Am
2
where
r
π 1 π2
L
h= +
−4 .
4
2
4
τ
Thus, the maximum phase margin can be obtained
r
p
2 L/τ
π
L
p
(14) φm,max = − 2
, and Aφ,max =
.
2
τ
π/2 − π 2 /4 − 4L/τ
Herein, the approach is hard to extend for PID controller. Thus, a simple
and straightforward estimation method is presented. Since, the stabilizing
region is assumed to be convex. Therefore, we would estimate the available
GPM region using the information of the stabilizing region. Every point
chosen randomly on the closure of the stabilizing region is used to calculate
the corresponding GPM using equations (6)-(9). Then, the available region
of GPM specification achieved by PID controller can be obtained. Figure 4
summarizes the approach.
A SURVEY OF PID CONTROLLER DESIGN
71
Figure 4. Concept of estimation of available GPM region.
Example 1: Estimation of Stabilizing Region. For Unstable plant
−0.2s
Gp (s) = es−1 with PI controller, firstly, we estimate the stabilizing region
of k is [0,10], which can be obtain by open-loop Bode plot. Then the range is
then partition to 100 elements uniformly (k l , l = 0, · · · , 100). For each value
k l , the corresponding stabilizing region of ki can be calculated by Routh
Hurwitz criterion. Finally, we take the union of these regions to obtain the
estimated stabilizing region. Figure 5 is the estimated stabilizing region of
−0.2s
unstable plant Gp (s) = es−1 with PI controller (restriction- ki > 0 and
k > 0). And, Fig. 6 gives the estimated available region for the unstable
−0.2s
plant Gp (s) = es−1 with PI controller.
10
9
8
7
6
ki
5
4
3
2
1
0
0
1
2
3
4
k
5
6
p
Figure 5. The stabilizing region of PI parameters.
7
8
72
CHING-HUNG LEE
45
40
35
25
m
P , deg.
30
20
Available GPM region
15
Estimated valid range
10
5
0
1
2
3
4
5
6
7
8
Am
Figure 6. Estimated available GPM region of unstable
e−0.2s
plant Gp (s) =
with PI controller.
s−1
4. Controller Design.
According to a survey of the state of process control system in 1989 conducted by Japan Electric Measuring Instrument Manufacture’s Association,
more than 90% of the control loops were of the PID type [45]. Design and
tuning of PID controller have been a large area since Ziegler and Nichols presented their methods in 1942 [48]. PID controllers are widely used and many
formulas have been derived. Here, we introduce the PID controller tuning
methods to achieve the GPM specifications. Despite the fact that many
PID tuning methods are available for achieving the specified GPM, they
can be divided into two categories. Firstly, approximation of tan−1 function is adopt to simplify the problem [4,9,18-26,45-49]. Secondly, inverse
function mapping was done by fuzzy neural network (FNN) [13,30,32-34].
4.1. Approximation Method. For system (2), some approximation were
taken in order to simplify the calculation of equations (6)-(9), i.e., Am and
φm [18-26]. For instance, a PI controller is employed to control the process
(2) which transfer function is given by
µ
(15)
GC (s) = KC
1
1+
sTI
¶
A SURVEY OF PID CONTROLLER DESIGN
73
Now, the tuning objective is to determine the controller parameters KC
and TI such that the given Am and φm are achieved. Note that the approach
is also available for the first order with time delay unstable process [22].
Firstly, for determining Am and φm , the loop transfer function should be
obtained
(16)
GC (s)GP (s) =
KC KP (1 + sTI ) −sL
e
.
sTI (1 + τ s)
Substituting equation (16) into equations (6)-(9)
(17)
1
π + tan−1 wP TI − tan−1 wP τ − wP L = 0
2
s
(18)
Am KC KP = wP TI
s
(19)
KC KP = wg TI
wP2 τ 2 + 1
wP2 TI2 + 1
wg2 τ 2 + 1
wg2 TI2 + 1
1
π + tan−1 wg TI − tan−1 wg τ − wg L.
2
For a given process (KP , τ, L) and specifications (Am , φm ), there are four
equations with four unknown variables. Thus, equations (17)-(20) can be
solved for the PI controller parameters (KC , TI ) and crossover frequencies
(wg , wp ) numerically but not analytically because of the presence of the
tan−1 function. However, an approximate analytical solution can be obtained, if we make the following approximation for the tan−1 function.
(20)
φm =


 1 πx
|x| ≤ 1
−1
4
(21)
tan x ≈
1
π

 π−
|x| > 1
2
4x
as shown in Fig. 7 (a). The identity
1
1
(22)
tan−1 x = π − tan−1 , for |x| > 1
2
x
is used in the expression equation (21) when |x| > 1, as Fig. 7(b).
Therefore, equations (17)-(20) can be rewritten as follows.
(23)
Am KC KP = wp τ
74
CHING-HUNG LEE
(b)
(a)
0.8
1.5
0.7
1.4
___ arctan (x)
...... Approximation
0.6
1.3
0.5
)
x(
n
at
c
a
1.2
)
x(
n
at
c
a
0.4
0.3
1
0.2
0.9
0.1
0.8
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
___ arctan (x)
...... Approximation
1.1
0.7
1
2
3
4
5
6
7
x
(a)
(b)
Figure 7. Approximation of tan−1 function: (a) |x| ≤ 1
(b) |x| > 1.
(24)
KC KP = wg τ
(25)
1
π
π
π−
+
− wp L = 0
2
4wp TI
4wp τ
(26)
φm =
1
π
π
π−
+
− wg L
2
4wg TI
4wg τ
Finally, solving for KC and TI in equations (23)-(26) gives
(27)
(28)
KC =
TI =
wp
Am KP
1
4wp2 L 1
2wp −
+
π
τ
where
(29)
1
Am φm + πAm (Am − 1)
2
wp =
.
(A2m − 1)L
Besides, PI controller (15) can be represented as velocity form
8
9
10
A SURVEY OF PID CONTROLLER DESIGN
(30)
75
µ
¶
∆t
∆e +
e
TI
∆u = KC
and then equation (27) can be set by fuzzy logic system [37,46,47].
In addition, other methods such as Ziegler-Nichols or ISTE were used to
find the PI controller parameters. Thus, equations (23)-(26) can be solved
to give
(31)
(32)
Am
φm
πτ
=
4KC KP L
Ã
1+
r
4L
4L
1−
+
πTI
πτ
1
KC KP L
π
= π−
+
2
τ
4KC KP
!
µ
¶
τ
1−
.
TI
Whatever the design method, knowledge of the robustness measures of
GPM is certainly useful. The advantage of using equations (31) and (32)
to obtain Am and φm is that numerical methods need not to used to solved
equations (17)-(20) for Am and φm . Since several approximations are used
to derive tuning equations (27) and (28). And then, the true GPM for the
chosen controllers are computed numerically from equations (17)-(20). The
results are given in Table 1. We observe that the largest error in Am is
19.81% and the largest error in φm is 10.43%.
In addition, for obtaining under-damped response for second-order plus
dead time plant
(33)
GP (s) =
KP
e−Ls , 0 < ςP < 1,
s2 + 2ςP wn s + wn2
the PID controller can be derived by the same approach [21]
(34)
KC =
2wP (πςP wn + πwP − 2wP2 L)
,
πAm KP
(35)
TI =
2wP (πςP wn + πwP − 2wP2 L)
πwn2
(36)
TD =
π
2wP (πςP wn + πwP − 2wP2 L)
where wP is the same as (29). Similarity of approximation approach [22],
it can be applied to the unstable process (12). Note that the frequency
76
CHING-HUNG LEE
response of unstable process is different from the stable one. Consequently,
the tan−1 function is modified as
(
(37)
tan
−1
x≈
x
1
1
π−
2
x
0≤x≤1
x>1
.
Similar derivation as previous description, the PI controller KC is the
same as equation (27) and TI can be obtained by solving the following
equation
(38)
TI =
1
π
1
wp − wp2 −
2
τ
.
4.1.1. Exact GPM Method. Unlike the previous methods, the process dynamics can be made to simplify the nonlinear problem encountered in computation [20,22,43]. For given user specification- GPM, a simple and straightforward controller design method that can simultaneously achieve exact
GPM for a general linear plant without any assumptions. Firstly, using
the critical point information and the gain margins specification, KP can
be chosen immediately. Based on this value and the phase margin specification, another point on the Nyquist plot can be searched and located. And
then, TI and TD (or KD and KI ) can be set from the imaginary parts of
the two points.
Note that there are five unknowns from equations (6)-(9). It is easily
found that the unknown number exceeds the equation number leading to an
infinite number of solutions. Thus, an additional equation was introduced
by considering the bandwidth of system. Denotes wc and wP to be the
open-loop and closed-loop bandwidth with
(39)
wP = αwc , α ∈ [0.5, 2].
Default value for α is set to be one and wp is available from ∠G(jwc ) =
−π. Thus, with (39), wP is readily determined and the unknowns is reduced
to four. Therefore, equations (17)-(20) can be solved
·
(40)
−1
KP = Re
Am GP (jwP )
¸
e−0.1s
(s + 1)
e−0.2s
(s − 1)
e−s
(s2 + s + 1)
Plant
0.68
2.5
45
KC
4.9087
3.0543
2.3453
1.7453
Specifications
Am
φm
3
45
5
60
3
35
4
30
1.14
TI
0.3520
0.5410
6.6120
4.5773
0.88
2.3
57
Result
TD
A∗m
φ∗m
3.3075 40.308
5.9904 58.14
3.2074 35.9332
4.0761 30.1350
8.0%
26.67%
Error
Error of Am Error of φm
10.25%
10.43%
19.81%
3.10%
6.913%
2.666%
1.902%
0.45%
Table 1. Results of approximated method [20-22].
A SURVEY OF PID CONTROLLER DESIGN
77
78
CHING-HUNG LEE
(41)
µ
KI = (Im [−1/Am GP (jwP )] wg −Im [−1 exp(jφm )/GP (jwg )] wP )
(42)
KD =
µ
Im [−1/Am GP (jwP )] Im [−1 exp(jφm )/GP (jwg )]
−
wg
wP
¶µ
wP
wg
−
wg
wP
wg
wP
−
wg
wP
¶−1
,
¶−1
and wg satisfies
·
(43)
Re
¸
− exp(jφm )
= KP .
GP (jwg )
Obviously, the design problem is solvable of and only if (43) yields a
solution for wg . The value may be identified by sweeping down from w = wP
towards w=0 until (43) holds. In addition, the existence condition for (43)
about Am and φm is given below
(44)
−π + tan−1
1 − Am cos φm
π
< ∠GP (jwP ) < − .
Am sin φm
2
Finally, if the solution exists, then equation (40) gives the controller
parameters.
4.1.2. Internal Model Control Method (IMC-PID). The well known internal model control PID formula (IMC-PID) [3,12,40] is also a tuning method
related system robustness and performance. If we relate the IMC-PID tuning formula to the GPM tuning formula, new insights into the IMC-PI
tuning formula and the choice of the GPM specifications can be obtained.
The IMC-PI tuning formula [3,12] for the first order with time delay
model is summarized as Table 2, where τcl is a user-specified parameter
that corresponds to the closed-loop speed response. When we substitute
the expressions
(45)
(46)
Am =
φm =
π
2
π(τcl + L)
2L
µ
¶
1
1−
Am
for the GPM specifications (Am , φm ) into the GPM-PI formula equations
(27) and (28), the IMC-PI in Table 2 is obtained. Therefore we can derive
A SURVEY OF PID CONTROLLER DESIGN
79
the IMC-PI formula from the GPM-PI formula through appropriate GPM
specifications.
The IMC-PI design method has only one tuning parameter, τcl , which
is related to the gain margin by equation (45). Once the gain margin is
obtained, the phase margin is given by equation (46). Since the IMC-PI
design only one tuning parameter, it can be expected that only one degree
of freedom is possible, since the GPM are not independent but are related
through equation (45) to (46). This approach was extended to a self-tuning
IMC-PID control [23]. The closed-loop transfer function with the PID controller is firstly approximately given as
(47)
1 − 12 sL
GC (s)GP (s)
≈
.
1 + GC (s)GP (s)
sτcl + 1
Similar to the previous discussion, (2), (47) and values of Table 2 can be
substituted into (17)-(20) such that Am , wg , wP , and φm are determined.
For guaranteeing the existence of solutions, a fine-tuning of the PID controller parameters are used. Then, the approximated function of tan−1
function is used again to solve the self-tuning IMC-PID controller with interval GPM.
Table 2. IMC-PID controller tuning method.
Process model
KP e−sL
1 + τs
KC
µ τ
KP
L
τcl +
2
¶
TI
TD
τ
L/2
Remarks:
(1) Obviously, equations (17)-(20) can be solved numerically but not
analytically because of the nonlinear tan−1 function. Thus, a piecewise linear approximation of it were used to solve approximately.
Methods of [18-26,43] also adopt an approximation, wP = αwC , to
help user to find the exact solution. These due to approximated
error and may have an unstable controller, see Table 3 and Fig. 8
[33]. For stable processes, controllers such as the IMC [12,23] and
approximated method [18-26] that are based on GPM cannot efficiently meet specifications within a 10% error margin owing to the
approximation of the tan−1 function. In addition, a similar controller based on approximation for an unstable process improves
80
CHING-HUNG LEE
performance but still can only meet the specifications within 5%
error [33].
(2) In addition, the approximate analytical solutions cannot be extended to the so-called test-batch process because of the limitation
on process form (should be first order or second order with timedelay). The test-batch processes as bellows are representative for
the dynamics of typical processes [3,40].
(48)
G1 (s) =
(49)
e−s
, T = 0.1, . . . , 10
(1 + sT )2
G2 (s) =
(50) G3 (s) =
(51) G4 (s) =
1
, n = 3, 4, 8
(1 + s)n
1
, α = 0.2, 0.5, 0.7,
(1 + s)(1 + αs)(1 + α2 s)(1 + α3 s)
1 − αs
, α = 0.1, 0.2, 0.5, 1.2.
(1 + s)3
(3) Also, the controller tuning method based on the model (2) typically
gives a controller gain that is too high [3].
(4) To cover the test-batch processes, it is necessary to develop a tuning
method for general system (1). Besides, the approximation error
should be reduced. Therefore, another approach using the FNN for
general processes is considered below. This approach yields high
accurate tuning formulas for controllers including P, PI, PD and
PID controllers of stable and unstable processes with time delay.
4.2. Fuzzy Neural Network (FNN) Approach. Considering the nthorder process with time-delay (1) and rewriting as
(52)
Gp (s) =
where n =
p
P
i=1
Kp (1 + wn1 s)n1 (1 + wn2 s)n2 · · · (1 + wnq s)nq −Ls
e
(1 + wd1 s)d1 (1 + wd2 s)d2 · · · (1 + wdp s)dp
di. Here, if wdi > 0 for all i, system (52) is stable. On the
other hand, if at least one of wdi is negative then the process is an unstable
plant. For the PID controller, the loop transfer function is then obtained:
A SURVEY OF PID CONTROLLER DESIGN
81
Table 3. Approximation results of unreasonable specifications.
Results of [22]
Am
φm
KC
3.0420 41.8603 2.3998
4.0559 42.0461 1.8035
5.0700 35.9405 1.4363
7.0952 -2.6003 1.0090
7.6038 -13.3516 0.9477
8.1117
∞
0.8934
7.6012 -12.6772 0.9398
8.1072 -20.2941 0.8796
8.6184
∞
0.8390
9.1249
∞
0.7909
Specifications
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
(3,40)
(4,40)
(5,35)
(7,15)
(7.5,15)
(8,15)
(7.5,10)
(8,5)
(8.5,10)
(9,5)
TI
-17.3642
-13.0428
-29.5354
8.5340
16.4203
101.3638
7.3770
6.6594
32.6130
20.7626
45
P2
P1
40
P3
35
phase Margin
30
25
20
P4
P5
P6
15
P8
5
0
P9
P7
10
1
2
3
4
5
6
Gain Margin
7
P10
8
9
10
Figure 8. Available region of (Am , φm ) for PI controller:
solid line, our result; dash-dotted line: result of [22] (P1 −
P10 are outside the estimated available region).
11
82
CHING-HUNG LEE
Gc (s)Gp (s) = e−Ls ×
Kc Kp (1 + sTI + TI TD s2 )(1 + wn1 s)n1 (1 + wn2 s)n2 · · · (1 + wnq s)nq
.
sTI (1 + wd1 s)d1 (1 + wd2 s)d2 · · · (1 + wdp s)dp
(53)
Substituting into equations (6)-(9)
1
π + tan−1 (wp wc1 ) + tan−1 (wp wc2 ) + n1 tan−1 (wp wn1 )
2
+ · · · + nq tan−1 (wp wnq ) − wp L − d1 tan−1 (wp wd1 )
−d2 tan−1 (wp wd2 ) − · · · − dp tan−1 (wp wdp ) = 0,
(54)
(55)
q
q
2 )n1 (1 + w 2 w 2 )n2 · · ·
2 )nq
(1 + wp2 wn1
(1 + wp2 wnq
p n2
q
q
q
Am Kc Kp = wp TI q
,
2
2
2 )d1 · · ·
2 )dp
1 + wp2 wc1
1 + wp2 wc2
(1 + wp2 wd1
(1 + wp2 wdp
q
(56)
q
q
q
2 )d1
2 )d2 · · ·
2 )dp
(1 + wg2 wd1
(1 + wg2 wd2
(1 + wg2 wdp
q
q
q
Kc Kp = wg TI q
,
2 ) (1 + w 2 w 2 ) (1 + w 2 w 2 )n1 · · ·
2 )nq
(1 + wg2 wc1
(1 + wg2 wnq
g c2
g n1
φm
(57)
=
1
π + tan−1 (wg wc1 ) + tan−1 (wg wc2 ) + n1 tan−1 (wg wn1 )
2
+ · · · + nq tan−1 (wg wnq ) − wg L − d1 tan−1 (wg wd1 )
−d2 tan−1 (wg wd2 ) − · · · − dp tan−1 (wg wdp ),
where wc1 and wc2 are the roots of (1 + TI s + TI TD s2 ). It is note that there
are five altogether five unknown, namely KC , TI , TD , wg , and wp in equations. (54)-(57). For a given process (Kp , wn1 , · · · , wnq , wd1 , · · · , wdp , L)
and specifications (Am ,φm ), equations (54)-(57) can be solved for the PID
controller parameters (Kc , TI , TD ) and crossover frequencies (wg , wp ) numerically but not analytically because of the presence of the tan−1 function.
4.2.1. Structure of the fuzzy neural network (FNN). The FNN system is
a network with fuzzy inference characteristics. It is a simple fuzzy logic
system implemented by using a multi-layer feedforward neural network. A
schematic diagram of the proposed FNN is shown in Fig. 9 [10,11,30-35].
The input-output representation is denoted as:
A SURVEY OF PID CONTROLLER DESIGN
83
Figure 9. Schematic diagram of fuzzy neural network
(FNN) systems.
(58)
yp (x) =
m
X
j=1
wj
m
Y
i=1
exp[−
(x − mij )2
]
2
σij
where m, σ and w are the mean, STD, and weight of the FNN, respectively.
Nodes in layer one are input nodes representing input linguistic variables.
Nodes in layer two are membership nodes. Here, the Gaussian function is
used as the membership function. Each membership node is responsible
for mapping an input linguistic variable into a possibility distribution for
that variable. The rule nodes reside in layer three. The last layer contains
the output variable nodes. The adjustment of parameters in the FNN can
84
CHING-HUNG LEE
Figure 10. Construction of the jth component of the FNN.
( Am ,φ m )
Equations (54)~(57)
( K C , TI , TD )
+
e
_
FNN
( Kˆ C , TˆI , TˆD )
Figure 11. Block diagram of function mapping using FNN.
A SURVEY OF PID CONTROLLER DESIGN
85
be divided into two categories, corresponding to the premise part and the
consequence part, of the fuzzy rules. In the premise part, we must initialize
the mean and the variance of Gaussian functions. In the consequence part,
the parameters are output singletons. These singletons are initialized with
small random values, as in a pure neural network. More details about the
FNN can be found in [10,11,30-35]. Its construction is directly based on the
fuzzy rules without adjustment. For example, if we encounter the jth fuzzy
rule described as follows:
(59) If x1 is Aj1 and x2 is Aj2 and . . . and xn is Ajn then y is βj .
Consider m fuzzy rules for generality which can be considered independently like dealing with the jth fuzzy rule in Fig. 10. The links between the
membership nodes and the rule nodes are the antecedent links, and those
between the rule nodes and the output nodes are the consequence links. For
each rule node, there is at most one antecedent link from a membership node
of a linguistic variable. However, all consequence links are fully connected
to the output nodes and interpreted directly as the strength of the output
action. Using the properties of universal approximation and high speed convergence, the FNN can be tuned to identify the relationship between GPM
and PID controllers, i.e., inverse function mapping of equations (54)-(57).
4.2.2. Training of the FNN for PID Controllers and GPM. Considering the
nonlinear couple equations (54)-(57), we find that there are five parameters
(wp , wg , Am , φm , KC , TI , TD ) in those four equations. If we are given
GPM specifications (Am ,φm ), it may not be possible to solve the five parameters because the equations are nonlinear. Now, let us consider another
aspect of these equations. First, we could give randomly controller parameters (KC ,TI ,TD ) as the input of equations. Using equations (54)-(57), we
could obtain the parameters (wp ,wg ,Am ,φm ) that correspond to the random
controller parameters (KC ,TI ,TD ), respectively. For example, in preparation for training the FNN, we gather much enough points (KC ,TI ,TD ) and
the corresponding (Am ,φm ) points respectively, and set them as the training data in Fig. 11, i.e., the input data is (Am ,φm ) and the output is
(KC ,TI ,TD ). This approach avoids the problem of no solution being yielded
by nonlinear equations (54)-(57), and reduces the overall task. Furthermore, this approach is useful for all processes (stable, unstable, high-order,
under-damped response, et al ).
For training the FNN, considering the single-output case for clarity, our
goal is to minimize the error function
86
CHING-HUNG LEE
1
[yP (k) − ŷ(k)]2 = 12 e2 (k)
2
where yP , ŷ denote the desired output (KC , TI , TD ) and estimated output
(K̂C , T̂I , T̂D ), respectively. Then, the gradient of the above error function
is
(60)
E(k) =
∂E
∂e(k)
∂ ŷ(k)
= e(k)
= −e(k)
∂W
∂W
∂W
where W = [m, σ, w] represents the weighting vector of the FNN. By using
the back-propagation algorithm [10,11,30-35], we can describe the update
law for the linear system as
(61)
W (k + 1)
(62)
=
=
µ
¶
∂E(k)
W (k) + ∆W (k) = W (k) + η −
∂W
¶
µ
∂ ŷ(k)
.
W (k) + ηe(k) −
∂W
Example 2: PID Controller for Different Plants. The simulation results of FNN approach are listed in Table 4. To illustrate the difference
between the tuning method of FNN and approximated method, the parameters of the controller are determined for both cases and listed in Tables 3-5.
These tables exhibit the results using FNN approach with different specifications. It is shown that the proposed FNN approach has better accuracy
than approximated method. Note that the given specifications in Table 5
are all inside the available region shown in Fig. 8. Here we choose two
specifications outside the available region to show the effectiveness of FNN
approach. The simulation results of unreasonable specifications are listed
in Table 5. Note that the method of literature [22] results negative phase
margins for these specifications, see Table 3. Table 5 also shows that the
FNN would still find suitable controller’s parameters close to the specifications and retain the stable of system (i.e., Am > 0, φm > 0) even though the
specifications are unreasonable. The FNN gives us an acceptable error for
specifications. It would still be regarded as “good tuning” based on FNN
tuning method.
4.2.3. PI-PD Controller. In literature [6,49], a special control scheme: PIPD controller was proposed to obtain significantly better performance. Here,
we extend the FNN based tuning method to this type controller. The socalled PI-PD controller scheme is shown in Fig. 12, where
1000e−0.0005s
s3 + 15s2 + 45s − 25
e−s
(s2 + s + 1)
e−0.2s
(s − 1)
Plant
Specifications
Am
φm
3
35
4
30
5
30
4
40
3
60
2
20
3
30
4
30
KC
2.3595
1.7557
1.5356
0.1012
0.76
0.3125
0.2288
0.1651
Result
TI
TD
4.2624
3.3538
11.4457
3.0254 0.2301
1.19
0.84
16.364
6.1378
3.1344
Am
3.0092
4.0118
4.9038
3.8737
2.93
2.1231
2.9158
3.9286
φm
34.0366
29.7902
30.7641
40.510
62.12
20.5021
27.0728
30.7463
Table 4. Different cases using FNN approach.
Error
Am
φm
0.306% 2.752%
0.295% 0.699%
1.924% 2.547%
3.157% 1.275%
2.33%
3.53%
6.155% 2.51%
2.807% 9.75%
1.785% 2.487%
A SURVEY OF PID CONTROLLER DESIGN
87
CHING-HUNG LEE
88
Approximation result
KC
TI
Am
φm
0.9999 5.0971
7.1238 -5.0841
0.9477 16.4203 7.6038 -13.3516
0.8934 101.3638 8.1117
∞
0.9398 7.3770
7.6012 -12.6772
0.8796 6.6594
8.1072 -20.2941
KC
1.3578
2.2576
1.7478
1.6937
1.1481
FNN
TI
23.8254
47.6725
34.7266
31.0468
16.3248
φm
29.3807
35.4362
37.4422
36.7577
17.1945
e−0.2s
.
s−1
result
Am
5.3392
3.2172
4.1529
4.2840
6.3001
Table 5. Result of specifications outside available region for Gp (s) =
Specifications (Am ,φm )
(7,10)
(7.5,15)
(8,15)
(7.5,10)
(8,5)
A SURVEY OF PID CONTROLLER DESIGN
(63)
89
GC2 (s) = Kf + sTD
µ
(64)
GC1 (s) = K
1 + sTI
sTI
¶
.
Obviously, this controller is a special type of two-degree of freedom controller with constant set point value [3]. Obviously, using the PI-PD controller, there are 6 unknown variables with 4 equations. Therefore, herein,
two in addition conditions are used to solve the PI-PD controllers. The first
one assumption is similar to literature [43], equation (39) is used and α is
set to be one. And the secondly is to minimize the performance index ISTE
Z∞
(65)
min J1 (θ) =
0

1
(te(θ, t)2 dt = min 
2πj

Zj∞
F (−s)F (s)ds
−j∞
where
F (s) = L [te(θ, t)] = −
d
E(s).
ds
Figure 12. PI-PD controller scheme.
The design procedure can be described as bellow: Firstly, calculate the
phase crossover frequencywp from ∠Gp (jwP ) = −π. And then set TI to be
a constant. From equation (7), we have
90
CHING-HUNG LEE
(66)
1
K
µ
jwp TI − (wp TI )2
1 + (wp TI )2
¶·
¸
1
+ (jwp TD + Kf ) = −Am .
Gp (jwp )
1+T (jw )
Let the closed-loop gain is denoted as T (s) and Gp (jwpp) = R1 (jwp ) +
jI1 (jwp ). Substitute into equation (66). Thus, the real and imaginary parts
can be represented as
−(wp TI )2 (R12 + I12 )Kf − (wp2 TI )(R12 + I12 )TD
= −Am K(1 + (wp TI )2 )(R12 + I12 ) + [(wp TI )2 R1 + (wp TI )I1 ]
(67)
(68) (wp TI )(R12 + I12 )Kf − (wp3 TI2 )(R12 + I12 )TD = −R1 (wp TI ) − I1 (wp TI )2 .
Define
M11 = −(wp Ti )2 (R12 + I12 ), M12 = −(wp2 Ti )(R12 + I12 )
M21 = (wp Ti )(R12 + I12 ), M22 = −(wp3 Ti2 )(R12 + I12 )
Const1 = −Am (1 + (wp Ti )2 )(R12 + I12 )
Const2 = [(wp Ti )2 R1 + (wp Ti )I1 ]
Const3 = −R1 (wp Ti ) − I1 (wp Ti )2 .
Therefore, equations (67) and (68) and solutions of Kf andTD are
·
(69)
M11
M21
(70)
Kf =
(71)
TD =
M12
M22
¸·
Kf
TD
¸
·
=
Const1 K + Const2
Const3
¸
M22 Const1 K + (M22 Const2 − M12 Const3 )
∆
−M21 Const1 K + (M11 Const3 − M21 Const2 )
∆
where
¯
¯
¯ M11 M12 ¯
¯
¯.
∆=¯
M21 M22 ¯
From definition equation of phase margin, we have
1
T (jwg )
(72)
µ
=
1
K
jwg TI
1 + jwg TI
=
−e−jφm .
¶µ
1
+ jTD wg + Kf
Gp (jwg )
¶
A SURVEY OF PID CONTROLLER DESIGN
91
Define
1 + T(jwg )
= R2 (jwg ) + jI2 (jwg ),
Gp (jwg )
and
3π
Td wg + I2
− tan−1 wg TI + tan−1
= −φm .
2
Kf + R 2
For simplification, the tan−1 approximation is used to solve the solution.
We thus obtain
(73)
(74)
(75)
¯
¶
µ ¯
¯ TD wg ¯
TD wg
1
¯
¯
≤1
π+
+
= −φm if ¯
wg TI
Kf
Kf ¯
¯
µ ¯
¶
¯ TD wg + I2 ¯
3
1
Kf
¯
¯
π+
−
= −φm if ¯
≥1 .
2
wg TI
Td wg
Kf + R2 ¯
¯ µ
¯
¶
¯1
¯
jwg TI
¯
¯
[j(T
w
+
I
)
+
(K
+
R
)]
D g
2
f
2 ¯ = 1.
¯ K 1 + jwg TI
Using the iterative method and let
(76)
M22 Const1 K + (M22 Const2 − M12 Const3 )
Kf
=
= R.
TD
−M21 Const1 K + (M11 Const3 − M21 Const2 )
Thus,
−
(77)
wg =
1
(π + φm ) +
TI R
r
1 2
4
) (π + φm )2 −
TI R
R
2
(
Substituting into (75) and solve K to find a new value of R. Iterative
calculation should be done for finding wg , K, R to meet E = R0 − R < ε,
where ε is the user defined accuracy. Thus, we have PI-PD controller parameters TI , K, Kf , TD . Finally, use Astrom’s recursive algorithm to find
the corresponding value of ISTE. Here, the value TI is chosen to be changeable variable for minimizing the ISTE. Table 6 and Fig. 13 show the result
2
. From Fig. 13, we have the better perfor unstable plant Gp (s) = 2
s −4
formance using the PI-PD controller comparison with PID controller even
if they are based on the same GPM.
CHING-HUNG LEE
92
PI-PD
PID
PI-PD
PID
PI-PD
PID
specifications
Am φm
2 20
3 30
4 35
Kf
TI
7.8404 0.1719
0.2742
7.2452 0.3333
0.3017
6.8670 0.2808
0.3251
result
TD
1.4845
0.8701
0.8980
1.0612
1.7908
1.0413
Am’
1.9177
2.0125
2.9432
3.0135
3.9596
3.8891
Φm’
19.4292
21.6699
29.2901
28.1395
33.5762
36.8343
J1
0.9403
0.9924
0.5200
0.5644
0.2190
0.3512
Table 6. PI-PD controller for unstable plant Gp (s) =
K
3.1748
2.8727
2.6585
2.4116
2.7023
2.1259
2
s2 −4 .
Error
Am
φm
4.115% 2.84%
1.02% 8.32%
1.89% 2.367%
0.4%
6.20%
1%
5.02%
2.773% 5.24%
A SURVEY OF PID CONTROLLER DESIGN
93
Step Response
From: U(1)
1.4
PID
1.2
To: Y(1)
Amplitude
1
PI-PD
PI-D
0.8
BackgroundResponseObjectLine
0.6
PI-PD
PID
PI-D
0.4
0.2
0
0
1.6
3.2
4.8
6.4
8
Time (sec.)
Figure 13. Comparison result of unstable plant Gp (s) =
s2
2
.
−4
4.2.4. Robust PID Controller Design. In the following, the purpose is to
determine the PID parameters (k, ki , kd ) for the given robust GPM specifications. Before describing our approach, the following example is introduced
to state the motivation.
Example 3: PI controller for plant with parameters uncertainties GP (s) =
e−[0.1:0.15]s
.
[0.5 : 1]s + [0.5 : 1]
e−0.1s
For the nominal plant GP (s) =
, we have a stabilizing PI controllers+1
(k = 6.1984, ki = 11.017) based on GPM specification (3, 45) with less 1%
e−0.15s
error [30]. If the plant parameters vary to G0P (s) =
, then the PI
s + 0.5
controller results with a large overshoot in step-response, (Am ,φm )=(1.54,
94
CHING-HUNG LEE
e−0.15s
, the PI con0.5s + 1
troller has an unstable result (Am ,φm )=(0.868, -14.929). For interval plant
e−[0.1:0.15]s
, the PI controller cannot guarantee the sysGP (s) =
[0.5 : 1]s + [0.5 : 1]
tem’s robustness and performance. Subsequently, a robust controller for
parametric uncertainty systems will develop to guarantee the good performance and robustness. Thus, problem formulation and some results of
parametric robust control are introduced bellow.
For meeting the theorem, plant (1) is rewritten as
23.71). If the plant parameters vary to G00P (s) =
0
(78)
0
bm0 sm + bm0 −1 sm −1 + · · · + b1 s + b0
Gp (s) =
.
sn0 + an0 −1 sn0 −1 + · · · + a1 s + a0
Here, we assume n0 > m0 , and the coefficients ai , bj are not exactly
known, but are only known to satisfy ai , bj ∈ R, 0 < ai ≤ ai ≤ ai , i =
0, 1, · · · , n0 − 1, 0 < bj ≤ bj ≤ bj , j = 0, 1, · · · , m0 . The plant is called
an interval plant family or interval system. Thus, the forward open loop
transfer function is
(79)
GO (s) = GC (s)GP (s) :=
N (s) ki + ks + kd s2
·
.
D(s)
s
Assume the uncertainty polynomial of degree n0 can be written as
(80)
0
P (s, A) = a0 + a1 s + a2 s2 + · · · + an sn
where the coefficients ai ∈ A, i = 1, · · · , n0 and vary in independent intervals
A := {ai ∈ R|0 < ai ≤ ai ≤ ai ,
i = 0, 1, · · · , n0 }.
Polynomial (80) is Hurwitz if all its zeros have negative real parts; the
family P is Hurwitz if all its members are Hurwitz. The Kharitonov theorem is introduced here that provides the stability condition for the interval
polynomial.
Theorem 1: (Kharitonov’s Theorem [6,7,17,30,33,41]) P (s,A) is Hurwitz
iff the following four Kharitonov polynomials (or extreme polynomials) are
Hurwitz
(81)
K11 (s) = a0 + a1 s + a2 s2 + a3 s3 + a4 s4 + a5 s5 + · · ·
(82)
K12 (s) = a0 + a1 s + a2 s2 + a3 s3 + a4 s4 + a5 s5 + · · ·
A SURVEY OF PID CONTROLLER DESIGN
(83)
K21 (s) = a0 + a1 s + a2 s2 + a3 s3 + a4 s4 + a5 s5 + · · ·
(84)
K22 (s) = a0 + a1 s + a2 s2 + a3 s3 + a4 s4 + a5 s5 + · · · .
95
Details for the theorem can be found in [6,7,17,30,33].
The Kharitonov theorem states that the stability of an interval polynomial can be determined by testing the stability of just four Kharitonov polynomials. In addition, the so-called Kharitonov rectangle contains the parameters varying in set A, i.e., the interval plant family is in the Kharitonov
rectangle at some frequency. Using the value set concept and the zero exclusion principle, the Kharitonov theorem is easily understood [7,17].
We now proceed to state a generalization of Kharitonov’s theorem [7,17,30,33].
Consider the interval plant (80) with the Kharitonov polynomials N1 (s),
N2 (s), N3 (s) and N4 (s) for the numerator and D1 (s), D2 (s), D3 (s) and
D4 (s) for the denominator. Then, the segments for numerator and denominator are defined (1 − λ)Ni (s) + λNj (s) and (1 − λ)Di (s) + λDj (s), where
λ ∈ [0, 1] and (i,j) ∈{(1,2),(1,3),(2,4),(3,4)}. Using these segments for the
numerator and denominator polynomials, we can obtain the following 32
extreme systems
(85)
GE (s) =
Ni (s)
(1 − λ)Nj (s) + λNk (s)
∪
(1 − λ)Dj (s) + λDk (s)
Di (s)
where λ ∈ [0, 1], i=1,2,3,4 and (j,k) ∈{(1,2),(1,3),(2,4),(3,4)}. Therefore,
for the 32 extreme systems, the Bode, Nyquist, and Nichols envelops of a
control plant with PID controller can be constructed to check the stability
of the interval plant.
Here we introduce the so-called robust GPM. This is used to be the specification for PID controller problem. The difference for the general used
GPM is the robust GPM which is calculated from the envelope of GPM of
the 32 extreme systems. Also, the FNN system will identify the relation
between PID controller parameters and robust GPM of the parametric interval plant. The purpose is to determine the PID parameters (k, ki , kd )
for the given robust GPM specifications. The problem of characterizing all
stabilizing PID controllers is to determine all the values of k, ki , and kd for
which the entire extreme systems is Hurwitz. By the way, we should first
find the Kharitonov 32 extreme systems. And then, use the previous results
of [30] to obtain a computational characterization of all PID controllers that
stabilize the interval family. Note that there are four parameters (λ, k, ki ,
and kd ) at the characteristic polynomial. At first, a rough range of these
96
CHING-HUNG LEE
parameters would be chosen (λ ∈ [0, 1]), it can be estimated by the robust
Bode plot without control. Thus, the stabilizing region and available region
can be obtained and the training patterns are selected as above.
Herein, consider the interval plant
(86)
GP (s, A) =
[0.0032 :
0.005]s3
[18 : 15]
.
+ [0.072 : 0.1]s2 + [1.28 : 1.305]s
We first plot the robust Bode plot of plant (86) for roughly choosing the
parameters range. Here, we choose the rough region as {0 < k < 1, 0 < ki <
5}. Figures 14 and 15 shows envelop of the Kharitonov 32 extreme systems
(robust Bode plot) and Movement of the Kharitonov rectangle of system
that guarantee the stability of system (86), that illustrates the effectiveness
of our approach. We also can demonstrate our result by the other graphic
tool, Nichol chart or Nyquist diagram.
Bode plots of the 32 K haritonov extrem al plants in dB
100
dB
50
0
-50
-100
-1
10
10
0
10
1
10
2
B ode plots of the 32 Kharitonov ex trem al plants in degree
-100
degree
-150
-200
-250
-300
-1
10
10
0
10
1
w---rad/s ec
Figure 14. Robust Bode plot of system (86) with PI controller.
10
2
A SURVEY OF PID CONTROLLER DESIGN
97
Im age polynom ial set of K haritonov s egment plant
50
40
30
20
Im ag
10
0
-10
-20
-30
-40
-50
-50
-40
-30
-20
-10
0
Real
10
20
30
40
50
Figure 15. Movement of the Kharitonov rectangle of system (86) with PI controller.
5. Conclusion Remarks
The survey was aimed at researchers currently working in the area of
PID controllers. This paper has provided a survey of PID controllers design
based on the GPM specification for a given stable, unstable, and interval
plants. The basic ideas, techniques, and results have been presented in language and notation familiar. Due to the highly nonlinear of definitions of
GPM equations, an analytical tuning method is not yet available. Approximation and FNN approaches have been introduced to treat this problem.
Therefore, they were used to automatically tune the PID controller parameters for GPM specifications so that neither numerical methods nor graphical
methods need be used. Applications of usual used process- first order with
time delay plant have been extended to general nth order plant. In addition, the robust PID controller is definition and presented by the FNN
approach. Finally, for obtaining the better performance, a novel PI-PD
98
CHING-HUNG LEE
controller scheme based on GPM was firstly proposed here. Simulation results are also presented to show the effectiveness. Thus, the GPM approach
is available for high performance and robustness requirement. In the future,
the GPM approach can be extended to the model predictive control, nonlinear control, and intelligent control, two-degree of freedom control, and
digital redesign.
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Department of Electrical Engineering, Yuan Ze University , Chungli, Taoyuan
320, Taiwan, R.O.C.
E-mail address: chlee@saturn.yzu.edu.tw