International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
PRECISE ANALYTICAL EXPRESSIONS FOR
MECHATRONICS SYSTEMS TIME DOMAIN
PERFORMANCE SPECIFICATIONS AND
VERIFICATION USING MATLAB
Farhan A. Salem 12
1
Mechatronics program, Department of Mechanical Engineering, Faculty of Engineering,
Taif University, 888, Taif, Saudi Arabia.
2
Alpha center for Engineering Studies and Technology Researches, Amman, Jordan.
Email: salem_farh@yahoo.com
ABSTRACT
The ultimate goal of this paper is to propose accurate analytical expressions for Mechatronics systems
performance specifications, that accurately reflect the actual system performance and can be used for
accurate performance specification, calculations, evaluation and verification. The derived analytical
expressions are intended for research purposes as well as for the application in educational process. The
proposed analytical expressions were analytically derived and verified using MATLAB software, where to
verify the accuracy of derived and suggested expressions the actual values of normalized ωnt versus ζ, a
long with derived and suggested analytical expressions was plotted and analyzed.
KEYWORDS
Performance specifications, Response, Analytical expression, Mechatronics systems.
1. INTRODUCTION
Most used formulae and expressions for performance specifications in texts e.g.
[1][2][3][4][5][7][8] lack accuracy, the determined performance specifications using these
expressions, rarely accurate compared with actual results and measurements since it is more
difficult to determine the exact analytical expressions of most used specifications and most
introduced expressions are rough approximation of actual values. A designer can often make a
linear approximation to a nonlinear system. Linear approximations simplify the analysis and
design of a system and are used as long as the results yield a good approximation to reality [4].
Mechatronics systems are supposed to operate with high accuracy and speed despite adverse
effects of system nonlinearities and uncertainties, therefore accuracy in Mechatronics systems
performance is of concern, and the need for precise analytical expressions for mechatronics
systems performance specifications is highly desired, such expressions are to be derived and
suggested.
The performance of systems, the form and properties of response, are determined by the locations
of its poles in Laplace domain. Many times, it is possible to identify a single pole, or a pair of
poles, as the dominant poles. In such cases, a fair idea of the control system's performance can be
obtained from the damping and natural frequency of the dominant poles [1]. The step response is
the measured reaction of the control system to a step change in the input, step response has
DOI:10.5121/ijitca.2013.3104
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
universal acceptance and popularity, because of simplicity of its form facilitates mathematical
analysis, modeling, and experimental verifications, as well as it is easy to generate and has
several measurement techniques for recording the time domain response. A typical step response
and its associated performance specifications of first and second order systems are shown in
Figure 1. The most used performance specifications are; Time constant T, Rise time TR, Settling
time Ts, Peak time, TP, Maximum overshoot MP, maximum undershoot Mu, Percent overshoot
OS%, Delay time Td, The decay ratio DR , Damping period TO and frequency of any oscillations
in the response, the swiftness of the response and the steady state error ess.
Figure 1. Second-order underdamped response specifications
2. PLANT BASIC EQUATIONS; PERMANENT MAGNET DC MOTOR
SIMPLIFIED MODELS:
Motion control systems takes input voltage as actuator input, and outputs linear or rotational
position/speed/acceleration/torque, the most used actuator for motion control systems is DC
motors. The PMDC motor is an example of electromechanical systems with electrical and
mechanical components, a simplified equivalent representation of DC motor's two components
are shown in Figure.2. Two PMDC motor simplified equations, first and second order, will be
used to apply suggested analytical expressions, for system's performance specifications with poles
and no zeros.
To obtain the DC motor electric component transfer function relating input armature current, ia
and voltage Vin, Applying Kirchoff’s law around the electrical loop by summing voltages
throughout the R-L circuit gives the next equations :
 di
V in = R ∗ i a + La  a
 dt
dθ

 + K b dt

Taking Laplace transform and rearranging, gives:
Vin (s) = RaIa+ LasIa+ Kbs θ


I a (s )
1
=

V
(
s
)
−
K
ω
(
s
)
L
s
+
R
[ in
] ( a
b
a )
(1)
The torque is given by:
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
T= j d2θ/dt2 = J dω/dt
(2)
The output torque developed by the motor ,Tm ,is related to the armature current, ia , by a torque
constant Kt, and given by the following equation:
Motor Torque = Tm = Kt* ia
(3)
Equating Eqs. (2),(3), separating armature current and taking Laplace transform gives:
Kt*ia= Jm d2θ/dt2
Ia(s) = Jm s2θ(s)/ Kt
(4)
Substituting in Eq. (4) in Eq. (1) and rearranging gives PMDC motor transfer function in terms of
output angle θ:
V in ( s ) = R a (
J m s 2θ ( s )
Kt
) + La s (
J m s 2θ ( s )
Kt
) + K b sθ
Vin (s) = Ra(Jm s2θ(s) / Kt) + Las(Jm s2θ(s)/ Kt) + Kbs θ
Kt
θ (s )
G angle (s ) =
=
3
V in (s )  La J m s + R a J m s 2 + K t K b s 
( 5)
To write transfer function in terms of output speed ω, we rewrite the torque Eq.(4) in terms of
output speed Kt*ia= Jm dω /dt , and repeat previous steps, this gives.
G speed (s ) =
ω (s )
V in (s )
=
Kt
 La J m s + R a J m s + K t K b 
2
( 6)
Based on the fact that, the PMDC motor response is dominated by the slow mechanical time
constant, this motivate us to assume motor inductance, (La =0) , this will result in the simplified
first and second order form of PMDC motor transfer function in terms input voltage, Vin and
output angle θ and speed , ω , respectively : Rearranging these first and second order equations
into standard first and second transfer function forms yields the following equation:
G angle (s ) =
θ (s )
V in (s )
G speed (s ) =
=
Kt
K
=
s ( R a J m s + K t K b ) s ( Js + b )
Kt
K t / Ra J m
ω (s )
K
=
=
=
V in (s ) (R a J m s + K t K b ) (s + K t K b / R a J m ) s + a
( 7)
( 8)
Using Esq. (7) and (8) we can built block diagram and simulink model of the PMDC motor
system shown in Figure 3(a)(b) ,with unity feedback , H(s)=1. The following nominal values for
the various parameters of a PMDC motor used: Vin=12 Volts; Jm=0.2 kg·m²; bm =0.3; Kt =0.23
N-m/A ; Kb=0.23 V-s/rad, Ra =1 Ohm ; and La=0.23 Henry .
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
m
ELECTRIC component of PMDC motor system
m
MECHANICAL component of PMDC motor system
Electromechanical PMDC motor system
Figure 2: Schematic of a simplified equivalent representation of the PMDC motor's
electromechanical components.
Figure 3(a) PMDC motor first order system
Figure 3(b) PMDC motor second order system
The closed loop transfer functions of simplified PMDC motor equations given by Eqs. (7) and
(8), can be manipulated and be rewritten in general form, to have the form:
T speed (s ) =
T angle (s ) =
ω (s )
V in (s )
=
K t / Ra J m
K
=
s + (K t K b + K t ) / R a J m s + a
( 9)
ωn2
K
K /J
=
=
Js 2 + bs + K s 2 + (b / J )s + ( K / J ) s 2 + 2ξωn s + ωn2
Where ζ = damping ratio = b
2 JK
(10 )
, ω n = undamped frequency = K / J,...J = R a *J m , b = ( K t *K b )
Most dynamic systems can be approximated as first or second order systems. The form and
properties of system's dynamic behavior depends on poles of the characteristic equation, and can
be described in terms of two constants, damping ratio ζ and undamped natural frequency ωn.
3.
PERFORMANCE
SYSTEM.
SPECIFICATIONS
OF
FIRST
ORDER
First order systems without zeros described by the transfer function given by Eq. (9), and systems
that can be approximated as first order systems, is characterized by time constant T, rise time TR,
settling time Ts and steady state error ess shown in Figure 4(b). When first order system is
subjected to unit step, R(s) = 1/s, the response for these systems is natural decay or growth
generated by the system pole, the Laplace transform of the step response is given by:
R (s ) *C (s ) =
1 K
s (s + a )
Taking the inverse transform, the step response is given by:
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
c (s ) = 1 − e −αt
(11)
3.1 Time constant T; the solution given by Eq.(11) has the exponential form eαt, when the real
part of α is negative, the plot of e-at has the form of natural decay or growth shown in Figure 4,
the value of time that makes the exponent of e equal –1 is called the time constant T, it is a
characteristic time that is used as a measure of speed of response and governs the approach to a
steady-state value after a long time, the larger the time constant is, the slower the system response
is. The system reaches 63.2% of its final value after time equals to one time constant and reaches
99.3% of its final value after time equal to five time constants.
e αt = e −1 ⇒ at = −1 ⇒ T =
1
a
(13)
Time constant of first order system, can be found using any of the following approaches; using
Eq.(13) or time for response c(t) to reach 63.2% (or 36.8% for exponential decrease) of its final
value, ( 1 − e − t / T = 1 − e − t / t = 0.632 =63.2%. ) or system reaches 99.32% of its final value after time
equals to 5T, geometrically the tangent drawn to the curve e-at at t = 0 intersects the time axis at
the value of time equal to the one time constant T. The slope of the tangent line at t=0 is given
by; slop=1/T, and the pole location in the s plane is given by; a = -1/T.
3.2 Rise time TR; is defined as the time required for the response curve to reach from 10% to
90% , 5% to 95% or 0% to 100% of its final value. Rise time is found by solving Eq. (11) for
the difference in time, e.g. rise time for 10% to 90% criterion is found by solving Eq. (11) for the
difference in time at c(t) = 0.9 and c(t) = 0.1.
T R = (1 − e −αt 90 ) − (1 − e −αt10 )
TR =
2.3026 0.1054 2.1972
−
=
a
a
a
(14)
The rise time can be measured in terms of the time constant, and given by:
t 90
t10
T R = (1 − e T ) − (1 − e T ) = 2.3026T − 0.1054T = 2.1972T
3.3 Settling time Ts is defined as the time required for the response curve to reach and stay
within a range about the final value of size specified by absolute percentage of final value, usually
2% or 5% [2]. For 2% criterion, settling time is found by equating Eq. (11) with 0.98, (that is 10.02 = 0.98) and solving for time, t:
0.98 = 1 − e −αt
ln 0.02
=t
−a * ln e
⇒ ln 0.02 = ln e −α t ⇒ ln 0.02 = −at ln e
⇒
Ts =
ln 0.2 3.9120
=
−a
a
(15 )
The settling time be measured in terms of the time constant, and given by:
−t
0.98 = 1 − e T
⇒
t = T S = −T ln(0.02) = 3.9120 *T
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
3.4 The steady-state error ess is defined as the difference between reference input r(t) and actual
final output c(t), it is the error after the transient response has decayed leaving only the steady
state response, and is given by:
e ss (s ) = r ( t ) − c ( t ) = 1 − 1 − e −αt = e −αt
(16 )
e ss = e (∞) = lim
s →0
sR (s )
1
=
1 + G (s ) 1 + lim G (s )
s →0
As t approaches infinity, e-at approaches zero and the steady-state error is to be:
ess =e(∞ )= limt →∞ [r(t)- y(t)]
3.5 DC Gain; is defined as the final value of the step response of a system, and is given
by:
1

DCgain = lim [sG (s ).R (s )] = lim sG (s ).  = lim [G (s ) ]
s →0
s →0
s  s →0

(17 )
Using the derived expressions given by Eqs. (13) by (17) to determine the performance
specifications of first order PMDC motor system described by Eq. (9), will result in response plot
shown in Figure 4(b), as well as the following performance specifications; System pole = -0.0566,
T =17.6741 second, 5T=88.3705 seconds, TR =38.8830 second ,Ts = 69.1569 second, final output
=20.3252, ess=0.
Step Response
25
20
Amplitude
15
10
5
Time constant,T
0
0
20
Rise time, Tr
40
60
Settling time, Ts
80
5T
100
120
Time (sec)
Figure 4(a) First order system response to step, and
performance specifications.
Figure 4(b) Performance specifications of first order
PMDC motor step response.
4. PERFORMANCE SPECIFICATIONS OF SECOND ORDER SYSTEM
For second order systems, and systems that can be approximated as second order systems, there
are four cases of response to consider; undamped, underdamped, critically damped and
overdamped response. We shall solve for response performance specifications considering these
cases , derive and suggest more accurate analytical expressions that can be used to determine
actual performance specifications values or values with minimum possible error. When second
order system is subjected to unit step input, R(s) = 1/s, the response depends on damping ratio ζ,
and undamped natural frequency ωn, where damping ratio determines how much the system
oscillates as the response decays toward steady state and undamped natural frequency ωn,
determines how fast the system oscillates during any transient response.
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
4.1 For underdamped case; 0< ζ<1 and two complex conjugate poles given by −ξωn
allow us to rewrite Eq.(10) to have the following form:
± j ωn 1 − ξ 2
,
ωn2
ωn2
C (s )
= 2
=
R (s ) s + 2ξωn s + ωn2 (s + ξωn + j 1 − ξ 2 )(s + ξωn − j 1 − ξ 2 )
The Laplace transform of the step response can be obtained, to obtain inverse Laplace transform
we need to expand by partial fractions and solve, this all gives:
ωn2
C (s ) 1
=
⇒
R (s ) s (s 2 + 2ξωn s + ωn2 )
=
=
s + 2ξωn
2
2
2
2
2
2
s + 2ξωn s + (ωn 1 − ξ )
s + 2ξωn
s + 2ξωn s + (ωn 1 − ξ )
+
s + 2ξωn
1
1
=
+
s s 2 + 2ξωn s + ωd 2 s
+
s + 2ξωn
1
1
=− 2
+
s
s + 2ξωn s + ωd 2 s
=
ωd
s + 2ξωn
ξω
1
−
− n
2
2
s (s + ξωn ) + ωd
ωd (s + ξωn ) 2 + ωd 2
=
ωd
s + 2ξωn
1
ξ
−
−
2 (s + ξω ) 2 + ω 2
s (s + ξωn ) 2 + ωd 2
1−ξ
n
d
Taking inverse Laplace transform, gives:
c (t ) = 1 − e −ξωn t (cos ωd t +
ξ
1−ξ 2
sin ωd t )
(18)
This can be rewritten to have the following forms:
c (t ) = 1 −
c (t ) = 1 −
1
1− ξ 2
e − ξωn t cos(ωd t − φ )
1
1− ξ
2
e −ξωn t sin(ωd t + cos −1 ξ )
(19 )
( 20)
Eq. (18),(19)and(20) show that the damped natural frequency ωd, given by ωd = ωn 1 − ζ 2 ,is the
frequency at which the system will oscillate if the damping is decreased to zero. The error signal
for this system is the difference between input r(t) and output c(t) and is given by:
e (t ) = e −ξωnt (cos ωd t +
ξ
1− ξ 2
sin ωd t )
( 21)
Eq.(18) shows that performance of second order system depends on damping ratio ζ and
undamped natural frequency ωn . Plots of the step response as functions of the normalized time
ωnt for various values of 0 ≤ ζ ≤1.5 are illustrated in Figure 5(a), the curves show that he response
becomes more oscillatory as ζ decreases in value, up to ζ=1, when ζ ≥ 1, the step response does
not exhibit any overshoot or oscillatory behavior, also when ζ between 0.5 and 0.8 the system
reaches final value more rapidly. Plots of the step response for various ωn are illustrated in Figure
5(b), the responses show that ωn has a direct effect on the rise time, delay time, and settling time
but does not affect the overshoot.
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
4.2 For undamped case; ζ=0, and two equal complex conjugate imaginary poles given by ±jω,
the response can be obtained by substituting ζ=0 in eq.(11), this gives:
c (t ) = 1 − (sin ωn t )
( 22)
This equation shows that ωn, is the system undamped natural frequency, at which the system will
oscillate if the damping is decreased to zero. Therefore, ωn, corresponds to the frequency of the
undamped sinusoidal response. Decreasing damping from zero will reduce ωn allowing observing
the damped frequency ωd. An increase in ζ would reduce damped natural frequency ωd, up to no
oscillator behavior.
4.3 For critically case; for ζ=1, the two equal real poles allow approximating the system by a
critically damped one pole, and given by:
C (s ) =
ωn2
1
⇒ c (t ) = 1 − e ωn t (1 + ωn t )
s (s + ωn ) 2
( 23)
This response can also be obtained by letting ζ approaches unity in eq.(12), using limit concepts.
4.4 For overdamped case; ζ>1, two distinct real poles, given by −ξωn ± j ωn ξ 2 −1 allow us to
rewrite eq.(10) to have the following form, when subjected to step input R(s):
ωn2
ωn2
C (s ) 1
1
=
=
R (s ) s ( s 2 + 2ξωn s + ωn2 ) s (s + ξωn + ωn ξ 2 −1)(s + ξωn − ωn ξ 2 −1)
Taking inverse Laplace transform gives:
c (t ) = 1 +
 e −a1t e −a2t 
−


a2 
2 ξ 2 −1  a1
ωn
( 24 )
Where: a1, a2 are system poles. When ζ is much greater than unity, i.e. ζ >> 1, then | a1| >> | a2|
,this response can be approximated, by neglecting one of systems two poles, the pole that is
farther from imaginary axis, to have the unit-step time response of the following form:
c (t ) = 1 − e
(
)
− ξ − ξ −1 ωn t
( 25)
Closed-Loop Step for various Omega
Closed-Loop Step for various zeta
n
1.8
1.6
1.6
zeta=0.1
zeta=0.2
1.4
zeta=0.3
1.4
1.2
zeta=0.5
zeta=0.6
1
zeta=0.7
1
Amplitude
Amplitude
zeta=0.4
1.2
zeta=0.8
zeta=0.9
0.8
zeta=1
zeta=1.1
0.6
0.8
omega =1
n
omega =2
0.6
n
zeta=1.2
omega =3
n
zeta=1.3
0.4
0.4
omega =4
n
zeta=1.4
zeta=1.5
omega =5
n
0.2
0.2
omega =6
n
omega =7
0
n
0
0.5
1
1.5
2
2.5
3
normalized time ,omegan*t (sec)
Figure 5(a) Plots of the step response for various
0≤ ζ ≤1.5 with ωn=10
0
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Figure 5(b) Plots of the step response for
various ωn with ζ =0.2.
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
5. DERIVING ANALYTICAL EXPRESSIONS FOR PERFORMANCE
SPECIFICATIONS OF UNDERDAMPED SECOND ORDER
SYSTEM.
It is more difficult to determine the exact analytical expressions of the delay time Td, rise time TR,
and settling time Ts. But in this paper, by applying mathematical concepts, curve fittings and trial
and error approach to Eq.(18) we will try to derive and suggest analytical expressions, with
minimum possible deviation at actual values, the correctness of derived analytical expressions, as
well as the correctness of used expression in different texts e.g. [1][2][3][4][5][7] will be verified
using MATLAB. MATLAB m.file was written, to determine specifications for suggested values
of damping ratio over the range 0 ≤ ζ ≤1.5, plots normalized times versus given ζ for derived as
well as most used specifications formulae and expressions in reference texts . The derived
analytical expressions are only accurate for second-order systems with no zeros and represent the
essential qualities of higher-order systems with two dominant poles.
The definitions for settling time and rise time are basically the same as the definitions for the
first-order response. All definitions are also valid for systems of order higher than second.
5.1 Time constant T
The definitions of second order system time constant basically the same as the definitions for the
first-order system. When the poles are complex quantity, (σ ± jω) , the transient has the form of
damped sinusoid [Aeσtsin(ωdt +Φ)], in the case the time constant is defined in terms of complex
pole real part σ, that characterizes the envelope Aeσt , see Figure 7(b), the time constant is equal
to :
T =
1
σ
=
1
( 26)
ζωn
Therefore, the larger the product of ξωn , the greater the instantaneous rate of decay of the
transient response.
5.2 Response maxima and minima.
5.2.1 Maximum overshoot Mp and minimum undershoot MU.
Maximum overshoot Mp, can be defined as the maximum value of the output, or it is the
magnitude of the overshoot after the first crossing of the steady-state value, or the amount by
which the system output response proceeds beyond the desired steady-state value. When output is
lower than the final value, the phenomenon is called undershoot MU.
Maximum overshoot = cmax(t) – css(∞)
5.2.2 Maxima time or Peak Time TP
Peak time TP is defined as the time required to reach the first maximum peak of the overshoot, it
is a criteria of speed of response. The peak time is determined by finding the time when the
derivative of Eq.(18) is equal to zero:
ξωd
dc (t )
ξ
= ξωn t * e −ξωn t (cos ωd t +
sin ωd t ) − e −ξωn t (− sin ωd t +
cos ωd t ) = 0
dt
1−ξ 2
1− ξ 2
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
= ξωn t * e −ξωn t (cos ωd t +
=(
ξ 2 ωn
1−ξ 2
ξ
1−ξ 2
+ 1)(sin ωd t ) = (
sin ωd t ) − e −ξωn t ( − sin ωd t + ξωn cos ωd t ) = 0
ξ 2ωn
1− ξ 2
+ 1)(sin ωn 1 − ξ 2 t ) = 0
= sin ωn 1 − ξ 2 t = 0 ⇒ ωn 1 − ξ 2 t = n π
The values, that makes dc (t ) = 0 , are 0,π,2π,3π,4π…=n π
dt
dc (t )
= sin ωn 1 − ξ 2 t = 0
dt
⇒
ωn 1 − ξ 2 t = n π
The overshoots and undershoots occur at periodic intervals, this is shown in Figure 6, the
overshoots occur at odd values of n, the undershoots occur at even values of n. The first
overshoot is the maximum overshoot, this corresponds to n = 1, the time at which the maximum
overshoot occurs is peak time TP , and is derived by further solving, to have the following form:
sin ωn 1 − ξ 2 t = 0 ⇒ ωn 1 − ξ 2 t = n π
⇒
t =TP =
nπ
ωn 1 − ξ 2
( 27 )
Since the peak time TP, corresponds to the first peak overshoot (n =1) , we have:
TP =
π
ωn 1 − ξ
2
=
π
ωd
Eq.(27) shows that the peak time is a function of both ζ and ωn. The exact magnitudes of the
overshoots and the undershoots can be determined by substituting peak time given by Eq. (27)
into Eq. (12), this gives the following:
c max or min (t ) = M P , or , M U , = 1 −
M P , or , M U = 1 −
e −ξ n π /
e −ξ n π /
1−ξ 2
1−ξ 2
1− ξ 2
1−ξ 2
sin(n π + φ ) = 1 − (−1) n −1 e −ξ nπ /
sin( n π + φ )
1−ξ 2
( 28)
The maximum overshoot is obtained by letting n= 1 in Eq. (28), this gives:
M P = e −ξπ /
1−ξ 2
⇒ M P = exp(−
ξπ
1−ξ 2
)
( 29)
This equation shows that, the maximum overshoot is a function of only the damping ratio ζ,
therefore it is used to evaluate the damping of the system, smoothness of response. The following
MATLAB code can be used to return the actual and estimated values of maximum overshoots for
given range of damping ratio 0≤ ζ≤ 1.5, as well as plot these values, the plot is shown in Figure 6
(a), analysis of this plot shows that the derived expression matches actual values with no error,
and can be used to determine accurate values of maximum overshoots.
>> zeta=0.1:0.01:1.5; omega_n=10; G={};
for i=1:length(zeta)
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
G{i}=tf(omega_n^2,[1,2*zeta(i)*omega_n, omega_n^2]);
end
[y,t]=step([G{:}]); Mp=max(y(:,:))-1'; Mp'; plot(zeta,Mp), hold
on; Mp_analy=[];
for i=1:length(zeta)
if(zeta(i)<1)
Mp_analy(i)=exp(-pi*zeta(i)/sqrt(1-zeta(i)*zeta(i)));
else
Mp_analy(i)=0;
end
end
plot(zeta,Mp_analy,'r.');
Comparing actual and analytical Mp for accuracy
0.8
Actual Mp
analyt. expression
0.7
0.6
0.5
Mp
0.4
0.3
0.2
0.1
0
-0.1
0
0.5
1
1.5
Zeta
Figure 6(a) Comparison between actual and analytical values of MP.
5.2.3 Percent maximum overshoot %OS.
Percent overshoot is the amount that the response overshoots the steady state final value at the
peak time, expressed as a percentage of the steady-state value, and can be derived as follows:
2
OS % = 100% *
c (T P ) − c (∞ )
e −ξπ / 1−ξ − 1
= 100% *
= OS % = 100 * e −ξπ /
c (∞)
1
1−ξ 2
( 30 )
Percent maximum overshoot %OS measures the closeness of the response to the desired response;
also it is a relative stability criterion, with 10% to 20% as an acceptable value. For given percent
maximum overshoot %OS, the damping ratio can be found by rearranging eq. (30) to have the
following form:
ξ=
− ln ( %OS / 100 )
π 2 + ln 2 ( %OS / 100 )
( 31)
For desired OS% less than 18% then ζ ≥ 0.479, and if 10% of OS% is acceptable then ζ ≥ 0.591.
The relationship between the percent maximum overshoot %OS and the damping ratio given in
Eq. (30) . Eq. (30) or (31) are plotted in Figure 6(b). This plot shows that it is good to increase
damping ratio, the decrease in the damping ratio ξ, leads to increase the overshoot and the time
response.
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
5.2.4 Minima Time TU:
The first undershoot is the minimum undershoot, this corresponds to n=2, in Eq. (28), the time at
which the minimum undershoot occurs is Minima Time TM, and is given by further solving, to
have the following form
2π
2π
TU =
=
2
ω
ωn 1 − ξ
d
Figure 6 (a) The overshoots and undershoots
occur at periodic intervals.
Figure 6(b) Percent overshoot as a function of
damping ratio, ζ [4]
5.2.5 The decay ratio DR;
Is defined as the exponential decay between successive peaks; the first maximum overshoot and
the second peak overshoot, It is the ratio of the second overshoot divided by the first, with 1/4 a
common design value; this specification is often used in process control industry. Referring to Eq.
(28), the second upper peak overshoot MP2, occurs at value n=3, the decay ratio is given by:
2
MP
e −3ξπ / 1−ξ
DR =
=
= e −2ξπ /
2
ξπ
ξ
−
/
1
−
M P2 e
1−ξ 2
( 32 )
The decaying ratio and maximum overshoot are functions of damping ratio only.
5.2.6 The damping factor Dζ
Is defined as the ratio of first maximum overshoot divided by the first undershoot, the maximum
overshoot is obtained by substituting n= 1 in Eq. (28), the first undershoot is obtained by
substituting n= 3, therefore the expression for damping factor is given by:
Dξ =
e −ξπ /
e −ξ 3π /
1−ξ 2
1−ξ
2
= e 2ξπ /
1−ξ 2
( 33)
5.2.7 The period, TO and frequency of any oscillations in the response.
The period of oscillation is the amount of time between two successive upper overshoot peaks; its
formula can be derived by subtracting time of first and second overshoots peaks. The period of
oscillation is given by:
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
TO = T P 2 − T P =
3π
ωn 1 − ξ
2
−
π
ωn 1 − ξ
2
=
2π
( 34 )
ωn 1 − ξ 2
The frequency of oscillation is given by:
ω = ωd =
2
1 ωn 1 − ξ
2π
2π
=
⇒ ωd =
=
= ωn 1 − ξ 2
2π
T
TO T P 2 −T P
( 35)
5.2.8 Settling time TS.
Settling time TS, is defined as the time required for the response curve to reach and stay within a
range about the final value of size specified by absolute percentage of final value, usually 2% or
5% [2]. In other words, the time the response curve takes to meet its desired final value; Ts is a
criterion of both of speed of response and stability, it reflects both response speed and damping,
settling time should be as small as possible because smaller values represent a faster response and
an ability to reduce costs. For 2% criterion, the settling time is the time for which response c(t) in
Eq. (18) reaches and stays within band of ±2% of the final steady-state value:
c (t ) = 1 −
1
1− ξ
2
e −ξωnTS cos(ωd t − φ ) = 0.02
The transient response decays is governed by the exponential term e −ζω t , assuming
cos(ωd t − φ ) = 1 , at the settling, and solving we have:
n
TS =
− ln(0.02 1 − ξ 2 )
ξωn
( 36 )
From this equation, the settling time for tolerance band of x per cent, may be obtained using the
following expression:
− ln(
TS =
x
1−ξ 2 )
100
ξωn
( 37 )
Plotting TS versus numerator of Eq.(36) with varying values of damping ζ= 0:0.01:1,( see Figure
7(a)) shows that the numerator will vary from 3.912 to 5.81, the corresponding to ζ numerator
value can then be divided over ζωn to obtain the value of settling time. analytical expression for
settling time can be obtained by analyzing and comparing both actual values of settling time
obtained from Figure 7(a), and results of running the below MATLAB code with defined tf and
zetas, this code will return the actual values of settling time for given range of damping ratio 0 ≤ ζ
≤1.5, as well as plot the actual settling time versus various values of damping ratio.
>>actual_Ts=[]; Q=(abs(y(:,:)-1)>0.05);
for i=1:length(zeta)
AA=t(Q(:,i)); actual_Ts(i)=AA(length(AA));
end
actual_Ts; plot(zeta,10*actual_Ts,'r'),hold on
The plot and results show that, for 0 ≤ ζ< 0.7, the unit-step response has a maximum overshoot
greater than 5%, and the response can enter the band between 0.95 and 1.05 for the last time from
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
either the top or the bottom. When ζ is greater than 0.7, the overshoot is less than 5%, and the
response can enter the band between 0.95 and 1.05 only from the bottom. It is difficult to obtain
exact analytical expression of the settling time TS, but using the envelope of the damped sinusoid
shown in Figure 7(b), it is possible to obtain an approximation for TS for 0 < ζ < 0.7, where when
the settling time corresponds to an intersection with the upper or the bottom envelope of c(t) , the
following corresponding relation are obtained:
1+
e
− ξωnTS
1−ξ 2
= Upper
⇔
1−
e
−ξωnTS
1− ξ 2
= Lower ⇒ 0 < ξ < 0.7
The same result for TS is obtained using either equation, solving for 5% TS, gives:
TS =
1
ξ
ln(0.05 1 − ξ 2 )
Plotting Ts versus numerator with varying values of ζ= 0:0.1:1 shows that the numerator will vary
from 3 to 4.951, but as ζ varies from 0 to 0.7, TS varies between 3.0 and 3.32. To obtain the
settling time, the corresponding to ζ numerator value can then be divided over corresponding ζ.
The following suggested simplified approach, can give an approximate expression for settling
time with minimum deviations from actual values; since the rate at which the transient response
decays is governed by the exponential term e −ζωn t , the settling time for the 2% and 5% criterion
can be calculated given by:
e −ξωnT S = 0.02 ⇒ T S =
e
− ξωnT S
= 0.05 ⇒ T S =
ln(0.02)
=
ξωn
ln(0.05)
ξωn
=
3.912
ξωn
2.9957
ξωn
⇒
⇒
0 < ξ < 0.7
( 38)
0.7 < ξ < 1.5
For ζ > 0.7 the value of TS is almost directly proportional to ζ, through curve analyzing, curve
fitting and trial and error, the following approximation can be suggested:
TS =
4.59ξ
ωn
⇒
ξ > 0.7
More better and more accurate expressions for ζ > 0.7 was chosen through curve analyzing, curve
fitting and trial and error, and given by:
TS =
TS =
(6.68ξ -1.9084)
ωn
(6.57ξ -1.64)
ωn
⇒
⇒
0.7 ≤ ξ ≤ 1.2
1.2 < ξ < 1.5
( 39 )
( 40 )
Different formulae and different approximations for the settling time appear in different texts,
Referring to [3] settling time is given by:
T S = 3.2 / ξωn
⇒ 0 < ξ < 0.7
T S = 4.5ξ / ωn
⇒ ξ > 0.7
( 41)
Referring to [5] settling time is given by:
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
T S = 4.6/ξωn
( 42)
Referring to [6] settling time is given by:
TS =
TS =
−0.5 ln((1 − ξ 2 ) / 400)
ξωn
(6.6ξ − 1.6)
⇒ ξ < 0.7
( 43)
⇒ ξ > 0.7
ωn
To verify the suggested expressions for settling time with the actual settling time and to compare
the results of suggested expressions with different expressions appear in different texts, we plot
normalized time ωnTR versus ζ , of all expressions and compare it with actual settling time. The
resulted plots are shown in Figure 7(c), these plots show that for ζ > 0.7 the actual settling times
rise smoothly as ζ increases, which results in slowing dawn the system response. For this range
the suggested two expressions given by Eqs.(39) and (40) are most accurate expressions that
allmostly fit the actual settling time with overage difference of 0.07 seconds. For 0.7 < ζ > 1.2 the
suggested expression given by Eq.(40) allmostly fit the actual values of settling time with overage
difference of 0.03 seconds. For 0.1 > ζ > 0.7, actual settling time peaks at times equal to, 0.12,
0.14, 0.16, 0.19, 0.24, 0.31, 0.44 and at 0.7, for the peak between 0.44 and at 0.7 the following
expression can be suggested:
⇒ 0.44 < ξ < 0.7
T S = 5.246/ω n
( 44)
For the peak between 0.31 and at 0.44 the following expression can be suggested:
T S = 7.8/ωn
⇒
0.31 < ξ < 0.44
( 45)
It would be difficult to model each peak exactly; the expression given by Eq.(38) can be
used to calculate the approximate values of settling time as well as maximum upper limit
for settling time at this range. A suggested expression for the lower limit of settling time
is given by:
T S = 2.5057/ξωn
⇒ 0 < ξ < 0.31
( 46)
The overage difference in time between the upper and lower limits is about ± 0.025, based on
this, following average expression can be suggested for 0.1 > ζ > 0.31:
T S ±0.025 = 2.7507/ξωn
⇒ 0 < ξ < 0.31
( 47 )
In Figure 7(d) both actual and suggested settling time curves are shown, these curves show that
the suggested analytical expression can be used to calculate the settling time for second order
systems, and systems that can be approximated as second order.
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
6
log(0.02.*sqrt(1-zeta. 2))
5.5
5
4.5
4
3.5
0
0.1
0.2
0.3
0.4
Figure 7(a) Ts versus
0.5
Zeta
0.6
− ln(0.02 1 − ξ 2 )
0.7
0.8
0.9
1
Figure 7(b) the envelope of the damped sinusoid
with ζ= 0:0.1:1.
Comparing analytical expressions for accuracy
Comparing analytical expressions for accuracy
30
35
30
25
25
Normalised tim e, Tr*Wn
Norm alis ed time, Tr*Wn
Actual Tr
Suggested
Actual Tr
Suggested
Used in most texts
Ref[6]
20
15
10
15
10
5
5
0
20
0
0
0.5
1
Zeta
Figure 7(c) comparing different analytical
expressions for TR with actual values.
1.5
0
0.5
1
1.5
Zeta
Figure 7(d) Curves of both actual and suggested settling
time.
5.2.9 Rise time TR.
Rise time TR is a measure of swiftness of response, it is the time required for the response curve to
reach from 10% to 90% , or 5% to 95% of its final value for critical and overdamped cases, or
0% to 100% of its final value for underdamped cases. An alternative measure is to represent the
rise time as the reciprocal of the slope of the step response at the instant that the response is equal
to 50% of its final value [3]. That is at delay time TD. The exact values of rise time for given
range of damping ratio, can be determined directly from the responses of Figure 5(a), or rise time
is found by solving Eq. (18) for the difference in time e.g. rise time for 10% to 90% is found by
solving Eq. (18) for the difference in time at c(t) = 0.9 and c(t) = 0.1. Also MATLAB code can be
written to return actual values of rise time, and plot it again given range of zetas, an example
code, for 10% to 90% is given next:
>>Q1=(y(:,:)>=0.1); Q2=(y(:,:)>=0.9); t0=[];t1=[];tr=[];
for i=1:length(zeta)
TQ1=t(Q1(:,i));TQ2=t(Q2(:,i)); t0(i)= TQ1(1); t1(i)= TQ2(1);
end
tr=t1-t0; plot(zeta,tr),hold
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
It is difficult to determine precise analytical expressions for rise time TR. Different approximate
formulae for the rise time appear in different texts. One reason for the different formulae is
because of different definitions of the rise time [7], as well as required accuracy. For
underdamped case; 0% to 100% of its final value, the rise time can be obtained by equating
Eq.(18) with unity and solve for time t, that is rise time:
c (t ) = 1 − e −ξωnT R (cos ωd T R +
ξ
1−ξ 2
sin ωd T R ) = 1
Since e −ξωnT R ≠ 0 , we have:
cos ωd T R +
tan ωd T R = −
TR =
ξ
1−ξ 2
1− ξ 2
sin ωd T R = 0 ⇒ tan ωd T R = −
ξ
ωn 1 − ξ 2
ω 
1
⇒TR =
tan −1  d 
ξωn
ωd
 ξωn 
π − φ π − tan −1 ( 1 − ξ 2 / ξ )
=
ωd
ωn 1 − ξ 2
→ 0 <ξ <1
( 48)
Where, referring to Figure 8 (a), ϕ is defined be the following Eqs.:
φ = cos −1 (ξ ) ⇔ φ = sin −1 ( 1 − ξ 2 ) ⇔ φ = tan −1 (
1− ξ 2
ξ
)
In the limit as ξ → 0 , this equation can be approximated as:
TR =
π −π / 2
π
=
ωn
2ωn
In the limit as ξ → 1 , this equation can be approximated as:
TR =
π −0
ωn 1 − ξ
2
=
π
ωn 1 − ξ 2
These equations imply that rise time increases as damping approaches unity. An approximation
techniques can be used to estimate approximate values; by plotting normalized time ωnTR versus
range of 0≤ ζ≤1.5, and then approximate the curve by a straight line or over the range of 0 < ζ < 1.
We first designate ωnTR as the normalized time variable and select a value for ζ. Using the
computer, we solve for the values of ωnTR that yield c(t) = 0.9 and c(t) = 0.1. Subtracting the two
values of ωnTR yields the normalized rise time, ωnTR, for that value of ζ, continuing in like
fashion with other values of ζ, and we obtain the results plotted in Figure 8(b)[4], for ωn=1, this
plot shows that increase in damping ratio leads to increase in the rise time that is not desirable .
Curve fitting can be applied to curve shown in Figure 8(b), to derive an approximate third order
approximation given by:
TR ≅
1.765ξ 3 − 0.417ξ 2 + 1.039ξ + 1
ωn
⇒ 0 < ξ < 0.9
( 49)
Quadratic approximation can result in the following expressions:
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
TR ≅
2.230ξ 2 − 0.078ξ 2 + 1.12
ωn
⇒ 0 < ξ < 0.9
( 50 )
2
TR ≅
2.917ξ − 0.4167ξ + 1
ωn
⇒ 0 <ξ <1
Referring to [3] the rise time Ts, for second order underdamped system, rise time can be
approximated as a straight line given by:
0.8 + 2.5ξ
TR ≅
ωn
( 51)
⇒ 0 < ξ <1
Referring to [4] the linear approximation of rise time is given by:
0.6 + 2.16ξ
TR =
ωn
⇒ 0.3 ≤ ξ ≤ 0.8
( 52 )
Referring to [5] rise time is given by:
TR =
2.2
( 53)
ξωn
Referring to [6] rise time is given by:
TR =
TR =
1.2 − 0.45ξ + 2.6ξ 2
ωn
4.7ξ − 1.2
ωn
⇒ ξ < 1.2
( 54 )
⇒ ξ > 1.2
These equations shows that rise time is proportional to ζ and inversely proportional ωn, Most of
these derived expression are rough approximations and mostly has huge deviation at actual
values, this is shown in Figure 8(c), analyzing actual curve, shown in Figure 8(b),shows that the
curve can be fit as ramp in some regions and of second order in others, applying curve fitting and
trial and error approaches, better and more accurate expressions, can be suggested for 0 ≤ ζ < 0.4,
for 0.4≤ ζ < 1.2, and for ζ > 1.2, suggested analytical expressions are given by:
TR =
TR =
TR =
1.2 − 0.2ξ + 3ξ 2
ωn
⇒ 0 < ξ < 0.4
1.26 − 0.51ξ + 2.58ξ 2
ωn
4.67ξ -1.2
ωn
⇒ 0.4 ≤ ξ < 1.2
( 55)
⇒ ξ > 1.2
Plotting actual rise time and a rise time obtained using suggested expressions, both versus
normalizes time, are shown in Figure 8(d), analysis of both plots show that the suggested
expressions match the actual values with maximum upper error of 0.06 seconds for 0.34 < ζ < 0.4
, and maximum upper error of 0.026 seconds for 0.6 < ζ < 0.7. The suggested expressions can be
used to analytically calculate rise time with error of ± 0.02 seconds.
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
5.2.10 Delay time TD, (Half final value time, T50):
Comparing analytical expressions for accuracy
Normalised time, Tr*Wn
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
Zeta
Figure8 (a) Definition of angle ϕ
Figure 8(b) Normalized rise time versus ζ, for second-order system
Comparing analytical expressions for accuracy
6
Comparing analytical expressions for accuracy
8
Actual Tr
5.5
Actual Tr
Suggested expression
7
5
6
Normalised Tr*Wn
Normalised Tr*Wn
4.5
5
4
3
2
3.5
3
2.5
1
0
4
2
0
0.5
1
1.5
1.5
Zeta
1
0
0.5
1
1.5
Zeta
Figure 8(c) plots of different approximation for
rise time.
Figure 8(d) Rise time plot of actual and using
suggested expressions.
It is defined as the time required for the response to reach half of final value the very first time
[2], Delay time can be determined directly from the responses of Figure 5(a), Another easier
approach is to plot normalized time ωnTD versus ζ, and then approximate the curve by a straight
line or approximate the curve over the range of 0 < ζ < 1. It is difficult to determine the exact
analytical expressions of the delay time, an approximation technique can be used, where we can
set Eq. (19) equal to 0.5 and solve for delay time t. based on the definitions of the delay time TD
and rise time TR, an analytical expression for delay time can be derived, also since the waveform
between 0.1c(s) to 0.9c(s) is not a linear line, the following expression with rough approximation
for delay time in terms of damping ratio and rise time, can be suggested:
T D = 0.67 *T R
( 56)
Softening this expression can be accomplished as follows, since response depends proportionally
on damping ratio and inversely on undamped natural frequency, also Figure 9 shows that the
delay time curve is of second order, adding these factors to Eq.(56), applying curve fittings and
trial and error method, the following expression for delay time in terms of damping ratio ζ ,
undamped natural frequency ωn and rise time TR, can be suggested:
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
can be suggested:
0.5.* ξ 2T R + 0.4ξ + 1.1
TD =
ωn
( 57 )
Applying the same approach, the following expression for delay time, in terms of damping ratio ζ
and undamped natural frequency ωn, can be suggested:
TD =
(1.2378 − 0.156ξ + 0.592ξ .2 )
( 58 )
ωn
Referring to [3], the delay time TD, for second order underdamped system, can be approximated
as a straight line given by:
TD ≅
1 + 0.7ξ
ωn
→ 0 <ξ <1
( 59 )
A better approximation by using a second-order approximation is given by:
TD ≅
1.1 + 0.125ξ + 0.469ξ 2
ωn
→ 0 <ξ <1
( 60 )
Equations show that delay time is proportional to ζ and inversely proportional ωn , a comparison
between actual and analytical values of delay time, using different expressions are shown in
Figure 9(a) . A comparison between delay time values, actual and obtained analytically obtained
using suggested expressions given by Eqs. (57) and (58), are shown in Figure 9(b), plots show
that both suggested expressions can be used to calculate delay time with overage error of 0.005
seconds, also expression based on rise time and damping ratio given by Eqs. (57) gives more
accurate results over given range of damping ratio, where expression given by Eqs. (58) gives
results with increasing error for all 0< ζ< 0.39.
Comparing analytical expressions for accuracy
2.5
Actual Delay time
Suggested, based on zeta & Tr
Suggested, based on zeta & Wn
Comparing analytical expressions for accuracy
Normalised time, Td*Wn
Actual Td
Suggested, based on zeta & Tr
Suggested, based on zeta & Wn
Ref.[3] first order
Ref.[3] second order
2
1.5
1
0
0.5
1
1.5
Zeta
Figure 9(a). Comparison between actual and
analytical delay time, using different expressions.
0
0.5
1
1.5
Zeta
Figure 9(b) a comparison between delay time actual and
analytical values.
Using the derived expressions for second order systems to determine the performance
specifications of second order PMDC motor system described by Eq. (9), will result in response
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International Journal of Information Technology, Control and Automation (IJITCA) Vol.3, No.1, January 2013
plot shown in Figure 10, as well as the following performance specifications; Poles = -0.1500 ±
0.4555i, T =6.6667 sec, 5T= 33.3335 sec, Mp =0.3554, OS% =35.5403%, Minima time
=13.7934 sec, Decay ratio =0.1263, Damping factor =7.9170, Period =13.7934, Frequency of
oscillation =0.4555, Settling time =26.0800 sec, Rise time =2.9837 sec , Delay time =2.6000 sec.
PMDC Second order step response
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
Time (sec)
Figure 10 Performance specifications of second order PMDC motor step response.
6. CONCLUSIONS
Most used formulae and expressions for performance specifications in texts lack accuracy, since
most are rough approximation of actual values. More accurate analytical expressions for most
performance specifications, with minimum deviation at actual values, were derived, introduced
and tested. The correctness and accuracy of derived suggested expressions were verified using
MATLAB. The derived analytical expressions are only accurate for second-order systems with no
zeros and represent the essential qualities of higher-order systems with one or two dominant
poles. Suggested expressions are intended to be used in systems dynamics analysis, design,
control and related sciences, as well as for the application in educational process.
7. REFERENCES.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Ashish Tewari, Modern Control Design with MATLAB and SIMULINK, John Wiley and sons,
LTD, 2002 England
Katsuhiko Ogata, modern control engineering, third edition, Prentice hall, 1997
Farid Golnaraghi Benjamin C.Kuo, Automatic Control Systems, John Wiley and sons INC .2010
Norman S. Nise control system engineering, Sixth Edition John Wiley & Sons, Inc,2011
Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini, Feedback Control of Dynamic
Systems, 4th ed., Prentice Hall, 2002.
http://courses.engr.illinois.edu/ece486/lab/estimates/estimates.html
Bill Goodwine, Engineering Differential Equations Theory and Applications, Springer 2011.
Dale E. Seborg, Thomas F. Edgar, Duncan A. Mellichamp ,Process dynamics and control,
second edition, Wiley 2004.
AUTHOR
Farhan Atallah Salem: B.Sc., M.Sc and Ph.D., in Mechatronics of production systems,
Moscow state Academy, He is author and co-author of books including automatic
control, Mechatronics systems design, CNC fundamentals, PLC fundamentals,
Industrial automation. Now he is ass. Professor in Taif University, Mechatronics
program, Dept. of Mechanical Engineering and gen. director of alpha center for
engineering studies and technology researches.
59