Control Systems
amarafie
Chapter5.pdf
ELEG 421 Control Systems
Transient and Steady State
Response Analyses
Dr. Ashraf A. Zaher
American University of Kuwait College of Arts and Science
Department of Electrical and Computer Engineering
Layout
2
Objectives This chapter introduces the analysis of the time response of different control systems under different scenarios. Only first and second order systems will be considered in details using analytical and numerical methods. Extension to higher order systems will be developed. Both transient and steady state responses will be evaluated. Stability analysis will be analyzed for different kinds of feedback, while investigating the effect of both proportional and derivative control actions on the performance of the closed-loop system. Finally systems types and steady state errors will be calculated for unity feedback.
Outcomes By the end of this chapter, students will be able to: evaluate both transient/steady state responses for control systems, analyze the stability of closed-loop LTI systems, investigate the effect of P and I control actions on performance, and understand dominant dynamics of higher order systems.
Dr. Ashraf Zaher
Introduction
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Test signals Transient response Steady state response Analytical techniques, and Numerical (simulation) techniques. Stability (definition and analysis methods), Relative stability, and Effect of P/I control actions on stability and performance. Summary of the used systems: First order systems, Second order systems, and Higher order systems.
Dr. Ashraf Zaher
Test Signals
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Impulse function: Used to simulate shock inputs, Laplace transform: 1.
Step function: Used to simulate sudden disturbances, Laplace transform: 1/s.
Ramp function: Used to simulate gradually changing inputs, Laplace transform: 1/s2.
Sinusoidal function(s): Used to test response to a certain frequency, Laplace transform: s/(s2+ω2) for cos(ωt) and ω/(s2+ω2) for sin(ωt).
White noise function: Used to simulate random noise, It is a stochastic signal that is easier to deal with in the time domain.
Total response:
C(s) = R(s)*TF(s) = Ctr(s) + Css(s) → c(t) = ctr(t) + css(t)
Fundamentals
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Definitions: Zeros (Z) of the TF Poles (P) of the TF Transient Response (Natural) Steady State Response (Forced) Total Response
Limits: Initial values Final values
Systems (?Zs): First order (one P) Second order (two Ps) Higher order!
More: Stability and relative stability Steady state errors (unity feedback)
First Order Systems
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TF: T: time constant Unit Step Response:
1 1
)( )(
+ =
TssR sC
)/1( 11
1 1
1 11
)( TssTs
T sTss
sC +
−= +
−= +
=
Ttetc /1)( −−=
632.01)( 1 =−== −eTtc
T e
Tdt tdc Tt
t
11)( / 0
== − =
01)0( 0 =−== etc
11)( =−=∞= −∞etc
First Order Systems
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Unit Ramp Response:
)/1( 1
1 11
)( 22 Ts T
s T
sTss sC
+ +−=
+ =
0)0( ==tc Tttc −=∞= )(
TtTeTttc /)( −+−=
Unit Impulse Response:
)/1( 11
1 1
)( TsTTs
sC +
= +
=
Ttc /1)0( == 0)( =∞=tc
Tte T
tc / 1
)( −=
First Order Systems
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Example: RC Circuit T = RC Input is ei and output is e0 Transient Response (Natural) Steady State Response (Forced) Total Response
Impulse: Step: Ramp: Other signals: current and power Effect of initial conditions
RCti e RC e
te /0 )( −=
)1()( /0 RCt
i eete −−=
)]1([)( /0 RCt
i eRCtete −+−=
)()()( 0 tetete iR −=
dt tde
cti R
te ti cR
)( )(
)( )( 00 ===
Second Order Systems
9 Dr. Ashraf Zaher 9 Dr. Ashraf Zaher
TF: ζ : damping ratio ωn: natural frequency
22
2
2)( )(
nn
n
sssR sC
ωζω ω
++ =
Characteristic equation:
Effect of damping: ζ = 0: no damping
0 < ζ < 1: under damping
ζ = 1: critical damping
ζ > 1: over damped
ζ < 0: unstable!
22 2 nn ss ωζω ++
2nd order: under damped case
10 Dr. Ashraf Zaher 10 Dr. Ashraf Zaher
Step Response: Time Response:
22
2
2 1
)( nn
n
sss sC
ωζω ω
++ =
222222 )()( )(1
2 21
)( ddnn
n
ss s
sss s
s sC
ωσ σ
ωσ σ
ωζω ζω
++ −
++ +
−= ++
+ −=
− −
− −=
− +
− −=
− +−=
− −
− −
−
2
1
2
2 1
2
2
1 tancos
1 1
1 tansin
1 1
sin 1
cos1)(
ζ ζ
ω ζ
ζ ζ
ω ζ
ω ζ
ζ ω
σ
σ
σ
t e
t e
ttetc
d
t
d
t
dd t
21 ζωω −= nd
nζωσ =
2nd order: under damped case
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Transient Response Specifications: Delay time, td Rise time, tr Peak time, tp Maximum overshoot, Mp Settling time, ts
πζζ
σσ
ω π ω βπ
−−
=
==
=
− =
21/
(5%) 3
(2%) 4
eM
t
t
t
p
s
d p
d r
2nd order: under damped case
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2nd order: critically damped case
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Step Response: Time Response:
2
2
)( 1
)( n
n
ss sC
ω ω +
=
22 )()( 11
)( 21
)( n
n
nn
n
ssss s
s sC
ω ω
ωω ω
+ −
+ −=
+ +
−=
( )tetc ntn ωω +−= − 11)(
0)0( ==tc 1)( =∞=tc
0=dω
Properties: No overshoot No damped frequency (no oscillation):
Initial and final values
2nd order: over damped case
14 Dr. Ashraf Zaher 14 Dr. Ashraf Zaher
Step Response: Time Response:
))(( 1
)( bsas
ab s
sC ++
=
bs aba
as abb
s sC
+ −
+ + −
−= )/()/(1
)(
( )btat aebe ab
tc −− − −
−= 1
1)(
0)0( ==tc 1)( =∞=tc
0=dω
Properties: No overshoot Sluggish (slower than the critically damped case):
Initial and final values
nab ω=
nba ζω2=+
( )1, 2 −±= ζζωnba
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2nd order: Comparison
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2nd order: Comparison
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2nd order: Comparison
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2nd order: Comparison
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2nd order: Impulse Response Time Response: Response type: Under damped Critically damped Over damped
22
2
2 )(
nn
n
ss sC
ωζω ω
++ =
20 Dr. Ashraf Zaher
2nd order: Impulse Response Under damped: Response:
Maximum: Critically damped: Response:
Maximum: Over damped
tetc d tn ω
ζ ω σ sin
1 )(
2
−
− =
t n
ntetc ωω −= 2)(
−
− =
−+−
−−− tt
n nn eetc ωζζωζζ
ζ ω 11
2
22
12 )(
d
t ω β
=max
− −
= 21
max ζ
βζ
ω ec n
n
t ω 1
max = nnec ωω 368.0 1
max == −
tmax = ? cmax = ?
21 Dr. Ashraf Zaher
Higher Order Systems Response C(s): Time Response: Break down the system into smaller systems (1st and 2nd) Dominant behavior (simplification) Use Laplace transform table Computer Simulations Matlab/Simulink
))...()(( ))...()((
)(1 )(
)( 21
21
n
m
pspsps zszszsK
sH sG
sC +++ +++
= +
=
Stability
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Definition: A system is stable if every bounded input produces a bounded output. In short, this is called BIBO stability. If an bounded input produces an unbounded output, the system is unstable. Stability condition(s) for LTI systems:
The natural response approaches zero as time approaches infinity All exponential terms have negative exponents Poles of the TF should be in the left half-plane
How to find the poles of the TF: Factorization (easy vs. difficult polynomials) Using Matlab (symbolic variables!) Routh-Hurwitz criterion
Routh’s Stability Criterion
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Closed-loop TF:
C/Cs equation:
Location of the system poles Roots of the c/cs equation
)( )(
...
... )( )(
1 1
10
1 1
10
sA sB
asasasa bsbsbsb
sR sC
nn nn
mm mm
=
++++ ++++
= −
− −
−
0... 1 1
10 =++++ − −
nn nn asasasa
Routh’s Stability Criterion
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Polynomials with +ve coefficients: Number of roots of the c/cs equation with +ve real parts = number of changes in sign of the coefficients of the first column of the array.
Routh’s Stability Criterion
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Example:
Two sign changes unstable! Matlab check: roots of s3 + 10 s2 + 31 s + 1030 = -13.4136 , 1.7068 ± 8.5950i
Routh’s Stability Criterion
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Special case I: Zero only in the first column Two sign changes unstable!
)5083.13429)(7020.15088.0)(6681.1( 10
35632 10
)( )(
2345
isiss
ssssssR sC
±−±++ =
+++++ =
Routh’s Stability Criterion
27 Dr. Ashraf Zaher
Special case II: Entire row is zero
Auxiliary polynomial:
Roots of P(s) No sign changes stable!
)2)(2)(7( 10
5684267 10
)( )(
2345
isiss
ssssssR sC
±±+ =
+++++ =
86)( 24 ++= sssP
0124 )( 3 =+= ss
ds sdP
)2)(2(86 24 isisss ±±=++
Relative Stability
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Definition: To find how close is the closed-loop system to instability.
How to find it? Shift the imaginary axis of the s-plane into the left Re-check for stability in the p-plane Repeat until the system is found to be unstable Find the final value of a
aps −=
Or, simply Factorize the closed-loop c/cs
equation and find the real-part of the nearest pole to the imaginary axis.
Routh’s Criterion Applications
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s4 1 3 k s3 3 2 0 s2 (7/3) k s1 2-(9/7)k s0 k
C/Cs equation: kssss ++++ 233 234
Range of k: 9
14 0 << k
Routh’s Criterion Applications
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s3 1 77 s2 18 k s1 (1386-k)/18 s0 k
C/Cs equation: ksss +++ 7718 23
Range of k: 13860 << k
Routh’s Criterion Applications
31 Dr. Ashraf Zaher
)3)(2( )20(
)( ++
+ =
sss sK
sG
s3 1 6+k s2 5 20k s1 6-3k s0 k
C/Cs equation: kskss 20)6(5 23 ++++
Range of k: 20 << k
Effect of I Control Action
32 Dr. Ashraf Zaher
P control action for plants with no internal integrators results in an offset that can be eliminated by I control action. Figure (a) shows that the control signal has a steady state value, even after the error goes to zero. Thus, I control action is used to remove offsets. In contrast, without I control action, P control actions results in a zero control signal, after the error vanishes. I control actions increase the order of the closed-loop system; thus, it might lead to instability and/or oscillatory performance
Effect of P Action (1st order systems)
33 Dr. Ashraf Zaher
)(
1 1
1 )(
)(1 1
)(
)(1 1
)( )(
1 )(
)()( )( )(
1
)(
sR
Ts K
sR sG
sE
sGsR sC
sR sCsR
sR sE
Ts K
sG
+ +
= +
=
+ =−=
− =⇒
+ =
KKTs Ts
ssEtee
sKTs
Ts sE
sstss + =
++ +
===
++ +
=
→→∞→ 1 1
1 1
lim)(lim)(lim
1 1
1 )(
00
Effect of I Action (1st order systems)
34 Dr. Ashraf Zaher
KTss Tss
sR sE
KTss K
sR sC
++ +
=
++ =
)1( )1(
)( )(
)1()( )(
0 1
1 )1(
lim)( 2
0 =
++ +
== → sKTs
Tss ssEe
sss
P, I, and PI Controllers (1)
35 Dr. Ashraf Zaher
p
p
KbsJssD sC
sD sE
KbsJssD sC
++ −=−=
++ =
2
2
1 )( )(
)( )(
1 )( )(
p
dd
p sss K
T s
T KbsJs
s ssEe −=
++ −
== → 20
lim)( p
d ssss K
T ec =−=
s T
sDTtd
sRtr
d d =⇒=
=⇒=
)( )(
0)( 0)(
P, I, and PI Controllers (2)
36 Dr. Ashraf Zaher
)/()( )(
)( )(
)/()( )(
23
23
ipp
ipP
TKsKbsJs s
sD sC
sD sE
TKsKbsJs s
sD sC
+++ −=−=
+++ =
0 )/(
lim)( 23 2
0 =
+++ −
== → s
T TKsKbsJs
s ssEe d
ipp sss
0=−= ssss ec
s T
sDTtd
sRtr
d d =⇒=
=⇒=
)( )(
0)( 0)(
P, I, and PI Controllers (3)
37 Dr. Ashraf Zaher
KbsJssD sC
sD sE
KbsJs s
sD sC
++ −=−=
++ =
23
23
1 )( )(
)( )( )( )(
0lim)( 23 2
0 =
++ −
== → s
T KbsJs
s ssEe d
sss 0=−= ssss ec
s T
sDTtd
sRtr
d d =⇒=
=⇒=
)( )(
0)( 0)(
Wrong Equations (the closed-loop system is unstable):
Effect of D Control Action
38 Dr. Ashraf Zaher
When P control action is used – alone – in the plant on the left figure, the resulting closed-loop will result in an oscillatory response. D control action has an anticipatory behavior; thus, it damps out the oscillations via responding to the rate of change of the error. D control action is used with either P, I control actions, or both, as it can’t remove offsets, as shown on the figure on the right.
PD Control Action
39 Dr. Ashraf Zaher
pd
pd
dp
KsKBJs BJss
sR sE
KsKBJs sKK
sR sC
+++ +
=
+++
+ =
)( )(
)( )(
)()( )(
2
2
p ss
ss
K B
ettr
etr
=⇒=
=⇒=
)(
0 1)(
p
dpd
JK KB
ζ J
K s
J KB
s 2
)(2 +=⇒+
+ +
Unity Feedback Systems
40 Dr. Ashraf Zaher
)(1 )(
lim)(lim)(lim
)(1 1
)( )(
1 )( )(
)(1 )(
)( )(
00 sG ssR
ssEtee
sGsR sC
sR sE
sG sG
sR sC
sstss + ===⇒
+ −=−=
+ =
→→∞→
Steady State Errors (unity feedback)
41 Dr. Ashraf Zaher
System type: the value of N in the above equation
)1)...(1)(1( )1)...(1)(1(
)( 21 +++
+++ =
sTsTsTs sTsTsTK
sG p
N mba
)(lim
)(lim
)0()(lim
2
0
0
0
sGsK
ssGK
GsGK
sa
sv
sp
→
→
→
=
=
==
Summary
42 Dr. Ashraf Zaher
First and second order systems can be effectively used to model the majority of existing control systems. Thus, understanding their performance for various input is very important. Stability analysis is very crucial for the design of control systems. Routh’s criterion is a powerful mathematical tool that can be used for such purpose. Transient and steady state responses can be evaluated for higher order systems via breaking them down into smaller first and second order subsystems. Knowing the systems type can simplify the determination of steady state errors for different control systems.
What is next? Analysis and design using root-locus, Analysis and design using frequency response, PID controllers.
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