Chapter 5
Time-Domain Analysis
of Control Systems
Automatic Control Systems, 9th Edition
F. Golnaraghi & B. C. Kuo
Section 5- 0, p. 253
Overview
• Find and discuss transient and steady state time response
of a simple control system.
• Develop simple design criteria for manipulating the time
response.
• Look at the effects of adding a simple gain or poles and
zeros to the system transfer function and relate them to
the concept of control.
• Look at simple proportional, derivative, and integral
controller design concepts in time domain.
5-1
Section 5- 1, p. 253
Sections 5-1 and 5-2
5-1 Introduction
•
Steady-state response
Time response:
Transient response
5-2 Typical Test Signals for the Time Response of
Control Systems
•
Step-Function Input:
5-2
Section 5- 2, p. 255
Ramp- and Parabolic-Function Inputs
• Ramp-Function Input:
• Parabolic-Function Input:
5-3
Section 5- 3, p. 256
5-3 The Unit-Step Response and
Time-Domain Specification
Steady-state error: r  yss
r: reference input
5-4
Section 5- 4, p. 258
5-4 Steady-State Error
• Definition: error 
steady-state error 
• Unity feedback system (H(s) = 1) 
5-5
Section 5- 4, p. 259
Velocity Control System
• A step input is used to control the system output that
contains a ramp in the steady state.
System transfer function:
Kt = 10 volts/rad/sec
• Closed-loop transfer function:
• Unit step response ( R(s)=1/s ):
• Steady-state error:
Kt = 10 volts/rad/sec  reference signal is 0.1t, not 1 (unit step input).
5-6
Section 5- 4, p. 260
Systematic Study of Steady-State Error
Three Types of Control Systems
• System with unity feedback: H(s) = 1
e=ry
• System with nonunity feedback,
but H(0) = KH = constant
 e = r/KH  y
• System with nonunity feedback
and H(s) has zeros at s = 0 of order N.
 E ( s) 
1
R( s)  Y ( s)
N
KH s
H (s)
K H  lim N
s 0 s
5-7
Section 5- 4, p. 260
Steady-State Error of Unity Feedback
• Steady-state error:
H ( s)  1
• ess depends on the number of poles G(s) has at s = 0.
System type (the type of the control system)
• Forward-path transfer function:
system type = j
5-8
Section 5- 4, p. 261
Unity Feedback with Step Func. Input
Step-function input: r (t )  Ru s (t )  R(s)  R / s
5-9
Section 5- 4, p. 262
Unity Feedback with Ramp Func. Input
Ramp-function input:
ess 
R
Kv
5-10
Section 5- 4, p. 263
Unity Feedback with Para. Func. Input
Parabolic-function input:
5-11
Section 5- 4, p. 264
Table 5-1
5-12
Section 5- 4, p. 265
Example 5-4-2
5-13
Section 5- 4, p. 265
Example 5-4-2 (cont.)
5-14
Section 5- 4, p. 266
Relationship between Steady-State Error
and Closed-Loop Transfer Function
State-State Error: H(0) = KH
•
Reference signal
• Error signal:
M ( s )  Y ( s ) / R( s )
• M(s) does not have any poles at s = 0:
5-15
Section 5- 4, p. 267
Steady-State Error: H(0)=KH
• Step-function input: R(s) = R/s
ess = 0 
or
• Ramp-function input: R(s) = R/s2
5-16
Section 5- 4, p. 268
Steady-State Error: H(0)=KH (cont.)
• Parabolic-function input: R(s) = R/s3
5-17
Section 5- 4, p. 268
Example 5-4-3
5-18
Section 5- 4, p. 269
Example 5-4-4
5-19
Section 5- 4, p. 269
Example 5-4-4 (cont.)
• Unit-step input: y(t)  1, ess 0
• Unit-ramp input: y(t)  t0.8, ess0.8
• Unit-parabolic input: y(t)0.5t2 0.8t11.2, ess0.8t+11.2
5-20
Section 5- 4, p. 270
Example 5-4-5
• Unit-step input:
 0.5
5-21
Section 5- 4, p. 270
Example 5-4-5 (cont.)
• Unit-ramp input:
 0.4
error:
• Unit-parabolic input:
error:
 0.4t + 2.6  
5-22
Section 5- 4, p. 270
Steady-State Error:
H(s) Has Nth-Order Zero at s = 0
• Reference signal:
R(s)/KHsN
K H  lim{H ( s) s N }
s 0
• Error signal:
R(s) = R/s
5-23
Section 5- 4, p. 271
Example 5-4-6
r (t )  tus (t ) / K H  0.5tus (t )
5-24
Section 5- 4, p. 272
Steady-State Error Caused by
Nonlinear System Elements
5-25
Section 5- 4, p. 273
Steady-State Error Caused by
Nonlinear System Elements (cont.)
5-26
Section 5- 5, p. 274
5-5 Time-Response of
a Prototype First-Order System
time constant
5-27
Section 5- 6, p. 275
5-6 Time-Response of
a Prototype Second-Order System
G(s)
Characteristic equation:
R(s) = 1/s (unit-step input)
5-28
Section 5- 6, p. 276
Unit-Step Responses
Figure 5-14 Unit-step response
of the prototype 2nd-order system
with various damping ratios.
5-29
Section 5- 6, p. 277
Damping Ratio and Damping Factor
• Characteristic equation:
• Unit-step response:
•  controls the rate of rise or decay of y(t).
 damping factor ( 1/  time constant )
• Damping ratio:
5-30
Section 5- 6, p. 278
Natural Undamped Frequency
• Natural undamped frequency: n
• Damped (or conditional) frequency: 
   cos1 
5-31
Section 5- 6, p. 279
Figure 5-16
5-32
Section 5- 6, p. 280
Classification of System Dynamics
5-33
Section 5- 6, p. 281
Step-Response Comparison (Fig. 5-18)
overdamped
critically overdamped
underdamped
5-34
Section 5- 6, p. 281
Step-Response Comparison (cont.)
undamped
negative overdamped
negative overdamped
5-35
Section 5- 6, p. 281
Maximum Overshoot
dy (t )
0
dt
y() is maximum when   1.
t = ,
The first overshoot is
the maximum overshoot
n=1
5-36
Section 5- 6, p. 282
Maximum Overshoot (cont.)
dy (t )
0
dt
5-37
Section 5- 6, p. 283
Maximum Overshoot (cont.)
The first overshoot (n = 1) is the maximum overshoot
5-38
Section 5- 6, p. 283
Maximum Overshoot (cont.)
5-39
Section 5- 6, p. 283
Delay Time
The time required for the step response to reach 50% of its
final value  Set y(t) = 0.5 and solve for t
• Approximation:
5-40
Section 5- 6, p. 284
Rise Time
The time for the step response to reach from 10 to 90% of its
final value.
• Approximation:
5-41
Section 5- 6, p. 285
Settling Time: 0 <  < 0.69
The time required for the step response to decrease and stay
within a specified percentage (2% or 5%) of its final value.
5-42
Section 5- 6, p. 287
Settling Time:  > 0.69
5-43
Section 5- 6, p. 287
Settling Time
5-44
Section 5- 7, p. 289
5-7 Speed and Position Control
of a DC Motor
• Open-loop response:
5-45
Section 5- 7, p. 290
Speed of Motor Shaft: Open-Loop
• La is very small
 e = La/Ra (motor electric-time constant) is neglected.
Keff = Km/(RaB+KmKb): motor gain constant
m = RaJm/(RaB+KmKb): motor mechanical time constant
5-46
Section 5- 7, p. 291
Time Response: Open-Loop
Superposition:
• TL(s)=0 (no disturbance and B=0) and Va(s)=A/s:
 (t)  A/Kb
• TL(s)=D/s and Va(s)=A/s :
 (t)  A/Kb RaD/KmKb
5-47
Section 5- 7, p. 291
Closed-Loop Response
5-48
Section 5- 7, p. 292
Speed of Motor Shaft: Closed-Loop
• La = 0 
• in = A/s and TL = D/s:
1/c
c: system mechanical-time constant
steady-state response:
5-49
Section 5- 7, p. 292
Position Control: Open-Loop
• (s) = (s)/s
• Open-loop transfer function:
for small La 
5-50
Section 5- 7, p. 293
Position Control: Closed-Loop
e = La/Ra  0
for small La
5-51
Section 5- 8, p. 293
5-8 Time-Domain Analysis of
a Position-Control System
• Attitude-control system of an aircraft:
 Forward path transfer function
(a third-order system)
electrical time constant:
mechanical time constant:
(a second-order approximation)
5-52
Section 5- 8, p. 294
Transfer Functions
• Forward-path transfer function:
• Closed-loop transfer function of unity-feedback control
system:
5-53
Section 5- 8, p. 295
Unit-Step Response
Closed-loop response
Forward-path
transfer function:
 = 0.2
 = 0.707
type 1 system
Step-error constant:
  1.0
ess = 0
5-54
Section 5- 8, p. 296
Comparison of Performance
5-55
Section 5- 8, p. 296
Root Loci
• Roots:
5-56
Section 5- 8, p. 298
Unit-Ramp Response
Closed-loop response
Forward-path
transfer function:
Ramp-error constant:
5-57
Section 5- 8, p.300
Time Response of a 3rd-Order System
• La = 0.003H  Forward-path transfer function:
Closed-loop transfer function:
Characteristic equation:
Poles:
5-58
Section 5- 8, p.302
Root Loci
5-59
Section 5- 8, p.303
Unit-Step Responses
s  3408.3,  j1097.3
5-60
Section 5- 9, p.304
5-9 Basic Control System and Effects of
Adding Poles and Zeros to Transfer Funcs.
Add a pole at s = 3008
Shift the pole at 361.2 to 400.26
Effect of adding a
pole at s = 3008:
push and bend the
complex-conjugate
portion of the root
loci of the 2nd-order
system toward the
right-half s-plane.
5-61
Section 5- 9, p.306
General Effect of the Addition of a Pole
• Forward-path transfer function: Add a new pole at s = 1/Tp
• Closed-loop transfer func.:
Increasing the maximum
overshoot of the closedloop system.
5-62
Section 5- 9, p.306
General Effect of the Addition of a Pole
As the value of Tp
increase, the pole 1/Tp
moves closer to the
origin in the s-plane, and
the maximum overshoot
increases. The rise time
also increases.
5-63
Section 5- 9, p.307
Addition of a Pole to the
Closed-Loop Transfer Function
• Closed-loop transfer function: Add a new pole at s = 1/Tp
The maximum overshoot decreases.
The rise time increases.
5-64
Section 5- 9, p.308
Addition of a Zero to the
Closed-Loop Transfer Function
• Closed-loop transfer function: Add a new zero at s = 1/Tz
The maximum overshoot increases.
The rise time decreases.
5-65
Section 5- 9, p.309
Effect of the Addition of a Zero
Reduce the rise time
Increase the maximum overshoot.
5-66
Section 5- 9, p.310
Addition of a Zero to the
Forward-Path Transfer Function
• Forward-path transfer function: Add a new zero at s = 1/Tz
Reduce the maximum overshoot
Improve the damping
5-67
Section 5- 10, p.311
5-10 Dominant Poles and Zeros of
Transfer Functions
• Insignificant poles: the real part of a pole is at least 5 to 10
times that of a dominant pole or a pair of complex
dominant poles.
5-68
Section 5- 10, p.313
Summary of Effects of Poles and Zeros
5-69
Section 5- 10, p.313
Relative Damping Ratio and
Neglecting the Insignificant Poles
• Relative damping ratio
 a system is higher than the 2nd order
dominant poles: 1j1  relative damping ratio = 0.707
• Neglecting the insignificant poles with consideration of the
steady-state response
s 10  1
5-70
Section 5- 11, p.314
5-11 Basic Control Systems
Utilizing Addition of Poles and Zeros
Example 5-11-1: Control System with PD Controller
• Forward-path transfer function:
 adding a simple zero at s = KP/KD
5-71
Section 5- 11, p.315
Example 5-11-1 (cont.)
• Closed-loop transfer function:
• Effects of PD controller:
– Adding a zero at s = KP/KD to the closed-loop transfer function.
– Increasing the damping term from 2 to 2+2KD.
• Characteristic equation:
5-72
Section 5- 11, p.316
Example 5-11-1 (cont.)
Upon increasing KD, the general
conclusion is that the PD controller
decreases the maximum overshoot,
the rise time, and the settling time.
5-73
Section 5- 11, p.317
Example 5-11-2
Control System with PI Controller:
• Forward-path transfer function:
• Effects of PI controller:
system type is increased by 1
– Adding a zero at s = KI/KP to the forward-path transfer function
– Adding a pole at s = 0 to the forward-path transfer function
5-74
Section 5- 11, p.317
Example 5-11-2 (cont.)
• Closed-loop transfer function:
• Characteristic equation:
Routh’s test  Stable: 0 < KI/Kp < 13.5
• The dominant pole: s = 1
• Approximation:
5-75
Section 5- 11, p.319
Example 5-11-2 (cont.)
5-76