Chapter 7 Stability and Steady-State Error Analysis
§ 7.1 Stability of Linear Feedback Systems
§ 7.2 Routh-Hurwitz Stability Test
§ 7.3 System Types and Steady-State Error
§ 7.4 Time-Domain Performance Indices
§ 7.1 Stability of Linear Feedback Systems (1)
• Basic Concepts:


(1) Equilibrium States, x  Constant (Usually x  0)
 
  
Nonlinear T - I System: x  F(x)  0  F(x)  0, Multiple equilibriu m states



Linear T - I System: x  Ax  0  Ax  0, Single equilibriu m state
EX : x  ax  bx  0, State  space form: x  x1 , x 1  x2
 x 1 
x   0  x  0, x  0 ,
 2
Since x  0 , x  0  x  0 in equilibriu m
(2) Stable System
1   x1 
 x 1   0

 x   b  a  x 
 2 
 2 
System response can restore to initial equilibrium state under small
disturbance.
(3) Meaning of Stable System
Energy sense – Stable system with minimum potential energy.
Signal sense – Output amplitude decays or grows with different meaning.
Lyapunov sense – Extension of signal and energy sense for state evolution
in state space.
§ 7.1 Stability of Linear Feedback Systems (2)
• Plant Dynamics:
Regular pendulum (Linear)
Inverted pendulum (Linear)
m
m
Equilibriu m State

l

  0

  0
g
m
m
b , b>0
Minimum potential energy
  b   g   0
ml 2
l
Positive spring ef fect
Pole - Zero diagram
b
g
( 2  2.0 ,  3.0)
ml
l
Natural response

0
l
b
b>0
g
Maximum potential energy
  b   g   0
ml 2
l
Negative spring effect
j
Pole - Zero diagram
b
g
( 2  2.0 ,  3.0)
ml
l
1/3
2

-1
 2
Natural response
1.0
j
1/3

-3
1

0
1.0
t
t
§ 7.1 Stability of Linear Feedback Systems (3)
• Closed-loop System:
d(t)
r(t) +
eb
y(t)
G(s)
b
H(s)
Natural behavior of a control system, r(t)=d(t)=0
-1
ea
G(s)
y(t)
H(s)
Equilibrium state ,ea  0 , y  0 ,y  0 , 
Initial relaxation system, I.C.=0
Characteristic equation and closed - loop poles
G(s)
N(s)

1  G(s)H(s) D(s)
Poles : D(s)  0
T(s) 
No general algebraic solution for 5th-order and above polynomial equation
(Abel , Hamilton)
§ 7.1 Stability of Linear Feedback Systems (4)
• Stability Problems:
Stabilization of unstable system
m
l

m
g
Destabilization Effect on stable system
Controller
and
Driver
, 
F
T
l
Controller
and
Driver
, 
G(s) : Unstable plant
Closed-loop : Stable
g

m
m
G(s) : stable plant
Closed-loop : Unstable
§ 7.1 Stability of Linear Feedback Systems (5)
• Stability Definition
(1) Asymptotic stability
Stable system if the transient response decays to zero
y(n) ( t )  an1y(n1) ( t )        a0 y( t )  0
I.C. y(0), y(1) (0) ,       , y(n1) (0)
lim y( t )  0
t 
(2) BIBO stability
Stable system if the response is bounded for bounded input signal
t
y(t )   g(t  ) u() d
0
u(t )  M  y(t )  N
The impulse response of a system is absolutely integrable.
g( t )
t
§ 7.1 Stability of Linear Feedback Systems (6)
(3) S-domain stability
System Transfer Function : T(s)
Stable system if the poles of T(s) all lies in the left-half s-plane.
j
stable
poles
Unstable
poles

Marginally stable/unstable
The definitions of (1), (2), and (3) are equivalent for LTI system.
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (1)
• Characteristic Polynomial of Closed-loop System
D(s)  ansn  an1sn1        a1s  a0, an  0
Hurwitz polynomial
All roots of D(s) have negative real parts.
stable system
Hurwitz’s necessary conditions: All coefficients (ai) are to be positive.
Define
D0 (s)  ansn  an2sn2   
D1(s)  an1sn1  an3sn3   

D0 (s)
1
 1s 
1
D1(s)
 2s 
3s 
, 1 
an
a
b
,  2  n1 ,  3  1 ,   
an1
b1
c1

Note: Any zero root has been removed in D(s).
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (2)
• Routh Tabulation (array)
an  2 an  4   
s n an
sn1 an1 an3 an5   
  1  an an 2
n2

b1  
, b2
b3   
s
b2
b1
a
a
a
n3
 n1  n1
c3   
c2
sn  3 c 1



s1
g1
h1
0
s0
  1  an an 4

 

a
a
a
n5
 n1  n1
an 5
an 3
  1 a
  1 a
c 1    n1
, c 2    n1

b
b
b
b
b
b
3
2
 1 1
 1 1

0
• Routh-Hurwitz Stability Criterion
(1) The polynomial D(s) is a stable polynomial if i are all positive, i.e.
an, an1, b1, c1,    h1 are all positive.
(2) The number of sign changes in an, an1, b1, c1,    h1 is equal to the
number of roots in the RH s-plane.
(3) If the first element in a row is zero, it is replaced by a small ε, ε >0,
and the sign changes when   0 are counted after completing the
array.
(4) If all elements in a row are zero, the system has poles in the RH plane
or on the imaginary axis.
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (3)
For entire row is zero
Identify the auxiliary polynomial
The row immediately above the zero row.
The original polynomial is with factor of auxiliary polynomial.
The roots of auxiliary polynomial are symmetric w.r.t. the origin:
j
4
3
2
possible cases (1~4)
of poles distribution
1

§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (4)
Ex: For a closed-loop system with transfer function T(s)
T(s)
T(s) 
N(s)
, order of D(s)  order of N(s)
D(s)
D(s)  a3s3  a2s2  a1s1  a0 , ai  0
Routh Array
s3
s2
s1
s0
a3
a2
a 2a1  a0a3
a2
a1
a0
0
a0
Stability condition : a0  0,
a2a1  a0a3
 0  a2a1  a0a3
a2
Ex: Find stability condition for a closed-loop system with
characteristic polynomial as s5  bs4  cs3  ds2  es  1  0
Sol: b, c, d, e  0, bcd  b  d2  b2e
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (5)
Ex: For a colsed-loop system with characteristic polynomial
D(s)  s5  2s4  3s3  6s2  5s  3
Determine if the system is stable
Sign of firstcolumn for   0,   0
Sol: Routh Table
s5

s4

7
2
s3

3
s2

s1

s0

s5
s4
1
2
3
6
s3
0ε
s2
6ε - 7
ε
42ε - 49 - 6ε 2
12ε -14
3
s1
s
0
5
3
tw osign changes  tw opoles in RHP
 unstable system
Ex: For D(s)  s3  2s2  4s  8, determine if the system is stable
1
Sol: Routh Table
D(s)  (2s2  8)( s  1)
s3
1
4
s2
2
8
s1
s
0
0
2
 (s2  4)(s  2)
2
 Auxiliary eq. : D(s)  2s  8
 (s  2j)(s  2j)(s  2)
poles : s  -2j, 2j, - 2
 stable system
j
2

2
2
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (6)
• Absolute and Relative Stability
Absolute Stability
Relative Stability
j
p-coordinate
j
s-coordinate
s=0



stability margin
Characteristic equation
D(s)  0
R-H Test on D(s)
Characteristic equation
set s  p - ,   0
 D(s)  D(p  )  D’ (p)  0
R-H Test on D ’(p)
§ 7.3 System Types and Steady-State Error (1)
• Steady-state error for unity feedback systems
R(s) +
E(s)
G(s)
Y(s)
Open - loop T.F. G(s)
Closed - loop T.F. T(s)
Error response
R(s)
T(s)
Y(s)
+
E(s)
E(s)  [1  T(s)]R(s)
Steady  state error : lim e( t )  e s.s.
t 
Use of final - value theorem :
(1) T(s) is stable
 1 
R(s)
(2) e( )  lim s[1  T(s)]R(s)  lim s 

s 0
s 0
1  G(s) 
For nonunity feedback systems
+
G(s)
H(s)

+
G'(s)
G' 
G
1  G(H-1)
§ 7.3 System Types and Steady-State Error (2)
• Fundamental Regulation and Tracking Error
Regulation s.s. error
1
R(s)  , Step input
s
1
e( ) 
1  lim G(s)
1
t
r(t)
s0
1
1  G( s)
e(t)
1
1  G(s)
e(t)
Tracking s.s. error
1
, Ramp input
s2
1
e() 
lim sG(s)
R(s) 
s0
1
t
r(t)
§ 7.3 System Types and Steady-State Error (3)
• Open-loop System Types
K(sm  bm1sm1  ......  b1s  b0 )  Tds
G(s)  N q
e , N  q  m for Td  0
q1
S (s  aq1s  ......  a1s  a0 )
a0 , b0  0, 1  G(s)  0
N: System Types
No. of pure integrators in open-loop transfer function
N  0, Type 0 system
N  1, Type 1 system
N  2, Type 2 system
.
.
.
.
.
.
DC Gain: limG(s)
s 0
Type 0 system DC Gain 
Kb0
(static gain)
a0
§ 7.3 System Types and Steady-State Error (4)
• Position Control of Mechanical Systems
(1) Command signal
r(t)
3
4
2
1
t
Region 1 and 3: Constant acceleration and deceleration
Region 2: Constant speed
Region 4: Constant position
(2) Error constants
K p  lim G(s) : position error constant
s0
K v  lim sG(s) : velocity error constant
s 0
K a  lim s2G(s) : acceleration error constant
s 0
§ 7.3 System Types and Steady-State Error (5)
(3) Systems control with non-zero steady-state position error
Constant position for Type 0 system
es.s. 
r0
1 Kp
position
Step signal
r0
es.s.
r , t  0
r0
 0, t  0
t
Constant velocity for Type 1 system
es.s.
v
 0
Kv
position
r  v 0t
v0
v t, t  0
r 0
 0, t  0
Kv
No velocity error in steady state
t
Constant acceleration for Type 2 system
es.s. 
a0
Ka
Ramp signal
position
r
a0
Ka
t
a0 2
t
2
Parabolic signal
 a0 2
 t ,t  0
r2
 0, t  0
§ 7.3 System Types and Steady-State Error (6)
•
Steady-state position errors for different types of system and input signal
Position command
Type of
Constant
Constant
Constant
position (r0 )
velocity (v 0 )
acceleration (a0 )
r  r0
r  v0t
r0
1  Kp


0
v0
Kv

0
0
a0
Ka
Open-loop System
Type 0
Type 1
Type 2
r
1 2
a0 t
2
Output positioning in feedback control is driven by the dynamic positional error.
System nonlinearities such as friction, dead zone, quantization will introduce
steady-state error in closed-loop position control.
§ 7.3 System Types and Steady-State Error (7)
Ex: Find the value of K such that there is 10% error in the steady state
+
G(s)
r(t)
1
G(s) 
K(s  1)
s(s  2)(s  3)
t
Sol: System G(s) is Type 1  s.s. error in ramp input
 e() 
1
 0.1  K v  10
Kv
For velocity error constant
K v  lim sG(s)  K 6
s0
 K  60
§ 7.4 Time-Domain Performance Indices (1)
• Performance of Control System
d(t)
Stability
Transient Response
Steady-state Error
r(t) +
-
e( t )
G1(s)
u( t )
G2(s)
y(t)
Internal States
x
• Performance Indices (PI)
Controller
A scalar function for quantitative measure of the performance specifications
of a control system.
T
P. I.   f(e(t), r(t), x(t), y(t))dt
0
error command state output
Use P.I. To trade off transient response and steady-state error with sufficient
stability margin.
§ 7.4 Time-Domain Performance Indices (2)
• Systems Control
(1) Classical control
Plant: Input-Output Model
T
T
P. I. : IAE   e(t)dt
ISE   e2 (t)dt
0
0
T
ITAE   t e(t)dt
0
T
ITSE   te2(t)dt
0
Controller: PID Control
KP
+
e( t )
KI / s
+
K Ds
 P-Proportional control:
 I-Integral control: e(t)

 D-Differential control:

(2) Modern control
Plant: State-space Model
P. I.: Usually Quadratic functional
Controller: States feedback control
u(t)  e(t)  Kx(t)
e( t )
KI / s
e( t )
KP
u( t )
u( t )
K Ds
u( t )
§ 7.4 Time-Domain Performance Indices (3)
• Optimal Control
Given: Plant model
Control configuration (Usually feedback)
Controller structure (Usually linear)
Design constraints
Objective: Minimize P. I. (P. I. to be selected)
Find: Optimal parameters in controller
Ex: Design optimal proportional control system
+
K
-
G(s)
proportional controller
Find optimal K to minimize P. I.
P. I.
K*
Optimal K
K