chapter 7 stability and steady-state error analysis § 7.1 stability of linear feedback systems §...
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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
• Basic Concepts:
§ 7.1 Stability of Linear Feedback Systems (1)
state mequilibriu Single 0,x A0xAx:System I-T Linear
states mequilibriu Multiple ,0)x(F 0)x(Fx :System I-T Nonlinear
211 xx , xx:form spaceState 0,bxxax:EX
mequilibriu in 0x0x , 0 xSince (2) Stable System
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.
, 0x ,0x 0x
x
2
1
2
1
2
1
x
x
ab
10
x
x
(1) Equilibrium States, 0)x(Usually Constantx
0
t
1.0
• Plant Dynamics: Regular pendulum (Linear)
effect spring Positive
0l
g
ml
b
energy potential Minimum
2
Inverted pendulum (Linear)
0
0
State mEquilibriu
effect spring Negative
0l
g
ml
b
energy potential Maximum
2
bb>0
l
g
mm
l
b , b>0 g
mm
diagram Zero-Pole diagram Zero-Pole
j
-1
2
2
1/3 j
1-3
1/3
Natural response 0
t
1.0
Natural response
§ 7.1 Stability of Linear Feedback Systems (2)
3.0)l
g , 2.0
ml
b(
2 3.0)
l
g , 2.0
ml
b(
2
Natural behavior of a control system, r(t)=d(t)=0
Equilibrium state
Initial relaxation system, I.C.=0
No general algebraic solution for 5th-order and above polynomial equation
(Abel , Hamilton)
• Closed-loop System:
G(s)y(t)
b
r(t) +
H(s)
eb
d(t)
G(s)y(t)
H(s)
ea-1
0D(s):Poles
)s(D
)s(N
G(s)H(s)1
G(s)T(s)
poles loop-closed and equationstic Characteri
§ 7.1 Stability of Linear Feedback Systems (3)
a,e 0 , y 0 ,y 0 ,
G(s) : Unstable plant
Closed-loop : Stable
Stabilization of unstable system Destabilization Effect on stable system
G(s) : stable plant
Closed-loop : Unstable
mm
l g
Controllerand
Driver
T
,
m m
lg
F
Controllerand
Driver
,
§ 7.1 Stability of Linear Feedback Systems (4)• Stability Problems:
• Stability Definition
The impulse response of a system is absolutely integrable.
(1) Asymptotic stability
Stable system if the transient response decays to zero
0)t(y lim
)0(y, , )0( y, y(0).C.I
0)t(ya)t(ya)t(y
t
1)(n(1)
0)1n(
1n)n(
(2) BIBO stability
Stable system if the response is bounded for bounded input signal
N)t(yM)t(u
d )(u )t(g)t(yt
0
§ 7.1 Stability of Linear Feedback Systems (5)
)t(g
t
(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
Unstablepolesstable
poles
Marginally stable/unstable
The definitions of (1), (2), and (3) are equivalent for LTI system.
§ 7.1 Stability of Linear Feedback Systems (6)
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
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (1)• Characteristic Polynomial of Closed-loop System
0a ,asasasa)s(D n011n
1nn
n
2n2n
nn0 sasa)s(D
3n3n
1n1n1 sasa)s(D
,c
b ,
b
a ,
a
a ,
s
1s
1s
)s(D
)s(D
1
13
1
1n2
1n
n1
3
2
11
0
Note: Any zero root has been removed in D(s).
(1) The polynomial D(s) is a stable polynomial if are all positive, i.e. are all positive.(2) The number of sign changes in 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 and the sign changes when 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) (2)
• Routh-Hurwitz Stability Criterion
1111nn h ,c ,b ,a ,a
i
1111nn h ,c ,b ,a ,a
ε, ε > 0,0
• Routh Tabulation (array)
31
5n1n
12
21
3n1n
11
5n1n
4nn
1n2
3n1n
2nn
1n1
bb
aa
b
1c ,
bb
aa
b
1c
aa
aa
a
1b ,
aa
aa
a
1b
ns na 2na 4na 1ns
2ns
3ns
1s0s
1na 3na 5na
1b 2b 3b
1c 2c 3c
1g
1h
0
0
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:
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (3)
j
1
234
possible cases (1~4)of poles distribution
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (4) Ex: For a closed-loop system with transfer function T(s)
T(s)
0a ,asasasaD(s)
N(s) of orderD(s) of order ,)s(D
)s(N)s(T
i01
12
23
3
Ex: Find stability condition for a closed-loop system with
characteristic polynomial as
Sol:
30122
30120 aaaa0
a
aaaa 0,a :conditionStability
ArrayRouth
01esdscsbss 2345
ebdbbcd ,0e ,d ,c ,b 22
3s 3a 1a2s1s
0s
2a 0a
2
3012
a
aaaa 0
0a
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (5) Ex: For a colsed-loop system with characteristic polynomial
3s5s6s3s2sD(s) 2345
Ex: For , determine if the system is stable
Sol:
system stable
2- 2j, -2j,s :poles
2)2j)(s2j)(s(s
2)4)(s(s
)1s2
1)(8s2()s(D
2
2
Determine if the system is stable
Sol:Table Routh 0 0, for column first of Sign
8s4s2sD(s) 23 Table Routh
j2
2
2
system unstable
RHP in poles twochanges sign two
82sD(s) :eq.uxiliary A 2
3s2s1s0s
5s4s
3s2s
1s0s
0
82
1 4
5
4
3
2
21
0
s 1 3 5
s 2 6 3
7s 0 ε
26ε -7
s 3ε
42ε - 49 - 6εs
12ε -14s 3
Characteristic equation
R-H Test on D(s)
§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (6)• Absolute and Relative Stability
0D(s)
Absolute Stability Relative Stability
Characteristic equation
R-H Test on D ’(p)
0(p)’D)p(DD(s)
0 ,-ps set
js-coordinate
jp-coordinate
stability margin
s=0
§ 7.3 System Types and Steady-State Error (1)• Steady-state error for unity feedback systems
G(s)+
H(s)
G(s)Y(s)R(s) + E(s)
)s(R)s(G1
1slim)s(R)]s(T1[slim)e( (2)
stable is T(s) (1)
:theorem value-final of Use
e)t(elim : error stateSteady
)s(R)]s(T1[)s(E
response Error
0s0s
.s.st
For nonunity feedback systems
T(s) T.F. loop-Closed
G(s) T.F. loop-Open
G'(s)+
G
G'1 G(H-1)
T(s)Y(s)R(s)
+E(s)
§ 7.3 System Types and Steady-State Error (2)• Fundamental Regulation and Tracking Error
Regulation s.s. error
Tracking s.s. error
e(t)r(t)
)s(G1
1
1
t
e(t)r(t)
)s(G1
1
1
t
G(s)lim1
1)e(
input Step ,s
1)s(R
0s
sG(s)lim
1)e(
input Ramp ,s
1)s(R
0s
2
§ 7.3 System Types and Steady-State Error (3)• Open-loop System Types
d
m m 1T sm 1 1 0
dN q q 1q 1 1 0
0 0
K(s b s ...... b s b )G(s) e , N q m for T 0
S (s a s ...... a s a )
a , b 0, 1 G(s) 0
N: System Types
No. of pure integrators in open-loop transfer function
N 0, Type 0 system
N 1, T
ype 1 system
N 2, Type 2 system
. .
. .
. .
0
s 0
0
DC Gain: limG(s)
Kb Type 0 system DC Gain (static gain)
a
§ 7.3 System Types and Steady-State Error (4)• Position Control of Mechanical Systems
constant error onaccelerati :)s(GslimK
constant errorvelocity :)s(sGlimK
constant error position :)s(GlimK
2
0sa
0sv
0sp
(1) Command signal
(2) Error constants
Region 1 and 3: Constant acceleration and deceleration
Region 2: Constant speed
Region 4: Constant position
r(t)
t
1
3 4
2
§ 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
p
0s.s. K1
re
Constant velocity for Type 1 system
0s.s.
v
ve
K
Constant acceleration for Type 2 system
a
0s.s. K
ae
20 t2
ar
a
0K
a
t
position
position
t
.s.se0r
tvr 0
v
0K
v
t
position
0t ,0
0t ,rr 0
0t 0,
0t t,vr 0
0t ,0
0t ,t2
ar
20
signal Step
signal Ramp
signalParabolic
No velocity error in steady state
§ 7.3 System Types and Steady-State Error (6)
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.
• Steady-state position errors for different types of system and input signal
Type 0
Type 1
Type 2
p
0
K1
r
0v
0
K
v
a
0
K
a
0
0
Constant
position (r )
r r 0
0
Constant
velocity (v )
r v t 21
2
0
0
Constant
acceleration (a )
r a t
0 0
Type of
Open-loop System
Position command
§ 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)+
)3s)(2s(s
)1s(KG(s)
Sol: System G(s) is Type 1 s.s. error in ramp input
10K1.0K
1)(e v
v
60K
6K)s(sGlimK 0sv
For velocity error constant
r(t)
t
1
§ 7.4 Time-Domain Performance Indices (1)
Stability
Transient Response
Steady-state Error
• Performance of Control System
• Performance Indices (PI)
A scalar function for quantitative measure of the performance specifications of a control system.
T
0P. I. f(e(t), r(t), x(t), y(t))dt
error command state output
Use P.I. To trade off transient response and steady-state error with sufficient stability margin.
G1(s) G2(s)r(t) y(t)
d(t)
+
-
)t(e )t(u
Controller
Internal States
x
§ 7.4 Time-Domain Performance Indices (2)• Systems Control
Plant: Input-Output Model(1) Classical control
(2) Modern control
Plant: State-space Model
P. I.: Usually Quadratic functional
Controller: States feedback control
T T 2
0 0
T T 2
0 0
P. I. : IAE e(t)dt ISE e (t)dt
ITAE t e(t)dt ITSE te (t)dt
u(t) e(t) Kx(t)
Controller: PID Control
P-Proportional control:
I-Integral control:
D-Differential control:
)t(e )t(uPK
)t(e )t(usKD
)t(e )t(us/KI
s/KI
PK
sKD
)t(e +
+
§ 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 controllerEx: Design optimal proportional control system
Find optimal K to minimize P. I.
K G(s)+
-
proportional controller
K
I. .P
K Optimal
*K