Basic Control Engineering Prof. Wonhee Kim Stability System response: - Stable: - Unstable: -...
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Basic Control Engineering Prof. Wonhee Kim Ch.6. Stability
Transcript of Basic Control Engineering Prof. Wonhee Kim Stability System response: - Stable: - Unstable: -...
Basic Control Engineering
Prof. Wonhee Kim
Ch.6. Stability
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StabilitySystem response:
- Stable:
- Unstable:
- Marginally stable:
A system is stable if every bounded input yields a bounded output.
The bounded-input, bounded-output (BIBO) definition of stability.
From input From system
naturallim 0t
c t
natural naturallim or limt t
c t c t
natural naturallim or limt t
c t a c t a
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Stability in Transfer Function
2
5c
0
2lim
5sc sC s
Stable systems have closed-loop transfer functions with poles only in the left half-plane
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Pole of Closed-loop System
G(s)R(s) C(s)
Open-loop control
P(s)+
-
Closed-loop control
R(s) C(s)
C sG s
R s
1
C s P sG s
R s P s
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Pole of Closed-loop System
3
3
3
1
3
3 23
13 2
3
3 2 3
C s G s
R s G s
s s s
s s s
s s s
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Pole of Closed-loop System
3
3
3
1
7
3 27
13 2
7
3 2 7
C s G s
R s G s
s s s
s s s
s s s
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Stability: Routh-Hurwitz criterion
A system is stable if there are no sign changes in the first column of the Routh table
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Stability: Routh-Hurwitz criterionExample 6.1)
Two poles in the right half plane
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Stability: Routh-Hurwitz criterionExample 6.2)
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Stability: Routh-Hurwitz criterion
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Stability: Routh-Hurwitz criterionExample 6.3)
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Stability: Routh-Hurwitz criterionExample 6.4)
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Stability for Closed-loop
P(s)+
-
R(s) C(s)
1 2
1 21
c c
c c
C s P s P s P s
R s P s P s P s
Pc1(s)
Pc2(s)
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Stability: PID Controller design
P(s)+
-
R(s) C(s)
2
2
1
1 11
P I DD P IPID
PID D P IP I D
N sK K K s
K s K s K N sC s P s P s s D s
N sR s P s P s sD s K s K s K N sK K K s
s D s
PPID(s)
1
PID P I DP s K K K ss
N sP s
D s
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Stability in State Space: Eigenvalue and eigenvector
Eigenvector
Eigenvector
For nonzero solution x
(a) Not eigenvector (b) Eigenvector
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Stability in State Space: Eigenvalue and eigenvector
1 2
2 4
Example)
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Stability in State Space
x x u
y x u
t A t B t
t C t D t
1 adj
det
i
i
I AY sT s C sI A B D C B D
U s I A
The system poles depend
on the eigenvalues of A
Example)
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Stability in State Space
x x u
y x u
t A t B t
t C t D t
Solution:
State transition matrix:
In particular, if the matrix A is diagonal, then
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Stability in State Space: State Feedback
x x u
y x u
t A t B t
t C t D t
u xt K t
x x x
x
t A t BK t
A BK t
K should be chosen such that (A-BK) is Hurwitz!
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Stability: Stats feedback controller design in time domain
Regulation)
Using final value theorem, steady-state response with step reference should be 1.
where Kdc is a dc gain for C(sI-(A-BK))-1B
x x u
y x
t A t B t
t C t
u xt K t Jr t
x x x
x
t A t BK t BKr t
A BK t BJr t
x x u
y x
t A t B t
t C t
1Y sT s C sI A B
U s
1Y sT s C sI A BK B J
R s
0 0 0
lim lim lim 1dcs s s
y sT s R s T s JK
1
dc
JK
