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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
3
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
4
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
7
Stability: Routh-Hurwitz criterion
A system is stable if there are no sign changes in the first column of the Routh table
13
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)
14
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
15
Stability in State Space: Eigenvalue and eigenvector
Eigenvector
Eigenvector
For nonzero solution x
(a) Not eigenvector (b) Eigenvector
17
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)
18
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