Stability Definitions
Stability: in the sense of Lyapunov (i.s. L).
Asymptotic (Internal) Stability: Zero-input
response.
Input-Output Stability or Bounded-InputBounded-Output (BIBO) Stability: Zerostate response.
Stability, Detectability &
Stabilizability
M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
1
Equilibrium State
2
Stability of Equilibrium
How sensitive is the system to small
perturbations in its equilibrium?
Solve
= equilibrium state
For nonlinear systems, multiple states
Stability of equilibrium state: depends on
behavior after a perturbation from the
equilibrium.
Unstable
Stable
Asymptotically
stable
3
4
Stability
Stable System
Definition: For any
there exists a
constant
such that
implies
• Can stay arbitrarily close to equilibrium by
starting sufficiently close to it.
• Unstable: not stable (cannot stay arbitrarily
close to the equilibrium.
0
:
:
= open ball of radius
= open ball of radius
5
6
Exponential Stability
Asymptotically Stable
Definition: There exist positive constants
such that
implies
Definition: Stable equilibrium and it is
such that
possible to choose
implies
• Global exponential stability: property holds
.
for any initial state
• Length of state vector decays faster than an
exponential function.
• For linear systems, decay is always
exponential.
→
• Converges to equilibrium by starting
sufficiently close to it.
• Globally asymptotically stable: converges
to equilibrium from any initial state.
7
8
Exponential Stability
Linear Time-invariant Systems
1
• For a nonsingular state matrix
if and only if
• Only one equilibrium point at the origin.
• For a singular state matrix
,
• Rank deficit=number of linearly independent
• Infinitely many equilibrium points on
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
9
10
Asymptotic (Internal) Stability for
LTI Systems
Stability for LTI Systesm
Theorem: LTI system is asymptotically stable if
the zero-input response converges to zero for any
initial state.
• Response is bounded for distinct eigenvalues on
the imaginary axis (stable).
• Response is unbounded for repeated
eigenvalues on the imaginary axis (unstable).
11
• LTI system is asymptotically stable if and only if
all eigenvalues are in the open LHP.
• LTI: asymptotic stability implies global
exponential stability.
12
State Trajectories
Example
3
Is the system asymptotically stable?
>> A=[3,-2,0;8,-3,-4;0,4,-9];
>> eig(A)
ans =
-1.0000
-3.0000
-5.0000
Yes. All poles in the open LHP.
2
Stable
1
0
Asymptotically stable
-1
-2
-1.5
-1
-0.5
0
0.5
1
13
Example
14
Input-output (BIBO) Stability
• Is the system stable?
>> A=[-5,5,-5;-20,20,-20;-20,20,-20];
>> eig(A)
ans =
0
-5
0
No. Repeated pole on the imaginary axis.
Definition For any bounded input, the output is
bounded.
• Definition can be generalized to time-varying
MIMO systems using ||.|| (norm) in place of |.|
• Definition can be generalized to distributed
parameter systems.
15
16
Contradiction
Theorem 1: BIBO Stability
Proof Theorem 1 BIBO Stability
A SISO LTI system is BIBO stable if and only
if its impulse response satisfies
• Remarks
• Condition can be generalized to time-varying
MIMO systems using ||.|| (norm) in place of |.|
• Condition can be generalized to distributed
parameter systems.
• Sufficiency (if)
Necessity (only if): Assume BIBO stable with condition
violated and let
17
18
Proof: Sufficiency
Theorem 2: BIBO Stability
,
LTI SISO system is BIBO stable if and only if all its
transfer function poles are in the open LFP.
Proof (Necessity)
The integral of
diverges for any transfer
function pole is in the closed RHP .
After pole-zero cancellation, poles =remaining
LHP poles (not all ) BIBO stability.
19
20
Relationship Between Internal
Stability BIBO Stability
Kalman Decomposition
U(s)
• Any system can be
decomposed into four
subsystems as shown
in the figure:
• Unobservable Mode
Y(s)
• BIBO stability is equivalent to open LHP poles
• Internal stability implies BIBO stability (since
poles are a subset of the eigenvalues).
• Some eigenvalues may cancel in the transfer
function and are not poles.
• BIBO stability does not , in general, imply internal
stability
• With no cancellation, {poles}={eigenvalues}
BIBO stability is equivalent to internal stability
Controllable
Observable
Y(s)
Uncontrollable
Observable
Us)
• Uncontrollable Mode
Controllable
Unobservable
Uncontrollable
Unobservable
21
22
Important Relations
Definitions
x2
Detectable: all unstable
modes are observable Unobservable
Subspace
(stable)
(i.e. all unobservable
modes are stable).
Stabilizable: all unstable
x2
modes are controllable Uncontrollable
Subspace
(i.e. all uncontrollable
(stable)
modes are stable).
Observable
Subspace
y
x1
Controllable
Subspace
u
x1
23
• Internally stable systems are stabilizable
and detectable (no unstable modes).
• Observable systems are detectable (no
unstable unobservable modes).
• Controllable systems are stabilizable (no
unstable uncontrollable modes).
• For minimal realizations, BIBO stability
and internal stability are equivalent
{poles}={eigenvalues}.
24
Example (continued)
Example
1
s 1
s 1
 2

s  1s  1 s  1 s  1
 1  0  BIBO stable
Controllable Form : without cancellation
1
s 1
s 1
 2

s  1s  1 s  1 s  1
 1  0  BIBO stable
Observable Form : without cancellation
G( s) 
G( s) 
0 1 
0 
 1 1 
A
B    C   1 1  O  


1 0 
1
 1  1
0 1 
1
 1  1
A
B    C  1 0  C  


1 0 
 1
 1 1 
 Controllable (phase var. form) but not observable.
 BIBO stable but not internally stable.
 Stabilizable but not detectable.
25
 Observable (observer form) but not controllable.
 BIBO stable but not internally stable.
 Detectable but not stabilizable.
26