bibo stability proof

8/12/2019 BIBO Stability Proof

1/7

1

Stability, Detectability &Stabilizability

M. Sami Fadali

Professor of Electrical EngineeringUniversity of Nevada

2

Stability Definitions

Stability: in the sense of Lyapunov (i.s. L).Asymptotic (Internal) Stability: Zero-input

response.Input-Output Stability or Bounded-Input-

Bounded-Output (BIBO) Stability: Zero-state response.

Equilibrium State

Solve= equilibrium state

For nonlinear systems, multiple statesStability of equilibrium state: depends on

behavior after a perturbation from theequilibrium.

3

Stability of EquilibriumHow sensitive is the system to small

perturbations in its equilibrium?

4

Unstable

Stable

Asymptoticallystable


2/7

StabilityDefinition: For any there exists aconstant such that implies

Can stay arbitrarily close to equilibrium bystarting sufficiently close to it.

Unstable: not stable (cannot stay arbitrarilyclose to the equilibrium.

5

Stable System

6

: = open ball of radius : = open ball of radius

0

Asymptotically Stable

Definition: Stable equilibrium and it is possible to choose such that

implies

Converges to equilibrium by startingsufficiently close to it.

Globally asymptotically stable: converges

to equilibrium from any initial state.

7

Exponential StabilityDefinition: There exist positive constants

such that implies

Global exponential stability: property holdsfor any initial state .

Length of state vector decays faster than anexponential function.

For linear systems, decay is alwaysexponential.

8


3/7

Exponential Stability

9

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Linear Time-invariant Systems

For a nonsingular state matrixif 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

10

Stability for LTI Systesm

Response is bounded for distinct eigenvalues onthe imaginary axis (stable).

Response is unbounded for repeatedeigenvalues on the imaginary axis (unstable).11 12

Asymptotic (Internal) Stability forLTI Systems

Theorem: LTI system is asymptotically stable ifthe zero-input response converges to zero for any

initial state.

LTI system is asymptotically stable if and only ifall eigenvalues are in the open LHP.

LTI: asymptotic stability implies globalexponential stability.


4/7


5/7

17

Theorem 1: BIBO StabilityA 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.

18

Proof Theorem 1 BIBO StabilityContradictio

Sufficiency (if)

Necessity (only if) : Assume BIBO stable with conditionviolated and let

19

Theorem 2: BIBO StabilityLTI SISO system is BIBO stable if and only if all itstransfer function poles are in the open LFP.

Proof (Necessity)

The integral of diverges for any transferfunction pole is in the closed RHP .

Proof: Sufficiency ,

After pole-zero cancellation, poles =remainingLHP poles (not all ) BIBO stability.

20


6/7

Kalman Decomposition

Any system can bedecomposed into foursubsystems as shownin the figure:

Unobservable Mode

Uncontrollable Mode

U (s) Y (s) Controllable

Observable

Y (s)Uncontrollable

Observable

Us ) Controllable

Unobservable

UncontrollableUnobservable

21 22

Relationship Between InternalStability BIBO Stability

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

23

Definitions

Detectable: all unstable

modes are observable(i.e. all unobservablemodes are stable).

Stabilizable: all unstablemodes are controllable(i.e. all uncontrollablemodes are stable).

x2

x1

ObservableSubspace Unobservable

Subspace(stable) y

x2

x1

ControllableSubspace Uncontrollable

Subspace(stable) u

24

Important Relations

Internally stable systems are stabilizableand detectable (no unstable modes).

Observable systems are detectable (nounstable unobservable modes).

Controllable systems are stabilizable (nounstable uncontrollable modes).

For minimal realizations, BIBO stability

and internal stability are equivalent{poles}={eigenvalues}.


7/7

25

Example

Controllable (phase var. form) but not observable.BIBO stable but not internally stable.Stabilizable but not detectable.

11

1111

10

0110

oncancellatiwithout:01

1

1

1

1

11

1)(2

OC B A

FormleControllab

stable BIBOss

s

ss

ssG

26

Example (continued)

Observable (observer form) but not controllable.BIBO stable but not internally stable.Detectable but not stabilizable.

1111

011

10110

oncancellatiwithout:01

11

11

111

)( 2

C C B A

FormObservable

stable BIBOss

s

ss

s

sG

bibo stability proof

Documents