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Stability

- To study BIBO stability for zero-state response, and marginal and asymptotic stabilities for zero-input response.

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Outline

- Stability
- Input-Output stability of LTI systems
- Internal stability
- Lyapunov Theorem
- Stability of LTV systems

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Stability

- Importance of stability:
- In addition to stability, one may further require other performances such as tracking, suppress noise, minimize energy consumption etc.
- For linear system,
response = zero-state response + zero-input response

I/O stability Internal stability

BIBO-stability Marginal and asymptotic stability

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Input-Output Stability

- Consider a SISO LTI system, as described by
- The description has implicitly assumed that the system is causal and relaxed at 0
- is said to be bounded if
Definition:System (S1) is called BIBO stable if every Bounded Input excites a Bounded Output (for zero initial state)

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Theorem:A SISO LTI System as described by (S1) is BIBO stable iff is absolutely integrable over , i.e., for some M

Remark:

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Theorem:A SISO system with proper rational function is BIBO stable iff every pole of has a negative real part.

Proof: has pole with multiplicity its partial fraction contains the factors

its inverse Laplace transform or the impulse response contains the factors

Since these terms are absolutely integrable iff

The result then follows.

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Example:Consider a positive feedback system with impulse response

the system is BIBO stable iff

Note that, the associated transfer function

is not a rational function.

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MIMO System

Theorem:A MIMO LTI system with impulse response matrix is BIBO stable is absolutely integrable over

Theorem:A MIMO system with proper rational transfer matrix is BIBO stable iff every pole of has a negative real part.

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State-Space Description

Consider

every pole of is an eigenvalue of A, (But eigenvalue

of A might not be a pole because of cancellation)

system is BIBO stable if

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Discrete-Time Case (I/O stability)

Consider a discrete-time SISO, LTI system described by

(S2)

where is impulse response sequence.

Definition: A system as described by (S2) is said to be BIBO-stable if every bounded input excites a bounded-output sequence (under zero-initial state)

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Theorem:A discrete-time SISO system as described by (S2) is BIBO stable iff is absolutely summable. i,.e.,

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Theorem:A discrete-time SISO system with proper rational transfer function is BIBO stable iff every pole of lies inside the unit disk.

Reason:If has a pole with multiplicity ,

its partial fraction contains the factors

the inverse Z-transform or the impulse response

sequence contains the factors

every such terms is absolutely summable iff

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Remark:In continuous-time case, is bounded or

But in discrete-time case, is absolutely summable

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Theorem:A MIMO discrete-time system with impulse response sequence matrix is BIBO stable iff every is absolutely summable.

Theorem:A MIMO discrete-time system with discrete proper rational transfer matrix is BIBO stable iff every pole of lies inside the unit disk.

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State-Space Description

- For
transfer matrix is

- pole of
- System is BIBO stable if is contained inside the open unit disk.

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Internal stability (For zero-input response)

- The zero-input response of
or

is called (marginal) stable if such that for all whenever

It is asymptotically stable if , in addition to stable,

as

Alternative definition of marginal stable: every finite initial state x(0) excites a bounded response

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Theorem:The equation is (i) stable and each eigenvalue in the j axis has index 1.

(ii) asymptotically stable

Reason: By considering the result from Jordan form and the result that the unique sol. of is

- Asymptotic stability BIBO stable.
- marginal stable.
- unstable.

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Example: Consider

- Not asymptotically stable
- The system is BIBO stable (with zero initial state)

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Discrete-Time Case (Internal Stability)

- Definition of marginally stable and asymptotically stable is similar to the continuous-time case.
Theorem: (i) is marginally stable and those eigenvalue with magnitude equal to 1 has index 1.

(ii) is asymptotically stable where

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Stability of LTV Systems

- Consider a SISO LTV system described by Similar to those of LTI system, the LTV system is BIBO stable iff for all and with
- For multivariable LTV case The system is BIBO stable iff for all and with

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- Consider a system described by
Impulse response matrix:

zero-state response:

The system is BIBO stable iff

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- Now, we consider the zero-input response of (S3), i.e., the response of
- The response is
- The response is marginal stable iff for all with
- The response is asymptotically stable iff it is stable and

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Question: the response of Consider the time-varying system . Does the condition imply asymptotic stability?

Example: Consider

- grows without bound Neither asymptotically stable nor marginally stable.

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- Recall that all stability properties in time-invariant case are invariant under any equivalence transformation.
- In time-varying case,
- BIBO stability is invariant under equivalence transformation since the impulse matrix is preserved under such a transformation.
- Marginal and asymptotic stability are not invariant under equivalence transformation since may be transformed into by an equivalence transformation.

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