- 376 Views
- Uploaded on
- Presentation posted in: General

Chapter 5 Stability

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Chapter 5

Stability

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

長庚大學電機系

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

長庚大學電機系

- 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

長庚大學電機系

- 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)

長庚大學電機系

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

Remark:

長庚大學電機系

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.

長庚大學電機系

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.

長庚大學電機系

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.

長庚大學電機系

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

長庚大學電機系

Example: Consider

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

長庚大學電機系

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)

長庚大學電機系

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

長庚大學電機系

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

長庚大學電機系

Remark:In continuous-time case, is bounded or

But in discrete-time case, is absolutely summable

長庚大學電機系

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.

長庚大學電機系

- For
transfer matrix is

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

長庚大學電機系

- 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

長庚大學電機系

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.

長庚大學電機系

Example: Consider

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

長庚大學電機系

- 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

長庚大學電機系

- 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

長庚大學電機系

- Consider a system described by
Impulse response matrix:

zero-state response:

The system is BIBO stable iff

長庚大學電機系

- 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

長庚大學電機系

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

Example: Consider

- grows without bound Neither asymptotically stable nor marginally stable.

長庚大學電機系

- 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.

長庚大學電機系