Chapter 5 Stability

1. 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

2. Outline • Stability • Input-Output stability of LTI systems • Internal stability • Lyapunov Theorem • Stability of LTV systems 長庚大學電機系

3. 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 長庚大學電機系

4. 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) 長庚大學電機系

5. 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

6. 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

7. 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

8. 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. 長庚大學電機系

9. 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 長庚大學電機系

10. Example: Consider • Not stable • The system is BIBO stable (with zero initial state) 長庚大學電機系

11. 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) 長庚大學電機系

12. 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

13. 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

14. Remark:In continuous-time case, is bounded or But in discrete-time case, is absolutely summable 長庚大學電機系

15. 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

16. State-Space Description • 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

17. 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 長庚大學電機系

18. 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

19. Example: Consider • Not asymptotically stable • The system is BIBO stable (with zero initial state) 長庚大學電機系

20. 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 長庚大學電機系

21. 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 長庚大學電機系

22. Consider a system described by Impulse response matrix: zero-state response: The system is BIBO stable iff 長庚大學電機系

23. 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

24. 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

25. 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