Revision:

1. Revision: What is the definition of ISS?

2. Lecture 13 Lp stability, The Small Gain Theorem

3. Recommended reading • Khalil Chapter 6 (2nd edition)

4. Outline: • Norms of signals • Lp stability • Relation of exponential and Lp stability • Small Gain Theorem • Summary

5. Norms of signals

6. Signals are functions of time u(t) u(t) 1/  t t Dirac function (t) Sinusoid

7. Measuring the “size” of signals • Amplitude (ISS and L1): • Energy (L2): • General Lp norm, p 2 [1,1):

8. Example ||u||22 Different signal norms measure various properties of the signal! u(t) 0.5 u2(t) ||u|| 0.25 t -0.5

9. Comments • If L norm of signal u is finite, we say that the signal is bounded and write: u  L • If L2 norm of signal is finite, we say that the signal has bounded energy and write u  L2 • If L1 norm of signal is finite, the signal is absolutely integrable and we write u  L1 • Most commonly used Lp norms are for p=1; p=2; and p=.

10. Examples

11. Input-output stability of systems

12. General setup: input-output stability  Is y  Lq ? If u  Lp y u Typically, we use p=q.

13. System properties • L stability captures: Bounded inputs ) bounded outputs u  L y  L • L2 stability captures: Bnd. energy inputs ) bnd. energy outputs u  L2 y  L2

14. Model of the system • We model systems via an operator H: • Example: linear systems with fixed x0

15. Extended Lp spaces (Lpe) • Truncated signals are defined as • Extended Lp space is defined as: • Example: u(t)=t satisfies u  L, u  L e

16. Causality • The system H is causal if for every  0: • In other words, the output at time t depends only on the values of the input up to time t. • We only consider causal systems!

17. Stability definitions • The system H: Lep Lep is Lp stable if there exists  K and  0 such that The system is finite gain Lp stable if there exist ,  0 such that • Minimum  is called the “gain” of the system.

18. Relation of Lp and exponential stability • Consider the systems • Question: 0 is exp. stable u is finite gain Lp stable for any x0?

19. The opposite does not always hold! • The following system is Lp stable for any p but it is not exp. stable: • Under certain conditions it is possible to conclude exponential stability from Lp stability for p  [1,).

20. Theorem • u is finite gain Lp stable for any p  [1,] and x0 if: • 0 is exp. stable: • f and h satisfy:

21. Comments • Linear systems always satisfy the conditions. • One can rely on converse theorems to conclude that Lyapunov conditions hold. • One can relax the conditions of the previous theorem in several directions: • Local results (small signal Lp stability) • Nonlinear gains

22. Small gain theorem

23. Feedback system 1 + e1 u1 y1 - + y2 2 e2 u2 +  We assume that the system is “well posed”

24. Small gain theorem • The system  is finite gain Lp stable if: • 1 is finite gain Lp stable with gain 1 • 2 is finite gain Lp stable with gain 2 • The small gain condition holds:

25. Comments: • Small gain theorem is very useful in robustness analysis. • Often it can be also used as a controller design tool. • A nonlinear version of ISS small gain theorem also exists. The small gain condition becomes:

26. Summary: • Lp stability can be used to capture a range of useful system properties: e.g. bounded input bounded output stability. • Exponential stability of unforced system and global linear bounds on f and h imply finite gain Lp stability. • Small gain theorem can be used to conclude stability of feedback interconnections – one of the most important tools in control engineering.

27. Next lecture: • L2 stability Homework: read Chapter 6 in Khalil

28. Thank you for your attention!