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Understanding ISS and Lp Stability: Key Concepts and the Small Gain Theorem

This lecture focuses on the definitions and properties of ISS (input-to-state stability) and Lp stability, as well as the Small Gain Theorem. We will explore various norms of signals, including L1, L2, and L∞, highlighting their relationship to system stability. The discussion includes how input-output stability is characterized in terms of these norms, the importance of causality in systems, and the implications of finite gain Lp stability. Attendees will gain insights into advanced concepts of stability that are crucial for robust control system design.

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Understanding ISS and Lp Stability: Key Concepts and the Small Gain Theorem

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Presentation Transcript

  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!

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