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Revision:. What is the definition of ISS?. Lecture 13. L p stability, The Small Gain Theorem. Recommended reading. Khalil Chapter 6 (2 nd edition). Outline:. Norms of signals L p stability Relation of exponential and L p stability Small Gain Theorem Summary. Norms of signals.
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Revision: What is the definition of ISS?
Lecture 13 Lp stability, The Small Gain Theorem
Recommended reading • Khalil Chapter 6 (2nd edition)
Outline: • Norms of signals • Lp stability • Relation of exponential and Lp stability • Small Gain Theorem • Summary
Signals are functions of time u(t) u(t) 1/ t t Dirac function (t) Sinusoid
Measuring the “size” of signals • Amplitude (ISS and L1): • Energy (L2): • General Lp norm, p 2 [1,1):
Example ||u||22 Different signal norms measure various properties of the signal! u(t) 0.5 u2(t) ||u|| 0.25 t -0.5
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=.
General setup: input-output stability Is y Lq ? If u Lp y u Typically, we use p=q.
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
Model of the system • We model systems via an operator H: • Example: linear systems with fixed x0
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
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!
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.
Relation of Lp and exponential stability • Consider the systems • Question: 0 is exp. stable u is finite gain Lp stable for any x0?
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,).
Theorem • u is finite gain Lp stable for any p [1,] and x0 if: • 0 is exp. stable: • f and h satisfy:
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
Feedback system 1 + e1 u1 y1 - + y2 2 e2 u2 + We assume that the system is “well posed”
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:
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:
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.
Next lecture: • L2 stability Homework: read Chapter 6 in Khalil