Transcription of 6.241J Course Notes, Chapter 13: Internal (Lyapunov) stability
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6.241J Course Notes, Chapter 13: Internal (Lyapunov) stability
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology1 1 cChapter 13 Internal (Lyapunov) stability Introduction We have already seen some examples of both stable and unstable systems. The objective of this Chapter is to formalize the notion of Internal stability for general nonlinear state-space models. Apart from de ning the various notions of stability , we de ne an entity known as a Lyapunov function and relate it to these various stability notions. Notions of stability For a general undriven system x_(t) f(x(t) 0 t)(CT ) ( ) x(k +1) f(x(k) 0 k)(DT ) ( ) we say that a point x is an equilibrium point from time t0 for the CT system above if f(x 0 t) 0 8t t0, and is an equilibrium point from time k0 for the DT system above if f(x 0 k) x 8k k0.
v alue; furthermore, ev ery alue whose real part equals 0 should ha e its geometric m ultiplicit y equal to its algebraic, i.e., all asso ciated Jordan blo c ks should b e of size 1. (V erify that the presence Jordan blo c ks greater than one for these imaginary-axis eigen v alues w ould lead to the state ariables gr owing p olynomial ly with ...
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