Lect_11 - Nonlinear Systems and Control Lecture # 11...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Nonlinear Systems and Control Lecture # 11 Exponential Stability & Region of Attraction p. 1/1 8 Exponential Stability: The origin of x = f ( x ) is exponentially stable if and only if the linearization of f ( x ) at the origin is Hurwitz Theorem: Let f ( x ) be a locally Lipschitz function defined over a domain D R n ; D . Let V ( x ) be a continuously differentiable function such that k 1 bardbl x bardbl a V ( x ) k 2 bardbl x bardbl a V ( x ) k 3 bardbl x bardbl a for all x D , where k 1 , k 2 , k 3 , and a are positive constants. Then, the origin is an exponentially stable equilibrium point of x = f ( x ) . If the assumptions hold globally, the origin will be globally exponentially stable p. 2/1 8 Proof: Choose c > small enough that { k 1 bardbl x bardbl a c } D V ( x ) c k 1 bardbl x bardbl a c c = { V ( x ) c } { k 1 bardbl x bardbl a c } D c is compact and positively invariant; x (0) c V k 3 bardbl x bardbl a k 3 k 2 V dV V k 3 k 2 dt V ( x ( t )) V ( x (0)) e ( k 3 /k 2 ) t p. 3/1 8 bardbl x ( t ) bardbl bracketleftbigg V ( x ( t )) k 1 bracketrightbigg 1 /a bracketleftBigg V ( x (0)) e ( k 3 /k 2 ) t k 1 bracketrightBigg 1 /a bracketleftBigg...
View Full Document

This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

Page1 / 18

Lect_11 - Nonlinear Systems and Control Lecture # 11...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online