lec6-1 - 1 Outline ¡ Motivation Review of vector and...

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Unformatted text preview: 1 Outline ¡ Motivation Review of vector and matrix norm s tems stems ¡ Review of vector and matrix norms ¡ Positive Definite Quadratic Functions ¡ Lyapunov Function for LTI Systems ¡ Examples ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Motivation ¡ Stability analysis of Linear Time-Invariant systems (LTI) is well understood stemsstems ¡ Lyapunov stability theory establishes a common framework for both LTI and nonlinear systems ¡ Lyapunov functions for an LTI system can be constructed algebraically ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys ¡ More linear algebraic tools are needed to study stability of LTI and related nonlinear systems 2 Lyapunov Candidates for LTI Systems ¡ System : dx/dt=f(x)=Ax s tems stems ¡ Lyapunov Function : ¡ How to choose P? ¡ Is Lyapunov stability equivalent to well known LTI stability? ) ( > = Px x x T V ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys ¡ Need more linear algebra tools to study Lyapunov stability for linear and ‘near’ linear systems stemsstems Basic Matrix Facts ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys MATHEMATICAL REVIEW 3 Symmetric and Skew Symmetric Matrices ¡ Matrix M is symmetric if M=M T Matrix M is skew symmetric if M= M T s tems stems ¡ Matrix M is skew-symmetric if M=-M T ¡ Examples: ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Modal Decomposition of a Symmetric Matrix M ¡ Let M be a symmetric matrix with eigenvalues λ 1 , λ 2 …, λ n , and (normalized) stemsstems 1 2 n eigenvectors v 1 ,v 2 ,…,v n . Then ¡ Eigenvalues λ 1 , λ 2 …, λ n are real ¡ The eigenvectors can be chosen so as to decompose M to with an orthonorma T U U M Λ = v v v = U ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys ¡ with an orthonormal ¡ i.e., U T U=I, and Λ = diag( λ 1 , λ 2 …, λ n ). Moreover [ ] n v v v L 2 1 = U max min λ ≤ ≤ λ x x Mx x T T 4 Matlab Example U'*D*U A = 1 2 3 s tems stems 1.0000 2.0000 3.0000 2.0000 4.0000 5.0000 3.0000 5.0000 6.0000 U'*U ans = 1.0000 0.0000 -0.0000 0.0000 1.0000 -0.0000 2 4 5 3 5 6 [U,D]=eig(A) U = 0.5910 -0.7370 0.3280-0.7370 -0.3280 0.5910 0.3280 0.5910 0.7370 ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys-0.0000 -0.0000 1.0000 D = 0.1709 0 0 -0.5157 0 0 11.3448 Justification of Modal Decomposition Result ¡ Mv i = λ i v i ⇒ M[v 1 … v n ]=[ λ 1 v 1 … λ n v n ] Letting U=[v v ] and stemsstems ¡ Letting U=[v 1 … v n ] and Λ =diag( λ 1 ,…, λ n ) then ¡ MU=U Λ⇒ M=U Λ U T ⇒Λ =U T MU ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 5 Matrix (Induced) Norm ¡ It is a measure of a matrix magnitude ¡ More precisely it is the maximum of the s tems stems ¡ More precisely, it is the maximum of the norm Ax over the norm of x: ¡ This definition holds for any vector x Ax A x max = ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys ¡ This definition holds for any vector...
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This note was uploaded on 09/22/2011 for the course ME 6402 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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lec6-1 - 1 Outline ¡ Motivation Review of vector and...

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