Lect_9 - Nonlinear Systems and Control Lecture # 9 Lyapunov...

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Nonlinear Systems and Control Lecture # 9 Lyapunov Stability – p.1/16
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Quadratic Forms V ( x ) = x T Px = n s i =1 n s j =1 p ij x i x j , P = P T λ min ( P ) b x b 2 x T Px λ max ( P ) b x b 2 P 0 (Positive semidefinite) if and only if λ i ( P ) 0 i P > 0 (Positive definite) if and only if λ i ( P ) > 0 i V ( x ) is positive definite if and only if P is positive definite V ( x ) is positive semidefinite if and only if P is positive semidefinite P > 0 if and only if all the leading principal minors of P are positive – p.2/16
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Linear Systems ˙ x = Ax V ( x ) = x T Px, P = P T > 0 ˙ V ( x ) = x T P ˙ x + ˙ x T Px = x T ( PA + A T P ) x def = x T Qx If Q > 0 , then A is Hurwitz Or choose Q > 0 and solve the Lyapunov equation PA + A T P = Q If P > 0 , then A is Hurwitz Matlab: P = lyap ( A , Q ) – p.3/16
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Theorem A matrix A is Hurwitz if and only if for any Q = Q T > 0 there is P = P T > 0 that satisfies the Lyapunov equation PA + A T P = Q Moreover, if A is Hurwitz, then P is the unique solution Idea of the proof: Sufficiency follows from Lyapunov’s theorem. Necessity is shown by verifying that P = i 0 exp( A T t ) Q exp( At ) dt is positive definite and satisfies the Lyapunov equation – p.4/16
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Linearization Consider ˙ x = f ( x ) . f ( x ) is continuously differentiable in D . By the mean value theorem: f i ( x ) = f i (0) + ∂f i ∂x ( z i ) x with z i = α i x , and 0 < α i < 1 .
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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.

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Lect_9 - Nonlinear Systems and Control Lecture # 9 Lyapunov...

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