{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–6. Sign up to view the full content.

Nonlinear Systems and Control Lecture # 9 Lyapunov Stability – p.1/16

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

View Full Document
Quadratic Forms V ( x ) = x T P x = n summationdisplay i =1 n summationdisplay j =1 p ij x i x j , P = P T λ min ( P ) bardbl x bardbl 2 x T P x λ max ( P ) bardbl x bardbl 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
Linear Systems ˙ x = Ax V ( x ) = x T P x, P = P T > 0 ˙ V ( x ) = x T P ˙ x + ˙ x T P x = x T ( P A + A T P ) x def = x T Qx If Q > 0 , then A is Hurwitz Or choose Q > 0 and solve the Lyapunov equation P A + A T P = Q If P > 0 , then A is Hurwitz Matlab: P = lyap ( A , Q ) – p.3/16

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

View Full Document
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 P A + 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 = integraldisplay 0 exp( A T t ) Q exp( At ) dt is positive definite and satisfies the Lyapunov equation – p.4/16
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 .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 16

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online