handout_07.pdf - Outline 1 Lyapunov stability Lyapunov functions and stability Examples Some extensions 2 Limit cycles Periodic orbits and their

handout_07.pdf - Outline 1 Lyapunov stability Lyapunov...

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Modeling Dynamics – 2019 Set 7 Department of Electrical Engineering Eindhoven University of Technology Siep Weiland Set 7 (TUE) Modeling Dynamics – 2019 Siep Weiland 1 / 44 Outline 1 Lyapunov stability Lyapunov functions and stability Examples Some extensions 2 Limit cycles Periodic orbits and their stability Poincar´ e-Bendixson’s theorem Li´ enard systems 3 Summary Set 7 (TUE) Modeling Dynamics – 2019 Siep Weiland 2 / 44 Lyapunov stability Set 7 (TUE) Modeling Dynamics – 2019 Siep Weiland 3 / 44 Lyapunov functions A Lyapunov function is a locally defined storage function. Definition Let x * be a fixed point of ˙ x = f ( x ). A Lyapunov function is a function V : S → R defined on an open region S containing x * such that V is continuous on S V is positive definite on S , i.e., V ( x * ) = 0 and V ( x ) > 0 for all x * 6 = x ∈ S V is differentiable along trajectories , i.e., V 0 ( x ) := h∇ V ( x ) , f ( x ) i exists and is continuous for all points x ∈ S . Here, V 0 ( x ) = h∇ V ( x ) , f ( x ) i = V x 1 ( x ) f 1 ( x ) + · · · + V x n ( x ) f n ( x ) is the inner product of V ( x ) and f ( x ) and thus depends on the system! Set 7 (TUE) Modeling Dynamics – 2019 Siep Weiland 4 / 44
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the Lyapunov stability theorem Theorem Let x * be an equilibrium point of the nonlinear system ˙ x = f ( x ) and suppose x * lies in the open set S . If there exists a Lyapunov function V : S → R with V 0 ( x ) 0 for all x ∈ S , then x * is stable . If there exists a Lyapunov function V : S → R with V 0 ( x ) < 0 for all x * 6 = x ∈ S , then x * is asymptotically stable . So, existence of a Lyapunov function with V 0 0 implies stability of x * . Doesn’t say how to find V . . . Think of V as a storage/energy function but now locally defined! Referred to as the Direct method to prove stability. Set 7 (TUE) Modeling Dynamics – 2019 Siep Weiland 5 / 44 Aleksandr Mikhailovich Lyapunov Aleksandr Mikhailovich Lyapunov Born on May 25, 1857 in St. Pe- tersburg Studied stability of rotating fluid masses Fellow student of Andrej Markov. Defended in 1892 his famous PhD thesis: “The General Problem of the Stability of Motion” Set 7 (TUE) Modeling Dynamics – 2019 Siep Weiland 6 / 44 proof of Lyapunov theorem (illustrative!) Idea behind proof: x * ε V γ δ x 0 Take ε > 0 and neighborhood B ε of x * . Take level set V γ of V that lies inside B ε . Choose δ such that B δ lies inside level set V γ Then x 0 B δ produces trajectory in V γ and therefore in B Set 7 (TUE) Modeling Dynamics – 2019 Siep Weiland 7 / 44 proof of Lyapunov theorem (illustrative!) Proof (See Khalil Thm 4.1): Let ε > 0. Choose 0 < r < ε such that B r := { x | k x - x * k ≤ r } ⊂ S . Let 0 < γ < min k x - x * k = r V ( x ) and define the level set V γ = { x B r | V ( x ) γ } . Then V γ is positive invariant since V 0 ( x ) 0 implies d d t V ( x ( t )) 0 and thus V ( x ( t )) V ( x (0)) γ for any x (0) ∈ V γ . Since V ( x ) is continuous at x * there exists δ > 0: k x 0 - x * k ≤ δ = V ( x 0 ) γ for every x 0 B δ ⊂ V γ . But then k x 0 - x * k ≤ δ x 0 ∈ V γ x ( t ; x 0 ) ∈ V γ B r for all time t 0. Since r < ε , this gives k x ( t ; x 0 ) - x * k ≤ ε .
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