Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions This lecture gives an introduction into basic methods for finding Lyapunov functions and storage functions for given dynamical systems 7.1 Convex search for storage functions The set of all real-valued functions of system state which do not increase along system trajectories is convea, i.e. closed under the operations of addition and multiplication by a positive constant. This serves as a basis for a general procedure of searching for Lyapunov functions or storage functions 7.1.1 Linearly parameterized storage function candidates Consider a system model given by discrete time state space equations r(t+1)=f(x(t),(t),y(t)=9(x(t),u(t) where r(tEXCR is the system state, w(t)EWCR is system input, y(EYCR is system output, and f: XxW+X, 9: XxWHY are given functions. A functional V: XH R is a storage function for system(7. 1)with supply rate o: Y x W HR if V((t+1))-v(a(t))<(y(t) (72) for every solution of(7. 1), i.e. if V(f(x,)-V(x)≤σ(9(正,),)V∈X,∈W I Version of September 26, 2003Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions1 This lecture gives an introduction into basic methods for finding Lyapunov functions and storage functions for given dynamical systems. 7.1 Convex search for storage functions The set of all real-valued functions of system state which do not increase along system trajectories is convex, i.e. closed under the operations of addition and multiplication by a positive constant. This serves as a basis for a general procedure of searching for Lyapunov functions or storage functions. 7.1.1 Linearly parameterized storage function candidates Consider a system model given by discrete time state space equations x(t + 1) = f(x(t), w(t)), y(t) = g(x(t), w(t)), (7.1) where x(t) ≤ X ∀ Rn is the system state, w(t) ≤ W ∀ Rm is system input, y(t) ≤ Y ∀ Rk is system output, and f : X ×W ∈� X, g : X ×W ∈� Y are given functions. A functional V : X ∈� R is a storage function for system (7.1) with supply rate ψ : Y × W ∈� R if V (x(t + 1)) − V (x(t)) → ψ(y(t)) (7.2) for every solution of (7.1), i.e. if V (f(¯x, w¯)) − V (¯x) → ψ(g(¯x, w¯), w¯) � x¯ ¯ ≤ X, w ≤ W. (7.3) 1Version of September 26, 2003