lec7_6243_2003 - Massachusetts Institute of Technology...

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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 1 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 R n is the system state, w ( t ) W R m is system input, y ( t ) Y R k 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 x, ¯ x ) ψ ( g w ) , ¯ x X, w W. (7.3) V ( f w )) V x, ¯ w ) ¯ ¯ 1 Version of September 26, 2003
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2 In particular, when ψ 0, this yields the definition of a Lyapunov function. Finding, for a given supply rate, a valid storage function (or at least proving that one exists) is a major challenge in constructive analysis of nonlinear systems. The most com- mon approach is based on considering a linearly parameterized subset of storage function candidates V defined by N ± x ) = φ q V q V = { V x ) , (7.4) q =1 where { V q } is a fixed set of basis functions, and φ k are parameters to be determined. Here every element of V is considered as a storage function candidate , and one wants to set up an efficient search for the values of φ k which yield a function V satisfying (7.3). Example 7.1 Consider the finite state automata defined by equations (7.1) with value sets X = { 1 , 2 , 3 } , W = { 0 , 1 } , Y = { 0 , 1 } , and with dynamics defined by f (1 , 1) = 2 , f (2 , 1) = 3 , f (3 , 1) = 1 , f (1 , 0) = 1 , f (2 , 0) = 2 , f (3 , 0) = 2 , g (1 , 1) = 1 , g w ) = 0 w ) x, ¯ x, ¯ = (1 , 1) . In order to show that the amount of 1’s in the output is never much larger than one third of the amount of 1’s in the input, one can try to find a storage function V with supply rate ψ w ) = w y, ¯ ¯ y. Taking three basis functions V 1 , V 2 , V 3 defined by 1 , x = k, ¯ V k x ) = 0 , x = k, ¯ the conditions imposed on φ 1 , φ 2 , φ 3 can be written as the set of six affine inequalities (7.3), two of which (with w ) = (1 , 0) and w ) = (2 , 0)) will be satisfied automatically, while x, ¯ x, ¯ the other four are x, ¯ φ 2 φ 3 1 at w ) = (3 , 0) , x, ¯ φ 2 φ 1 → − 2 at w ) = (1 , 1) , x, ¯ φ 3 φ 2 1 at w ) = (2 , 1) , x, ¯ φ 1 φ 3 1 at w ) = (3 , 1) .
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