In particular, when o=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 y defined by ={V(2)=∑nV( where (Va) is a fixed set of basis functions, and Tk 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 Th 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,∫(2,1)=3,f(3,1)=1,f(1,0)=1,f(2,0)=2,f(3,0)=2, 9(1,1)=1,g(,)=0V(,)≠(1,1) In order to show that the amount of 1s in the output is never much larger than one third of the amount of 1s in the input, one can try to find a storage function V with supply rate (, Taking three basis functions Vi, V2, V3 defined by 1,=k 0,≠k the conditions imposed on T1, T2, T3 can be written as the set of six affine inequalities(7.3) two of which(with(I, a)=(1, 0)and(C, )=(2, 0)) will be satisfied automatically, while the other four are ≤1at(x,)=(3,0), 72-n1≤-2at(x,如)=(1,1) 73-72≤1at(x,)=(2,1) 73≤1at(x,)=(3,1) Solutions of this linear program are given by� 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 V = {V (¯x) = φqVq(¯x), (7.4) q=1 where {Vq} is a fixed set of basis functions, and φk 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 φk 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(¯x, w¯) = 0 � (¯x, w¯ =) ≡ (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(¯y, w¯ ¯ ) = w − 3¯y. Taking three basis functions V1, V2, V3 defined by 1, x¯ = k, Vk(¯x) = 0, x¯ =≡ k, the conditions imposed on φ1, φ2, φ3 can be written as the set of six affine inequalities (7.3), two of which (with (¯x, w¯) = (1, 0) and (¯x, w¯) = (2, 0)) will be satisfied automatically, while the other four are φ2 − φ3 → 1 at (¯x, w¯) = (3, 0), φ2 − φ1 → −2 at (¯x, w¯) = (1, 1), φ3 − φ2 → 1 at (¯x, w¯) = (2, 1), φ1 − φ3 → 1 at (¯x, w¯) = (3, 1). Solutions of this linear program are given by φ1 = c, φ2 = c − 2, φ3 = c − 1