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Consider the nonlinear dynamical system G given by (6.1) and (6.2) and the nonlinear controller Gc given by

where Ac ∈ R nc×nc , Bc ∈ R nc×m, S ∈ R nc×nc is skew symmetric, and diag(y) is a diagonal matrix whose entries on the diagonal are the components of y. Show that if G is passive and zero-state observable and the triple (Ac,Bc,) is strictly positive real and self-dual (see Problem 5.33), then the negative feedback interconnection of G and Gc is asymptotically stable.

Problem 5.33
Let

0 and B = C T. Show that a self-dual realization can be obtained from the change of coordinates z = P 1/2x, where P satisfies (5.151) and (5.152), and x is the internal state of the realization of G(s)

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