\(H_ \infty\) optimal control for symmetric linear systems (Q689908)

Let \(G(s)\) be a real-rational proper transfer matrix of a time-invariant linear system. Denote by \(Q\) and \(R\) orthogonal representations of a finite group \(A\) given on the input space \(U\) and output space \(Y\), respectively. \(G(s)\) is called symmetric (or invariant) with respect to \(A\) if \(R(g) G(s)= G(s) Q(g)\), for any \(g\) from \(A\). The standard \(H_\infty\)-control problem is called \(A\)-symmetric with respect to \(A\) (or invariant) if the plant transfer matrix \(P\) is \(A\)-symmetric. The model matching problem is called symmetric if its three transfer matrices are symmetric. Providing, first, that a symmetric standard problem can be reduced to a symmetric model matching problem, and, second, that the latter has a symmetric \(H_\infty\)-optimal solution, it is shown that the symmetric standard problem has a symmetric optimal solution. Two numerical computation approaches are proposed to solve an \(A\)- symmetric \(H_\infty\)-standard control problem. The first one uses the \(A\)-symmetry of the two Riccati equations to get an \(A\)-symmetric central solution, while the second one uses block-diagonalization of \(A\)- symmetric matrices and decomposition of the problem into subproblems of lower dimensions.