Dimension reduction of multivariate linear systems under \(H^ \infty\) constraints (Q1914350)

Given a multi-input multi-output linear system \(\dot y(t) = Ay(t) + Bw(t)\), \(z(t) = Cy(t)\) with minimal realization \((A,B,C)\) of order \(n\). The input \(w(t)\) is zero mean white noise, and \(z(t)\) is the output. The problem is to approximate it by a reduced system \((A_r, B_r, C_r)\) with \(A_r\) of size \(r<n\) and producing an output \(z_r(t)\). The objective is to minimize the functional \[ J = \lim_{t \to \infty} M \biggl\{ \bigl |z(t) - z_r(t) \bigr |^2 \biggr\} \] where \(M\) denotes mathematical expectation and \(|\cdot |\) is the Euclidean vector norm. In this paper, the problem is solved under several constraints. In the simplest problem, one has only to satisfy the constraint \(|H_r(s) |_\infty \leq \gamma\) where \(H_r\) is the transfer function of the reduced system, \(|\cdot |_\infty\) is the usual \(H_\infty\)-norm for matrix valued functions (supremum on the imaginary axis of the largest singular value), and \(\gamma\) is a given positive number. It might be required that in addition to this, the reduced system retains a certain number of the poles of the original system. Another problem is to minimize \(J\), keeping a number of the original poles, but now requiring that \(|E |_\infty \leq \gamma\) where \(E = H - H_r\) is the difference between the transfer functions of the original and the reduced system. To solve the simplest problem, an upper bound is constructed for the values of \(J\) reachable under the given constraint on \(H_r\). It is obtained by solving a modified matrix Riccati equation. This upper bound is then minimized. To solve the problems where some poles are prescribed, the same technique is used, now projecting on the appropriate subspaces so that the poles are preserved. Some comments are given to the choice of the bounds \(\gamma\), but numerical results are not available.