Problem 8.2 Non-iterative computation of optimal value in H∞ control

Abstract

We consider an n-th order generalized linear system Σ characterized by the following state-space equations: Σ :where x is the state, u is the control input, w is the disturbance input, y is the measurement output, and h is the controlled output of Σ. For simplicity, we assume that D 11 = 0 and D 22 = 0. We also let Σ P be the subsystem characterized by the matrix quadruple (A, B, C 2 , D 2 ) and Σ Q be the subsystem characterized by (A, E, C 1 , D 1 ). The standard H ∞ optimal control problem is to find an internally stabilizing proper measurement feedback control law,such that when it is applied to the given plant (1), the H ∞ -norm of the resulting closed-loop transfer matrix function from w to h, say T hw (s), is minimized. We note that the H ∞ -norm of an asymptotically stable and proper continuous-time transfer matrix T hw (s) is defined aswhere w and h are, respectively, the input and output of T hw (s).

H<inf>&#x221E;</inf>-control for descriptor systems - A structured matrix pencils approach

“…For standard state space systems, where the dynamics of the system are modelled by a linear constant coefficient ordinary differential equation, the analysis of this problem is well studied and numerical methods have been developed and integrated in control software packages such as [2], [6], [12]. These methods work well for a wide range of problems in computing close to optimal (suboptimal) controllers, but the exact computation of the optimal value γ in H ∞ control is considered a challenge [7]. In order to avoid some of the numerical difficulties that arise when approaching the optimum, in [4], [5] several improvements of the previously known methods were presented.…”