Probability-one homotopy algorithms for full- and reduced-orderH2/H∞ controller synthesis

Abstract

Homotopy algorithms for both full- and reduced-order LQG controller design problems with an H∞ constraint on disturbance attenuation are developed. The H∞ constraint is enforced by replacing the covariance Lyapunov equation by a Riccati equation whose solution gives an upper bound on H 2 performance. The numerical algorithm, based on homotopy theory, solves the necessary conditions for a minimum of the upper bound on H 2 performance. The algorithms are based on two minimal parameter formulations : Ly, Bryson and Cannon's 2 x 2 block parametrization and the input normal Riccati form parametrization. An overparametrization formulation is also proposed. Numerical experiments suggest that the combination of a globally convergent homotopy method and a minimal parameter formulation applied to the upper bound minimization gives excellent results for mixed-norm H 2 /H∞ synthesis. The nonmonotonicity of homotopy zero curves is demonstrated, proving that algorithms more sophisticated than standard continuation are necessary.