Probability‐one homotopy algorithms for full- and reduced‐order H2/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 H2 performance. The numerical algorithm, based on homotopy theory, solves the necessary conditions for a minimum of the upper bound on H2 performance. The algorithms are based on two minimal parameter formulations: Ly, Bryson and Cannon's 2 × 2 block parametrization and the input normal Riccati form parametrization. An over-parametrization 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 H2/H∞ synthesis. The non-monotonicity of homotopy zero curves is demonstrated, proving that algorithms more sophisticated than standard continuation are necessary.