Homotopy algorithms for H²/H^∞ control analysis and synthesis

Abstract

The problem of finding a reduced order model, optimal in the H² sense, to a given system model is a fundamental one in control system analysis and design. The addition of a H^∞ constraint to the H² optimal model reduction problem results in a more practical yet computationally more difficult problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H^∞ model reduction problems are very difficult.

For both problems homotopy algorithms based on several formulations—input normal form; Ly, Bryson, and Cannon's 2 X 2 block parametrization; a new nonminimal parametrization—are developed and compared here. For the H² optimal model order reduction problem, these numerical algorithms are also compared with that based on Hyland and Bernstein's optimal projection equations.

Both the input normal form and Ly form are very efficient compared to the over parametrization formulation and the optimal projection equations approach, since they utilize the minimal number of possible degrees of freedom. However, they can fail to exist or be very ill conditioned. The conditions under which the input normal form and the Ly form become ill conditioned are examined.

The over-parametrization formulation solves the ill conditioning issue, and usually is more efficient than the approach based on solving the optimal projection equations for the H² optimal model reduction problem. However, the over-parametrization formulation introduces a very high order singularity at the solution, and it is doubtful whether this singularity can be overcome by using interpolation or other existing methods.

Although there are numerous algorithms for solving Riccati equations, there still remains a need for algorithms which can operate efficiently on large problems and on parallel machines and which can be generalized easily to solve variants of Riccati equations. This thesis gives a new homotopy-based algorithm for solving Riccati equations on a shared memory parallel computer. The central part of the algorithm is the computation of the kernel of the Jacobian matrix, which is essential for the corrector iterations along the homotopy zero curve. Using a Schur decomposition the tensor product structure of various matrices can be efficiently exploited. The algorithm allows for efficient parallelization on shared memory machines.

The linear-quadratic-Gaussian (LQG) theory has engendered a systematic approach to synthesize high performance controllers for nominal models of complex, multi-input multioutput systems and hence it is a breakthrough in modern control theory. 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 X 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 with a minimal parameter formulation applied to the upper bound minimization gives excellent results for mixed-norm synthesis.