A comparison of homotopies for alternative formulations of the L 2 optimal model order reduction problem

Abstract A fundamental problem in control systems theory is finding a reduced order model that is optimal in the L 2 sense to a given (full order) system model. The numerical solution of this problem is challenging and the global convergence properties of homotopy methods are advantageous. A number of homotopy-based approaches have been developed. The primary numerical issues are the number of degrees of freedom in the homotopy parameter vector, the well-posedness of the finite-dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. This paper develops two new homotopy algorithms for optimal model reduction and uses several examples to compare their performance with the performance of two previous algorithms. The results show that the numerical well-conditioning is inversely related to the algorithmic efficiency and that the relative performance of a given algorithm is problem dependent.

[1]  P. Kabamba.  Balanced gains and their significance for L^{2} model reduction , 1985 .

[2]  D. Bernstein,et al.  Homotopy Approaches to the H2 Reduced Order Model Problem , 1991 .

[3]  D. Bernstein,et al.  Homotopy Methods for Solving the Optimal Projection Equations for the H2 Reduced Order Model Problem , 1992 .

[4]  Arthur E. Bryson,et al.  Design of low-order compensators using parameter optimization , 1985, Autom..

[5]  P. Kabamba,et al.  Balanced forms: Canonicity and parameterization , 1985, 1985 24th IEEE Conference on Decision and Control.

[6]  J. Hickin,et al.  Model reduction for linear multivariable systems , 1980 .

[7]  R. Skelton,et al.  Component cost analysis of large scale systems , 1983 .

[8]  D. Wilson.  Optimum solution of model-reduction problem , 1970 .

[9]  L. Watson,et al.  HOMPACK: a suite of codes for globally convergent homotopy algorithms. Technical report No. 85-34 , 1985 .

[10]  E. Collins,et al.  A parameterization of minimal plants , 1994, IEEE Trans. Autom. Control..

[11]  D. Bernstein,et al.  The optimal projection equations for fixed-order dynamic compensation , 1984 .

[12]  Layne T. Watson,et al.  Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms , 1987, TOMS.

[13]  D. Bernstein,et al.  The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore , 1985 .

[14]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[15]  A. Morgan,et al.  Computing singular solutions to polynomial systems , 1992 .

[16]  A. Morgan,et al.  A power series method for computing singular solutions to nonlinear analytic systems , 1992 .

[17]  Robert Skelton,et al.  Model reductions using a projection formulation , 1987, 26th IEEE Conference on Decision and Control.

[18]  Emmanuel G. Collins,et al.  A Homotopy Algorithm for Maximum Entropy Design , 1993, 1993 American Control Conference.

[19]  A. Morgan,et al.  Computing singular solutions to nonlinear analytic systems , 1990 .

[20]  Robert E. Skelton,et al.  Comments on "Balanced gains and their significance for L 2 model reduction" , 1986 .

[21]  Emmanuel G. Collins,et al.  An input normal form homotopy for the L2 optimal model order reduction problem , 1994, IEEE Trans. Autom. Control..