An approximate approach to H/sup 2/ optimal model reduction

Abstract

This paper deals with the problem of computing an H H H 2 optimal reduced-order model for a given stable multivariable linear system. By way of orthogonal projection, the problem is formulated as that of minimizing the H H H 2 model-reduction cost over the Stiefel manifold so that the stability constraint on reduced-order models is automatically satisfied and thus totally avoided in the new problem formulation. The closed form expression for the gradient of the cost over the manifold is derived, from which a gradient flow results as an ordinary differential equation (ODE). A number of nice properties about such a flow are established. Furthermore, two explicit iterative convergent algorithms are developed from the flow; one has a constant step-size and the other has a varying step-size and is much more efficient. Both of them inherit the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model reduction cost is decreasing to minima along the iterates. A procedure for closing the gap between the original and modified problem is proposed. In the symmetric case, the two problems are shown to be equivalent. Numerical examples are presented to illustrate the effectiveness of the proposed algorithms as well as convergence. . His current research interests include the areas of control and signal processing. James Lam (S'87-M'88-SM'99) received the first class B.Sc. degree in mechanical engineering from the University of Manchester in 1983. He then obtained the M.Phil. and Ph.D. degrees in the area of control engineering from the University of Cambridge in 1985 and 1988, respectively.

Routh--type $L_2$ model reduction revisited

“…However, these procedures are intrinsically nonlinear, strongly depend on the initial conditions, do not even retain the steady-state value of the step response and, in some cases, might give rise to unstable models of stable systems [70]. These drawbacks justify the search for alternative simpler and more robust techniques, even if they lead to constrained optima or near-optima in the L 2 sense [24,49,68]. Such a path is followed in this paper by suitably combining some classic control-theory tools.…”

“…Besides the reduction methods based on the conservation of first-order information indices (e. g., coefficients of suitable series expansions), such as the classic Padé technique and its numerous variants [8,9] that are characterised by remarkable computational simplicity and ease of implementation, the methods based on second-order information indices (e. g., principal components, Hankel singular values, impulse-response energies) [16]- [19], [27,31,32,43,45,60,63], and on suitable quadratic criteria, such as the L 2 norm of the error [7,14], [21]- [24], [29], [33]- [35], [49,62], [66]- [68], [70,72], have enjoyed an increasing popularity since the late Seventies and early Eighties, and dedicated software has been developed for their implementation.…”

H2 Optimal Model Reduction of Coupled Systems on the Grassmann Manifold

“…For the system (2.4), when E = I, there is an existing model reduction method [32] on the Riemannian manifold, which is stated in the following theorem.…”

“…Since the transfer function of the reduced system is only related to the subspaces spanned by the transformation matrices, the H 2 model reduction error is seen as the cost function defined on the Grassmann manifold, and the H 2 optimal model reduction problem is treated as a minimization problem on the Grassmann manifold. Then some optimization techniques are employed to solve the minimization problem [15,32].…”