Controller reduction via stable factorization and balancing

Abstract

A controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper. For a certain right coprime factorization of an LQG designed controller K(s) = N(s)D-'(s), we approximate using a balancing technique the pair [D(s), N(S)]T by a low-order pair [D,(s), N,(S)]T defining a factorization of the reduced-order controller K ,(s) = N ,(s)D~'(s). Weshow that reducing the controller order in this way is motivated in a natural way, which leads to the expectation of both good stability properties and good accuracy of approximation of closed-loop behaviour. This is also demonstrated in some examples.

Finite‐dimensional approximation for optimal fixed‐order compensation of distributed parameter systems

“…However, with ne = 1, such controllers were found to be destabilizing. Based upon the results in [13], this was not unexpected. We employ an approximation scheme recently proposed by Ito and Kappel in [26J.…”

“…In fact, with the second approach, this may occur even when a suitable controller is known to exist. For example, as is shown in [13], controller reduction techniques may even destabilize the closed-loop system.…”

“…First, since such methods are not predicated on the minimization of a performance index, prospects for convergence are slim. And, second, controllerreduction methods have not proven to be reliable in producing stabilizing compensators (see, for example, [13]). …”

Gramian-Based Model Reduction for Data-Sparse Systems

“…In state-space this corresponds to a zero steady-state error. Zero steady-state errors can be obtained by a singular perturbation approximation (SPA) to the original system [35,47], also called balanced residualization. Assume the realization of the system (1) is minimal (otherwise use balanced truncation to reduce the order to the McMillan degree of the system) and balanced.…”

Structured coprime factor model reduction based on LMIs☆

“…To avoid this problem, in this paper we seek structured model reduction methods in the coprime factor description of a system. In the absence of structure, such methods have been developed by Anderson & Liu (1989); Liu & Anderson (1986); McFarlane & Glover (1990); Meyer (1990);El-Zobaidi & Jaimoukha (1998), and can be applied even to unstable systems; they are briefly reviewed in Section 2. Particularly attractive are reductions based on normalized coprime factorizations, since robustness results on closed-loop stability are available.…”