CONTROLLER REDUCTION FOR NONLINEAR PLANTS—AN L 2 APPROACH

Abstract

SUMMARY This paper extends the H = balanced truncation approach of Mustafa—Glover for linear plants to input aƒne nonlinear plants. Nonlinear H = balanced truncation is used to obtain a reduced order controller. Conditions which ensure that this controller stabilizes the full order plant are derived. This is done by relating the model reduction problem to a robust stabilization problem with unstructured perturbation. In addition an upper bound on the performance of the closed loop system, with respect to the ‚ 2 gain, is obtained. When specialized to linear plants this bound reduces to Mustafa—Glover’s result. ( 1997 by John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control, Vol. 7, 475—505 (1997) When modern controller design algorithms are applied to high order plants the resulting controller may have order high enough to create problems with its implementation. The controller reduction problem is concerned with reducing the order of the controller so as to maintain the stability of the control system and preserve the control system performance. Early research on this problem revealed that reducing the order of a full order stabilizing controller using open loop model reduction techniques was not suƒcient to preserve closed loop stability. Moreover this drawback was also encountered when the low order controller was obtained by designing it to control a reduced order approximation of the plant. Recently in References 1 and 2 this diƒculty has been overcome, in the linear case, by using H = closed loop balancing,2 to reduce the order of the plant. The controller obtained for this reduced order plant will then be of low order and still achieve both stability and an H = bound when used with the reduced order plant. The preservation of stability when this controller is used with the full order plant is then obtained by resorting to robust stability theory.3 This is done by viewing the change in the plant resulting from replacing the reduced order plant by the full order plant as an H = bounded unstructured uncertainty. The nonlinear balancing method and its use in model reduction was introduced in Reference 4 asymptotically stable plants and extended to unstable plants in Reference 5—7. The plant/controller reduction procedure based on nonlinear H = balancing was studied in References 6 and 7. Properties of the H = singular value functions and some properties of the reduced order „his paper was recommended for publication by editor I. Postlethwaite