Uncertain Discrete-time Singularly Perturbed Systems Robust H_ ¡Þ Control

Small parasitic parameters are frequently encountered in many physical sys-tems. such as motor control systems. electronic circuits, and so on. Singularly perturbed systems are dynamic systems with multiple time-scales. they are usually described by state-space models in which a small positive parameter multiplies the time derivatives of some of the states.Hx control of singularly perturbed systems has been investigated by many researchers and various approaches have been proposed. In the past decade, linear matrix inequality (LMI) technique has been extensively exploited to solve control problems.In particular, based on the LMI technique, state feedback H∞controller design methods are given in Fridman (2006) for continuous-time singularly perturbed systems with norm-bounded uncertainties. With pole-placement constraints, Lin and Li (2006) presents a sufficient condition for designing robust H∞dynamic output feedback controller. However. for the discrete-time case, there has been little LMI-based formulation for control synthesis.a linear discretetime singularly perturbed system can be represented by several models. However, they can all be classified under two categories:slow sampling rate model and fast sampling rate model (Naidu 1988). An LMI approach has been proposed to solve the H∞control problem of fast sampling discrete-time singularly perturbed systems.This paper studies the problem of H∞control of slow sampling discrete-time singularly perturbed systems. A method for designing H∞controller is given in terms of solutions to a set of linear matrix inequalities. Furthermore, a method of evaluating the upper bound of singular perturbation parameter (?) with meeting a prescribed H∞performance bound requirement is also given.The main conclusions in this paper are as follows(1)A method for designing H∞controller is given in terms of solutions to a set of linear matrix inequalities. A sufficient condition, which ensures the existence of state feedback controllers such that the resulting closed-loop system is asymptotically stable while satisfying a prescribed H∞norm bound, is obtained. A numerical example is given to illustrate the effectiveness of the proposed method.(2) A method of evaluating the upper bound of singular perturbation parameter (?)is given in terms of solutions to a set of linear matrix inequalities. A sufficient condition on searching for the allowable upper bound e* is obtained, such that the discrete-time singularly perturbed system is asymptotically stable and satisfies a prescribed Hx norm bound for any singular perturbation parameter∈∈(0,∈*]. A numerical example is given to illustrate the effectiveness of the proposed method.(3)For the discrete-time singularly perturbed systems with polytopic uncertain-ties, the results are extended to robust H∞control synthesis.