On H_∞ Model Reduction For Discrete-Time SSystems Using LMIs Billinear Transforamtion

This paper firstly summarized the development of system reduction; and then extended the LMIs-based continuous-time H∞model reduction problems via bilinear transformation and obtained a new model reduction for discrete-time LTI systems; In addition , the discrete-time H∞model redution problems was adressed as well ,which is nonetheless interesting to develop LMI techniques for purely discrete–time systems and the goal is not to purse heuristic numerical optimization procedures but to obtain"analytic"results by means of simple algebraic manipulations related to LMIs; Another motivation of present work is the recent intensive studies on model reduction of linear time-varying discrete-time systems, which can become effctive even for model reduction of LTV systems.The main innovations are outlined as follows:1. For discrete-time LTI systems, extended the LMIs-based continuous-time H∞model reduction problems via bilinear transformation, and obtained an equivalent proposition of the discrete-timeγ-suboptimal H∞model reduction. In addition, the bounds of H∞cost were given, thus the infimum of which was gained. Finally, given the constraints and computer implementation of MATLAB-LMI, an algorithm which can construct discrete-time H∞optimal reduced-order model was proposed.2. For H∞model reduction problem for LTI discrete-time systems. A more conciser proof than the previous studies was adressed where the analysis of the Hankel operaters plays a central role. for the well-known lower bounds on the approximation error, given in terms of the Hankel singular values of the system, via simple algebraic manipulations related to LMIs.Moreover, the moment reducing the system order by the multiplicity of the small Hankel singular value , turned the optimization problems subject to bilinear matrix inequalities (BMIs) into LMIs via constructing a pair of constant matrics based on banlanced realization,thus showed the optimal reduced-order models can be constructed by LMI optimizition.Finally, a numerical example show the effctiveness of the related approaches.The last part is the summary and perspective of this paper.