10766 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 67, NO. 12, DECEMBER 2020 A Novel Scheme of Nonfragile Controller Design for Periodic Piecewise LTV Systems Xiaochen Xie , Member, IEEE, James Lam , Fellow, IEEE, and Ka-Wai Kwok , Senior Member, IEEE Abstract—In this article, a novel nonfragile controller de- sign scheme is developed for a class of periodic piece- wise systems with linear time-varying subsystems. Two types of norm-bounded controller perturbations, includ- ing additive and multiplicative ones, are considered and partially characterized by periodic piecewise time-varying parameters. Using a new matrix polynomial lemma, the problems of nonfragile controller synthesis for peri- odic piecewise time-varying systems (PPTVSs) are made amenable to convex optimization based on the favorable property of a class of matrix polynomials. Depending on selectable divisions of subintervals, sufficient conditions of the stability and nonfragile controller design are proposed for PPTVSs. Case studies based on a multi-input multi- output PPTVS and a mass-spring-damper system show that the proposed control schemes can effectively guarantee the close-loop stability and accelerate the convergence under controller perturbations, with more flexible periodic time-varying controller gains than those obtained by the existing methods. Index Terms—Matrix polynomial, nonfragile control, peri- odic systems, time-varying systems. I. INTRODUCTION P ERIODIC characteristics, models, and tasks are present or required in a multitude of fields, including but not lim- ited to aerospace, mechatronics, networks, and signal process- ing [1]–[5]. Among them, linear periodic systems play important roles, given their convenience in tackling linear time-varying (LTV) systems similarly to linear time-invariant (LTI) ones [6]. Aimed at the stability and robust performance of linear periodic systems, research efforts have been focused on the Floquet– Lyapunov theory for periodic differential equations [7] and lifting-techniques based discrete-time periodic applications [8] over the past decades. Recent studies [9], [10] have revealed the Manuscript received August 1, 2019; revised November 15, 2019; ac- cepted December 10, 2019. Date of publication January 1, 2020; date of current version August 18, 2020. This work was supported in part by the National Natural Science Foundation of China under Grant 61973259, in part by the Innovation and Technology Commission of Hong Kong via the Innovation and Technology Fund (ITF) under Grant UIM/353, and in part by the Research Grants Council of Hong Kong under Grant 17206818, Grant 17202317, and Grant 17200918. (Corresponding author: Ka-Wai Kwok.) The authors are with the Department of Mechanical Engineer- ing, The University of Hong Kong, Pokfulam, Hong Kong (e-mail: xcxie@connect.hku.hk; james.lam@hku.hk; kwokkw@hku.hk). Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2019.2962439 efficiency of periodic piecewise models as approximations of continuous-time periodic systems that do not necessarily have closed-form expressions. Unlike the numerical computational approaches such as the Floquet–Lyapunov transformation and the monodromy matrix, periodic piecewise systems are less complicated and capable of overcoming the difficulties in con- troller synthesis brought by continuous-time periodic dynamics, making the related problems more amenable to convex optimiza- tion tools. The investigations on periodic piecewise systems are in- cipiently based on the model formulation consisting of sev- eral LTI subsystems, namely, the periodic piecewise linear system (PPLS). Motivated by the decomposition of periodic dynamics [11] and the theory of switched systems [12], the stability analysis and stabilizing controller design of PPLSs have been achieved by using Lyapunov functions with peri- odic piecewise matrices [9]. The periodic control scheme for PPLSs has been improved in [13] to provide time-varying controller gains under the framework of finite-time stability. Furthermore, H ∞ control and guaranteed cost control schemes are established for time-delay PPLSs as well as delay-free PPLSs in [14]–[16]. A peak-to-peak filter is designed for PPLSs with polytopic uncertainties in [17]. In [18], a ma- trix polynomial-based time-varying controller is proposed for PPLSs. For PPLSs with positivity, the stability and L 1 -gain are analyzed in [19]. Despite the simplicity of PPLSs, the time-invariant subsys- tem formulations may result in a loss of system dynamics. In practice, it is preferable to use time-varying models to represent practical systems, such as mechanical systems with periodic time-varying stiffness, loads or motions, and process plants involving periodic variables [20], [21]. Hence, the model of periodic piecewise time-varying system (PPTVS) is recom- mended. Compared with PPLSs, PPTVSs consist of a number of time-varying subsystems, leading to approximations that may be more desirable to preserve periodic dynamics. On the other hand, the time-varying dynamics in PPTVSs also lead to noncon- vex conditions especially during controller synthesis, bringing more difficulties for complicated cases with uncertainties or perturbations. For many engineering applications, controller perturbations commonly exist and result in the degradation of control performance [22], giving rise to the need of nonfragile control schemes, which have received extensive attention in the related fields including multivariable systems [23], switched systems [24], [25], time-delay systems [26], [27], sampled-data 0278-0046 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: The University of Hong Kong Libraries. Downloaded on August 26,2020 at 08:24:20 UTC from IEEE Xplore. Restrictions apply.