1088 IEEE TRAKSACTIONS ON AUTOMATIC CONTROL. VOL. zyxw AC-30. NO. 11. NOVEMBER zy 1985 Robust Control of Linear Time-Invariant Plants Using Periodic Compensation zy PRAMOD P. KHARGONEKAR, MEMBER. IEEE, KAMESHWAR POOLLA, AND ALLEN TANNENBAUM Abstract-This paper considers the use and design of linear periodic time-varying controllers for the feedback control of linear time-invariant discrete-time plants. We will show that for a large class of robustness problems, periodic compensators are superior to time-invariant ones. We will give explicit design techniques which can be easily implemented. In the context of periodic controllers, we also consider the strong and simultaneous stabilization problems. Finally, we show that for the problem of weighted sensitivity minimization for linear time-invariant plants, time-varying controllers offer no advantage over the time- invariant ones. I. INTRODUCTION I N this paper we study the use and design of linear periodic time-varying controllers for feedback control of finite-dimensional linear time-invariant (LTI) plants. We will show that time-varying controllers are superior to time-invariant ones for a large class of control problems. We shall be particularly concerned with the key questions of robust stabilization and sensitivity minimization, and shall stress explicit design tech- niques. Throughout most of this work, we deal with periodically varying discrete-time plants. A frequency domain approach for discrete-time periodic time-varying systems appears in the work of Davis [3]. Jury and Mullin zyxwvutsr [ 1 11. Meyer and Burrus [IS]. However. we have been particularly influenced by the book of Sz. Nag;-Foias [IS] (see especially ch. zyxwvutsrq 5) whose ideas lead to a “categorical” equivalence between periodic discrete-time sys- tems and certain kinds of LTI systems. Essentially. an m-input, p- output, N-periodic discrete-time system can be treated as an nlN- input, pN-output. LTI discrete-time system. Many of the results which. we obtain in this paper are derived using this LTI system representation for periodic time-varying systems together with some recent results of Khargonekar and Tannenbaum [ 121. In [12],the authors havestudied and solved certain kindsof robust feedback system design problems. In particular. in the context of LTI compensators, these authors show that for a discrete-time plant P(z) having both zeros and poles outside the closed unit disk, the maximal attainable gain margin is bounded. Indeed, they also derive explicit formulas for the maximal attainable gain margin in terms of unstable poles and zeros. However, we will show in this paper that by using time-varying controllers, itispossibleinseveralinteresting casesto signifi- cantly improve gain and phase margins for a discrete-time LTI plant P(s). The time-varying controllers we use are periodic with period less than or equal to the dimension plus one of the plant. In point of fact in most cases 2-periodic controllers suffice. Moreover. these controllers can be explicitly computed. Manuscript received June 21. 1981; revised April 15. 1985. Paper recommended by Associate Editor. T. L. Johnson. This &ork was supported in part by theNational Science Foundation under Grant ECS-8200607 and ECS-8400832. P. P. Khargonekar is with the Department of Electrical Engineering. University of Minnesota. Minneapolis. MN 55455. K. Poolla is with the Coordinated Science Laboratoq and the Department of Electrical Engineering. University of Illinois. Urbana. 1L 61801. A. Tannenbaum is with the Department of Mathematics. Ben Gurion University of the Negev, Beer Sheva. Israel. and the Department of Electrical Engineering. McGill University, Montreal, P.Q.. Canada. The basic reason why we may expect improvement in robust- ness via the use of periodic compensation is seen through the representation of periodic systems as LTJ systems (see Section 11). Briefly. given an LTI p X zyxw m plant P(z), we can regard P(z) as defining an N-periodic system. and represent it by a zy pN x rnN transfer matrix (Section 11). This transformation (for z N suffi- ciently large) has the effectof removing blocking zeros, and based on the work of [ 121 (see Section I11 for details), allows us to construct periodic controllers which in many cases drastically improve the robustness of the feedback system. In this paper we will also examine the problems of simultaneous stabilization and stabilization with a stable controller. Youla er z ai. [22] have proved the beautiful result that a continuous-time (respectively. discrete-time) LTI plant can be stabilized by a stable LTI controller if and only if a certain interlacing property involving the real right half plane (respectively. complement of the unit disk) poles and blocking zeros ofthe plant is satisfied. We show that any LTI plant can bestabilized by a stable periodic time-varying controller. The problem of simultaneous stabiliza- tion for LTI plants is thefollowing.Given n LTI plants Pl(z), PI&), ., P&), find (if possible) onecontroller that stabilizes each of the plants. This problem has been studied by [20], [17], [IO] withtherestrictionthatthe controller be time-invariant. It will be seen that by using periodic time-varying controllers, it is possible to stabilizeany finite collection of discrete-time time- invariant plants, and in point of fact, with a stable controller. Recently. Zarnes and Francis 1241. have formulated and solved the important problem of weighted sensitivity minimization. (See, also. 191, [2]. 141. and the references cited therein.) Khargonekar and Tannenbaum [12] have shown that certain robust system design problems (e.g.. gain/phase margin optimization problems. robust stabilization problem of Kimura [13]) and the sensitivity minimization problem are equivalent (in a precise mathematical sense). if one considers LTI controllers. We show in this paper that the minimal sensitivity cannot be improved by the use of arbitrary (not necessarily periodic) time-varying feedback. This is in contrast with such robustness properties asgaidphase margins, which in certain cases can be very significantly improved using periodic time-varying controllers. Thus. if one considers the more general class of time-varying controllers, the problems of robustness and sensitivity minimization appear to be dichotomous. The use of time-varying controllers for the control oftime- invariant plants has been known to be quite useful in some ‘ instances. For example, Anderson and Moore [ 1 J and Wang 1211 have shown that time-invariant plants with unstable decentralized fixed modes cannot be stabilized using decentralized time- invariant controllers. and yet can be stabilized by decentralized time-varying controllers. Our results provide further evidence that the use of time-varying controllers can be advantageous in many other control problems. 11. TRANSFER FUNCTIONS FOR PERIODIC SYSTEMS In this section we briefly discuss the basic elements of a transfer functiontheory for periodic, discrete-time, linear, time-varying systems. Similar theories have been developed in the system 0018-9286/85/1100-1088$01 .OO 0 1985 IEEE Authorized licensed use limited to: Georgia Institute of Technology. Downloaded on June 14,2010 at 19:17:15 UTC from IEEE Xplore. Restrictions apply.