298 IEEE zyxwvutsrq TRANSACTIONS ON AUTOMATIC CONTROL, VOL. zyxwvu 38. NO. 2, FEBRUARY zyxw 1993 The Graph Topology for Linear Plants with Applications to Nonlinear Robust Stabilization Antonio M. Pascoal, R. Ravi, and Pramod P. Khargonekar Abstract-In this note, we define the graph topology for the set of finite-dimensional linear time-varying plants that are internally stabiliz- able by output feedback. This follows some earlier results of Vidyasagar, Schneider, and Francis on time-invariant systems. We consider feedback connections of linear time-varying plants and general nonlinear time- varying controllers, and show that the graph topology is the weakest topology in which feedback stabilization is a robust property. Using this result, we derive necessary and sufficient conditions for robust stahiliz- ability of families of linear time-varying plants with parametric uncer- tainty. I. INTRODUCTION In this note, we address the problem of robust stabilization of families of finite dimensional linear time-varying (FDLTV) plants. The fundamental underlying question is: given a nominal linear time-varying plant zyxwvutsrqpon Po and a nonlinear time-varying (NLTV) stabilizing feedback controller K, what is the nature of the perturbations allowed in Po such that i) feedback stability is not compromised, and ii) the input-output behavior of the resulting feedback system is “close” to the nominal one? zyxwvutsrq An elegant answer to this question can be found in [12] and [13] for a broad class of feedback systems containing linear plants and controllers. In particular, a very readable account of the subject for the case where the plants and controllers are linear time-invariant (LTI) can be found in [13]. The graph topology for unstable plants was introduced in [13], and shown to be the weakest topology in which closed-loop stability is a robust property [13, theorem 2.11. A more general result concerning the robustness of stabilization under simultaneous plant and con- troller perturbations was also given [13, theorem 2.21. The methodology developed in [12] and [13] to prove the main results on robust stabilization is algebraic in nature, and relies on concepts from the areas of topology and feedback system analy- sis using the factorization approach [3], [14]. Motivated by this observation, we combine the ideas from [12], [13] and the ab- stract algebraic setting for feedback systems described in [3] and [SI to define the graph topology for the set of finite-dimensional linear time-uaryingplants that are stabilizable by output feedback. We consider general connections of linear time-iurying plants and nonlinear time-varying controllers, and show that the graph topology is the weakest topology in which nonlinear feedback Manuscript received June 28, 1991; revised January 24, 1992. This work was supported in part by the National Science Foundation under Grant ECS-8451519, by grants from Honeywell, Boeing, and General Electric, by the United States Air Force Office of Scientific Research under Grant AFOSR-88-0020, by the Commission of European Commu- nities under Grant MAST PL-890186, and by a University of Minnesota Doctoral Dissertation Fellowship award to R. Ravi. A. M. Pascoal is with CAPS-Complexo zyxwvutsr L and the Department zyxwvut of Electrical Engineering, Instituto Superior Tecnico, AV. Rovisco Pais. 1096 Lisbon Codex, Portugal. R. Ravi is with the Control Systems Laboratory GE CR & D, Schenec- tady, NY 12301. P. P. Khargonekar is with the Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109. IEEE Log Number 9203098. stabilization is robust (Theorem 3.10). Using this result, we derive necessary and sufficient conditions for robust stabilizabil- ity of families of linear time-varying plants with parametric uncertainty (Theorem 3.12). In particular, Theorem 3.12 shows that if there exists one (possible nonlinear) controller that ro- bustly stabilizes a given family of plants in the vicinity of a nominal plant Po, then every controller that stabilizes P,, will have the same property. Closely related to the results described here is the problem of defining a metric that generates the graph topology. This has been solved for the linear time-invariant case with the introduction of the gap [16] and graph metrics [131, which can be shown to be equivalent. The gap metric is usually preferred for quantitative robustness analysis since the gap distance between two systems can be effectively computed by performing normalized coprime factorizations for the systems and solving two (two-block) H,-optimization problems [5]. Using some recent results on normalized coprime factorizations 1113, gap metric computation [4], and H,-optimal control [lo], it is straightforward to define the gap metric for finite dimensional linear time-varying plants and prove that it generates the topol- ogy introduced in this note. The reader is referred to [ll] for further details on this issue. The note is organized as follows. Section I1 contains a brief exposition of an algebraic setting for the analysis of feedback systems, and Section 111 is devoted to the problem of robust stabilization of linear time-varying plants using nonlinear con- trollers. This section brings out the importance of the algebraic and topological aspects of feedback stability, and contains the main results on the graph topology. 11. FEEDBACK SYSTEM STABILITY: AN ALGEBRAIC SETTING This section is a brief summary of a general algebraic setting for the study of feedback systems containing linear plants and nonlinear controllers. In order to make the results general and independent of any particular representation of the controllers involved, a very broad set-up is adopted in which nonlinear time-varying systems are simply viewed as input-output opera- tors acting between suitably defined spaces [15]. This type of description fits naturally into the fractional representation ap- proach to feedback system analysis developed in [3] and [14], and will be used in the sequel as the major tool to study feedback system stability. We start with an introduction to the mathematical formalism required for the study of feedback systems, both from an inter- nal and an input-output point of view. zyxw A. Internal and Input-Output Descriptions We denote by zyxwvu R (respectively R,) the set of real (respectively positive real) numbers. The symbol RP denotes the Euclidean space of p-tuples of real numbers. Let X be the linear space of functions zyxwvut f mapping R+ to R”. For any T E R , n, denotes the projection operator defined for every f in X by (lI,f)(t) =f(t) when t I r, and 0 otherwise. Let L,[O, E; RP) denote the Hilbert space of Lebesgue measurable functions in X, endowed with the usual norm 11. [I1, and define the extended space L,,[O, zy x; [w”) := {f E x:n,f E L,[0, x; RP) for all finite T in R+}. In the sequel we compress the notation L,,[O, x; RP) and L,[0, x; RP) to Lfe, and Lf, respectively. This notation will be further simplified to L,, and L, whenever the dimension p is not relevant. 0018-9286/93$03.00 0 1993 IEEE