922 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004 Simplex Methods for Nonlinear Uncertain Sliding-Mode Control Giorgio Bartolini, Elisabetta Punta, Senior Member, IEEE, and Tullio Zolezzi Abstract—We develop a new analysis of the behavior of sim- plex control methods applied to multiple-input–multiple-output nonlinear control systems under uncertainties. According to such sliding-mode control methods the control vector is constrained to belong to a finite set of (fixed or varying) vectors, with an appro- priate switching logic to guarantee the specified sliding condition. Bounded uncertainties acting on the nominal system are allowed. The proposed sliding control methodology relies on the knowledge of the nominal system only. We prove rigorously the convergence of these methods to the sliding manifold in a finite time under explicit quantitative conditions on the system parameters and the available bounds of the uncertainty. Application to a robotic problem is discussed and a nonlinear example is presented. Index Terms—Multiple-input sliding-mode control, nonlinear uncertain control systems, simplex methods. I. INTRODUCTION I N RECENT literature, hybrid systems have been proposed as a general framework suitable to describe complex, natural as well as artificial, processes. Even if the definition of hybrid systems is not unique, generally speaking, they can be charac- terized as systems the behavior of which is described by contin- uous-time signals which evolve interlaced with discrete-valued ones. In this sense they can represent the point of convergence of discrete-events systems and continuous-time systems theories. As far as the analysis of hybrid systems behavior is con- cerned, a rather general formalization of such systems is by rep- resenting them as continuous time differential inclusions. These are more appropriate to describe uncertainties and allow a more flexible existence theory than standard ode’s, see [1]–[3]. Their right-hand side is modified in correspondence to a discrete set of events, possibly the outcome of a decision process. This new class of systems appears to be a very promising area of investi- gation for both applications and theory. In the recent literature some basic concepts and tools (Lyapunov stability, invariant set, uniform stability, etc.) in nonlinear systems analysis [4] have been extended to this more general framework (see [5]–[7] and the cited literature). Manuscript received July 18, 2002; revised June 24, 2003 and December 22, 2003. Recommended by Associate Editor P. Tomei. This work was supported in part by MURST, P.R.I.N. “Guidance and Control of Underwater Vehicles,” by CNR, “Progetto Giovani,” codice CNRG004E90, linea di ricerca: Fluido- dinamica Industriale, and by MURST Programma Cofinanziato “Controlli in retroazione e controlli ottimali.” G. Bartolini is with the Department of Electrical and Electronic Engineering, University of Cagliari, 09123 Cagliari, Italy (e-mail: giob@dist.unige.it). E. Punta is with the Institute of Intelligent Systems for Automation, Na- tional Research Council of Italy (ISSIA-CNR), 16149 Genoa, Italy (e-mail: punta@ge.issia.cnr.it). T. Zolezzi is with the Department of Mathematics, University of Genoa, 16146 Genoa, Italy (e-mail: zolezzi@dima.unige.it). Digital Object Identifier 10.1109/TAC.2004.829617 Hybrid systems formalization could also be used to describe control synthesis procedures for large scale continuous time pro- cesses. Even if it is, in principle, possible to find a solution to a control problem for such systems, it is often reasonable to partition the state–space into subregions and to associate with each one of them a suitable control vector attained through a simplified synthesis procedure. A wide variety of possibilities is offered to the designer, in the choice of the subregions and the corresponding control structures which are allowed to range within a set of either well established or ad hoc control method- ologies. Sliding-mode control methods, to which the name variable structure systems [8] is often associated, can be considered as a synthesis procedure leading to hybrid systems. Sliding-mode control techniques design simple control laws which constrain the system motion on suitably chosen mani- folds. The sliding motion is guaranteed despite uncertainties and strong nonlinearities, and is characterized by good properties (e.g., invariance, perfect tracking). Intrinsic to this methodology is the discontinuity of the control law on the boundaries of the regions in which the state space is suitably partitioned. The way in which this partition is accomplished and the choice of the rel- evant control law to attain a sliding motion in finite time are sup- ported by a generalization of the Lyapunov approach involving nonsmooth functions [9]–[12]. The motion along the sliding manifold is described ideally by Filippov’s solution concept, see [9] and [13]. In real implementations, the control law on the sliding manifold commutes at high frequency among the ele- ments of a discrete set of vectors, all separated from zero to guar- antee robustness. This behavior, usually called chattering phe- nomenon, is considered the main drawback of the method. Ac- tual development of output feedback methods [14]–[16] allows one to remove chattering by enforcing sliding motion, possibly using robust observer-differentiators, by means of higher order derivatives of the control signals which turn out to be continuous [17]. This idea is the basis of the second order sliding-mode ap- proach recently developed by some of the authors (see [18] and the cited bibliography). This paper is concerned mainly with the development of a special multiple-input control technique called vector simplex sliding-mode control. This method, introduced in [19] as an open-loop bang–bang control strategy in an optimal control context, has been applied as a discontinuous feedback control law in [20] with the twofold aim of minimizing the number of vector control structures re- quired to force the state trajectories on the sliding manifold (reaching condition) and to define unambiguously the represen- tation of the sliding motion (at least for systems affine in the 0018-9286/04$20.00 © 2004 IEEE