Robust constrained model predictive control using linear matrix inequalities,

The primary disadvantage of current design techniques for model predict,ive control (MPC) is their inability to explicitly deal with model uncertainty. In t,liis paper, we address the robustness issue in MPC by directly incorporating the description of plant. uncertainty in the MPC problem formulation. The plant uncertainty is expressed in the t,ime-domain by allowing the state-space matrices of the discrete-time plant, t,o be arbitrarily time-varying and belonging to a polytope. The existence of a feedback control law minimizing an upper bound on the infinite horizon objective function and sat.isfying the input and output constraints is reduced to a convex opt.imization over linear matrix inequalities (LMIs). It is shown that for the plant uncertaint,y described by the polyt.ope, t,he feasible receding horizon state feedback control design is robustly stabilizing. 1. Iiitroductioii Model predict.ive cont,rol (MPC), also known as receding horizon control, is beconiing an increasingly popn1a.r cont.ro1 design method in the process industries. It. involves solnt,ion of a.n optimiza1,ion problem on-line (typically a linear or quadratic program) in order to determine opt,imal fnt.nre control inpnt,s. The first control move is implemented and at, t.he next sampling time, system measurements are nsed to resolve the opt,imiza.tion problem. The principal short,corning of exist,ing design techniqnes for MPC is their inability to deal explicit,ly wit.1~ plant, model uncertainty. Thus, nearly all known formulations of MPC as also the resolts on st,ability (e.g., [ 1 4 ] , [ 1 G ] ) make the assnmption tha.t a single linear time-invariant. (LTI) model describes the plant behaviour accnra.tely. Snch cont.rol systems might perform very poorly when implemented on a physical system which is not exactly described by the model (for example, see [19]). Garcia and Morari [G, 7, 81 suggested t,hat. the filt.er in MAC and IMC can be nsed t,o make nnconst,ra.ined MPC robust. Campo and Morari [4] modified the on-line optimization problem (linear program) for a single plant, t,o a “min-max” problem over a set, of p1ant.s. Allwright, a.nd Papavasilion [ I ] have explored the computational issnes of t.his “min-max” algorithm. Genreli and Nikolaon [9] showed how a. set of SISO systems can be robustly stabilized, given npper and lower bounds on the plant. impulse responses. Zheng and hiorari [19] have proposed a model predictive controller for uncertain Finite Impulse Response (FIR) models. Zafirion [18] has developed necessary/snfficient. condit,ions for robnst. st.abilit,y of MPC based on the cont.raction mapping principle. Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125. Current affiliation: Institute for Systems Research, University of Maryland. (818)395-4186. fax (818)568-8743. e-mail: MORARICOCALTECH .EDU ’To whom all correspondenre shoiild be aclclressed: phonr Clearly, what is needed is a computationally inexpensive scheme for robust MPC which allows consideration of a fairly general model uncertainty description and which at the same time guarantees stability for the class of systems described by the uncert,aint,y set. In the past few years, the important role of Linear Matrix Inequalities (LhlIs) in systems and control has increasingly been recognized [3] by researchers. Until recently, only a handful of these LMIs could be solved. The breakthrough came in 1988 when Nesterov and Nemirovski [13] developed interior point, met.hods t,hat, apply directly to convex problems involving LMIs. The implication of these developments is that problems whose solnt.ions can be computed numerically with reasonable compnt,at.ional effort must be considered “solved” even though no analytical solutions exist for these problems. The goal of this paper is to show that the infinite horizon MPC problem wit,li input, and output constraints and plant uncert,aint,y can be formulated as a convex optimization problem involving LMIs. The paper is organized as follows: In Section 2, we present. some preliminaries. In Section 3, we formulate the robnst constrained MPC problem with state-feedback as an LMI problem. We show that a feasible solution of the problem is robnst.ly stabilizing. In Section 4, we present examples illustrating t,he design procedure and finally, in Section 5 , we present. concluding remarks and comments on computational issnes. 2. Background Consider the discret,e time-varying linear system z(k + 1) = A(k)z (k ) + B(k)u(k) ( 1 ) y(k) = C(k)z (k ) (2) which represents the dynamics of the plant under consideration. Here, u ( k ) E X”“ is the manipulated variable, z(k) E ‘R”= is t,he state of the plant. The plant uncertainty is c a p tured by allowing the state-space matrices A(k) E 7L”” X n r , B ( k ) E RnrXnu, C(k) E ‘ R n Y x n r to be arbitrarily timevarying and lying in a polytope, i.e., [A(k ) B ( k ) C(k ) ] E R = Co{[A1 BI C l ] , . . , [ A L BL CL]}. In other words, [ A ( k ) B ( k ) C ( k ) ] = Cf‘L,,ai(k)[Ai Bi Ci] where a i ( k ) 2 0, X f = l a i ( k ) = 1. This description is also suitable for a nonlinear discrete time-varying system z(k + 1 ) = f ( x ( k ) ; u ( k ) , k ) , y(k) = g ( z ( k ) , k) if the Jacobian [g 31 E ~2 (see Liu et al. [11] for equivalence of a nonlinear syst.em and a time-varying linear system). Similarly, SISO finit.e impulse response uncertainty can be translated to such a state-space uncertainty. 2.1. Modcl Predictive Control The goal of model predictive control (MPC) is to calculate, a t each sampling t.ime I;, m manipulated variables u(k + i l k ) , i=O.l,. . . , m. 1, t,o minimize the quadratic objective J p ( k ) = Cf,,(z(k+ZJk)TQ1~(k+ilk)+g(k+ilL)TRffl(k+iJk)) subject