Stochastic MPC with Imperfect State Information and Bounded Controls

Abstract

This paper addresses the problem of output feedback Model Predictive Control for stochastic linear systems, with hard and soft constraints on the control inputs as well as soft constraints on the state. We use the so-called purified outputs along with a suitable nonlinear control policy and show that the resulting optimization program is convex. We also show how the proposed method can be applied in a receding horizon fashion. Contrary to the state feedback case, the receding horizon implementation in the output feedback case requires the update of several optimization parameters and the recursive computation of the conditional probability densities of the state given the previous measurements. Algorithms for performing these tasks are developed.

Stochastic receding horizon control with output feedback and bounded controls

“…Therefore, we restrict attention to a subclass of causal feedback policies for which the optimization problem is tractable. Guided by our earlier approach in [20,16,22,21] and given a control horizon N c 1 and a prediction horizon N N c , we would like to periodically minimize the cost 4…”

“…Since taking expectation of a convex function retains convexity [12], we conclude that the cost V t = E Yt X T t QX t + U T t RU t is convex in (η t , Θ t ). Similarly, the constraints (17) and (22) are convex in (η t , Θ t ) as they are a composition of convex and affine functions [12]. SOCP Formulation: Substituting the augmented dynamics (5) into the objective function (6), we have that…”

“…In particular, if we have full state information available, the control policy proposed in this article reduces to that proposed in [20] and to a special case of the policies proposed in [16] where the vector space is spanned by a single function. However, none of the earlier results, including those in [22], are able to deal with recursive feasibility and stability for the setup of this article. Moreover, the control policy structure in this article is different from that in [22].…”

“…However, none of the earlier results, including those in [22], are able to deal with recursive feasibility and stability for the setup of this article. Moreover, the control policy structure in this article is different from that in [22].…”

“…Moreover, in contrast to chance constraints where a bound is imposed on the probability of constraint violation, expectation-type constraints tend to give rise to convex optimization problems under weak assumptions [17,26,27]. We can augment problem (23) with the constraints (30) and (31) J t (5), (15), (16), (17), (22), (30), (31) . (32) Notice that the constraints (30) and (31) are not necessarily feasible at time t for any choice of parameters α t and β t .…”