Input-to-state stability of robust receding horizon control with an expected value cost

Abstract This paper is concerned with the stability of a class of receding horizon control laws for constrained linear discrete-time systems subject to bounded state disturbances and compact and convex state and input constraints. The paper considers the class of finite horizon feedback control policies parameterized as affine functions of the system state, calculation of which has recently been shown to be tractable via a convex reparameterization. When minimizing the expected value of a finite horizon quadratic cost, we show that the value function is convex. When solving this optimal control problem at each time step and implementing the result in a receding horizon fashion, we provide sufficient conditions under which the closed-loop system is input-to-state stable (ISS). This paper proposes a class of robust receding horizon control (RHC) laws for constrained linear discrete-time systems subject to bounded state disturbances. We consider a class of affine feedback control policies parameterized as affine functions of the system state, calculation of which has been shown to be tractable via a convex reparameterization In In this paper we present a generalization of these results, using the expected value of a quadratic cost. We demonstrate that, for systems with compact and convex state and input constraints and disturbance sets, the resulting value function is convex and lower semicontinuous when optimizing over affine feedback policies, and provide conditions under which input-to-state stability can be established for such systems using convex Lyapunov functions. Since the constraints and disturbance sets we consider are not necessarily polytopic, the proofs differ substantially from those required in the polytopic case. This generalization is of particular interest, for example, in the case where the disturbance set is 2−norm bounded, and the resultant optimization problem can be solved as a tractable second-order cone program (SOCP), but for which no stability proof exists at present. Notation: E [·] is the expectation operator. Given vectors x and y and matrices A and B, x := √ x x, vec(x, y) := [x y ] , A F := trace(A A) and A⊗B is the Kronecker product of A and B. Given a set X, intX is its interior and linX is its linear hull