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This note describes a model predictive control (MPC) formulation for discrete-time linear systems with hard constraints on control and state variables, under the assumption that the solution of the associated quadratic program is neither optimal nor satisfies the inequality constraints. This is common in embedded control applications, for which real-time constraints and limited computing resources dictate restrictions on the possible number of on-line iterations that can be performed within a sampling period. The proposed approach is rather general, in that it does not refer to a particular optimization algorithm, and is based on the definition of an alternative MPC problem that we assume can only be solved within bounded levels of suboptimality, and violation of the inequality constraints. By showing that the inexact solution is a feasible suboptimal one for the original problem, asymptotic or exponential stability is guaranteed for the closed-loop system. Based on the above general results, we focus on a specific dual accelerated gradient-projection method to obtain a stabilizing MPC law that only requires a predetermined maximum number of on-line iterations.

Model Predictive Controllers (MPC) designed for sampled-data systems can be shown to violate the constraints in continuous time. A reformulation of the initial problem will guarantee constraint satisfaction throughout the intersample period. Polytopic inclusions of the continuous trajectory are used in this technical note to establish additional constraints leading to a linearly constrained quadratic optimization problem. Continuous time asymptotic stability and continuous-time positive invariance are proven for the reformulated problem.

This paper proposes a dual fast gradient-projection method for solving quadratic programming problems that arise in linear model predictive control with general polyhedral constraints on inputs and states. The proposed algorithm is quite suitable for embedded control applications in that: (1) it is extremely simple and easy to code; (2) the number of iterations to reach a given accuracy in terms of optimality and feasibility of the primal solution can be estimated quite tightly; (3) the computational cost per iteration increases only linearly with the prediction horizon; and (4) the algorithm is also applicable to linear time-varying (LTV) model predictive control problems, with an extra on-line computational effort that is still linear with the prediction horizon.