Abstract-This paper considers high-speed control of constrained linear parameter-varying systems using model predictive control. Existing model predictive control schemes for control of constrained linear parameter-varying systems typically require the solution of a semi-definite program at each sampling instance. Recently, variants of explicit model predictive control were proposed for linear parameter-varying systems with polytopic representation, decreasing the online computational effort by orders of magnitude. Depending on the mathematical structure of the underlying system, the constrained finite-time optimal control problem can be solved optimally, or close-to-optimal solutions can be computed. Constraint satisfaction, recursive feasibility and asymptotic stability can be guaranteed a-priori by an appropriate selection of the terminal state constraints and terminal cost. The paper at hand gathers previous developments and provides new material such as a proof for the optimality of the solution, or, in the case of close-to-optimal solutions, a procedure to determine a bound on the suboptimality of the solution.Index Terms-Explicit model predictive control, linear parameter-varying systems, constrained control, optimal control, dynamic programming.

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a b s t r a c tIn this article we present a parametric branch and bound algorithm for computation of optimal and suboptimal solutions to parametric mixed-integer quadratic programs and parametric mixed-integer linear programs. The algorithm returns an optimal or suboptimal parametric solution with the level of suboptimality requested by the user. An interesting application of the proposed parametric branch and bound procedure is suboptimal explicit MPC for hybrid systems, where the introduced user-defined suboptimality tolerance reduces the storage requirements and the online computational effort, or even enables the computation of a suboptimal MPC controller in cases where the computation of the optimal MPC controller would be intractable. Moreover, stability of the system in closed loop with the suboptimal controller can be guaranteed a priori.

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