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SUMMARYThis contribution addresses the problem of discrete time receding horizon quadratic control for plants whose input is restricted to belong to a finite set. We also study the dynamics of the resulting closed-loop system. Based upon the geometry of the underlying quadratic programme, a finitely parametrized expression for the control law is derived, which makes use of vector quantizers. Alternatively, the control law can be formulated by means of a polyhedral partition of the state space, which is closely connected with the partition induced when considering saturation-like constraints. Exact analytic expressions for the partition can be developed, therefore avoiding the need for on-line optimization. The closed-loop system, comprising controller and plant, exhibits highly nonlinear dynamics, due to the finite set restriction. Asymptotic stability only holds for very special cases. In general, this notion is too strong. Nevertheless, ultimate boundedness of state trajectories is often achieved. Tools for determining positively invariant sets, hence ensuring ultimate boundedness, are presented.

This paper characterises the geometric structure of receding horizon control (RHC) of linear, discrete-time systems, subject to a quadratic performance index and linear constraints. The geometric insights so obtained are exploited to derive a closed-form solution for the case where the total number of constraints is less than or equal to the number of degrees of freedom, represented by the number of control moves. The solution is shown to be a partition of the state space into regions for which an analytic expression is given for the corresponding control law. Both the regions and the control law are characterised in terms of the parameters of the open-loop optimal control problem that underlies RHC and can be computed off line. The solution for the case where the total number of constraints is greater than the number of degrees of freedom is addressed via an algorithm that iteratively uses the off-line solution and avoids on-line optimisation.

We derive a closed-form global analytical solution for Model Predictive Control (MPC) of linear, discretetime systems, subject to a quadratic performance index and hard magnitude constraints at the system input. The solution is shown to be a partition of the state space in regions for which an analytic expression is given for the corresponding control law. Both the regions and the control law are characterised in terms of the parameters of the open-loop optimal control problem that underlies MPC. The result exploits the geometric properties of quadratic programming.

In this paper, the problem of state and input constrained control is addressed, with multidimensional constraints. We obtain a local description of the boundary of the admissible subset of the state space where the state and input constraints can be satisfied \emph{for all times}. This boundary is made of two disjoint parts: the subset of the state constraint boundary on which there are trajectories pointing towards the interior of the admissible set or tangentially to it; and a barrier, namely a semipermeable surface which is constructed via a minimum-like principle.Comment: 36 pages, 8 figures, submitte

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