Nuno P. Faísca

An algorithm for robust explicit/multi-parametric model predictive control

A new algorithm for robust explicit/multi-parametric Model Predictive Control (MPC) for uncertain, linear discrete-time systems is proposed. Based on previous work on Dynamic Programming (DP), multi-parametric Programming and Robust Optimization, the proposed algorithm features, (i) a DP reformulations of the MPC optimization problem, (ii) a robust reformulation of the constraints, and (iii) a multi-parametric programming step, where the control variables are obtained as explicit functions of the state variable, such that the state and input constraints are satisfied for all admissible values of the uncertainty. A key feature of the proposed procedure is that, as opposed to previous methods, it only solves a convex multi-parametric programming problem for each stage of the DP procedure.

Explicit/multi-parametric model predictive control (MPC) of linear discrete-time systems by dynamic and multi-parametric programming

This work presents a new algorithm for solving the explicit/multi-parametric model predictive control (or mp-MPC) problem for linear, time-invariant discrete-time systems, based on dynamic programming and multi-parametric programming techniques. The algorithm features two key steps: (i) a dynamic programming step, in which the mp-MPC problem is decomposed into a set of smaller subproblems in which only the current control, state variables, and constraints are considered, and (ii) a multi-parametric programming step, in which each subproblem is solved as a convex multi-parametric programming problem, to derive the control variables as an explicit function of the states. The key feature of the proposed method is that it overcomes potential limitations of previous methods for solving multi-parametric programming problems with dynamic programming, such as the need for global optimization for each subproblem of the dynamic programming step.

A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems

In this paper, we outline the foundations of a general global optimisation strategy for the solution of multilevel hierarchical and general decentralised multilevel problems, based on our recent developments on multi-parametric programming and control theory. The core idea is to recast each optimisation subproblem, present in the hierarchy, as a multi-parametric programming problem, with parameters being the optimisation variables belonging to the remaining subproblems. This then transforms the multilevel problem into single-level linear/convex optimisation problems. For decentralised systems, where more than one optimisation problem is present at each level of the hierarchy, Nash equilibrium is considered. A three person dynamic optimisation problem is presented to illustrate the mathematical developments.

Explicit robust model predictive control

Abstract Abstract Explicit robust multi–parametric feedback control laws are designed for constrained dynamic systems involving uncertainty in the left-hand side(LHS) of the underlying MPC optimization model. Our proposed procedure features: (i) a robust reformulation/optimization step, (ii) a dynamic programming framework for the model predictive control (MPC) problem formulation, and (iii) a multi-parametric programming solution step.

Global optimization of multi-parametric MILP problems

In this paper, we present a novel global optimisation approach for the general solution of multi-parametric mixed integer linear programs (mp-MILPs). We describe an optimisation procedure which iterates between a (master) mixed integer nonlinear program and a (slave) multi-parametric program. Moreover, we explain how to overcome the presence of bilinearities, responsible for the non-convexity of the multi-parametric program, in two classes of mp-MILPs, with (i) varying parameters in the objective function and (ii) simultaneous presence of varying parameters in the objective function and the right-hand side of the constraints. Examples are provided to illustrate the solution steps.

A framework for multi-parametric programming and control — an overview

In this paper we present an overview of the recent developments in multi-parametric programming and its specific use for control. An unifying optimisation framework to solve general multi-parametric programming problems is described, with special focus given to the important class of model predictive control problems.

Towards the Design of Parametric Model Predictive Controllers for Non-linear Constrained Systems

The benefits of parametric programming for the design of optimal controllers for constrained systems are widely acknowledged, especially for the case of linear systems. In this work we attempt to exploit these benefits and further extend the theoretical contributions to multi-parametric Model Predictive Control (mp-MPC) for non-linear systems with state and input constraints. The aim is to provide an insight and understanding of multi-parametric control and its benefits for non-linear systems and outline key issues for ongoing research work.

A global parametric programming optimisation strategy for multilevel problems

Abstract In this paper, we outline the foundation of a general global optimisation strategy for the solution of multilevel hierarchical and general decentralised multilevel problems based on our recent developments in multiparametric programming theory. The core idea is to recast each optimisation subproblem in the multilevel hierarchy as a multiparametric programming problem and the transform the multilevel problem into a single-level optimisation problem. For decentralised systems, where more than one optimisation problem is present at each level of the hierarchy, Nash equilibrium is considered. A three person dynamic optimisation problem is presented to illustrate the mathematical developments.

A Multi-Parametric Optimization Strategy for Multilevel Hierarchical Control Problems.

Abstract In this work the three–level hierarchical control problem and the decentralised control problem are investigated and a general optimisation strategy is developed for solving these problems based on recent developments on multi–parametric programming. The main idea is to recast each optimisation subproblem in the multilevel hierarchy as a multi–parametric programming problem and then transform the multilevel problem into a single-level optimisation problem. This allows for the control policies (decisions) at each level of the multilevel optimisation problem to be obtained as explicit functions of the state of the dynamic systems involved in each level and the control policies of the higher levels. A three person dynamic optimisation problem is presented to illustrate the mathematical developments.

Robust dynamic programming via multi-parametric programming

Abstract In this work; we present a new algorithm for solving complex multi-stage optimisation problems involving hard constraints and uncertainties; based on dynamic and multi-parametric programming. Each echelon of the dynamic programming procedure; typically employed in the context of multi-stage optimisation models; is interpreted as a robust multi-parametric optimisation problem; with the present states and future decision variables being the parameters; while the present decisions the corresponding optimisation variables. This reformulation significantly reduces the dimension of the original problem; essentially to a set of lower dimensional multi-parametric programs; which are sequentially solved. Furthermore; the use of sensitivity analysis circumvents non-convexities that naturally arise in constrained dynamic programming problems. The application of the proposed novel framework to robust constrained optimal control is highlighted.