Richard Oberdieck

Explicit hybrid model-predictive control

This article presents an algorithm for the exact solution of explicit hybrid model-predictive control problems of time-invariant, discrete-time mixed logical dynamical systems. Using multiparametric programming, this control problem is formulated as a multiparametric mixed-integer quadratic programming problem where the initial states of the system are treated as parameters. In conjunction with decomposition type or branch-and-bound type approaches, the proposed solution strategy first creates affine relaxations of nonconvex critical regions that result from the comparison of two quadratic objective functions. These relaxations are then used to generate an affine outer approximation for the critical regions. Upon termination, the bounded space of the initial states is partitioned into possibly nonconvex critical regions and corresponding optimal control laws.

A branch and bound method for the solution of multiparametric mixed integer linear programming problems

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm.

Multi-objective optimization with convex quadratic cost functions: A multi-parametric programming approach

Abstract In this note we present an approximate algorithm for the explicit calculation of the Pareto front for multi-objective optimization problems featuring convex quadratic cost functions and linear constraints based on multi-parametric programming and employing a set of suitable overestimators with tunable suboptimality. A numerical example as well as a small computational study highlight the features of the novel algorithm.

Explicit model predictive control: A connected-graph approach

Abstract The ability to solve model predictive control (MPC) problems of linear time-invariant systems explicitly and offline via multi-parametric quadratic programming (mp-QP) has become a widely used methodology. The most efficient approaches used to solve the underlying mp-QP problem are either based on combinatorial considerations, which scale unfavorably with the number of constraints, or geometrical considerations, which require heuristic tuning of the step-size and correct identification of the active set. In this paper, we describe a novel algorithm which unifies these two types of approaches by showing that the solution of a mp-QP problem is given by a connected graph, where the nodes correspond to the different optimal active sets over the parameter space. Using an extensive computational study, as well as the explicit MPC solution of a combined heat and power system, the merits of the proposed algorithm are clearly highlighted.

Offset-Free Explicit Hybrid Model Predictive Control of Intravenous Anaesthesia

The paper describes strategies for the control of intravenous depth of anaesthesia for the induction and maintenance phase, based on a detailed compartmental model composed of a pharmacokinetic and a pharmacodynamic model. The system can be described in a piece-wise affine fashion, leading to a hybrid model predictive control problem, which is solved explicitly via the solution of a multi-parametric mixed integer quadratic programming problem. Two model predictive control strategies are presented: an explicit hybrid model predictive strategy and a robust explicit hybrid model predictive strategy that uses a robust reference tracking algorithm. The control strategies are successfully tested on a set of 7 patients.

On unbounded and binary parameters in multi-parametric programming: applications to mixed-integer bilevel optimization and duality theory

In multi-parametric programming an optimization problem is solved as a function of certain parameters, where the parameters are commonly considered to be bounded and continuous. In this paper, we use the case of strictly convex multi-parametric quadratic programming (mp-QP) problems with affine constraints to investigate problems where these conditions are not met. Based on the combinatorial solution approach for mp-QP problems featuring bounded and continuous parameters, we show that (i) for unbounded parameters, it is possible to obtain the multi-parametric solution if there exists one realization of the parameters for which the optimization problem can be solved and (ii) for binary parameters, we present the equivalent mixed-integer formulations for the application of the combinatorial algorithm. These advances are combined into a new, generalized version of the combinatorial algorithm for mp-QP problems, which enables the solution of problems featuring both unbounded and binary parameters. This novel approach is applied to mixed-integer bilevel optimization problems and the parametric solution of the dual of a convex problem.

Explicit hybrid model predictive control strategies for intravenous anaesthesia

Abstract In this work, we first present a piece-wise affine model for intravenous anaesthesia, based on which a hybrid explicit/multiparametric model predictive control strategy is developed. To deal with the inter- and intra-patient variability, an estimation strategy, the multiparametric moving horizon estimator, and different robust algorithms such as Offset Correction, State-Output Correction and Prediction Output Correction are further designed and implemented simultaneously with the hybrid multiparametric model predictive control. Simulation results for a set of 12 virtually generated patients for the regulation of the depth of anaesthesia by means of the Bispectral Index with Propofol as the anaesthetic, demonstrate the validity and usefulness of the proposed advanced control and estimation strategies.

A control strategy for periodic systems – application to the twin-column MCSGP

Abstract In this work we present advanced multi-parametric control strategies for the “Multicolumn Countercurrent Solvent Gradient Purification” (MCSGP) process, which is a continuous chromatographic separation process, governed by a periodic operation profile, linked to the intensification of monoclonal antibody production. We demonstrate a seamless, step-by-step procedure for the development of multi-parametric controllers as part of our recently introduced PAROC framework and software platform. The designed controller assures optimal operating conditions, avoiding pertubations, under continuous operation, while capturing the periodic nature of the process.

A framework for hybrid multi-parametric model-predictive control with application to intravenous anaesthesia

Abstract In this paper we present a framework for the development of hybrid multi-parametric model predictive controllers applied to intravenous anaesthesia. A step-by-step procedure is described featuring a piece-wise model for anaesthesia describing the induction and maintenance phase, a recently developed multi-parametric mixed-integer quadratic programming solver and a novel hybrid control strategy tested for a set of 12 patients.

Parallel computing in multi-parametric programming

Abstract In multi-parametric programming, an optimization problem is solved as a function of certain bounded parameters. Hence it requires the exploration of the corresponding parameter space, a procedure which inherently leads to independent subproblems to be solved for each part of the parameter space. This characteristic is used to develop a parallelization strategy for many classes of multi-parametric programming algorithms. The trade-off between information overhead and independence of each machine is addressed explicitly through the introduction of a user-defined parameter. This novel approach is applied to a geometrical multi-parametric quadratic programming algorithm; a computational study as well as the application to a combined heat and power heat recovery subsystem show the benefits of the developed approach.