Bert Pluymers

The efficient computation of polyhedral invariant sets for linear systems with polytopic uncertainty

In this paper the concept of maximal admissable set (MAS), introduced by Gilbert et al. (1991) for linear time-invariant systems, is extended to linear systems with polytopic uncertainty under linear state feedback. It is shown that by constructing a tree of state predictions using the vertices of the uncertainty polytope and by imposing state and input constraints on these predictions, a polyhedral robust invariant set can be constructed. The resulting set is proven to be the maximal admissable set. The number of constraints defining the invariant set is shown to be finite if the closed loop system is quadratically stable (i.e. has a quadratic Lyapunov function). An algorithm is also proposed that efficiently computes the polyhedral set without exhaustively exploring the entire prediction tree. This is achieved through the formulation of a more general invariance condition than that proposed in Gilbert et al. (1991) and by the removal of redundant constraints in intermediate steps. The efficiency and correctness of the algorithm is demonstrated by means of a numerical example.

Interpolation based MPC for LPV systems using polyhedral invariant sets

Guaranteeing asymptotic stability and recursive constraint satisfaction for a set of initial states that is as large as possible and with both a minimal control cost and computational load can be identified as a common objective in the model predictive control (MPC) community. General interpolation (Rossiter et al., 2004, Bacic et al, 2003) provides a favourable trade oil between these different aspects, however, in the robust case, this requires on-line semi-definite programming (SDP), since one typically employs ellipsoidal invariant sets. Recently, (Pluymers et al., 2005) have proposed an efficient algorithm for constructing the robust polyhedral maximal admissible set (Gilbert et al., 1991) for linear systems with polytopic model uncertainty. In this paper a robust interpolation based MPC method is proposed that makes use of these sets. The algorithm is formulated as a quadratic program (QP) and is shown to have improved feasibility properties, efficiently cope with non-symmetrical constraints and give better control performance than existing interpolation based robust MPC algorithms.

A SIMPLE ALGORITHM FOR ROBUST MPC

Abstract The majority of recent works on robust MPC either require very burdensome online computations or are restricted to relatively small feasible regions. This paper builds on a recent work which demonstrated that one could compute the maximal admissible set for a linear parameter varying system and shows how this set can be used as the terminal region and hence allows the definition of an MPC algorithm requiring only quadratic programming, but with a maximal region of attraction and guaranteed convergence for the robust case.

Robust triple mode MPC

This paper reviews triple mode predictive control for LTI systems, and proposes a new algorithm for robust triple mode predictive control for constrained linear systems described by polytopic uncertainty models. The approach significantly enlarges the feasibility region compared to robust dual mode approaches. The efficacy of the approach is demonstrated with numerical examples

Min-max feedback MPC using a time-varying terminal constraint set and comments on: Efficient robust constrained model predictive control with a time-varying terminal constraint set

Abstract In this paper a robust MPC scheme using a time-varying terminal constraint set for input-constrained systems with a polytopic uncertainty description is proposed. The new scheme is a refinement of the algorithm published in “Efficient robust constrained model predictive control with a time-varying terminal constraint set” [Z. Wan, M.V. Kothare, Systems Control Lett. 48 (2003) 375–383]. The original result contains an error in the stability proof which is solved in this paper by introducing within-horizon feedback [P.O.M. Scokaert, D.Q. Mayne, IEEE Trans. Automat. Control 43(8) (1998) 1136–1142]. in the on-line algorithm. The new scheme is proven to be robustly asymptotically stabilizing but at the cost of an increased computational complexity.

A minimal model for glycemia control in critically ill patients.

In this paper we propose a modified minimal model to be used for glycemia control in critically ill patients. For various reasons the Bergman minimal model is widely used to describe glucose and insulin dynamics. However, since this model is mostly valid in a rather restrictive setting, it might not be suitable to be used in a model predictive controller. Simulations show that the new model exhibits a similar glycemia behaviour but clinically more realistic insulin kinetics. Therefore it is potentially more suitable for glycemia control. The designed model is also estimated on a set of critically ill patients giving promising results

Vibro-acoustic analysis procedures for the evaluation of the sound insulation characteristics of agricultural machinery cabins

Abstract Over the last few years, customer demands regarding acoustic performance, along with the tightening of legal regulations on noise emission levels and human exposure to noise, have made the noise and vibration properties into important design criteria for agricultural machinery cabins. In this framework, both experimental analysis procedures for prototype testing as well as reliable numerical prediction tools for early design assessment are compulsory for an efficient optimization of the cabin noise and vibration comfort. This paper discusses several numerical approaches, which are based on the finite element and boundary element method, in terms of their practical use for airborne sound insulation predictions. To illustrate the efficiency and reliability of the various vibro-acoustic analysis procedures, the numerical procedures are applied for the case of a harvester drivers cabin and validated with experimental results.

Constrained linear MPC with time-varying terminal cost using convex combinations

Recent papers (IEEE Transactions on Automatic Control 48(6) (2003) 1092-1096, Automatica 38 (2002) 1061-1068, Systems and Control Letters 48 (2003) 375-383) have introduced dual-mode MPC algorithms using a time-varying terminal cost and/or constraint. The advantage of these methods is the enlargement of the admissible set of initial states without sacrificing local optimality of the controller, but this comes at the cost of a higher computational complexity. This paper delivers two main contributions in this area. First, a new MPC algorithm with a time-varying terminal cost and constraint is introduced. The algorithm uses convex combinations of off-line computed ellipsoidal terminal constraint sets and uses the associated cost as a terminal cost. In this way, a significant on-line computational advantage is obtained. The second main contribution is the introduction of a general stability theorem, proving stability of both the new MPC algorithm and several existing MPC schemes (IEEE Transactions on Automatic Control 48(6) (2003) 1092-1096, Automatica 38 (2002) 1061-1068). This allows a theoretical comparison to be made between the different algorithms. The new algorithm using convex combinations is illustrated and compared with other methods on the example of an inverted pendulum.

An adaptive input–output modeling approach for predicting the glycemia of critically ill patients

In this paper we apply system identification techniques in order to build a model suitable for the prediction of glycemia levels of critically ill patients admitted to the intensive care unit. These patients typically show increased glycemia levels, and it has been shown that glycemia control by means of insulin therapy significantly reduces morbidity and mortality. Based on a real-life dataset from 15 critically ill patients, an initial input-output model is estimated which captures the insulin effect on glycemia under different settings. To incorporate patient-specific features, an adaptive modeling strategy is also proposed in which the model is re-estimated at each time step (i.e., every hour). Both one-hour-ahead predictions and four-hours-ahead simulations are executed. The optimized adaptive modeling technique outperforms the general initial model. To avoid data selection bias, 500 permutations, in which the patients are randomly selected, are considered. The results are satisfactory both in terms of forecasting ability and in the clinical interpretation of the estimated coefficients.

Comments on: "Efficient robust constrained model predictive control with a time varying terminal constraint set" by Wan and Kothare

Abstract We present an algorithm that modifies the original formulation proposed in Wan and Kothare [Efficient robust constrained model predictive control with a time-varying terminal constraint set, Systems Control Lett. 48 (2003) 375–383]. The modified algorithm can be proved to be robustly stabilizing and preserves all the advantages of the original algorithm, thereby overcoming the limitation pointed out recently by Pluymers et al. [Min–max feedback MPC using a time-varying terminal constraint set and comments on “Efficient robust constrained model predictive control with a time-varying terminal constraint set”, Systems Control Lett. 54 (2005) 1143–1148].