Max Model Predictive Control Research Articles

Min–max model predictive control as a quadratic program

The implementation of min–max model predictive control for constrained linear systems with bounded additive uncertainties and quadratic cost functions is dealt with. This type of controller has been shown to be a continuous piecewise affine function of the state vector by geometrical methods. However, no algorithm for computing the explicit solution has been given. Here, it is shown that the min–max optimisation problem can be expressed as a multi-parametric quadratic program, and so, the explicit form of the controller may be determined by standard multi-parametric techniques.

A Decomposition Algorithm for Feedback Min–Max Model Predictive Control

An algorithm for solving feedback min–max model predictive control for discrete-time uncertain linear systems with constraints is presented in this note. The algorithm is based on applying recursively a decomposition technique to solve the min–max problem via a sequence of low complexity linear programs. It is proved that the algorithm converges to the optimal solution in finite time. Simulation results are provided to compare the proposed algorithm with other approaches.

Input to state stability of min–max MPC controllers for nonlinear systems with bounded uncertainties

Min–max model predictive control (MPC) is one of the control techniques capable of robustly stabilize uncertain nonlinear systems subject to constraints. In this paper we extend existing results on robust stability of min–max MPC to the case of systems with uncertainties which depend on the state and the input and not necessarily decaying, i.e. state and input dependent bounded uncertainties. This allows us to consider both plant uncertainties and external disturbances in a less conservative way. It is shown that the input-to-state practical stability (ISpS) notion is suitable to analyze the stability of worst-case based controllers. Thus, we provide Lyapunov-like sufficient conditions for ISpS. Based on this, it is proved that if the terminal cost is an ISpS-Lyapunov function then the optimal cost is also an ISpS-Lyapunov function for the system controlled by the min–max MPC and hence, the controlled system is ISpS. Moreover, we show that if the system controlled by the terminal control law locally admits certain stability margin, then the system controlled by the min–max MPC retains the stability margin in the feasibility region.

Approximate robust dynamic programming and robustly stable MPC

We present a technique for approximate robust dynamic programming that is suitable for linearly constrained polytopic systems with piecewise affine cost functions. The approximation method uses polyhedral representations of the cost-to-go function and feasible set, and can considerably reduce the computational burden compared to recently proposed methods for exact robust dynamic programming [Bemporad, A., Borrelli, F., & Morari, M. (2003). Min–max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9), 1600–1606; Diehl, M., & Björnberg, J. (2004). Robust dynamic programming for min–max model predictive control of constrained uncertain systems. IEEE Transactions on Automatic Control, 49(12), 2253–2257]. We show how to apply the method to robust MPC, and give conditions that guarantee closed-loop stability. We finish by applying the method to a state constrained tutorial example, a parking car with uncertain mass.

Explicit solution of min–max MPC with additive uncertainties and quadratic criterion

Min–max model predictive control (MMMPC) is one of the strategies proposed to control plants subject to bounded uncertainties. This technique is very difficult to implement in real time because of the computation time required. Recently, the piecewise affine nature of this control law has been proved for unconstrained linear systems with quadratic performance criterion. However, no algorithm to compute the explicit form of the control law was given. This paper shows how to obtain this explicit form by means of a constructive algorithm. An approximation to MMMPC in the presence of constraints is presented based on this algorithm.

Min–max MPC algorithm for LPV systems subject to input saturation

In this paper, a new model predictive control (MPC) algorithm is developed for polytopic linear parameter-varying (LPV) systems subject to input saturation. A set invariance condition for discrete-time LPV systems with input saturation is identified and the invariant set is determined by solving an LMI optimisation problem, which directly incorporates input saturation. Based on this set invariance condition, a min–max MPC algorithm is proposed for the LPV systems with input saturation. A gain-scheduling MPC algorithm is also proposed for the design of a parameter-dependent controller. Numerical examples demonstrate the effectiveness of the algorithms.

Robust Dynamic Programming for Min–Max Model Predictive Control of Constrained Uncertain Systems

We address min-max model predictive control (MPC) for uncertain discrete-time systems by a robust dynamic programming approach, and develop an algorithm that is suitable for linearly constrained polytopic systems with piecewise affine cost functions. The method uses polyhedral representations of the cost-to-go functions and feasible sets, and performs multiparametric programming by a duality based approach in each recursion step. We show how to apply the method to robust MPC, and give conditions guaranteeing closed loop stability. Finally, we apply the method to a tutorial example, a parking car with uncertain mass.

Efficient implementation of constrained min–max model predictive control with bounded uncertainties: a vertex rejection approach

Min–Max Model Predictive Control (MMMPC) is one of the strategies used to control plants subject to bounded additive uncertainties. The implementation of MMMPC suffers a large computational burden due to the NP-hard optimization problem that has to be solved at every sampling time. This paper shows how to overcome this by transforming the original problem into a reduced min–max problem in which the number of extreme uncertainty realizations to be considered is significantly lowered. Thus, the solution is much simpler. In this way, the range of processes to which MMMPC can be applied is considerably broadened. A simulation example is given in the paper.

Real-Time Application of Scheduling Quasi-Min−Max Model Predictive Control to a Bench-Scale Neutralization Reactor

Scheduling quasi-min-max model predictive control is an MPC algorithm to control nonlinear systems. In this paper, real-time application of the scheduling quasi-min-max MPC algorithm to a bench-scale pH neutralization reactor is presented. The control performance is compared with multilinear model-based MPC and a scheduling IMC-PID controller.

Implementation of min–max MPC using hinging hyperplanes. Application to a heat exchanger

Min–max model predictive control (MMMPC) is one of the few control techniques able to cope with modelling errors or uncertainties in an explicit manner. The implementation of MMMPC suffers a large computational burden due to the numerical min–max problem that has to be solved at every sampling time. This fact severely limits the range of processes to which this control structure can be applied. An implementation scheme based on hinging hyperplanes that overcome these problems is presented here. Experimental results obtained when applying the controller to the heat exchanger of a pilot plant are given.

Feedback min‐max model predictive control using a single linear program: robust stability and the explicit solution

AbstractIn this paper we introduce a new stage cost and show that the use of this cost allows one to formulate a robustly stable feedback min–max model predictive control problem that can be solved using a single linear program. Furthermore, this is a multi‐parametric linear program, which implies that the optimal control law is piecewise affine and can be explicitly pre‐computed so that the linear program does not have to be solved on‐line. We assume that the plant model is known, is discrete‐time and linear time‐invariant, is subject to unknown but bounded state disturbances and that the states of the system are measured. Two numerical examples are presented; one of these is taken from the literature, so that a direct comparison of solutions and computational complexity with earlier proposals is possible. Copyright © 2004 John Wiley & Sons, Ltd.