I. Necoara

Model predictive control for perturbed continuous piecewise affine systems with bounded disturbances

Continuous piecewise-affine systems are a powerful tool for describing or approximating both nonlinear and hybrid systems. In this paper, we extend the model predictive control (MPC) framework for continuous piecewise-affine systems that we have developed previously to deterministic uncertainty. We show that the resulting MPC optimization problem can be transformed into a sequence of linear optimization problems (LP), which can be solved very efficiently.

Model predictive control for uncertain max–min-plus-scaling systems

We extend the model predictive control (MPC) framework that has been developed previously to a class of uncertain discrete event systems that can be modeled using the operations maximization, minimization, addition and scalar multiplication. This class encompasses max-plus-linear systems, min-max-plus systems, bilinear max-plus systems and polynomial max-plus systems. We first consider open-loop min-max MPC and we show that the resulting optimization problem can be transformed into a set of linear programming problems. Then, min-max feedback model predictive control using disturbance feedback policies is presented, which leads to improved performance compared to the open-loop approach

Finite-Horizon Min–Max Control of Max-Plus-Linear Systems

In this note, we provide a solution to a class of finite-horizon min-max control problems for uncertain max-plus-linear systems where the uncertain parameters are assumed to lie in a given convex and compact set, and it is required that the closed-loop input and state sequence satisfy a given set of linear inequality constraints for all admissible uncertainty realizations. We provide sufficient conditions such that the value function is guaranteed to be convex and continuous piecewise affine, and such that the optimal control policy is guaranteed to be continuous and piecewise affine on a polyhedral domain.

Stable Model Predictive Control for Constrained Max-Plus-Linear Systems

Discrete-event systems with synchronization but no concurrency can be described by models that are “linear” in the max-plus algebra, and they are called max-plus-linear (MPL) systems. Examples of MPL systems often arise in the context of manufacturing systems, telecommunication networks, railway networks, parallel computing, etc. In this paper we provide a solution to a finite-horizon model predictive control (MPC) problem for MPL systems where it is required that the closed-loop input and state sequence satisfy a given set of linear inequality constraints. Although the controlled system is nonlinear, by employing results from max-plus theory, we give sufficient conditions such that the optimization problem that is performed at each step is a linear program and such that the MPC controller guarantees a priori stability and satisfaction of the constraints. We also show how one can use the results in this paper to compute a time-optimal controller for linearly constrained MPL systems.

CONTROL OF PWA SYSTEMS USING A STABLE RECEDING HORIZON METHOD

Abstract In this paper we derive stabilization conditions for the class of piecewise affine (PWA) systems using the linear matrix inequality (LMI) framework. We take into account the piecewise structure of the system and therefore the matrix inequalities that we solve are less conservative. Using the upper bound of the infinite-horizon quadratic cost as a terminal cost and constructing also a convex terminal set we show that the receding horizon control stabilizes the PWA system. We derive also an algorithm for enlarging the terminal set based on a backward procedure; therefore, the prediction horizon can be chosen shorter, removing some computations off-line.

Robustly stabilizing MPC for perturbed PWL systems

This paper deals with robustly stable model predictive control (MPC) of the class of piecewise linear systems. A piecewise linear feedback controller, that stabilizes the nominal system, is derived from linear matrix inequalities. Further, an algorithm is designed for constructing a polyhedral robustly positively invariant set for the system. First, a minmax feedback MPC scheme with known mode, based on a dual-mode approach that stabilizes the system, is presented. The second robustly stable MPC scheme is based on a semi-feedback controller, but this time the mode of the system is unknown.

Robust hybrid MPC applied to the design of an adaptive cruise controller for a road vehicle

Hybrid model predictive control (MPC) for piece-wise affine (PWA) systems is used to solve a control problem for the tracking of a road vehicle. The study originates from the design of an adaptive cruise controller (ACC), that aims to track the velocity transmitted by a leading vehicle. In addition we consider disturbances due to model mismatch and to state measurements. The design specifications and corresponding constraints related to safety and comfort lead to nontrivial control problem. We present the results of a method that merges the use of hybrid MPC for tracking and regulation to a final equilibrium state, for which we compute the terminal cost and constraint set both in the deterministic and the perturbed case

Worst-case optimal control of uncertain max-plus-linear systems

In this paper the finite-horizon min-max optimal control problem for uncertain max-plus-linear (MPL) discrete-event systems is considered. We assume that the uncertain parameters lie in a given convex and compact set and it is required that the input and state sequence satisfy a given set of linear inequality constraints. The optimal control policy is computed via dynamic programming using tools from polyhedral algebra and multi-parametric linear programming. Although the controlled system is nonlinear, we provide sufficient conditions, which are usually satisfied in practice, such that the value function is guaranteed to be convex, continuous and piecewise affine, and the optimal control policy is continuous and piecewise affine on a polyhedral domain

Min-Max Model Predictive Control for Uncertain Max-Min-Plus-Scaling Systems

We extend the model predictive control (MPC) framework that has been developed previously to a class of uncertain discrete event systems that can be modeled using the operations maximization, minimization, addition and scalar multiplication. This class encompasses max-plus-linear systems, min-max-plus systems, bilinear max-plus systems and polynomial max-plus systems. We first consider open-loop min-max MPC and we show that the resulting optimization problem can be transformed into a set of linear programming problems. Then, min-max feedback model predictive control using disturbance feedback policies is presented, which leads to improved performance compared to the open-loop approach

Stabilization of Max-Plus-Linear Systems Using Receding Horizon Control — The Unconstrained Case

Abstract Max-plus-linear (MPL) systems are a class of discrete-event systems that can be described by models that are linear in the max-plus algebra. MPL systems arise in the context of e.g. manufacturing systems, telecommunication networks, railway networks, and parallel computing. We derive a receding horizon control scheme for MPL systems that guarantees a priori stability (in the sense of boundedness of the normalized state) of the closed-loop system in the unconstrained case. We also discuss the main properties of the resulting receding horizon controllers.