Min-max Model Predictive Control Research Articles

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

In this paper we extend the classical min–max model predictive control framework to a class of uncertain discrete event systems that can be modelled using the operations maximization, minimization, addition and scalar multiplication, and that we call max–min-plus-scaling (MMPS) systems. Provided that the stage cost is an MMPS expression and considering only linear input constraints then the open-loop min–max model predictive control problem for MMPS systems can be transformed into a sequence of linear programming problems. Hence, the min–max model predictive control problem for MMPS systems can be solved efficiently, despite the fact that the system is non-linear. A min–max feedback model predictive control approach using disturbance feedback policies is also presented, which leads to improved performance compared to the open-loop approach.

Discussion on Min-max Model Predictive Control of Nonlinear Systems: A Unifying Overview on Stability

Discussion on Min-max Model Predictive Control of Nonlinear Systems: A Unifying Overview on Stability

Scanning and tracking with independent cameras—a biologically motivated approach based on model predictive control

This paper presents a framework for visual scanning and target tracking with a set of independent pan-tilt cameras. The approach is systematic and based on Model Predictive Control (MPC), and was inspired by our understanding of the chameleon visual system. We make use of the most advanced results in the MPC theory in order to design the scanning and tracking controllers. The scanning algorithm combines information about the environment and a model for the motion of the target to perform optimal scanning based on stochastic MPC. The target tracking controller is a switched control combining smooth pursuit and saccades. Min-Max and minimum-time MPC theory is used for the design of the tracking control laws. We make use of the observed chameleon's behavior to guide the scanning and the tracking controller design procedures, the way they are combined together and their tuning. Finally, simulative and experimental validation of the approach on a robotic chameleon head composed of two independent Pan-Tilt cameras is presented.

Fuzzy Constrained Min-Max Model Predictive Control Based on Piecewise Lyapunov Functions

This paper proposes two novel stable fuzzy model predictive controllers based on piecewise Lyapunov functions and the min-max optimization of a quasi-worst case infinite horizon objective function. The main idea is to design state feedback control laws that minimize the worst case objective function based on fuzzy model prediction, and thus to obtain the optimal transient control performance, which is of great importance in industrial process control. Moreover, in both of these predictive controllers, piecewise Lyapunov functions have been used in order to reduce the conservatism of those existent predictive controllers based on common Lyapunov functions. It is shown that the asymptotic stability of the resulting closed-loop discrete-time fuzzy predictive control systems can be established by solving a set of linear matrix inequalities. Moreover, the controller designs of the closed-loop control systems with desired decay rate and input constraints are also considered. Simulations on a numerical example and a highly nonlinear benchmark system are presented to demonstrate the performance of the proposed fuzzy predictive controllers.

On input-to-state stability of min–max nonlinear model predictive control

In this paper we consider discrete-time nonlinear systems that are affected, possibly simultaneously, by parametric uncertainties and other disturbance inputs. The min–max model predictive control (MPC) methodology is employed to obtain a controller that robustly steers the state of the system towards a desired equilibrium. The aim is to provide a priori sufficient conditions for robust stability of the resulting closed-loop system using the input-to-state stability (ISS) framework. First, we show that only input-to-state practical stability can be ensured in general for closed-loop min–max MPC systems; and we provide explicit bounds on the evolution of the closed-loop system state. Then, we derive new conditions for guaranteeing ISS of min–max MPC closed-loop systems, using a dual-mode approach. An example illustrates the presented theory.

Constrained robust model predictive control for time-delay systems with polytopic description

This paper considers an infinite-horizon min–max model predictive control (MPC) problem for time-delay discrete systems with a polytopic uncertainty description, which has rarely been addressed in the literature. Several new solutions to this problem are proposed which are inter-complementary. If there is an input delay, then the single state feedback solution requires all the system matrices to be stable. This deficiency can be removed by adding free control moves before a modified state feedback law. If there is no input delay, then a computationally efficient off-line state feedback solution can be applied. In order to obtain desirable region of attraction while at the same time preserve reasonably low on-line computational effort, one can choose the standard solution. Simulation results are given to illustrate these new solutions.

Formulation of Closed-Loop Min–Max MPC as a Quadratically Constrained Quadratic Program

In this note, we show that min-max model predictive control (MPC) for linearly constrained polytopic systems with quadratic cost can be cast as a quadratically constrained quadratic program (QCQP). We use the rigorous closed loop formulation of min-max MPC, and show that any such min-max MPC problem with convex costs and constraints can be cast as a finite dimensional convex optimization problem, with the QCQP arising from quadratic costs as a special case. At the base of the proof is a lemma showing the convexity of the dynamic programming cost-to-go, which implies that the worst case on an infinite polytopic set is assumed on one of its finitely many vertices. As the approach is based on a scenario tree formulation, the number of variables in this problem grows exponentially with the horizon length. Fortunately, the QCQP is tree structured, and can thus be efficiently solved by specially tailored interior-point methods whose computational costs are linear in the number of variables. The new formulation as a tree sparse QCQP promises to facilitate online solution of the rigorous min-max MPC problem with quadratic costs

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.

Application of an explicit min-max MPC to a scaled laboratory process

Min-max model predictive control (MMMPC) requires the on-line solution of a min-max problem, which can be computationally demanding. The piecewise affine nature of MMMPC has been proved for linear systems with quadratic performance criterion. This paper shows how to move most computations off-line obtaining the explicit form of this control law by means of a heuristic algorithm. These results are illustrated with an application to a scaled laboratory process with dynamics fast enough to preclude the use of numerical solvers.

RBF-ARX MODEL-BASED ROBUST MPC FOR NONLINEAR SYSTEMS

An integrated modeling and robust model predictive control (MPC) approach is proposed for a class of nonlinear systems. First, the nonlinear system is identified off-line by a RBF-ARX model possessing linear ARX model structure and state-dependent Gaussian RBF neural network type coefficients. On the basis of the RBF-ARX model, a combination of a local linearization model and a polytopic uncertain linear parameter-varying (LPV) model are built to approximate the present and the future system's nonlinear behavior respectively. Subsequently, based on the approximate models, a min-max robust MPC algorithm with input constraint is designed for the nonlinear systems. The closed loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and the feasibility of the linear matrix inequalities (LMIs). Simulation study to a NOx decomposition process illustrates the effectiveness of the modeling and robust MPC approaches proposed in this paper.

EXPLICIT MIN-MAX MODEL PREDICTIVE CONTROL OF CONSTRAINED NONLINEAR SYSTEMS WITH MODEL UNCERTAINTY

This paper presents an approximate multi-parametric nonlinear programming approach to explicit solution of constrained nonlinear model predictive control (MPC) problems in the presence of model uncertainty. The case of time-invariant parameter uncertainty is considered. The explicit MPC controller is based on an orthogonal search tree structure of the state space partition and is designed by solving a min-max optimization problem. It is robust in the sense that all constraints are satisfied for all possible values of the uncertain parameters. The approach is applied to design an explicit min-max MPC controller for a continuous stirred tank reactor, where the heat transfer coefficient is an uncertain parameter.

EFFICIENT OFF-LINE SOLUTIONS TO ROBUST MODEL PREDICTIVE CONTROL USING ORTHOGONAL PARTITIONING

The main limitation of many robust model predictive control (MPC) schemes is the formidable real-time computational complexity. In this paper, a new algorithm for computing efficient approximate solutions to the min-max MPC problem for discrete-time polytopic systems is proposed. It is shown that the resulting control profile is, in fact, piecewise affine (PWA) defined on an orthogonal partition of the state space. This explicit structure is exploited for efficient real-time implementation via binary search trees. Conditions for robust exponential stability of the closed-loop system can be derived in terms of linear matrix inequality (LMI) constraints.

MIN-MAX MODEL PREDICTIVE CONTROL AS A QUADRATIC PROGRAM

This paper deals with the implementation of min-max model predictive control for constrained linear systems with bounded additive uncertainties and quadratic cost functions. 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. In this paper, we show that the min-max optimization 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.

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.