Eduardo F. Camacho

Generalized Predictive Control

This chapter describes one of the most popular predictive control algorithms: Generalized Predictive Control (GPC). The method is developed in detail, showing the general procedure to obtain the control law and its most outstanding characteristics. The original algorithm is extended to include the cases of measurable disturbances and change in the predictor. Close derivations of this controller such as CRHPC and Stable GPC are also treated here, illustrating the way they can be implemented.

Constrained Model Predictive Control

The control problem was formulated in the previous chapters considering all signals to possess an unlimited range. This is not very realistic because in practice all processes are subject to constraints. Actuators have a limited range of action and a limited slew rate, as is the case of control valves limited by a fully closed and fully open position and a maximum slew rate. Constructive or safety reasons, as well as sensor range, cause bounds in process variables, as in the case of levels in tanks, flows in pipes, and pressures in deposits. Furthermore, in practice, the operating points of plants are determined to satisfy economic goals and lie at the intersection of certain constraints. The control system normally operates close to the limits and constraint violations are likely to occur. The control system, especially for longrange predictive control, has to anticipate constraint violations and correct them in an appropriate way. Although input and output constraints are basically treated in the same way, as is shown in this chapter, the implications of the constraints differ. Output constraints are mainly due to safety reasons and must be controlled in advance because output variables are affected by process dynamics. Input (or manipulated) variables can always be kept in bound by the controller by clipping the control action to a value satisfying amplitude and slew rate constraints.

Brief Guaranteed state estimation by zonotopes

This paper presents a new approach to guaranteed state estimation for non-linear discrete-time systems with a bounded description of noise and parameters. The main result is an algorithm to compute a set that contains the states consistent with the measured output and the given noise and parameters. This set is represented by a zonotope. The size of the zonotope is minimized each sample time by an analytic expression or by solving a convex optimization problem. Interval arithmetic is used to calculate a guaranteed trajectory of the process state. Two examples have been provided to clarify the algorithm.

Brief paper: MPC for tracking piecewise constant references for constrained linear systems

In this paper, a novel model predictive control (MPC) for constrained (non-square) linear systems to track piecewise constant references is presented. This controller ensures constraint satisfaction and asymptotic evolution of the system to any target which is an admissible steady-state. Therefore, any sequence of piecewise admissible setpoints can be tracked without error. If the target steady state is not admissible, the controller steers the system to the closest admissible steady state. These objectives are achieved by: (i) adding an artificial steady state and input as decision variables, (ii) using a modified cost function to penalize the distance from the artificial to the target steady state (iii) considering an extended terminal constraint based on the notion of invariant set for tracking. The control law is derived from the solution of a single quadratic programming problem which is feasible for any target. Furthermore, the proposed controller provides a larger domain of attraction (for a given control horizon) than the standard MPC and can be explicitly computed by means of multiparametric programming tools. On the other hand, the extra degrees of freedom added to the MPC may cause a loss of optimality that can be arbitrarily reduced by an appropriate weighting of the offset cost term.

Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties

In this paper a robust model predictive control (MPC) for constrained discrete-time nonlinear system with additive uncertainties is presented. This controller uses a terminal cost, terminal constraint and nominal predictions. The terminal region and constraints on the states are computed to get robust feasibility of the closed loop system for a given bound on the admissible uncertainties. Furthermore, it is proved that the closed-loop system is input-to-state stable with relation to the uncertainties. Therefore, the closed-loop system evolves towards a compact set where it is ultimately bounded. In case of decaying uncertainties, the closed-loop system is asymptotically stable. The convergence of the closed loop system is guaranteed despite the suboptimality of the solution.

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.

Input-to-State Stability: A Unifying Framework for Robust Model Predictive Control

This paper deals with the robustness of Model Predictive Controllers for constrained uncertain nonlinear systems. The uncertainty is assumed to be modeled by a state and input dependent signal and a disturbance signal. The framework used for the analysis of the robust stability of the systems controlled by MPC is the wellknown Input-to-State Stability. It is shown how this notion is suitable in spite of the presence of constraints on the system and of the possible discontinuity of the control law.

Constrained generalized predictive control

A generalized predictive controller for systems with constrained input and output signals is presented. The optimum values of the future control signals are obtained by transforming the quadratic optimization problem into a linear complementarity problem. A simple algorithm to decrease the number of constraints and a modification to a standard algorithm for solving the linear complementarity problem are proposed in order to reduce the amount of computation required. >

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

Min-max model predictive control (MPC) is one of the few techniques suitable for robust stabilization of uncertain nonlinear systems subject to constraints. Stability issues as well as robustness have been recently studied and some novel contributions on this topic have appeared in the literature. In this survey, we distill from an extensive literature a general framework for synthesizing min-max MPC schemes with ana priori robust stability guarantee. First, we introduce a general predictionmodel that covers a wide class of uncertainties, which includes bounded disturbances as well as state and input dependent disturbances (uncertainties). Second, we extend the notion of regional input-to-state stability (ISS) in order to fit the considered class of uncertainties. Then, we establish that the standard min-max approach can only guarantee practical stability. We concentrate our attention on two different solutions for solving this problem. The first one is based on a particular design of the stage cost of the performance index, which leads to aH∞ strategy, while the second one is based on a dual-mode strategy. Under fairly mild assumptions both controllers guarantee ISS of the resulting closed-loop system.Moreover, it is shown that the nonlinear auxiliary control law introduced in [29] to solve theH∞ problem can be used, for nonlinear systems affine in control, in all the proposed min-max schemes and also in presence of state-independent disturbances. A simulation example illustrates the techniques surveyed in this article.

Improving the robustness of dead-time compensating PI controllers

Abstract This paper describes a PI controller with dead-time compensation that presents robust behaviour. The formulation is based on a Smith predictor structure plus the addition of a filter acting on the error between the output and its prediction in order to improve robustness. The controller is very simple, and the filter needs no adjustment, since it is directly related to the plant dead-time. Simulations and experimental results show that this controller can improve the performance of related algorithms.