T. Alamo

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.

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.

Randomized Strategies for Probabilistic Solutions of Uncertain Feasibility and Optimization Problems

In this paper, we study two general semi-infinite programming problems by means of a randomized strategy based on statistical learning theory. The sample size results obtained with this approach are generally considered to be very conservative by the control community. The first main contribution of this paper is to demonstrate that this is not necessarily the case. Utilizing as a starting point one-sided results from statistical learning theory, we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy. In particular, we show that the number of required samples grows with the accuracy parameter epsiv as 1/epsivln 1/epsiv , and this is a significant improvement when compared to the existing bounds which depend on 1/epsiv2ln 1/epsiv2. Secondly, we present new results for optimization and feasibility problems involving Boolean expressions consisting of polynomials. In this case, when the accuracy parameter is sufficiently small, an explicit bound that only depends on the number of decision variables, and on the confidence and accuracy parameters is presented. For convex optimization problems, we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence. Thirdly, we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived.

MPC FOR TRACKING OF PIECE-WISE CONSTANT REFERENCES FOR CONSTRAINED LINEAR SYSTEMS

Abstract Model predictive control (MPC) is one of the few techniques which is able to handle with constraints on both state and input of the plant. The admissible evolution and asymptotically convergence of the closed loop system is ensured by means of a suitable choice of the terminal cost and terminal contraint. However, most of the existing results on MPC are designed for a regulation problem. If the desired steady state changes, the MPC controller must be redesigned to guarantee the feasibility of the optimization problem, the admissible evolution as well as the asymptotic stability. In this paper a novel formulation of the MPC is proposed to track varying references. This controller ensures the feasibility of the optimization problem, constraint satisfaction and asymptotic evolution of the system to any admissible steady-state. Hence, the proposed MPC controller ensures the offset free tracking of any sequence of piece-wise constant admissible set points. Moreover this controller requires the solution of a single QP at each sample time, it is not a switching controller and improves the performance of the closed loop system.

Technical communique: MPC for tracking with optimal closed-loop performance

In the recent paper [Limon, D., Alvarado, I., Alamo, T., & Camacho, E.F. (2008). MPC for tracking of piece-wise constant references for constrained linear systems. Automatica, 44, 2382-2387], a novel predictive control technique for tracking changing target operating points has been proposed. Asymptotic stability of any admissible equilibrium point is achieved by adding an artificial steady state and input as decision variables, specializing the terminal conditions and adding an offset cost function to the functional. In this paper, the closed-loop performance of this controller is studied and it is demonstrated that the offset cost function plays an important role in the performance of the model predictive control (MPC) for tracking. Firstly, the controller formulation has been enhanced by considering a convex, positive definite and subdifferential function as the offset cost function. Then it is demonstrated that this formulation ensures convergence to an equilibrium point which minimizes the offset cost function. Thus, in case of target operation points which are not reachable steady states or inputs for the constrained system, the proposed control law steers the system to an admissible steady state (different to the target) which is optimal with relation to the offset cost function. Therefore, the offset cost function plays the role of a steady-state target optimizer which is built into the controller. On the other hand, optimal performance of the MPC for tracking is studied and it is demonstrated that under some conditions on both the offset and the terminal cost functions optimal closed-loop performance is locally achieved.

Stochastic Programming Applied to Model Predictive Control

Many robust model predictive control (MPC) schemes are based on min-max optimization, that is, the future control input trajectory is chosen as the one which minimizes the performance due to the worst disturbance realization. In this paper we take a different route to solve MPC problems under uncertainty. Disturbances are modelled as random variables and the expected value of the performance index is minimized. The MPC scheme that can be solved using Stochastic Programming (SP), for which several efficient solution techniques are available. We show that this formulation guarantees robust constraint fulfillment and that the expected value of the optimum cost function of the closed loop system decreases at each time step.