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The goal of this note is to highlight that the reformulation of saturation as constraints, typical of the computation of the maximal output admissible sets used in the reference governor literature, is not without loss of generality and may introduce a relevant amount of conservativeness. The note also proposes a modified reference governor able to overcome this limitation.

In this paper, we propose a novel robust model predictive controller for tracking periodic signals for linear systems subject to bounded additive uncertainties based on nominal predictions and constraint tightening. The proposed controller joins optimal periodic trajectory planning and robust control for tracking in a single optimization problem and guarantees that the perturbed closed-loop system converges asymptotically to a neighborhood of an optimal reachable periodic trajectory while robustly satisfying the constraints. In addition, the closed-loop system maintains recursive feasibility even in the presence of sudden changes in the target reference.

In this paper we propose a novel economic robust predictive controller for periodic operation. The proposed controller joins dynamic and economic trajectory planning and robust predictive controller for tracking in a single layer taking into account bounded disturbances, algebraic constraints and periodic operation. We study the closed-loop system properties of the proposed controller and provide a design procedure that guarantees that the perturbed closed-loop system converges asymptotically to the optimal economic reachable periodic trajectory, constraint satisfaction and recursive feasibility. The proposed controller has been applied to control a cluster of interconnected micro-grids. Each nano-grid is connected to an electric utility and has a renewable energy source, a cluster of batteries and a metal hydride based hydrogen storage system. The cluster must satisfy a periodic energy demand while maximizing the profit of the energy sold to the electric utility taking into account time varying prices.

In this paper a new model predictive controller for tracking arbitrary periodic references is presented. The proposed controller is based on a single layer that unites dynamic trajectory planning and control. A design procedure to guarantee that the closed loop system converges asymptotically to the optimal admissible periodic trajectory while guaranteeing constraint satisfaction is provided. In addition, the constraints of the optimization problem solved by the controller do not depend on the reference, allowing for sudden changes in the reference without loosing feasibility. The properties of the proposed controller are demonstrated with a simulation example of a ball and plate system. MINECO-Spain and FEDER under project DPI2013-48243-C2-2-R University of Seville under contracts 2014/425 and 2014/758 European Research Council under the European Unions Seventh Framework Programme (FP/2007- 2013)/ ERC Grant Agreement n. 307608

Summary In model predictive control (MPC), the input sequence is computed, minimizing a usually quadratic cost function based on the predicted evolution of the system output. In the case of nonlinear MPC (NMPC), the use of nonlinear prediction models frequently leads to non-convex optimization problems with several minimums. This paper proposes a new NMPC strategy based on second order Volterra series models where the original performance index is approximated by quadratic functions, which represent a lower bound of the original performance index. Convexity of the approximating quadratic cost functions can be achieved easily by a suitable choice of the weighting of the control increments in the performance index. The approximating cost functions can be globally minimized by convex optimization techniques in order to compute the input sequence. The minimization of the performance index is carried out by an iterative optimization procedure, which guarantees convergence to the solution. Furthermore, for a nominal prediction model, asymptotic stability for the proposed NMPC strategy can be shown. In the case of considering an estimation error in the prediction model, input-to-state practical stability is assured. The control performance of the NMPC strategy is illustrated by experimental results. Copyright © 2014 John Wiley & Sons, Ltd.

This study presents an analysis about state estimation and dead-time compensation effect over constrained control systems with bounded disturbances and dead-time. It is shown that input-to-state stability and constraint satisfaction can be guaranteed by using an equivalent dead-time free system with measurable states and modified disturbances. This result may be useful to simplify synthesis and analysis of a given constrained control strategy. A linear output feedback control scheme and a tube-based model predictive control strategy are used as motivating examples.

In this paper we propose to apply a novel robust predictive controller for tracking periodic references following a constraint tightening procedure to an uncertain discrete time algebraic-differential linear model of a large scale drinking water network. The system considered has been obtained from the water balance equations of a section of Barcelona's drinking water network taking into account bounded additive perturbations. In this model, we assume that a prediction of the water demand is available and that the prediction error is bounded. The controller guarantees that the perturbed closed-loop system converges asymptotically to the optimal reachable periodic trajectory, robust constraint satisfaction and recursive feasibility. To demonstrate the main properties of the controller three different simulation scenarios have been considered.

This paper deals with the design of robust predictive controllers for uncertain constrained nonlinear systems. The considered controller is based on nominal predictions and can be posed as a standard MPC. Robust feasibility is ensured by imposing more conservative constraints in the optimization problem to take into account explicitly the effect of the uncertainty. This is done by means of the calculation of a sequence of sets, which can be easily obtained exploiting the uniform continuity of the model function. Sufficient conditions to prove input-to-state stability of the proposed controller are presented. The proposed approach can be considered an enhanced and updated formulation of a previous approach presented by the authors aimed to the reduction of the conservativeness.

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 an a priori robust stability guarantee. First, we introduce a general prediction model 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 a Hoo 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 the H8 problem can be used,for nonlinear systems affine incontrol, 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. .

This paper presents a robust model predictive controller with terminal cost weighting based on a filtered Smith predictor structure. A special state-space representation is applied in order that the states are composed by past control actions and predicted outputs up to the nominal delay. This strategy can be used to control multiple dead-time multivariable processes is such way that: (i) a terminal cost is applied to ensure nominal stability and (ii) the filtered Smith predictor structure is used to improve robustness. The proposed formulation avoids the high order model representation of long dead-time processes. A simulation example is used to illustrate the usefulness of the proposed approach.

This work proposes a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude saturating actuators. The nonlinear effects introduced by the saturations in the closed-loop system are taken into account by using a generalized sector condition, which leads to theoretical conditions for solving the problem directly in the form of linear matrix inequalities.

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.

The domain of attraction of a given non-linear system constitutes a zone of safe operation that can avoid unnecessary operational restrictions. In this paper, an alternative approach to the estimation of the domain of attraction of a saturated linear system is presented. Given a system with m saturated control inputs, we show how to choose a linear difference inclusion (LDI) in such a way that the conservativeness in the estimation is reduced. For that purpose, an LMI problem with 2 m + m constraints must be solved. In this paper, an algorithm that estimates the domain of attraction of the non-linear system is provided. Moreover, sufficient conditions to guarantee that the proposed algorithm obtains the greatest domain of attraction for the linear difference inclusion are given. Some illustrative examples are presented.

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.

The usual way to guarantee stability of model predictive control (MPC) strategies is based on a terminal cost function and a terminal constraint region. This note analyzes the stability of MPC when the terminal constraint is removed. This is particularly interesting when the system is unconstrained on the state. In this case, the computational burden of the optimization problem does not have to be increased by introducing terminal state constraints due to stabilizing reasons. A region in which the terminal constraint can be removed from the optimization problem is characterized depending on some of the design parameters of MPC. This region is a domain of attraction of the MPC without terminal constraint. Based on this result, it is proved that weighting the terminal cost, this domain of attraction of the MPC controller without terminal constraint is enlarged reaching (practically) the same domain of attraction of the MPC with terminal constraint; moreover, a practical procedure to calculate the stabilizing weighting factor for a given initial state is shown. Finally, these results are extended to the case of suboptimal solutions and an asymptotically stabilizing suboptimal controller without terminal constraint is presented.

In this paper, the notion of SNS-domain of attraction is applied to the synthesis of saturated controllers. The notion of SNS-domain was first introduced in the framework of estimation of the domain of attraction of a discrete saturated control system. The main contribution of this paper is an algorithm that allows to obtain a saturated control law for a linear system maximizing the domain of attraction. The parameters of the controller are obtained in such a way that the size of the corresponding polyhedric SNS-domain of attraction is maximized. It is well known that, for single input systems, the greatest ellipsoidal invariant set can be obtained by means of a control law that does not saturate in the corresponding ellipsoidal set. In this paper it is shown how saturated control laws yield to greater domain of attractions when polyhedric invariant sets are considered. That is, the algorithm proposed in this paper provides a controller with a domain of attraction that contains any pre-specified ellipsoidal control invariant set obtained by means of a non saturated control law.

This paper shows how interval arithmetic can be applied to the computation of control invariant sets. The paper reviews some concepts in invariant set theory and presents recursive algorithms to compute sequences of control invariant sets. These ideas can be used to design stable controllers subject to state and control constraints. To prove set inclusion and inversion, interval arithmetic is also used.

In this note, an efficient way of implementing a constrained min-max predictive controller is presented. The new approach modifies the objective function in such a way that the resulting min-max problem can be solved in polynomial time. Different modifications are proposed. The main contribution of the note is to provide a robust constrained min-max predictive controller that can be implemented in real time. The new controller stabilizes the uncertain system.

A robust MPC for constrained discrete-time nonlinear systems with additive uncertainties is presented. The proposed controller is based on the concept of reachable sets, that is, the sets that contain the predicted evolution of the uncertain system for all possible uncertainties. If processes are nonlinear these sets are very difficult to compute. A conservative approximation based on interval arithmetic is proposed for the online computation of these sets. This technique provides good results with a computational effort only slightly greater than the one corresponding to the nominal prediction. These sets are incorporated into the MPC formulation to achieve robust stability. By choosing a robust positively invariant set as a terminal constraint, a robustly stabilising controller is obtained. Stability is guaranteed in the case of suboptimality of the computed solution. The proposed controller is applied to a continuous stirred tank reactor with an exothermic reaction.

This paper presents a method for enlarging the domain of attraction of nonlinear model predictive control (MPC). The usual way of guaranteeing stability of nonlinear MPC is to add a terminal constraint and a terminal cost to the optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated Lyapunov function. The domain of attraction of the controller depends on the size of the terminal region and the control horizon. By increasing the control horizon, the domain of attraction is enlarged but at the expense of a greater computational burden, while increasing the terminal region produces an enlargement without an extra cost. In this paper, the MPC formulation with terminal cost and constraint is modified, replacing the terminal constraint by a contractive terminal constraint. This constraint is given by a sequence of sets computed off-line that is based on the positively invariant set. Each set of this sequence does not need to be an invariant set and can be computed by a procedure which provides an inner approximation to the one-step set. This property allows us to use one-step approximations with a trade off between accuracy and computational burden for the computation of the sequence. This strategy guarantees closed loop-stability ensuring the enlargement of the domain of attraction and the local optimality of the controller. Moreover, this idea can be directly translated to robust MPC. Ministerio de Ciencia y Tecnología DPI2002-04375-c03-01

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.

This paper presents a dead-time compensation strategy to control constrained nonlinear systems with single-input delay by using a dead-time free model. Prediction error and disturbance effects are analyzed in order to guarantee input-to-state stability (ISS) and robust constraint satisfaction in the presence of bounded disturbances. It is shown robust constraint satisfaction and ISS are ensured by imposing tighter constraints and a modified ISS condition to the dead-time free model. These modified specifications are derived in terms of a K function description in order to reduce Lipschitz constant conservatism. A numerical comparison based on the CSTR reactor model is presented to illustrate the advantage of the proposed approach.

In this paper we consider periodic optimal operation of constrained periodic linear systems. We propose an economic model predictive controller based on a single layer that unites dynamic real time optimization and control. The proposed controller guarantees closed-loop convergence to the optimal periodic trajectory that minimizes the average operation cost for a given economic criterion. A priori calculation of the optimal trajectory is not required and if the economic cost function is changed, recursive feasibility and convergence to the new periodic optimal trajectory is guaranteed. The results are demonstrated with two simulation examples, a four tank system, and a simplified model of a section of Barcelona's water distribution network.

Model Predictive Control (MPC) is the most used advanced control technique in process industries, since it ensures stability, constraints satisfaction and convergence to the setpoint. The optimal setpoint is calculated by the Real Time Optimizer (RTO), minimizing the economic objective taking into account the operational limits of the plant. Since RTO employs complex stationary nonlinear models to perform the optimization and a larger sampling time than the controller, the economic setpoints calculated by the RTO may be inconsistent for the MPC layer and the economic performance of the overall controller may be worse than expected. The aim of this work is to propose an MPC controller that explicitly integrates the RTO into the MPC control layer. The proposed strategy is based on the MPC for tracking; the optimization problem to be solved only requires one evaluation of the gradient of the economic cost function at each sampling time. Based on this gradient, a second order approximation of the economic function is obtained and used in the MPC optimization problem resulting in a convex optimization problem. Recursive feasibility and convergence to the optimal equilibrium point is ensured.

This paper is devoted to the design of a predictive controller for constrained linear systems to track periodic references. The only assumption on the dynamics of the reference is that it is periodic and its period is known. It is also assumed that the reference signal is a priori known by the controller. Inspired in the hierarchical control scheme based on the trajectory planification, the ideas of the MPC for tracking [Limon et al., 2008] are extended to this case. The proposed predictive controller has the future sequence of inputs and an artificial reference as decision variables. The cost function is divided into two terms: one penalizes the tracking error with the artificial reference and other penalizes the deviation of the artificial reference to the reference to be tracked. Stability is ensured thanks to the addition of two constraints: a terminal constraint on the predicted trajectory and a constraint that enforces the artificial reference to be periodic. It is proved that the proposed controller is recursively feasible and the controlled system satisfies the hard constraints, is asymptotically stable and converges to the best possible reachable trajectory. The properties of the proposed controller are illustrated in an example. Spanish Ministry of Science and Innovation and FEDER DPI2008-05818 Spanish Ministry of Science and Innovation and FEDER DPI2010-21589-C05-01

This paper deals with the problem of robust tracking of target sets using a model predictive control (MPC) law. Real industries applications often require a control strategy in which some system outputs are controlled within specified ranges or zones (zone control), while some others variables – possibly including input variables – are steered to fixed target or setpoint. From a theoretical point of view, the control objective of this kind of problem can be seen as a target set (in the output space) instead of a target point, since inside the zones there are no preferences between one point or another. This problem is particularly interesting in case of additive disturbances which might push the outputs out of the zones. In this work, a stable robust MPC formulation for constrained linear systems, based on nominal predictions is presented. The main features of this controller are the use of nominal predictions, restricted constraints and the concept of distance from a point to a set as offset cost function. The controller ensures both recursive feasibility and local optimality. The properties of the controller are shown in a simulation test, in which we consider a subsystem of an industrial FCC system. Fil: Ferramosca, Antonio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentina Fil: Limon, D.. Universidad de Sevilla. Departamento de Ingeniería de Sistemas y Automática. Escuela Superior de Ingenieros Industriales; España Fil: González, Alejandro Hernán. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentina Fil: Alvarado, I.. Universidad de Sevilla. Departamento de Ingeniería de Sistemas y Automática. Escuela Superior de Ingenieros Industriales; España Fil: Camacho, E. F.. Universidad de Sevilla. Departamento de Ingeniería de Sistemas y Automática. Escuela Superior de Ingenieros Industriales; España

Model predictive control (MPC) is one of the few techniques which is able to handle constraints on both state and input of the plant. The admissible evolution and asymptotic convergence of the closed-loop system is ensured by means of suitable choice of the terminal cost and terminal constraint. 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 optimisation problem, the admissible evolution as well as the asymptotic stability. Recently, a novel MPC has been proposed to ensure the feasibility of the optimisation problem, constraints satisfaction and asymptotic evolution of the system to any admissible target steady-state. A drawback of this controller is the loss of a desirable property of the MPC controllers: the local optimality property. In this article, a novel formulation of the MPC for tracking is proposed aimed to recover the optimality property maintaining all the properties of the original formulation.

The stability of equilibria in discrete-time systems where the model function may be discontinuous is studied. Due to discontinuities, underlying properties of systems with continuous model functions can be lost. This fact has led, in the recent years, to the need of reviewing the foundation of stability theory. In this paper, peculiarities of discontinuous systems are illustrated by means of examples. Stability definitions are given and relations among them is studied: the strong local boundedness of the model function is demonstrated to have a crucial role for the equivalence of different stability notions. The results are used to characterize stability in terms of Lyapunov functions: necessary and sufficient conditions for stability are provided.

This paper presents a new robust MPC formulation based on nominal predictions that improves other popular robust MPC formulations such as Chisci et al. (2001); Mayne et al. (2005). As in Chisci et al. (2001), the proposed controller ensures robust feasibility by means of tighter constraints, however, the condition on the terminal constraint has been notably relaxed, leading to less conservative controllers. Another remarkable property is that the calculation of (an approximation of) the minimum robust positively invariant set is not required, which makes its design more simple than the tube-based MPC proposed in Mayne et al. (2005). Furthermore, input-to-state stability of the closed loop system is proved. These properties are demonstrated in an illustrative example.

This paper deals with the problem of tracking target sets using a model predictive control (MPC) law. Some MPC applications require a control strategy in which some system outputs are controlled within specified ranges or zones (zone control), while some other variables – possibly including input variables - are steered to fixed target or set-point. In real applications, this problem is often overcome by including and excluding an appropriate penalization for the output errors in the control cost function. In this way, throughout the continuous operation of the process, the control system keeps switching from one controller to another, and even if a stabilizing control law is developed for each of the control configurations, switching among stable controllers not necessarily produces a stable closed loop system. From a theoretical point of view, the control objective of this kind of problem can be seen as a target set (in the output space) instead of a target point, since inside the zones there are no preferences between one point or another. In this work, a stable MPC formulation for constrained linear systems, with several practical properties is developed for this scenario. The concept of distance from a point to a set is exploited to propose an additional cost term, which ensures both, recursive feasibility and local optimality. The performance of the proposed strategy is illustrated by simulation of an ill-conditioned distillation column. Fil: Ferramosca, Antonio. Universidad Tecnológica Nacional. Facultad Regional Reconquista; Argentina Fil: Limon, Daniel. Universidad de Sevilla. Departamento de Ingenieria de Sistemas y Automatica. Escuela Superior de Ingenieros Industriales; España Fil: González, Alejandro Hernán. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Santa Fe. Instituto de Desarrollo Tecnológico Para la Industria Química (i); Argentina Fil: Odloak, Darci. Universidade de Sao Paulo; Brasil Fil: Camacho, Eduardo. Universidad de Sevilla. Departamento de Ingenieria de Sistemas y Automatica. Escuela Superior de Ingenieros Industriales; España

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.

Copia personal In the recent paper [Limon, D., Alvarado, I., Álamo, 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. Ministerio de Eduación y Ciencia DPI2007-66718-C04-01 Ministerio de Eduación y Ciencia DPI2005-04568

This paper shows how interval arithmetic can be used to design stabilizing robust MPC controllers. Interval arithmetic provides a suitable framework to obtain a tractable procedure to calculate an outer bound of the range of a given nonlinear function. This can be used to calculate a guaranteed outer bound on the predicted sequence of reachable sets. This allows us to consider the effect of the uncertainties in the prediction and to formulate robust dual-mode MPC controllers with ensured admissibility and convergence. Interval arithmetic can also be used to estimate the state when only outputs are measurable. This method provides a guaranteed outer bound on the set of states consistent with the output measurements. Generalizing the controllers based on reachable sets, a novel robust output feedback MPC controller is also proposed.

In this paper, an alternative approach to the computation of control invariant sets for piecewise affine systems is presented. Based on two approximation operators, two algorithms that provide outer and inner approximations of the maximal robust control invariant set are presented. These algorithms can be used to obtain a robust control invariant set for the system. An illustrative example is presented.

This paper presents a novel formulation of a robust model predictive controller (RMPCT) to track piecewise constant references. The real plant is assumed to be modelled as a linear system with additive bounded uncertainties on the states. Under mild assumptions, the proposed MPC can steer the uncertain system in an admissible evolution to any admissible steady state, that is, under any change of the set point. This allows us to reject constant disturbances compensating the effect of then, changing the setpoint. Feasibility of the proposed controller for any admissible setpoint is achieved by adding an artificial steady state as decision variable. Robust constraint satisfaction is guaranteed by tube-based approach and considering nominal predictions. Robust stability and convergence to (a neighborhood of) the desired steady state is ensured by considering a modified cost function and an extended terminal constraint. The cost function penalizes the tracking error with the artificial reference and the deviation between the artificial and desired steady state; the terminal constraint restricts the terminal state and the artificial steady state. The optimization problem to be solved is a quadratic programming problem, which allows explicit implementations. In order to demonstrate the benefits of the proposed controller, this has been tested on a real positioning plant consisting in a linear motor driving a cart. The fast dynamics of the systems requires an explicit calculation of the controller, which has been implemented by means of a search tree strategy. The experimental results show the robust tracking and the admissible evolution of the closed loop system.

This paper presents a novel formulation of a robust output feedback model predictive controller to track piecewise constant references (output feedback RMPCT). The real plant is assumed to be modelled as a linear system with additive bounded uncertainties on the states. Under mild assumptions, the proposed MPC can steer the uncertain system in an admissible evolution to any admissible steady state, that is, under any change of the set point. The proposed output feedback controller consists of a stable state estimator and a recently developed robustly stabilizing, tube based, model predictive control for tracking. Feasibility of the proposed controller for any admissible setpoint is achieved by adding an artificial steady state as decision variable. Robust constraint satisfaction is guaranteed by tube-based approach and considering nominal predictions. Robust stability and convergence to (a neighborhood of) the desired steady state is ensured by considering a modified cost function and an extended terminal constraint. The cost function penalizes the tracking error with the artificial reference and the deviation between the artificial and desired steady state; the terminal constraint restricts the terminal state and the artificial steady state. The optimization problem to be solved is a quadratic programming problem, which allows explicit implementations. Robust stability is guaranteed under mild conditions which allows one to consider tracking specifications and disturbance rejection conditions. Moreover, a simple method to achieve offset-free control, compensating the steady tracking error of the outputs, is presented.

Min-max model predictive control (MPC) is an appealing strategy due to its performance and its capability to ensure robust satisfaction of the constraints. The associated control technique requires the solution of a differential game which is an NP-hard problem. In this paper a relaxed formulation of the min-max MPC for constrained nonlinear systems is presented. In the proposed MPC, the maximization problem is replaced by the simple evaluation of an appropriate sequence of disturbances. This reduces dramatically the computational burden of the optimization problem and produces a solution that does not differ much from the one obtained with the original min-max problem. Moreover, the proposed predictive control inherits the convergence and the domain of attraction of the standard min-max strategy.

The aim of this chapter is the presentation of a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude saturating actuators. Two synthesis objectives are considered: the maximization of the domain of attraction of the closed-loop system under some time-domain performance constraints; and the guarantee that the trajectories of the system are bounded in the presence of Ł 2 -bounded disturbances, while ensuring an overbound for the Ł 2 gain from the disturbance to the regulated output. The nonlinear effects introduced by the saturations in the closed-loop system are taken into account by using a generalized sector condition, which allows to propose theoretical conditions to solve the synthesis problems directly in the form of linear matrix inequalities (LMIs). From these conditions, convex optimization problems are proposed in order to compute the controller aiming at the maximization of the basin attraction of the closed-loop system, as well as aiming at ensuring a level of Ł 2 disturbance tolerance and rejection. Numerical examples are provided to illustrate the application of the proposed methodology.

In this work, a new technique for the computation of ellipsoidal invariant sets for continuous-time linear systems controlled by a saturating linear control law is presented. New sufficient conditions to guarantee that an ellipsoid is a contractive invariant set for the closed-loop system is presented. The contractive nature of the invariant set ensures asymptotic stability of the controlled system. The main contributions of the paper are the following: the proposed sufficient condition is expressed in form of linear matrix inequalities constraints. The presented method includes (and consequently improves) previous results on this topic. The computational complexity of the proposed approach is analyzed. Illustrative examples are given.

This paper considers the problem of state estimation for repetitive nonlinear systems. Taking the repetitive nature of the process into account a new state estimation scheme is proposed, which from repetition to repetition iteratively improves the estimate. The scheme combines ideas from iterative learning control and moving horizon state estimation. The state estimate during every repetition is based on approximately minimizing the deviation between the measured and estimated output. Stability and iterative improvements of the state estimates are ensured by enforcing a sufficient contraction of the deviation between the measured and estimated output over the considered estimation window. As shown, under the contraction constraints the state estimation scheme ensures asymptotic convergence of the state estimation error in the nominal case, provided that the system satisfies an uniform reconstructability condition.

This paper deals with the application of a model predictive controller (MPC) for tracking to a real system: a linear motor. The MPC formulation used in this paper in a novel approach to track piecewise constant references recently presented by the authors. The main novelties of this controller are: (i) the addition of an artificial steady state and input as decision variables, (ii) the minimization of a modified cost function which penalizes the error with the artificial steady and the distance from the artificial to the desired steady state and (iii) the usage of an extended terminal constraint. The obtained MPC ensures the admissible convergence to any admissible set point for any feasible initial state. The optimization problem associated to this MPC technique results to be a quadratic programming problem parametrized in the current state and the desired steady state. Thus, well-known techniques to compute the control law explicitly based on multiparametric programming can be used, which allows us to apply this control technique to fast systems. We have applied an explicit MPC for tracking to a synchronous linear motor with permanent magnets using a sampling time of 10 ms. This system has been modelled as a linear system with constraints in the input and states. The control objective is to steer the motor at any admissible position in an admissible evolution. The explicit controller has been coded as a search tree for its implementation on a dSPACE Card. The obtained experimental results are satisfactory and show the properties of the predictive controller

In this paper, a new concept of invariance for saturated linear systems is presented. This new notion of invariance, denoted SNS-invariance, has a number of geometrical properties that makes its use suitable for the estimation of the domain of attraction of saturated systems. The notion of SNS-domain of attraction, that serves as an estimation of the domain of attraction of a saturated system, is introduced. It is shown that, in case of single input saturated systems, any contractive set is contained in the SNS-domain of attraction. A simple algorithm that converges to the SNS-domain of attraction is presented. Some illustrative examples are given.

This paper is concerned with the design of stabilizing model predictive control (MPC) laws for constrained linear systems. This is achieved by obtaining a suitable terminal cost and terminal constraint using a saturating control law as local controller. The system controlled by the saturating control law is modelled by a linear difference inclusion. Based on this, it is shown how to determine a Lyapunov function and a polyhedral invariant set which can be used as terminal cost and constraint. The obtained invariant set is potentially larger than the maximal invariant set for the unsaturated linear controller, O∞. Furthermore, considering these elements, a simple dual MPC strategy is proposed. This dual-mode controller guarantees the enlargement of the domain of attraction or, equivalently, the reduction of the prediction horizon for a given initial state. If the local control law is the saturating linear quadratic regulator (LQR) controller, then the proposed dual-mode MPC controller retains the local infinite-horizon optimality. Finally, an illustrative example is given.

The closed loop formulation of the robust MPC has been shown to be a control technique capable of robustly stabilize uncertain nonlinear systems subject to constraints. This control technique mitigates the conservativeness derived from the open loop formulations but it is computationally too demanding or even untractable. This paper provides a way of overcoming the complexity of this technique. This is done by means of a sequence of sets. It is shown that it suffices to obtain a sequence of robust controllable sets. This sequence can be computed off-line using any tractable conservative procedure for the computation of the robust one-step set. Incorporating this sequence as a stabilizing constraint in the optimization problem, robust stability is guaranteed in case of suboptimal solutions or open loop approaches. Then, the optimization problem can be considerably simplified, yielding tractable approaches at the expense of performance. The proposed technique has been applied to a CSTR model.

In this paper, a unified approach to the estimation of the domain of attraction of a saturated linear system is presented. We show how to choose a linear difference inclusion (LDI) in such a way that the conservativeness in the estimation is reduced. We provide an algorithm that estimates the domain of attraction of the nonlinear system. Under mild assumptions, the proposed algorithm obtains a polyhedral invariant set that equals the maximal domain of attraction for the linear difference inclusion. An alternative approach, that obtains a sequence of invariant sets with a reduced number of constraints, is also proposed.

In this paper, an efficient way of implementing a constrained min-max predictive controller is presented. The new approach modifies the objective function in such a way that the resulting min-max problem can be solved in polynomial time. Different modifications are proposed. The main contribution of the paper is to provide a robust constrained min-max predictive controller that can be implemented in real time. The new controller stabilizes the uncertain system.