brief paper convergence properties of constrained linear system under mpc control law using affine disturbance feedback this paper shows new convergence properties of constrained linear discrete time system with bounded disturbances under model predictive control mpc law the mpc control law is obtained using an affine disturbance feedback parametrization with an additional linear state feedback term this parametrization has the same representative ability as some recent disturbance feedback parametrization but its choice together with an appropriate cost function results in a different closed loop convergence property more exactly the state of the closed loop system converges to a minimal invariant set with probability one deterministic convergence to the same minimal invariant set is also possible if a less intuitive cost function is used numerical experiments are provided that validate the results

robust model predictive control of constrained linear systems with bounded disturbances this paper provides a novel solution to the problem of robust model predictive control of constrained linear discrete time systems in the presence of bounded disturbances the optimal control problem that is solved online includes uniquely the initial state of the model employed in the problem as a decision variable the associated value function is zero in a disturbance invariant set that serves as the origin when bounded disturbances are present and permits a strong stability result namely robust exponential stability of the disturbance invariant set for the controlled system with bounded disturbances to be obtained the resultant online algorithm is a quadratic program of similar complexity to that required in conventional model predictive control

model predictive control for stochastic max plus linear systems with chance constraints the topic of this paper is model predictive control mpc for max plus linear systems with stochastic uncertainties the distribution of which is supposed to be known we consider linear constraints on the inputs and the outputs due to the uncertainties these linear constraints are formulated as probabilistic or chance constraints i e the constraints are required to be satisfied with a predefined probability level the proposed chance constraints can be equivalently rewritten into a max affine i e the maximum of affine terms form if the linear constraints are monotonically nondecreasing as a function of the outputs based on the resulting max affine form two methods are developed for solving the chance constrained mpc problem for stochastic max plus linear systems method 1 uses boole s inequality to convert the multivariate chance constraint into univariate chance constraints for which the probability can be computed more efficiently method 2 employs chebyshev s inequality and transforms the chance constraint into linear constraints on the inputs the simulation results for a production system example show that the two proposed methods are faster than the monte carlo simulation method and yield lower closed loop costs than the nominal mpc method

robust event triggered mpc with guaranteed asymptotic bound and average sampling rate we propose a robust event triggered model predictive control mpc scheme for linear time invariant discrete time systems subject to bounded additive stochastic disturbances and hard constraints on the input and state for given probability distributions of the disturbances acting on the system we design event conditions such that the average frequency of communication between the controller and the actuator in the closed loop system attains a given value we employ tube mpc methods to guarantee robust constraint satisfaction and a robust asymptotic bound on the system state moreover we show that instead of a given periodically updated tube mpc scheme an appropriate event triggered mpc scheme can be applied with the same guarantees on constraints and region of attraction but with a reduced number of average communications

robust output feedback model predictive control of constrained linear systems this paper provides a solution to the problem of robust output feedback model predictive control of constrained linear discrete time systems in the presence of bounded state and output disturbances the proposed output feedback controller consists of a simple stable luenberger state estimator and a recently developed robustly stabilizing tube based model predictive controller the state estimation error is bounded by an invariant set the tube based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the center of which is the solution of a nominal disturbance free system and the cross section of which is also invariant satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system the complexity of the resultant controller is similar to that required for nominal model predictive control

brief paper stochastic tube mpc with state estimation an output feedback model predictive control mpc strategy for linear systems with additive stochastic disturbances and probabilistic constraints is proposed given the probability distributions of the disturbance input the measurement noise and the initial state estimation error the distributions of future realizations of the constrained variables are predicted using the dynamics of the plant and a linear state estimator from these distributions a set of deterministic constraints is computed for the predictions of a nominal model the constraints are incorporated in a receding horizon optimization of an expected quadratic cost which is formulated as a quadratic program the constraints are constructed so as to provide a guarantee of recursive feasibility and the closed loop system is stable in a mean square sense all uncertainties in this paper are taken to be bounded in most control applications this gives a more realistic representation of process and measurement noise than the more traditional gaussian assumption

stability analysis of quadratic mpc with a discrete input alphabet we study stability of model predictive control mpc with a quadratic cost function for lti systems with a discrete input alphabet since this kind of systems may present a steady state error the focus is on practical stability i e ultimate boundedness of solutions to derive sufficient conditions for practical stability and characterize the ultimately invariant set we analyze the one step horizon solution and adapt tools used for convex mpc formulations

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

controlling chaos in unidimensional maps using constant feedback we demonstrate that a constant feedback can control chaotic oscillations in one dimensional discrete maps this is a simple method of controlling chaos as the feedback signal does not require a priori knowledge of the dynamics of the system and it generates the desired behavior by simply varying the strength of the feedback we illustrate this method with an application to the quadratic and exponential logistic maps

brief a receding horizon approach to the nonlinear h control problem the receding horizon rh methodology is extended to the design of a robust controller of h type for nonlinear systems using the nonlinear analogue of the fake h algebraic riccati equation we derive an inverse optimality result for the rh schemes for which increasing the horizon causes a decrease of the optimal cost function this inverse optimality result shows that the input output map of the closed loop system obtained with the rh control law has a bounded l 2 gain robustness properties of the nonlinear h control law in face of dynamic input uncertainty are considered

brief constrained receding horizon predictive control for nonlinear systems the paper concerns the receding horizon predictive control of constrained nonlinear systems and presents an algorithm which relies on the online solution of a simple linear program lp use is made of a finite control horizon in conjunction with a terminal inequality constraint and a predicted cost that includes a terminal penalty term the optimization procedure is based on predictions made by linearized incremental models at points of a given seed trajectory and the effects of linearization error are taken into account to give a bound on the predicted tracking error the algorithm is posed in the form of an lp and the proper selection of the terminal penalty term of the predicted cost guarantees the asymptotic stability the results of the paper are illustrated by means of a simple example

efficient robust optimization for robust control with constraints this paper proposes an efficient computational technique for the optimal control of linear discrete time systems subject to bounded disturbances with mixed linear constraints on the states and inputs the problem of computing an optimal state feedback control policy given the current state is non convex a recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program while the reparameterized problem is theoretically tractable the number of variables is quadratic in the number of stages or horizon length n and has no apparent exploitable structure leading to computational time of mathcal o n 6 per iteration of an interior point method we focus on the case when the disturbance set is norm bounded or the linear map of a hypercube and the cost function involves the minimization of a quadratic cost here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem that is the policy optimization problem may be decomposed into a set of coupled finite horizon control problems this decomposition can then be formulated as a highly structured quadratic program solvable by primal dual interior point methods in which each iteration requires mathcal o n 3 time this cubic iteration time can be guaranteed using a riccati based block factorization technique which is standard in discrete time optimal control numerical results are presented using a standard sparse primal dual interior point solver that illustrate the efficiency of this approach

robust self triggered mpc for constrained linear systems we propose a robust self triggered control algorithm for constrained linear discrete time systems subject to additive disturbances based on mpc at every sampling instant the controller provides both the next sampling instant as well as the inputs that are applied to the system until the next sampling instant by maximizing the inter sampling time subject to bounds on the mpc value function the average sampling frequency in the closed loop system is decreased while guaranteeing an upper bound on the performance loss when compared with an mpc scheme sampling at every point in time robust constraint satisfaction is achieved by tightening input and state constraints based on a tube mpc approach moreover a compact set in the state space which is a parameter in the mpc scheme is shown to be robustly asymptotically stabilized

robust model predictive control of nonlinear systems with unmodeled dynamics and bounded uncertainties based on neural networks this paper presents a neural network approach to robust model predictive control mpc for constrained discrete time nonlinear systems with unmodeled dynamics affected by bounded uncertainties the exact nonlinear model of underlying process is not precisely known but a partially known nominal model is available this partially known nonlinear model is first decomposed to an affine term plus an unknown high order term via jacobian linearization the linearization residue combined with unmodeled dynamics is then modeled using an extreme learning machine via supervised learning the minimax methodology is exploited to deal with bounded uncertainties the minimax optimization problem is reformulated as a convex minimization problem and is iteratively solved by a two layer recurrent neural network the proposed neurodynamic approach to nonlinear mpc improves the computational efficiency and sheds a light for real time implementability of mpc technology simulation results are provided to substantiate the effectiveness and characteristics of the proposed approach

on dynamic output feedback robust mpc for constrained quasi lpv systems this paper considers the dynamic output feedback robust model predictive control mpc of a quasi linear parameter varying quasi lpv system with bounded noise in our previous works for the unknown true state either its ellipsoidal bounds or its polyhedral bounds were solely applied in the main optimisation problem the recursive feasibility of the main optimisation problem was guaranteed by a simple refreshment of the ellipsoidal bound but might be lost by applying the polyhedral bounds this paper shows how and to what extent the recursive feasibility can be restored when the polyhedral bounds are still utilised first we propose a new approach which at each sampling time utilises either the ellipsoidal bound or the polyhedral bound in the main optimisation problem the latter being used if and only if it is contained in the former then we show the sufficient conditions under which the approaches based on polyhedral bounds preserve the property of recursive feasibility a numerical example is

robust triple mode mpc this paper reviews triple mode predictive control for linear time invariant systems and then proposes a new algorithm for robust triple mode predictive control for constrained linear systems described by polytopic uncertainty models the approach significantly enlarges the feasibility region compared to recently proposed robust dual mode approaches the efficacy of the approach is demonstrated with numerical examples

inverse minimax optimality of model predictive control policies we present novel results linking model predictive control mpc and minimax optimal control theory specifically we show that for linear discrete time systems with constraints and disturbance inputs the closed loop optimal solutions of a particular class of minimax optimal control problems are a class of typical mpc policies we also present conditions which ensure that the inverse optimal mpc policies achieve a prescribed regional l2 gain from the disturbance input

brief robust receding horizon predictive control for systems with uncertain dynamics and input saturation a receding horizon predictive control method which assures stability for systems with model uncertainty and input saturation is derived by extending earlier work in two important respects i ellipsoidal invariant sets are replaced by polyhedral invariant sets and ii the constraint that the state lie in an invariant set is invoked n steps ahead the new algorithm allows for a considerable enlargement of the stabilizable set and provides extra degrees of freedom with which to improve performance

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

an approach to output feedback mpc of stochastic linear discrete time systems in this paper we propose an output feedback model predictive control mpc algorithm for linear discrete time systems affected by a possibly unbounded additive noise and subject to probabilistic constraints in case the noise distribution is unknown the probabilistic constraints on the input and state variables are reformulated by means of the chebyshev cantelli inequality the recursive feasibility is guaranteed the convergence of the state to a suitable neighbor of the origin is proved under mild assumptions and the implementation issues are thoroughly addressed two examples are discussed in detail with the aim of providing an insight into the performance achievable by the proposed control scheme

worst case formulations of model predictive control for systems with bounded parameters two different predictive control formulations are developed based on minimization of the worst case quadratic cost for systems with bounded parameters the two formulations differ on the assumptions made about the future inputs in optimizing the current input one assumes open loop control while the other considers closed loop control their closed loop properties such as asymptotic stability are examined we then focus on a moving average model with an integrator and derive computationally simpler suboptimal algorithms

probabilistic tubes in linear stochastic model predictive control this paper considers constrained control of linear systems with additive and multiplicative stochastic uncertainty and linear input state constraints both hard and soft constraints are considered and bounds are imposed on the probability of soft constraint violation assuming the plant parameters to be finitely supported a method of constraint handling is proposed in which a sequence of tubes corresponding to a sequence of confidence levels on the predicted future plant state is constructed online around nominal state trajectories a set of linear constraints is derived by imposing bounds on the probability of constraint violation at each point on an infinite prediction horizon through constraints on one step ahead predictions a guarantee of the recursive feasibility of the online optimization ensures that the closed loop system trajectories satisfy both the hard and probabilistic soft constraints the approach is illustrated by a numerical example

infinite horizon performance bounds for uncertain constrained systems we present a new method to bound the performance of controllers for uncertain linear systems with mixed state and input constraints and bounded disturbances we take as a performance metric either an expected value or minimax discounted cost over an infinite horizon and provide a method for computing a lower bound on the achievable performance of any causal control policy in either case our lower bound is compared to an upper performance bound provided by restricting the choice of controller to one that is affine in the observed disturbances and we show that the two bounds are closely related in particular the lower bounds have a natural interpretation in terms of affine control policies that are optimal for a problem with a restricted disturbance set we show that our performance bounds can be computed via solution of a finite dimensional convex optimization problem and provide numerical examples to illustrate the efficacy of our method

output feedback receding horizon control of constrained systems this paper considers output feedback control of linear discrete time systems with convex state and input constraints which are subject to bounded state disturbances and output measurement errors we show that the non convex problem of finding a constraint admissible affine output feedback policy over a finite horizon to be used in conjunction with a fixed linear state observer can be converted to an equivalent convex problem when used in the design of a time varying robust receding horizon control law we derive conditions under which the resulting closed loop system is guaranteed to satisfy the system constraints for all time given an initial state estimate and bound on the state estimation error when the state estimation error bound matches the minimal robust positively invariant mrpi set for the system error dynamics we show that this control law is time invariant but its calculation generally requires solution of an infinite dimensional optimization problem finally using an invariant outer

optimization over state feedback policies for robust control with constraints this paper is concerned with the optimal control of linear discrete time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input it is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence this implies that a broad class of constrained finite horizon robust and optimal control problems where the optimization is over affine state feedback policies can be solved in a computationally efficient fashion using convex optimization methods this equivalence result is used to design a robust receding horizon control rhc state feedback policy such that the closed loop system is input to state stable iss and the constraints are satisfied for all time and all allowable disturbance sequences the cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance free state and input sequences the value of the receding horizon control law can be calculated at each sample instant using a single tractable and convex quadratic program qp if the disturbance set is polytopic or a tractable second order cone program socp if the disturbance set is given by a 2 norm bound

tube mpc scheme based on robust control invariant set with application to lipschitz nonlinear systems abstract the paper presents a tube model predictive control mpc scheme of continuous time nonlinear systems based on robust control invariant sets with respect to unknown but bounded disturbances the cost functional of the optimization problem is not necessarily quadratic the scheme has the same online computational burden as the standard mpc with guaranteed nominal stability robust stability as well as recursive feasibility is guaranteed if the optimization problem is feasible at the initial time instant in particular we consider a scheme to obtain robust control invariant sets for a class of lipschitz nonlinear systems and to show the effectiveness of the proposed schemes by a simple example

robust time optimal control of constrained linear systems a version of dynamic programming which computes level sets of the value function rather than the value function set itself is used to design robust non linear controllers for linear discrete time dynamical systems subject to hard constraints on controls and states the controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time for the robust regulator problem the control invariant terminal set is a neighborhood preferably small of the origin for the robust tracking problem the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero two non linear controllers which utilize the level sets of the value function are described the first requires the controller to solve on line a modest linear program whose dimension is approximately the same as that of the control variable the second decomposes each level set into a set of simplices a piecewise linear control law affine in each simplex is then constructed

robust model predictive control using tubes a form of feedback model predictive control mpc that overcomes disadvantages of conventional mpc but which has manageable computational complexity is presented the optimal control problem solved on line yields a tube and an associated piecewise affine control law that maintains the controlled trajectories in the tube despite uncertainty computational complexity is linear rather than exponential in horizon length asymptotic stability of the controlled system is established

min max feedback model predictive control for constrained linear systems min max feedback formulations of model predictive control are discussed both in the fixed and variable horizon contexts the control schemes the authors discuss introduce in the control optimization the notion that feedback is present in the receding horizon implementation of the control this leads to improved performance compared to standard model predictive control and resolves the feasibility difficulties that arise with the min max techniques that are documented in the literature the stabilizing properties of the methods are discussed as well as some practical implementation details

inherent robustness properties of quasi infinite horizon nonlinear model predictive control we consider inherent robustness properties of model predictive control mpc for continuous time nonlinear systems with input constraints and terminal constraints we show that mpc with a nominal prediction model and persistent but bounded disturbances has some degree of inherent robustness when the terminal control law and the terminal penalty matrix are chosen as the linear quadratic control law and the related lyapunov matrix respectively we emphasize that the input constraint sets can be any compact set rather than convex sets and our results do not depend on the continuity of the optimal cost function or of the control law in the interior of the feasible region

systems with persistent disturbances predictive control with restricted constraints predictive regulation of linear discrete time systems subject to unknown but bounded disturbances and to state control constraints is addressed an algorithm based on constraint restrictions is presented and its stability properties are analysed

spaceability of strict order integrability abstract in this paper sharp conditions on a measure space are provided in order that the subset of functions in the corresponding lebesgue space l p which are in no other l q contains except for zero an infinite dimensional closed subspace this completes earlier work of several authors the main result is derived from a new general criterion for spaceability on function spaces a further application to spaceability of the class of non absolutely continuous functions is supplied

a one step approach to computing a polytopic robust positively invariant set a procedure and theoretical results are presented for the problem of determining a minimal robust positively invariant rpi set for a linear discrete time system subject to unknown bounded disturbances the procedure computes via the solving of a single lp a polytopic rpi set that is minimal with respect to the family of rpi sets generated from a finite number of inequalities with pre defined normal vectors

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

parameterized robust control invariant sets for linear systems theoretical advances and computational remarks we characterize a family of parametrized robust control invariant sets for linear discrete time systems subject to additive but bounded state disturbances the existence of a member of the introduced family of parametrized robust control invariant sets can be verified by solving a tractable convex optimization problem in the linear convex case which reduces to the standard linear or convex quadratic programme in the linear polytopic case the developed method can also be utilized to detect and obtain an implicit representation of local control lyapunov functions in the linear convex case from the solution of a single and tractable convex optimization problem the offered method permits for the computation of polytopic robust control invariant sets and local control lyapunov functions of indirectly controlled and limited complexity in the linear polytopic case

cooperative distributed stochastic mpc for systems with state estimation and coupled probabilistic constraints a cooperative distributed stochastic model predictive control cdsmpc algorithm is given for multiple dynamically decoupled subsystems with additive stochastic disturbances and coupled probabilistic constraints for which states are not measurable cooperation between subsystems is promoted by a scheme in which a local subsystem designs hypothetical plans for others in some cooperating set and considers the weighted costs of these subsystems in its objective to achieve satisfaction of coupled probabilistic constraints in a distributed way only one subsystem is permitted to optimize at each time step in addition by using a lifting technique and the probabilistic information on additive disturbances measurement noise and the state estimation error a set of deterministic constraints is constructed for the predictions of nominal models recursive feasibility with respect to both local and coupled probabilistic constraints is guaranteed and stability for any choice of update sequence and any structure of cooperation is ensured numerical examples illustrate the efficacy of the proposed algorithm

h infinity set membership observer design for discrete time lpv systems this paper presents a new set membership observer design method for discrete time linear parameter varying systems the real process is assumed to be perturbed by unknown but bounded disturbances

a framework for multiple robust explicit mpc controllers for linear systems this article presents a multi mode explicit controller for constrained linear systems with bounded disturbances using a switching strategy based on model predictive control mpc in the proposed approach the system switches among several mpc controllers having different performance levels the switching is done so as to achieve increasing levels of performance as time evolves reaching the desired controller in finite time steps the conditions needed for switching and robust convergence of the multi mode mpc controllers are provided compared with standard robust explicit mpc implementations the proposed approach has the flexibility of having a large domain of attraction a good asymptotic behaviour and a small number of partitions

robust mpc strategy with optimized polytopic dynamics for linear systems with additive and multiplicative uncertainty abstract polytopic dynamics have been introduced into the predictions in model predictive control mpc and have been optimized so as to maximize invariant ellipsoids for systems subject to multiplicative uncertainty this work was recently extended to the case of mixed additive and multiplicative uncertainty and sufficient conditions were provided to guarantee the invariance of an ellipsoid additionally under the assumption that the multiplicative uncertainty is known during a prediction horizon n this extension was used as the terminal control law of an overall robust mpc strategy that deployed an affine in the disturbance policy this assumption was needed to enable the handling of constraints in the overall mpc strategy the aims of this paper are to establish the necessity and sufficiency of the relevant invariance conditions of the polytopic dynamics and to propose a robust mpc strategy that uses polytopic sets to describe the evolution of predicted variables and therefore does not make an assumption that future multiplicative uncertainty is known the number of constraints and free variables of the resulting online optimization grows linearly with n the benefits of the strategy are illustrated by numerical simulations

convex liftings based robust control design this paper presents a new approach for control design of constrained linear systems affected by bounded additive disturbances and polytopic uncertainties this method hinges on so called convex liftings which emulate control lyapunov function by providing a constructive framework for optimization based control implementation it will be shown that this method can guarantee the recursive feasibility and robust stability finally a numerical example will be presented to illustrate this method

brief paper constrained linear systems with hard constraints and disturbances an extended command governor with large domain of attraction this paper considers a nonlinear feedback control policy that is an extension of those provided by command governors and reference governors as in these control approaches it applies to discrete time linear systems with hard constraints and set bounded disturbances the control policy retains the main properties of traditional governors such as straightforward direct implementation and finite settling time response to arbitrarily specified set points its principal advantage over traditional governors is a significantly larger domain of attraction that may compete in size with those obtained by dynamic programming connections to model predictive control are made numerical examples illustrate advantageous features of the proposed approach