reach controllability of single input affine systems on a simplex we study the reach control problem rcp for a single input affine system with a simplicial state space we extend previous results by exploring arbitrary triangulations of the state space particularly allowing the set of possible equilibria to intersect the interior of simplices in the studied setting it is shown that closed loop equilibria nevertheless only arise on the boundary of simplices this allows to define a notion of reach controllability which quantifies the effect of the control input on boundary equilibria using reach controllability we obtain necessary and sufficient conditions for solvability of rcp by affine feedback

suboptimal target control for hybrid automata using model predictive control in this paper a computationally efficient controller is proposed for the target control problem when the system is modelled by hybrid automata the design is carried out in two stages first we compute off line the shortest switching path which has the minimum discrete cost from an initial set to the given target set next a controller is derived which successfully drives the system from any given initial state in the initial set to the target set while minimizing a cost function the model predictive control mpc technique is used when the current state is not within a guard set otherwise the mixed integer predictive control mipc technique is employed an on line semi explicit control algorithm is derived by combining these two techniques when the system is subject to additive bounded disturbance it is shown that the proposed on line control algorithm holds if tighter constraints on the original nominal state and controller are imposed finally as an application of the proposed control procedure the high speed and energy saving control problem of the cpu processing is considered

necessary and sufficient conditions for reachability on a simplex in this paper we solve the general problem of designing a feedback controller to reach a set of facets of an n dimensional simplex in finite time for a system evolving with linear affine dynamics necessary and sufficient conditions are presented in the form of bilinear inequalities on the vertices of the simplex by exploiting the structure of the problem the bilinear inequalities are converted to a series of linear programming problems

stabilising model predictive control for discrete time fractional order systems in this paper a model predictive control scheme is proposed for constrained fractional order discrete time systems we prove that constraints are satisfied and we prescribe conditions for the origin to be an asymptotically stable equilibrium point of the controlled system a finite dimensional approximation of the original infinite dimensional dynamics is employed for which the approximation error can become arbitrarily small the approximate dynamics is used to design a tube based model predictive controller which steers the system state to a neighbourhood of the origin of controlled size stability conditions are finally derived for the mpc controlled system which are computationally tractable and account for the infinite dimensional nature of the fractional order system and the state and input constraints the proposed control methodology guarantees asymptotic stability of the discrete time fractional order system satisfaction of the prescribed constraints and recursive feasibility

nonlinear control of constrained linear systems via predictive reference management a method based on conceptual tools of predictive control is described for solving set point tracking problems wherein pointwise in time input and or state inequality constraints are present it consists of adding to a primal compensated system a nonlinear device called command governor cg whose action is based on the current state set point and prescribed constraints the cg selects at any time a virtual sequence among a family of linearly parameterized command sequences by solving a convex constrained quadratic optimization problem and feeds the primal system according to a receding horizon control philosophy the overall system is proved to fulfill the constraints be asymptotically stable and exhibit an offset free tracking behavior provided that an admissibility condition on the initial state is satisfied though the cg can be tailored for the application at hand by appropriately choosing the available design knobs the required online computational load for the usual case of affine constraints is well tempered by the related relatively simple convex quadratic programming problem

reach control on simplices by continuous state feedback this paper studies a theoretical problem of whether continuous state feedback and affine feedback are equivalent from the point of view of making an affine system defined on a simplex reach a prespecified facet in finite time we show that the two classes of feedbacks are equivalent as a byproduct new necessary and sufficient conditions for solvability based more directly on the problem data are obtained

distributed model predictive control of linear systems with stochastic parametric uncertainties and coupled probabilistic constraints this paper investigates the distributed stochastic model predictive control dsmpc for multiple constrained dynamically decoupled subsystems subject to stochastic uncertainties in the parameters to obtain a computationally tractable formulation for real control applications a spectral method called generalized polynomial chaos expansions gpces is utilized to propagate the stochastic parametric uncertainties through the system model by using gpces combined with the probabilistic information on the uncertainties both local probabilistic constraints and coupled probabilistic constraints are converted explicitly into deterministic convex second order cone constraints the constraints can achieve satisfaction of coupled probabilistic constraints in a distributed way by permitting a single subsystem to optimize a local cost function at each time step while freezing the plans of others the proposed gpces based dsmpc algorithm guarantees recursive feasibility with respect to both local and coupled prob

probabilistic set invariance and ultimate boundedness the notions of invariant sets and ultimate bounds are important concepts in the analysis of dynamical systems and very useful tools for the design of control systems several approaches have been reported for the characterisation of these sets including constructive methods for their computation and procedures to obtain different approximations however there are shortcomings in those concepts in the sense that no general probability distributions can be considered for the disturbances affecting the system which for example precludes the assumption of gaussian distributions insofar as they are not bounded motivated by those shortcomings we propose in this paper the novel concepts of probabilistic ultimate bounds and probabilistic invariant sets which extend the notions of invariant sets and ultimate bounds to consider containment in probability and have the important feature of allowing stochastic noises with more general distributions including the ubiquitous gaussian distribution to be considered we introduce some key definitions for these sets establish their main properties and develop methods for their computation a numerical example illustrates the main ideas

characterization of the solution to a constrained h optimal control problem this paper obtains an explicit solution to a finite horizon min max optimal control problem where the system is linear and discrete time with control and state constraints and the cost quadratic the disturbance is negatively costed as in the standard h problem and is constrained the cost is minimized over control policies and maximized over disturbance sequences so that the solution yields a feedback control it is shown that under certain conditions the value function is piecewise quadratic and the optimal control policy piecewise affine being quadratic and affine respectively in polyhedra that partition the domain of the value function

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

on the strongly damped wave equation we prove the existence of the universal attractor for the strongly damped semilinear wave equation in the presence of a quite general nonlinearity of critical growth when the nonlinearity is subcritical we prove the existence of an exponential attractor of optimal regularity having a basin of attraction coinciding with the whole phase space as a byproduct the universal attractor is regular and of finite fractal dimension moreover we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient

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

contraction based nonlinear model predictive control formulation without stability related terminal constraints contraction based nonlinear model predictive control nmpc formulations are attractive because they generally require short prediction horizons and there is no need for the terminal set computation and reinforcement that are common requirements to guarantee stability however the inclusion of the contraction constraint in the definition of the underlying optimization problem often leads to non standard features such as a need for the multi step open loop application of control sequences or the use of multi step memorization of the contraction level which may cause unfeasibility in the presence of unexpected disturbances in this study we propose a new contraction based nmpc formulation where no contraction constraint is explicitly involved the convergence of the resulting closed loop behavior is proved under mild assumptions an illustrative example is provided in order to demonstrate the relevance of the proposed formulation

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

robust economic model predictive control using stochastic information in this paper we develop a new tube based robust economic mpc scheme for linear time invariant systems subject to bounded disturbances with given distributions by using the error distribution in the predictions of the finite horizon optimal control problem we can incorporate stochastic information in order to improve the expected performance while being able to guarantee strict feasibility for this new framework we can provide bounds on the asymptotic average performance of the closed loop system moreover a constructive approach is presented in order to find an appropriate terminal cost leading to a slight degradation of the bound on the guaranteed average performance

local control lyapunov functions for constrained linear discrete time systems the minkowski algebra approach this technical note utilizes minkowski algebra of convex sets to characterize a family of local control lyapunov functions for constrained linear discrete time systems local control lyapunov functions are induced by parametrized contractive invariant sets underlying contractive invariant sets belong to a family of minkowski decomposable convex sets and are in fact parametrized by a basic shape set and linear transformations of system matrices and a set of design matrices corresponding local control lyapunov functions can be detected by solving a single tractable convex optimization problem which in case of polyhedral constraints reduces to a single linear program the a priori complexity estimate of the characterized local control lyapunov function is provided for some practically relevant cases an illustrative example and relevant numerical experience are also reported

finite dimensional exponential attractors for semilinear wave equations with damping abstract we consider the initial value problem for a class of second order evolution equations that includes among others the 3d sine gordon equation with damping and the 3d klein gordon type equations with damping we show the existence of a set with finite fractal dimension that contains the global attractor and attracts all smooth solutions at an exponential rate

global existence for energy critical waves in 3 d domains neumann boundary conditions we prove that the defocusing quintic wave equation with neumann boundary conditions is globally well posed on h 1 l 2 for any smooth compact domain r 3 the proof relies on one hand on l p estimates for the spectral projector and on the other hand on a precise analysis of the boundary value problem which turns out to be much more delicate than in the case of dirichlet boundary conditions

a control problem for affine dynamical systems on a full dimensional polytope given an affine system on a full dimensional polytope the problem of reaching a particular facet of the polytope using continuous piecewise affine state feedback is studied necessary conditions and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the polytope special attention is paid to affine systems on full dimensional simplices in this case the necessary and sufficient conditions are equivalent and a constructive procedure yields an affine feedback control law that solves the reachability problem under consideration

controlling a class of nonlinear systems on rectangles in this paper we focus on a particular class of nonlinear affine control systems of the form xdot f x bu where the drift f is a multi affine vector field i e affine in each state component the control distribution b is constant and the control u is constrained to a convex set for such a system we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant we then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time if the control constraints are polyhedral we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle this work is motivated by the need to construct discrete abstractions for continuous and hybrid systems in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs we show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators

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

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

brief paper squaring the circle an algorithm for generating polyhedral invariant sets from ellipsoidal ones this paper presents a new geometrical approach to the computation of polyhedral robustly positively invariant pi sets for general possibly discontinuous nonlinear discrete time systems possibly affected by disturbances given a b contractive ellipsoidal set e the key idea is to construct a polyhedral set that lies between the ellipsoidal sets be and e a proof that the resulting polyhedral set is contractive and thus pi is given and a new algorithm is developed to construct the desired polyhedral set the problem of computing polyhedral invariant sets is formulated as a number of quadratic programming qp problems the number of qp problems is guaranteed to be finite and therefore the algorithm has finite termination an important application of the proposed algorithm is the computation of polyhedral terminal constraint sets for model predictive control based on quadratic costs

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

reachable controllable sets and stabilizing control of constrained linear systems we consider discrete time linear systems with constrained controls we present an algorithm for computing an external representation of the n step reachable set and controllable set it is an improvement of previous methods in particular it is shown that for constructing the set of initial states that can be driven to the origin asymptotically one can restrict the search to a polyhedral set in the unstable subspace of the autonomous system which yields a drastic reduction in the computational burden application to the synthesis of stabilizing feedback controllers is discussed

asymptotic regularity for some dissipative equations abstract this paper is devoted to proving some asymptotic regularity for both reaction diffusion equation with a polynomially growing nonlinearity of arbitrary order and strongly damped wave equation with critical nonlinearity which excel the sharp regularity allowed by the corresponding stationary equations equilibrium points based on this regularity the existence of the finite dimensional global and exponential attractors can be obtained easily

brief paper feedback and invariance under uncertainty via set iterates we examine discrete time control systems under non parametric disturbances sets which a given control feedback makes invariant under the disturbance are analyzed via lifting the feedback operation to the space of sets properties of being an attractor of the disturbed dynamics and being a minimal invariant set are derived from the corresponding notions of the set dynamics yielding in turn useful characterizations and error estimates for numerical algorithms which detect the minimal invariant sets concrete numerics for some examples of practical feedback rules are offered

brief paper on a reachability problem for affine hypersurface systems on polytopes this paper studies the problem for an affine hypersurface system with n 1 inputs to reach a polytopic target set starting from inside a polytope in the state space we present a solution which begins with a characterization of solvability by open loop control and concludes with a procedure to synthesize a feedback control our emphasis is on methods of subdivision triangulation and covers which explicitly account for the capabilities of the control system in contrast with the previous literature the partition methods are guaranteed to yield a correct feedback synthesis assuming the problem is solvable by open loop control

robust nonlinear model predictive control with variable block length robust model predictive controllers for discrete time nonlinear systems are proposed in this technical note the algorithms first search for an open loop controller with some block length and then try to improve it in a closed loop fashion by solving minimax problems on line it is proved that the controllers are capable of making a subsequence of the state converge into a target set in the presence of bounded disturbances

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

exponential attractors for weakly damped wave equation with sub quintic nonlinearity abstract in this paper we study the existence of exponential attractors for the weakly damped wave equation u t t u t u f u g in a bounded smooth domain of r 3 based on the recent extension of the strichartz estimate for the case of a bounded domain we establish an exponential attractor in a natural energy space for the case of sub quintic nonlinearity the results obtained essentially improve and complete some previous works

reach control problem for linear differential inclusion systems on simplices this technical note concerns the reach control problem for the dynamical systems represented by linear differential inclusions the goal is to derive conditions for the trajectories of a differential inclusion defined on a full dimensional simplex to reach exit facets in finite time using affine feedback to achieve this goal an invariance condition is firstly derived for lipschitz differential inclusions as an application of the obtained invariance condition the reach control problem rcp is solved for two classes of differential inclusions norm bounded linear differential inclusion and polytypic linear differential inclusion in the sense of strong and weak reachability respectively

technical communique a synthesis approach for output feedback robust constrained model predictive control this paper addresses the synthesis approach for output feedback robust model predictive control for systems with polytopic description bounded state disturbance and measurement noise off line it calculates a sequence of output feedback laws based on the state estimators by solving linear matrix inequality lmi optimization problems on line at each sampling time it chooses an appropriate output feedback law from this sequence the primary contribution is to present a rigorous method to guarantee satisfaction of input state constraints a numerical example is given to illustrate the effectiveness of the controller

brief paper on the computation of convex robust control invariant sets for nonlinear systems in this paper we provide a method to compute robust control invariant sets for nonlinear discrete time systems a simple criterion to evaluate if a convex set in state space is a robust control invariant set for a nonlinear uncertain system is presented the criterion is employed to design an algorithm for computing a polytopic robust control invariant set the method is based on the properties of dc functions i e functions which can be expressed as the difference of two convex functions since the elements of a wide class of nonlinear functions have dc representation or at least admit an arbitrarily close approximation the method is quite general the algorithm requires relatively low computational resources

striped parameterized tube model predictive control a modification of the parameterized tube model predictive control ptmpc strategy for linear systems with additive disturbances is proposed which reduces the dependence of the number of optimization variables on horizon length from quadratic to linear by using a triangular striped prediction structure unlike ptmpc which assumes a fixed linear terminal feedback law for predictions the proposed prediction scheme allows disturbance compensation to extend beyond the initial n step prediction horizon the resulting scheme can potentially outperform ptmpc in terms of the size of the domain of attraction and allows for a longer horizon n for the same computational demand

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

reachability of a set of facets for linear affine systems with n 1 inputs this note provides new necessary and sufficient conditions for an n dimensional linear affine system with n 1 inputs to reach an exit facet or set of exit facets of a simplex the conditions reduce the original np hard necessary and sufficient conditions to a set of at most n lp problems

low complexity polytopic invariant sets for linear systems subject to norm bounded uncertainty we propose a novel algorithm to compute low complexity polytopic robust control invariant rci sets along with the corresponding state feedback gain for linear discrete time systems subject to norm bounded uncertainty additive disturbances and state input constraints using a slack variable approach we propose new results to transform the original nonlinear problem into a convex lmi problem whilst introducing only minor conservatism in the formulation through numerical examples we illustrate that the proposed algorithm can yield improved maximal minimal volume rci set approximations in comparison with the schemes given in the literature

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

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

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

robust explicit model predictive control via regular piecewise affine approximation this paper proposes an explicit model predictive control design approach for regulation of linear time invariant systems subject to both state and control constraints in the presence of additive disturbances the proposed control law is implemented as a piecewise affine function defined on a regular simplicial partition and has two main positive features first the regularity of the simplicial partition allows one to efficiently implement the control law on digital circuits thus achieving extremely fast computation times moreover the asymptotic stability or the convergence to a set including the origin of the closed loop system can be enforced a priori rather than checked a posteriori via lyapunov analysis

on maximal robust positively invariant sets in constrained nonlinear systems abstract in this technical communique we study the maximal robust positively invariant set for state constrained continuous time nonlinear systems subjected to a bounded disturbance extending results from the theory of barriers we show that this set is closed and that its boundary consists of two complementary parts one of which we name the invariance barrier which consists of trajectories that satisfy the maximum principle

moving horizon estimation error dynamics and bounding error sets for robust control in this work we present a method for the derivation of the estimation error dynamics the bounding set of the estimation error and the state estimate dynamic equations of the constrained moving horizon estimator mhe

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

invariant approximations of the minimal robust positively invariant set this note provides results on approximating the minimal robust positively invariant mrpi set also known as the 0 reachable set of an asymptotically stable discrete time linear time invariant system it is assumed that the disturbance is bounded persistent and acts additively on the state and that the constraints on the disturbance are polyhedral results are given that allow for the computation of a robust positively invariant outer approximation of the mrpi set conditions are also given that allow one to a priori specify the accuracy of this approximation

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

reachability and control synthesis for piecewise affine hybrid systems on simplices in this paper we consider the synthesis of control laws for piecewise affine hybrid systems on simplices the construction is based on the solution to the control to facet problem at the continuous level and on dynamic programming at the discrete level the construction is given as an explicit algorithm using only linear algebra and reach set computations for automata no numerical integration is required the method is conservative in that it may fail to find a control law where one exists but one cannot hope for a sharp algorithm for control synthesis since reachability for piecewise affine hybrid systems is undecidable

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

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

optimal move blocking strategies for model predictive control this paper presents a systematic methodology for designing move blocking strategies to reduce the complexity of a model predictive controller for linear systems with explicit optimisation of the blocking structure using mixed integer programming given a move blocked predictive controller with a terminal invariant set constraint for stability combined with an input parameterisation to preserve recursive feasibility two different optimisation problems are formulated for blocking structure selection the first problem calculates the maximum achievable reduction in the number of input decision variables and prediction horizon length subject to the controller s region of attraction containing a specified subset of the state space then for a given fixed horizon length and block count determined by hardware capabilities the second problem seeks to maximise the volume of an inner approximation to the region of attraction numerical examples show that the resulting blocking structures are able to optimally reduce controller complexity and improve region of attraction volume

control to facet by piecewise affine output feedback the control to facet problem plays an important role in the design of feedback controllers for piecewise affine hybrid systems on polytopes in the literature necessary conditions and sufficient conditions for solvability by static state feedback exist in this paper we extend these results to the case of continuous piecewise affine static output feedback for the construction of a controller a triangulation of the output polytope is made which satisfies additional conditions to guarantee compatibility with the induced subdivision of the state polytope in the state feedback case the use of this special type of triangulation is not required

feasible parallel update distributed mpc for uncertain linear systems sharing convex constraints a distributed mpc approach for linear uncertain systems sharing convex constraints is presented the systems which are dynamically decoupled but share constraints on state and or inputs optimize once in parallel at each time step and exchange plans with neighbours thereafter coupled constraint satisfaction is guaranteed despite the simultaneous decision making by extra constraint tightening in each local problem necessary and sufficient conditions are given on the margins for coupled constraint satisfaction and a simple on line scheme for selecting margins is proposed that satisfies the conditions robust feasibility and stability of the overall system are guaranteed by use of the tube mpc concept in conjunction with the extra coupled constraint tightening

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