constrained linear quadratic regulation the paper is a contribution to the theory of the infinite horizon linear quadratic regulator lqr problem subject to inequality constraints on the inputs and states extending an approach first proposed by sznaier and damborg 1987 a solution algorithm is presented which requires solving a finite number of finite dimensional positive definite quadratic programs the constrained lqr outlined does not feature the undesirable mismatch between open loop and closed loop nominal system trajectories which is present in the other popular forms of model predictive control mpc that can be implemented with a finite quadratic programming algorithm the constrained lqr is shown to be both optimal and stabilizing the solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution inherent to the approach is the removal of a tuning parameter the control horizon which is present in other mpc approaches and for which no reliable tuning guidelines are available two examples are presented that compare constrained lqr and two other popular forms of mpc the examples demonstrate that constrained lqr achieves significantly better performance than the other forms of mpc on some plants and the computational cost is not prohibitive for online implementation

a strong law of large numbers for fuzzy random sets abstract an slln for fuzzy random variables in rp including other convergence theorems has been proved by klement puri and ralescu the present paper is motivated by the definitions and properties developed by puri and ralescu and concerned with an slln for banach spaced valued fuzzy random sets as an easy application of the previous results

cones and decomposition of sub and supermartingales abstract we study sub and supermartingales with values in the class f c of all convex fuzzy subsets of a banach space having compact support the terms sub and super are understood with respect to a partial order not necessarily inclusion in this general framework orders with respect to which a doob type decomposition can hold are naturally associated to convex subcones with a special property termed f c properness decomposition theorems into the sum or difference of a martingale and a compensator are proven in the sub case the martingale need exist only in a superspace of f c

the attractor for a nonlinear reaction diffusion system in an unbounded domain in this paper the quasi linear second order parabolic systems of reaction diffusion type in an unbounded domain are considered our aim is to study the long time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions to this end we give a systematic study of given parabolic systems and their attractors in weighted sobolev spaces dependence of the hausdorff dimension of attractors on the weight of the sobolev spaces is considered 2001 john wiley sons inc

explicit hybrid model predictive control this article presents an algorithm for the exact solution of explicit hybrid model predictive control problems of time invariant discrete time mixed logical dynamical systems using multiparametric programming this control problem is formulated as a multiparametric mixed integer quadratic programming problem where the initial states of the system are treated as parameters in conjunction with decomposition type or branch and bound type approaches the proposed solution strategy first creates affine relaxations of nonconvex critical regions that result from the comparison of two quadratic objective functions these relaxations are then used to generate an affine outer approximation for the critical regions upon termination the bounded space of the initial states is partitioned into possibly nonconvex critical regions and corresponding optimal control laws

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

global existence for reaction diffusion systems modelling ignition the pair of parabolic equations with a 0 and b 0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f of particular interest is the case where which appears in the frank kamenetskii approximation to arrhenius type reactions we show here that for a large choice of the nonlinearity f u v in 1 2 including the model case 3 the corresponding initial value problem for 1 2 in the whole space with bounded initial data has a solution which exists for all times finite time blow up might occur though for other choices of function f u v and we discuss here a linear example which strongly hints at such behaviour

on the structure of the set of active sets in constrained linear quadratic regulation the constrained linear quadratic regulation problem is solved by a continuous piecewise affine function on a set of state space polytopes it is an obvious question whether this solution can be built up iteratively by increasing the horizon i e by extending the classical backward dynamic programming solution for the unconstrained case to the constrained case unfortunately however the piecewise affine solution for horizon n is in general not contained in the piecewise affine law for horizon n 1 we show that backward dynamic programming does in contrast result in a useful structure for the set of the active sets that defines the solution essentially every active set for the problem with horizon n 1 results from extending an active set for horizon n if the constraints are ordered stage by stage consequently the set for horizon n 1 can be found by only considering the constraints of the additional stage furthermore it is easy to detect which polytopes and affine pieces are invariant to increasing the horizon and therefore persist in the limit n several other aspects of the structure of the set of active sets become evident if the active sets are represented by bit tuples there exists for example a subset of special active sets that generates a positive invariant and persistent i e horizon invariant set around the origin it is very simple to identify these special active sets and the positive invariant and persistent region can be found without solving optimal control or auxiliary optimization problems the paper briefly discusses the use of these results in model predictive control some opportunities for uses in computational methods are also briefly summarized

technical communique on the facet to facet property of solutions to convex parametric quadratic programs in some of the recently developed algorithms for convex parametric quadratic programs it is implicitly assumed that the intersection of the closures of two adjacent critical regions is a facet of both closures this will be referred to as the facet to facet property it is shown by an example whose solution is unique that the facet to facet property does not hold in general consequently some existing algorithms cannot guarantee that the entire parameter space will be explored a simple modification applicable to several existing algorithms is presented for the purpose of overcoming this problem numerical results indicate that compared to the original algorithms for parametric quadratic programs the proposed method has lower computational complexity for problems whose solutions consist of a large number of critical regions

enumeration based approach to solving parametric linear complementarity problems this paper presents a solution method for parametric linear complementarity problems plcp that relies on an enumeration technique to discover all feasible bases the enumeration procedure is based on evaluating all possible combinations of active constraints and testing for feasibility although the enumeration approach is known to grow exponentially in the number of constraints the formulation of the plcp allows incorporation of cheap rank tests to quickly prune the infeasible directions in the exploration the motivation for the development of the enumeration based plcp solver is that it represents a direct method to solve parametric linear and quadratic optimization problems as well as their mixed integer counterparts these types of problems often arise in the field of model predictive control for linear and hybrid systems the enumeration based plcp solver offers another alternative to compute explicit solutions in the field of hybrid model predictive control that can be extremely effective in some important cases

fast algorithm for a constrained infinite horizon lq problem this paper presents an efficient algorithmic solution to the infinite horizon linear quadratic optimal control problem for a discrete time siso plant subject to bound constraints on a scalar variable the solution to the corresponding quadratic programming problem is based on the active set method and on dynamic programming it is shown that the optimal solution can be updated after inclusion or removal of an active constraint by a simple procedure requiring in the order of kn operations n being the system order and k the time at which the constraint is included or removed

on the convergence of fuzzy martingales the purpose of this paper is to prove the convergence theorem for fuzzy martingales without assuming that their values are lipschitz or continuous fuzzy numbers on rn this approach allows us to deduce many results on convergence of fuzzy numbers and fuzzy random variables from the relevant results on real numbers and real valued random variables that appear as their support functions based on the concept of fuzzy downward martingale and applications of convergence theorem of fuzzy martingales the strong law of large numbers for fuzzy random variables is proved here the convergence is in the uniform metric but not the separable metric on fuzzy number space

competition in ramped turing structures abstract stationary pattern selection and competition in the uniform brusselator in two 2d and three 3d dimensions are reviewed including reentrant hexagonal and striped zig zag phases influences of linear or chain like profiles of the pool chemicals on this selection are presented in the form of numerical experiments the relation with the recent experimental patterns obtained with the cima reaction is discussed

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

blowup in reaction diffusion systems with dissipation of mass we prove possible blowup in finite time of the solutions to reaction diffusion systems which preserve nonnegativity and for which the total mass of the components is nonincreasing in time two natural properties in applications this is done by presenting explicit counterexamples constructed with the help of formal computation software several partial results of global existence had been obtained previously in the literature our counterexamples explain a posteriori why extra conditions were needed negative results of independent interest are also provided as a by product for linear parabolic equations in nondivergence form and with discontinuous coefficients and for nonlinear hamilton jacobi evolution equations

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

note on sums of independent fuzzy random variables this note extends the results in feng fuzzy sets and systems 123 2001 11 about sums of independent fuzzy random variables by means of a convergence criterion in uniform hausdorff metric

attractors of reaction diffusion systems in unbounded domains and their spatial complexity the nonlinear reaction diffusion system in an unbounded domain is studied it is proven that under some natural assumptions on the nonlinear term and on the diffusion matrix this system possesses a global attractor in the corresponding phase space since the dimension of the attractor happens to be infinite we study its kolmogorov s entropy upper and lower bounds of this entropy are obtained moreover we give a more detailed study of the attractor for the spatially homogeneous rde in n in this case a group of spatial shifts acts on the attractor in order to study the spatial complexity of the attractor we interpret this group as a dynamical system with multidimensional time if n 1 acting on a phase space it is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy in order to clarify the nature of this chaos we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic and even lipschitz continuous embedding of this system into the spatial shifts on the attractor finally we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution 2003 wiley periodicals inc

on borel measurability and large deviations for fuzzy random variables fuzzy random variables are a special case of measurable mappings taking on values in a non separable metric space due to that non separability the two main definitions in the literature turn out not to be equivalent we give an easy necessary and sufficient condition for their equivalence an intermediate result of independent value is a characterization of separable subsets of that metric space several probabilistic applications are proposed notably a generalization to this setting of the large deviations principle of bolthausen

a strong law of large numbers for fuzzy random variables abstract a limit of a sequence of fuzzy numbers is defined and its some properties are shown based on these concept and properties an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown

an algebraic approach to bounded controllability of linear systems in this paper we present an algebraic approach to the proof that a linear system with matrices a b is null controllable using bounded inputs if and only if it is null controllable with unbounde

a unified framework for the study of anti windup designs we present a unified framework for the study of linear time invariant lti systems subject to control input nonlinearities the framework is based on the following two step design paradigm design the linear controller ignoring control input nonlinearities and then add anti windup bumpless transfer awbt compensation to minimize the adverse eflects of any control input nonlinearities on closed loop performance the resulting awbt compensation is applicable to multivariable controllers of arbitrary structure and order all known lti anti windup and or bumpless transfer compensation schemes are shown to be special cases of this framework it is shown how this framework can handle standard issues such as the analysis of stability and performance with or without uncertainties in the plant model the actual analysis of stability and performance and robustness issues are problems in their own right and hence not detailed here the main result is the unification of existing schemes for awbt compensation under a general framework

attractor and spatial chaos for the brusselator in rn abstract brusselator is an important dynamical system which can be described by a reaction diffusion system in this paper it is proved that this reaction diffusion system possesses a global attractor a in the corresponding phase space the upper and lower bounds of kolmogorov e entropy of a are obtained moreover we give a more detailed study of spatial chaos of the attractor a for the brusselator in r n we interpret a group of spatial shifts as a dynamical system which acts on the attractor a by using the technique of unstable manifolds it is proved that this dynamical system is chaotic in order to clarify the nature of this chaos we construct the lipschitz continuous homeomorphic embedding of a typical model dynamical system whose chaotic behavior is evident into the spatial shifts on the attractor a this typical dynamical system generalizes the symbolic system it was first introduced by zelik

ultimate boundedness control for uncertain discrete time systems via set induced lyapunov functions in this note linear discrete time systems affected by both parameter and input uncertainties are considered the problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence is investigated it is shown that the problem has a solution if and only if there exists a certain lyapunov function which does not belong to a preassigned class of functions e g the quadratic ones but it is determined by the target set in which ultimate boundedness is desired one of the advantages of this approach is that we may handle systems with control constraints no matching assumptions are made for systems with linearly constrained uncertainties it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets the resulting compensator may be implemented as a linear variable structure control

linear machine solution to point location problem linear machine has been recently proposed as an elegant solution for solving the point location problem arising in multi parametric programming mp p based online optimization linear machine associates a linear discriminant function with each polytopic region in the parametric space the solution to the point location problem is then obtained by simply evaluating these discriminant functions and finding their maximum value in this technical note we rigorously establish the correctness of the linear machine generation procedure and identify a necessary condition for existence of linear machine a modified procedure involving systematic subdivision of the parametric space is proposed when this condition is not satisfied analysis of complexity and storage requirements along with computational experiments on a large sized example indicate that linear machine can be an efficient tool for solving the point location problem

convergence in distribution for level continuous fuzzy random sets in this paper we give the relationships between tightness and convergence in distribution for level continuous fuzzy random sets to this end we first obtain some characterizations of compact subsets of the space of level continuous fuzzy sets in r p and establish the tightness for level continuous fuzzy random sets

an improved algorithm for combinatorial multi parametric quadratic programming the goal of multi parametric quadratic programming mpqp is to compute analytic solutions to parameter dependent constrained optimization problems e g in the context of explicit linear mpc we propose an improved combinatorial mpqp algorithm that is based on implicit enumeration of all possible optimal active sets and a simple saturation matrix pruning criterion which uses geometric properties of the constraint polyhedron for excluding infeasible candidate active sets in addition techniques are presented that allow to reduce the complexity of the discussed algorithm in the presence of symmetric problem constraints performance improvements are discussed for two example problems from the area of explicit linear mpc

explicit model predictive control a connected graph approach abstract the ability to solve model predictive control mpc problems of linear time invariant systems explicitly and offline via multi parametric quadratic programming mp qp has become a widely used methodology the most efficient approaches used to solve the underlying mp qp problem are either based on combinatorial considerations which scale unfavorably with the number of constraints or geometrical considerations which require heuristic tuning of the step size and correct identification of the active set in this paper we describe a novel algorithm which unifies these two types of approaches by showing that the solution of a mp qp problem is given by a connected graph where the nodes correspond to the different optimal active sets over the parameter space using an extensive computational study as well as the explicit mpc solution of a combined heat and power system the merits of the proposed algorithm are clearly highlighted

the laws of large numbers for fuzzy random variables the new attempt of weak and strong law of large numbers for fuzzy random variables is discussed in this paper by proposing the convergence in probability and convergence with probability one for fuzzy random variables we first consider the limit properties of fuzzy numbers by invoking the hausdorff metric and then we extend it to the convergence in probability and convergence with probability one for fuzzy random variables we provide the notion of weak and strong convergence in probability and weak and strong convergence with probability one for fuzzy random variables finally we come up with the weak and strong law of large numbers for fuzzy random variables in weak and strong sense

decomposition theorems for fuzzy supermartingales and submartingales doob decomposition and riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales the concepts of two types of riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale resp supermartingale has doob decomposition resp riesz decomposition are discussed in detail

brief paper a predictive controller with artificial lyapunov function for linear systems with input state constraints this paper copes with the problem of satisfying input and or state hard constraints in set point tracking problems stability is guaranteed by synthesizing a lyapunov quadratic function for the system and by imposing that the terminal state lies within a level set of the function procedures to maximize the volume of such an ellipsoidal set are provided and interiorpoint methods to solve on line optimization are considered

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

a novel approach to multiparametric quadratic programming multiparametric mp programming pre computes optimal solutions offline which are functions of parameters whose values become apparent online this makes it particularly well suited for applications that need a rapid solution of online optimization problems in this work we propose a novel approach to multiparametric programming problems based on an enumeration of active sets and use it to obtain a parametric solution for a convex quadratic program qp to avoid the combinatorial explosion of the enumeration procedure an active set pruning criterion is presented that makes the enumeration implicit the method guarantees that all regions of the partition are critical regions without any artificial cuts and further that no region of the parameter space is left unexplored

a dynamic programming approach to solving constrained linear quadratic optimal control problems abstract the solution of a constrained linear quadratic regulator problem is determined by the set of its optimal active sets we propose an algorithm that constructs this set of active sets for a desired horizon n from that for horizon n 1 while it is not obvious how to extend the optimal feedback law itself for horizon n 1 to horizon n a simple relation between the optimal active sets for two successive horizon lengths has recently been established essentially we show how to use this recent result to improve the efficiency of existing active set enumeration algorithms

stability of model predictive control with mixed constraints we derive stability conditions for model predictive control mpc with hard constraints on the inputs and soft constraints on the outputs for an infinitely long output horizon we show that with state feedback mpc is globally asymptotically stabilizing if and only if all the eigenvalues of the open loop system are in the closed unit disk with output feedback we show that the results hold if all the eigenvalues are strictly inside the unit circle the online optimization problem defining mpc can be posed as a finite dimensional quadratic program even though the output constraints are specified over an infinite horizon

convergence theorems for fuzzy random variables and fuzzy martingales in this paper we discuss the properties of fuzzy random variables and fuzzy conditional expectation some extended results of dominated convergence theorems for fuzzy random variables are proved we define the concept of a right closed fuzzy martingale and give the necessary and sufficient conditions of right closability for fuzzy martingales the convergence theorem for fuzzy martingales is also extended

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

on the stabilizing property of siorhc it is shown that the stabilizing property of siorhc stabilizing i o receding horizon control holds for general stabilizable discrete time linear plants irrespective of the condition that the plant has no poles at the origin

brief an algorithm for multi parametric quadratic programming and explicit mpc solutions explicit solutions to constrained linear model predictive control problems can be obtained by solving multi parametric quadratic programs mp qp where the parameters are the components of the state vector we study the properties of the polyhedral partition of the state space induced by the multi parametric piecewise affine solution and propose a new mp qp solver compared to existing algorithms our approach adopts a different exploration strategy for subdividing the parameter space avoiding unnecessary partitioning and qp problem solving with a significant improvement of efficiency

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

approximating integrably bounded fuzzy random variables in terms of the generalized hausdorff metric abstract consider the class of the mappings from a euclidean space to the unit interval 0 1 that is the class of the fuzzy sets of this space which are upper semicontinuous the closure of their supports is compact and the inverse images of the singleton are nonempty sets when the metric d which generalizes the hausdorff metric is defined on the preceding class the resulting metric space is complete but nonseparable consequently for this nonseparability given a probability space and an integrably bounded fuzzy random variable associated with it one cannot ensure the existence of a sequence of simple fuzzy random variables d converging almost surely to the former one however we are going to show in this paper that for any given integrably bounded fuzzy random variable we can guarantee the existence of a sequence of integrably fuzzy random variables whose level functions are simple random compact sets and which d converges at every point to the former variable on the basis of this result we will obtain in this paper a constructive definition of integrably bounded fuzzy random variables and an operational way to compute their fuzzy expected values

the strong law of large numbers for fuzzy random variables abstract sequences of independent and identically distributed fuzzy random variables are considered it is shown that the strong law of large numbers holds also for fuzzy random variables this result is used to give a consistent estimator for the expectation of a fuzzy random variable

a two phase algorithm for the multiparametric linear complementarity problem abstract a new two phase method for solving the multi parametric linear complementarity problem mplcp with sufficient matrices is presented in the first phase an initial feasible solution to mplcp which satisfies certain criteria is determined in the second phase the set of feasible parameters is partitioned into polyhedral regions such that the solution of the mplcp as a function of the parameters is invariant over each region the worst case complexity of the presented algorithms matches that of current methods for nondegenerate problems and is lower than that of current methods for degenerate problems

online constraint removal we show how to use a lyapunov function to accelerate mpc for linear discrete time systems with linear constraints and quadratic cost our method predicts in the current time step which constraints will be inactive in the next time step these constraints can be removed from the online optimization problem of the next time step the criterion for the detection of inactive constraints is based on the decrease of the lyapunov function along the trajectory of the controlled system the criterion is simple easy to implement in existing mpc algorithms and its computational cost is small

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

convergence of set valued and fuzzy valued martingales the purpose of this paper is to prove the convergence theorems of set valued and fuzzy valued martingales in kuratowski mosco sense without assuming that their values are compact or of compact level sets

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space