polynomial time probabilistic controllability analysis of discrete time piecewise affine systems this paper proposes a probabilistic approach to the controllability analysis for discrete time piecewise affine pwa systems three kinds of randomized algorithms which are based on random sampling of the mode sequence and or the initial state for determining with a probabilistic accuracy if the system is controllable are presented a positive one sided error algorithm a negative one sided error algorithm and a two sided error algorithm it is proven that these are polynomial time algorithms with respect to several variables of the problem it is also shown with some examples for which it is hopeless to check the controllability in a deterministic way that these algorithms are efficient

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

brief paper identification of dynamic systems using piecewise affine basis function models piecewise affine pwa basis function autoregressive exogenous bpwarx models are proposed in this paper for nonlinear black box identification a bpwarx model is a weighted sum of pwa basis bpwa functions which are the minimum or maximum of n 1 affine functions in n dimensions since the bpwa functions have a universal representation capability for continuous pwa functions the bpwarx models provide better accuracy than the hinging hyperplane arx hharx models with the same number of parameters and the same order of computational complexity when using a modified gauss newton algorithm to build the models from input output data

technical communique who needs qp for linear mpc anyway conventional mpc uses quadratic programming qp to minimise on line a cost over n linearly constrained control moves however stability constraints often require the use of large n thereby increasing the on line computation rendering the approach impracticable in the case of fast sampling here we explore an alternative that requires a fraction of the computational cost which increases only linearly with n and propose an extension which in all but a small class of models matches to within a fraction of a percent point the performance of the optimal solution obtained through qp the provocative title of the paper is intended to point out that the proposed approach offers a very attractive alternative to qp based mpc

reduced explicit constrained linear quadratic regulators in this note it is studied how structural properties of certain linear systems can be exploited to derive reduced dimension multiparametric quadratic programs that lead to explicit piecewise linear feedback solutions to the state and input constrained linear quadratic regulation problem the reduced dimensionality typically results in suboptimal controllers of lower complexity with associated computational advantages in the online implementation at the heart of the methods are state space projections based on the singular value decomposition

set membership approximation theory for fast implementation of model predictive control laws in this paper the use of set membership sm methodologies is investigated in the approximation of model predictive control mpc laws for linear systems such approximated mpc laws are derived from a finite number n of exact control moves computed off line properties in terms of guaranteed approximation error closed loop stability and performance are derived assuming only the continuity of the exact predictive control law these results are achieved by means of two main contributions at first it will be shown that if the approximating function enjoys two key properties i e fulfillment of input constraints and explicit evaluation of a bound on the approximation error which converges to zero as n increases then it is possible to guarantee the boundedness of the controlled state trajectories inside a compact set their convergency to an arbitrary small neighborhood of the origin and satisfaction of state constraints moreover the guaranteed performance degradation in terms of maximum state trajectory distance can be explicitly computed and reduced to an arbitrary small value by increasing n then two sm approximations are investigated both enjoying the above key properties the first one minimizes the guaranteed approximation error but its on line computational time increases with n the second one has higher approximation error but lower on line computational time which is constant with n when the off line computed moves are suitably chosen the presented approximation techniques can be systematically employed to obtain an efficient mpc implementation for fast processes the effectiveness of the proposed techniques is tested on two numerical examples

corrigendum corrigendum to the explicit linear quadratic regulator for constrained systems automatica 38 1 2002 3 20 we apologize that example 7 1 as publishedin bemporad morari dua andpistikopoulos 2002 is incorrect due to a miscalculation of the weight matrix p on the terminal state

approximate explicit constrained linear model predictive control via orthogonal search tree solutions to constrained linear model predictive control problems can be precomputed off line in an explicit form as a piecewise linear state feedback on a polyhedral partition of the state space avoiding real time optimization we suggest an algorithm that can determine an approximate explicit piecewise linear state feedback by imposing an orthogonal search tree structure on the partition this leads to a real time computational complexity that is logarithmic in the number of regions in the partition and the algorithm yields guarantees on the suboptimality asymptotic stability and constraint fulfillment

a predictive control scheme for systems with variable time delay this article deals with the robust control design problem for linear time invariant systems affected by variable feedback delay constraints on the system state are considered in the system description the starting point is the construction of a predictive control law which guarantees the existence of a nonempty robust positive invariant rpi set with respect to the closed loop dynamics subsequently an iterative algorithm is proposed in order to obtain an approximation of the maximal rpi set the problem can be treated in the framework of piecewise affine systems due to the explicit formulations of the control law obtained via multiparametric programming

order restricted bayesian estimation of the age composition of a population of atlantic cod abstract the management of a commercial fishery often requires estimates of the age composition these estimates are typically based on age and length data obtained from sampling the commercial landings from the fishery and the catches from a research vessel survey we use data from annual research cruises of georges bank conducted by the canadian department of fisheries and oceans to show how inferences for atlantic cod gadus morhua can be improved traditionally two phase stratified sampling is used with fish length or weight as the stratification variable letting pi denote the proportion of fish belonging to length stratum i and ij denote the proportion of fish belonging to age class j in stratum i we use bayesian methods to estimate p j i ij the proportion of fish that are age j specifications of smoothness expressed as unimodal order relations among the ij within and between the length strata are incorporated into the prior distributions uncertainty about both the locations

optimal two phase stratified sampling for estimation of the age composition of a fish population the provision of biological advice for the management of a commercial fish stock requires among other things an estimate of the age composition this estimate is usually obtained from length and age information collected by sampling the commercial landings associated with that fish stock the sampling methods used and the types of problems encountered are discussed by quinn et al 1983 and stevenson 1983 the final stage of the sample design is typically a two phase stratified sample of a large catch from a commercial vessel quinn et al 1983 stevenson 1983 southward 1976 at the first phase a large simple random sample srs of size n is selected and each fish is assigned to a stratum consisting of all fish having lengths in a specified range denote by nf the number of sampled fish in length stratum i at the second phase a srs of size ni is chosen from stratum i i 1 i each of these ni fish is aged with nij fish found to be agej j 1 j the problems are i estimation of ipj j 1 j where p j is the population proportion of age j fish ii given n and in i 1 i how to select ini i 1 i and iii for a specified cost function and given budget how to select n smith and sedransk 1982 provide methods for i and ii for ii their approach is flexible in that currently used or proposed allocation schemes are shown to be optimal under appropriate specifications in the current paper we i review the smith sedransk methodology ii show how to determine the optimal value of n and iii exhibit potential gains from using two phase sampling in section 3 we give a numerical illustration of the

lagrange stability and performance analysis of discrete time piecewise affine systems with logic states in this paper we consider discrete time piecewise affine systems with boolean inputs outputs and states and show that they can be represented in a logic canonical form where the logic variables influence the switching between different submodels but not the continuous valued dynamics we exploit this representation for studying lagrange stability and developing performance analysis procedures based on linear matrix inequalities moreover by using arguments from dissipativity theory for non linear systems we generalize our approach to solve the h analysis problem

mixtures of empirical distributions an iterative application of the age length key it is well known that an age length key is capable of providing biased estimates of age distributions when length distributions are applied to age length keys constructed from sampling in a different year from that in which the length data were collected in separate unpublished papers fukuda and chikuni far seas fisheries research laboratory fisheries agency of japan 1974 proposed a method of iterating an age length key that corrects this problem for a length distribution collected in any year their method provides an estimate of the age distribution corresponding to this length distribution which is consistent with an age length key constructed from the estimated age distribution and previously estimated length at age distributions we show that this method is an application of the em algorithm to mixtures of empirical distributions also we give conditions for which these estimates of age distribution are unique maximum likelihood estimates the method is used to estimate age distributions of the catch of greenland turbot reinhardtius hippoglossoides by small japanese freezer trawlers in the eastern bering sea

computational complexity certification for real time mpc with input constraints based on the fast gradient method this paper proposes to use nesterov s fast gradient method for the solution of linear quadratic model predictive control mpc problems with input constraints the main focus is on the method s a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre specified suboptimality is obtained for any possible state of the system we investigate cold and warm starting strategies and provide an easily computable lower iteration bound for cold starting and an asymptotic characterization of the bounds for warm starting moreover we characterize the set of mpc problems for which small iteration bounds and thus short solution times are expected the theoretical findings and the practical relevance of the obtained lower iteration bounds are underpinned by various numerical examples and compared to certification results for a primal dual interior point method

an eigenspace projection clustering method for inexact graph matching in this paper we show how inexact graph matching that is the correspondence between sets of vertices of pairs of graphs can be solved using the renormalization of projections of the vertices as defined in this case by their connectivities into the joint eigenspace of a pair of graphs and a form of relational clustering an important feature of this eigenspace renormalization projection clustering epc method is its ability to match graphs with different number of vertices shock graph based shape matching is used to illustrate the model and a more objective method for evaluating the approach using random graphs is explored with encouraging results

model predictive control based on linear programming the explicit solution we study model predictive control mpc schemes for discrete time linear time invariant systems with constraints on inputs and states that can be formulated using a linear program lp in particular we focus our attention on performance criteria based on a mixed 1 norm namely 1 norm with respect to time and norm with respect to space first we provide a method to compute the terminal weight so that closed loop stability is achieved we then show that the optimal control profile is a piecewise affine and continuous function of the initial state and briefly describe the algorithm to compute it the piecewise affine form allows to eliminate online lp as the computation associated with mpc becomes a simple function evaluation besides practical advantages the availability of the explicit structure of the mpc controller provides an insight into the type of control action in different regions of the state space and highlights possible conditions of degeneracies of the lp such as multiple optima

cardinality constrained linear quadratic optimal control as control implementation often incurs not only a variable cost associated with the magnitude or energy of the control but also a setup cost we consider a discrete time linear quadratic lq optimal control problem with a limited number of control implementations termed in this technical note the cardinality constrained linear quadratic optimal control cclq we first derive a semi analytical feedback policy for cclq problems using dynamic programming dp due to the exponential growth of the complexity in calculating the action regions however dp procedure is only efficient for cclq problems with a scalar state space recognizing this fact we develop then two lower bounding schemes and integrate them into a branch and bound bnb solution framework to offer an efficient algorithm in solving general cclq problems adopting the devised solution algorithm for cclq problems we can solve efficiently the linear quadratic optimal control problem with setup costs

suboptimal receding horizon estimation via noise blocking abstract for discrete time linear systems we propose a suboptimal approach to constrained estimation so that the associated computation burden is reduced this is achieved by enforcing a move blocking mb structure in the estimated process noise sequence pns we show that full information estimation fie and receding horizon estimation rhe with mb are both stable in the sense of an observer the techniques in proving stability are inspired by those that have been proposed for standard rhe to be specific stability results are mainly achieved by i carefully embellishing the general assumptions for standard rhe to accommodate the mb requirement ii exploiting the principle of optimality as well as convexity of the quadratic programs qps associated with fie and rhe iii relying on the fact that the kalman filter is the best linear estimator in the least squares sense a crucial requirement in achieving stability for mb rhe is that the segment structure ss of the pns of mb fie for the optimization steps within the receding horizon i e steps between t n and t 1 has to be enforced in the mb rhe optimization as a result the mb rhe strategy becomes a dynamic estimator with a periodically varying computational complexity the theoretical results have been illustrated with examples

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

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

on constrained infinite time linear quadratic optimal control abstract this work presents a solution to the infinite time linear quadratic optimal control itlqoc problem with state and control constraints it is shown that a single finite dimensional convex program of known size can yield this solution properties of the resulting value function with respect to initial conditions are also established and are shoen to be useful in determining the aforementioned problem size an example illustrating the method is finally presented

performance bounds for linear stochastic control we develop computational bounds on performance for causal state feedback stochastic control with linear dynamics arbitrary noise distribution and arbitrary input constraint set this can be very useful as a comparison with the performance of suboptimal control policies which we can evaluate using monte carlo simulation our method involves solving a semidefinite program a linear optimization problem with linear matrix inequality constraints a convex optimization problem which can be efficiently solved numerical experiments show that the lower bound obtained by our method is often close to the performance achieved by several widely used suboptimal control policies which shows that both are nearly optimal as a by product our performance bound yields approximate value functions that can be used as control lyapunov functions for suboptimal control policies

complexity reduction of explicit model predictive control via separation the problem of reducing complexity of explicit mpc feedback laws for linear systems is considered we propose to simplify controllers defined by continuous piecewise affine pwa functions by employing separating functions if a state resides in a region where the optimal control action attains a saturated value the optimal control move is determined from the sign of the separator thus instead of storing all regions only the unconstrained regions and the separator are needed we propose several approaches to construct separators with different efficacy and properties

real time heuristic search abstract we apply the two player game assumptions of limited search horizon and commitment to moves in constant time to single agent heuristic search problems we present a variation of minimax lookahead search and an analog to alpha beta pruning that significantly improves the efficiency of the algorithm paradoxically the search horizon reachable with this algorithm increases with increasing branching factor in addition we present a new algorithm called real time a for interleaving planning and execution we prove that the algorithm makes locally optimal decisions and is guaranteed to find a solution we also present a learning version of this algorithm that improves its performance over successive problem solving trials by learning more accurate heuristic values and prove that the learned values converge to their exact values along every optimal path these algorithms effectively solve significantly larger problems than have previously been solvable using heuristic evaluation functions

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

explicit sub optimal linear quadratic regulation with state and input constraints optimal feedback solutions to the infinite horizon lqr problem with state and input constraints based on receding horizon real time quadratic programming are well known in this paper we develop an explicit solution to the same problem eliminating the need for real time optimization it is shown that the resulting feedback controller is piecewise linear this explicit functional structure is exploited for efficient real time implementation a suboptimal strategy based on a suboptimal choice of a finite horizon and imposing additional limitations on the allowed switching between active constraint sets on the horizon is suggested in order to address the computer memory and processing capacity requirements of the explicit solution

minimal conjunctive normal expression of continuous piecewise affine functions continuous piecewise affine pwa functions arise in many aspects of control for this kind of function we propose the minimal conjunctive normal expression cne the cne can be expressed as the minimum of a collection of terms each of which is the maximum of a set of affine functions the minimal cne is defined to contain the smallest number of parameters analogous to boolean algebra we propose implicants and prime implicants for continuous pwa functions after obtaining all prime implicants the problem of finding minimal cnes can then be cast as a binary programming problem a sharp bound on the number of boolean variables in the binary programming problem is given in two worked examples minimal cnes are derived for given continuous pwa functions

stabilizing polynomial approximation of explicit mpc a given explicit piecewise affine representation of an mpc feedback law is approximated by a single polynomial computed using linear programming this polynomial state feedback control law guarantees closed loop stability and constraint satisfaction the polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage

clipping based complexity reduction in explicit mpc the idea of explicit model predictive control mpc is to characterize optimal control inputs as an explicit piecewise affine pwa function of the initial conditions the function however is often too complex and either requires too much processing power to evaluate on line or consumes a prohibitive amount of memory the paper focuses on the memory issue and proposes a novel method of replacing a generic continuous pwa function by a different function of significantly lower complexity in such a way that the simple function guarantees the same properties as the original the idea is based on eliminating regions of the pwa function over which the function attains a saturated value an extensive case study is presented which confirms that a significant reduction of complexity is achieved in general

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

an eigendecomposition approach to weighted graph matching problems an approximate solution to the weighted graph matching problem is discussed for both undirected and directed graphs the weighted graph matching problem is that of finding the optimum matching between two weighted graphs which are graphs with weights at each arc the proposed method uses an analytic instead of a combinatorial or iterative approach to the optimum matching problem using the eigendecompositions of the adjacency matrices in the case of the undirected graph matching problem or hermitian matrices derived from the adjacency matrices in the case of the directed graph matching problem a matching close to the optimum can be found efficiently when the graphs are sufficiently close to each other simulation results are given to evaluate the performance of the proposed method

technical communique a logarithmic time solution to the point location problem for parametric linear programming the optimiser of a multi parametric linear program plp is a piecewise affine function defined over a polyhedral subdivision of the set of feasible states once this affine function has been pre calculated the optimal solution can be computed for a particular parameter by determining the region that contains it this is the so called point location problem in this paper we show that this problem can be written as an additively weighted nearest neighbour search that can be solved in time linear in the dimension of the state space and logarithmic in the number of regions it is well known that linear model predictive control mpc problems based on linear control objectives e g 1 or norm can be posed as plps and on line calculation of the control law involves the solution to the point location problem several orders of magnitude sampling speed improvement are demonstrated over traditional mpc and closed form mpc schemes using the proposed scheme

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

solution of the input constrained lqr problem using dynamic programming the input constrained lqr problem is addressed in this paper i e the problem of finding the optimal control law for a linear system such that a quadratic cost functional is minimised over a horizon of length n subject to the satisfaction of input constraints a global solution i e valid in the entire state space for this problem and for arbitrary horizon n is derived analytically by using dynamic programming the scalar input case is considered in this paper solutions to this problem and to more general problems state constraints multiple inputs have been reported recently in the literature for example approaches that use the geometric structure of the underlying quadratic programming problem and approaches that use multi parametric quadratic programming techniques the solution by dynamic programming proposed in the present paper coincides with the ones obtained by the aforementioned approaches however being derived using a different approach that exploits the dynamic nature of the constrained optimisation problem to obtain an analytical solution the present result complements the previous methods and reveals additional insights into the intrinsic structure of the optimal solution

ultra fast stabilizing model predictive control via canonical piecewise affine approximations this paper investigates the use of canonical piecewise affine pwa functions for approximation and fast implementation of linear mpc controllers the control law is approximated in an optimal way over a regular simplicial partition of a given set of states of interest the stability properties of the resulting closed loop system are analyzed by constructing a suitable pwa lyapunov function the main advantage of the proposed approach to the implementation of mpc controllers is that the resulting stabilizing approximate mpc controller can be implemented on chip with sampling times in the order of tens of nanoseconds

irredundant lattice representations of continuous piecewise affine functions in this paper we revisit the full lattice representation of continuous piecewise affine pwa functions and give a formal proof of its representation ability based on this we derive the irredundant lattice pwa representations through removal of redundant terms and literals necessary and sufficient conditions for irredundancy are proposed besides we explain how to remove terms and literals in order to ensure irredundancy an algorithm is given to obtain an irredundant lattice pwa representation in the worked examples the irredundant lattice pwa representations are used to express the optimal solution of explicit model predictive control problems and the results turn out to be much more compact than those given by a state of the art algorithm

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

a patchy approximation of explicit model predictive control the explicit solution of multi parametric optimisation problems mpop has been used to construct an off line solution to relatively small and medium sized constrained control problems the control design principles are based on receding horizon optimisation and generally use linear prediction models for the system dynamics in this context it can be shown that the optimal control law is a piecewise linear pwl state feedback defined over polytopic cells of the state space however as the complexity of the related optimisation problems increases the memory footprint and implementation of such explicit optimal solution may be burdensome for the available hardware principally due to the high number of polytopic cells in the state space partition in this article we provide a solution to this problem by proposing a patchy pwl feedback control law which intend to approximate the optimal control law the construction is based on the linear interpolation of the exact solution at the vertices of a feasibl

brief multiplier theory for stability analysis of anti windup control systems we apply the passivity theorem with appropriate choice of multipliers to develop a general framework for analyzing stability of anti windup bumpless transfer awbt schemes for appropriate choices of the multipliers we reduce the resulting sufficient conditions for stability to equivalent linear matrix inequalities lmis we show that a number of previously reported attempts to analyze awbt stability using seemingly diverse techniques such as the popov circle and off axis circle criteria the optimally scaled small gain theorem generalized m upper bound and describing functions can all be generalized within the framework of this paper

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

projection free parallel quadratic programming for linear model predictive control a key component in enabling the application of model predictive control mpc in fields such as automotive aerospace and factory automation is the availability of low complexity fast optimisation algorithms to solve the mpc finite horizon optimal control problem in architectures with reduced computational capabilities in this paper we introduce a projection free iterative optimisation algorithm and discuss its application to linear mpc the algorithm originally developed by brand for non negative quadratic programs is based on a multiplicative update rule and it is shown to converge to a fixed point which is the optimum an acceleration technique based on a projection free line search is also introduced to speed up the convergence to the optimum the algorithm is applied to mpc through the dual of the quadratic program qp formulated from the mpc finite time optimal control problem we discuss how termination conditions with guaranteed degree of suboptimality can be enforced and how the algorithm

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

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

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

brief paper constrained robust model predictive control via parameter dependent dynamic output feedback this paper considers output feedback robust model predictive control for the quasi linear parameter varying quasi lpv system with bounded disturbance the so called quasi lpv means that the varying parameters of the linear system are known at the current time but unknown in the future the control law is parameterized as a parameter dependent dynamic output feedback and the closed loop stability is specified by the notion of quadratic boundedness an iterative algorithm is proposed for the on line synthesis of the control law via convex optimization a numerical example is given to illustrate the effectiveness of the controller

brief paper an integrated perturbation analysis and sequential quadratic programming approach for model predictive control computationally efficient algorithms are critical in making model predictive control mpc applicable to broader classes of systems with fast dynamics and limited computational resources in this paper we propose an integrated formulation of perturbation analysis and sequential quadratic programming inpa sqp to address the constrained optimal control problems the proposed algorithm combines the complementary features of perturbation analysis and sqp in a single unified framework thereby leading to improved computational efficiency and convergence property a numerical example is reported to illustrate the proposed method and its computational effectiveness

multi resolution explicit model predictive control delta model formulation and approximation this paper deals with the explicit solution and approximation of the constrained linear finite time optimal control problem for systems with fast sampling rates to this aim the recently developed explicit model predictive control empc is reformulated and characterized using the operator to enjoy its promising advantages compared to the time shift operator using the proposed model empc formulation a systematic method is proposed for first designing a low complexity approximate empc solution and then improving its closed loop action without first determining an exact optimal solution that might be of prohibitive complexity it is shown that the stability and feasibility of the proposed sub optimal solution is guaranteed

analytical expression of explicit mpc solution via lattice piecewise affine function an analytical expression of the explicit solution to linear model predictive control mpc is proposed by the introduction of a lattice piecewise affine pwa function a systematic procedure is developed for building a lattice pwa representation from a continuous explicit mpc solution obtained by a multi parametric program a simple method is presented to remove the redundant parameters in the lattice expression of mpc control laws the effectiveness of this approach is supported by the study of three benchmark mpc problems the proposed analytical expression provides a very efficient and practically viable method for implementing the explicit mpc solutions regarding its online calculation and memory space requirements

technical communique evaluation of piecewise affine control via binary search tree we present an algorithm for generating a binary search tree that allows efficient computation of piecewise affine pwa functions defined on a polyhedral partition this is useful for pwa control approaches such as explicit model predictive control as it allows the controller to be implemented online with small computational effort the computation time is logarithmic in the number of regions in the pwa partition

relaxing dynamic programming the idea of dynamic programming is general and very simple but the curse of dimensionality is often prohibitive and restricts the fields of application this paper introduces a method to reduce the complexity by relaxing the demand for optimality the distance from optimality is kept within prespecified bounds and the size of the bounds determines the computational complexity several computational examples are considered the first is optimal switching between linear systems with application to design of a dc dc voltage converter the second is optimal control of a linear system with piecewise linear cost with application to stock order control finally the method is applied to a partially observable markov decision problem pomdp

comments on analytical expression of explicit mpc solution via lattice piecewise affine function automatica 45 2009 910 917 the simplification conditions for lattice representation of the pwa control law in the explicit mpc suggested in the aforementioned article are shown by means of a counterexample not to be sufficient in view of this a modified lemma is proposed in this note for ensuring the sufficiency

bayesian estimation of finite population parameters in categorical data models incorporating order restrictions this note describes a bayesian method for estimation of finite population parameters in general population surveys where acceptable regression type models are typically unavailable a categorical data model is adopted as in ericson 1969 section 4 however specifications of smoothness are incorporated into the prior distribution these smoothness conditions are expressed as unimodal or possibly multi modal order relations among the category probabilities emphasis is placed on posterior inference about the finite population mean of independent interest is the methodology for evaluating the posterior moments and probabilities using monte carlo integration with importance sampling

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

minimising conservatism in infinite horizon lqr control this paper studies the formulation of the constrained infinite horizon linear quadratic regulator control law cihlqr results from recent studies in this area are extended to show that conditions used in the standard formulation of the cihlqr law are not necessary but merely sufficient through the use of a novel proof it is shown that for a general siso system with input constraints and certain conditions imposed saturated lqr provides the same control sequence as cihlqr it is further shown that saturated lqr is equivalent to the cihlqr in the case of first order systems subject to both state and control constraints finally the region of constrained stabilisability is characterised for the case of open loop unstable first order systems

computation of the constrained infinite time linear quadratic regulator this paper presents an efficient algorithm for computing the solution to the constrained infinite time linear quadratic regulator clqr problem for discrete time systems the algorithm combines multi parametric quadratic programming with reachability analysis to obtain the optimal piecewise affine pwa feedback law the algorithm reduces the time necessary to compute the pwa solution for the clqr when compared to other approaches it also determines the minimal finite horizon email protected s such that the constrained finite horizon lqr problem equals the clqr problem for a compact set of states s the on line computational effort for the implementation of the clqr can be significantly reduced as well either by evaluating the pwa solution or by solving the finite dimensional quadratic program associated with the clqr for a horizon of email protected s

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

brief paper fast large scale model predictive control by partial enumeration partial enumeration pe is presented as a method for treating large linear model predictive control applications that are out of reach with available mpc methods pe uses both a table storage method and online optimization to achieve this goal versions of pe are shown to be closed loop stable pe is applied to an industrial example with more than 250 states 32 inputs and a 25 sample control horizon the performance is less than 0 01 suboptimal with average speedup factors in the range of 80 220 and worst case speedups in the range of 4 9 39 2 compared to an existing mpc method small tables with only 25 200 entries were used to obtain this performance while full enumeration is intractable for this example

brief paper using hash tables to manage the time storage complexity in a point location problem application to explicit model predictive control the online computational burden of linear model predictive control mpc can be moved offline by using multi parametric programming so called explicit mpc the solution to the explicit mpc problem is a piecewise affine pwa state feedback function defined over a polyhedral subdivision of the set of feasible states the online evaluation of such a control law needs to determine the polyhedral region in which the current state lies this procedure is called point location its computational complexity is challenging and determines the minimum possible sampling time of the system a new flexible algorithm is proposed which enables the designer to trade off between time and storage complexities utilizing the concept of hash tables and the associated hash functions the proposed method solves an aggregated point location problem that overcomes prohibitive complexity growth with the number of polyhedral regions while the storage processing trade off can be optimized via scaling parameters the flexibility and power of this approach is supported by several numerical examples

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

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

technical communique improving off line approach to robust mpc based on nominal performance cost this paper gives two alternative off line synthesis approaches to robust model predictive control rmpc for systems with polytopic description in each approach a sequence of explicit control laws that correspond to a sequence of nested asymptotically invariant ellipsoids are constructed off line in order to accommodate a wider class of systems nominal performance cost is chosen to substitute the worst case one in an existing technique in the design of control law for a larger ellipsoid the second approach further incorporates the knowledge of control laws associated with all smaller ellipsoids so as to further improve feasibility and optimality the effectiveness of the alternative approaches is illustrated by a simulation example

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

bayesian analysis of constrained parameter and truncated data problems using gibbs sampling abstract constrained parameter problems arise in a wide variety of applications including bioassay actuarial graduation ordinal categorical data response surfaces reliability development testing and variance component models truncated data problems arise naturally in survival and failure time studies ordinal data models and categorical data studies aimed at uncovering underlying continuous distributions in many applications both parameter constraints and data truncation are present the statistical literature on such problems is very extensive reflecting both the problems widespread occurrence in applications and the methodological challenges that they pose however it is striking that so little of this applied and theoretical literature involves a parametric bayesian perspective from a technical viewpoint this perhaps is not difficult to understand the fundamental tool for bayesian calculations in typical realistic models is multidimensional numerical integration which often is problem

heuristically enhanced feedback control of constrained discrete time linear systems recent advances in computer technology have spurred new interest in the use of feedback controllers based upon on line minimization for the control of constrained linear systems still the use of computers in the feedback loop has been hampered by the fact that the amount of time available for computation in most sampled data systems is not enough to achieve a complete solution using conventional algorithms several ad hoc techniques have been proposed but their applicability is restricted by the lack of supporting theory in this paper we present a theoretical framework to analyze the stability of the closed loop system resulting from the use of on line optimization in the feedback loop using these results we show that a suboptimal algorithm based upon the use of heuristic search techniques yields asymptotically stable systems provided that enough computation power is available to solve at each sampling interval an optimization problem considerably simpler than the original the controller presented in this paper is valuable for situations where the customary approaches of using pontryagin s minimum principle or storing a family of extremal curves are not applicable due to limitations in the computational resources available

a measure of similarity between graph vertices applications to synonym extraction and web searching we introduce a concept of similarity between vertices of directed graphs let ga and gb be two directed graphs with respectively na and nb vertices we define an nb times na similarity matrix s whose real entry sij expresses how similar vertex j in ga is to vertex i in gb we say that sij is their similarity score the similarity matrix can be obtained as the limit of the normalized even iterates of sk 1 bskat btska where a and b are adjacency matrices of the graphs and s0 is a matrix whose entries are all equal to 1 in the special case where ga gb g the matrix s is square and the score sij is the similarity score between the vertices i and j of g we point out that kleinberg s hub and authority method to identify web pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them in analogy to kleinberg we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix potential applications of our similarity concept are numerous we illustrate an application for the automatic extraction of synonyms in a monolingual dictionary

asymptotically exact stabilisation for constrained discrete takagisugeno systems via set invariance given a takagisugeno ts system this paper proposes a novel methodology to obtain the state feedback controller guaranteeing asymptotically as a polya related complexity parameter grows the largest membership shape independent possible domain of attraction with contraction rate performance based on polyhedral contractive sets from constrained linear systems literature the resulting controller is valid for any realisation of the memberships as usual in most ts literature for a finite complexity parameter an inner estimate of such largest set is obtained the frontier of such approximation can be understood as the level set of a polyhedral control lyapunov function convergence of a proposed iterative algorithm is asymptotically necessary and sufficient for ts system stabilisation for a high enough value of the complexity parameter any conceivable shape independent lyapunov controller design procedure will yield a proven domain of attraction smaller or equal to the algorithm s output

a gradient type algorithm optimizing the coupling between matrices in this paper we consider the problem of maximizing the coupling between the isometric projections of two square matrices of dimensions m and n this coupling is defined as an inner product between the matrices this is a non convex optimization problem with isometry constraints on the variables the optimization set is an equinormed set and we develop a gradient type algorithm to solve the problem numerical experiments and an application to graph matching are also presented c 2007 elsevier inc all rights reserved

piecewise linear lq control for systems with input constraints abstract a piecewise linear control plc law based on lq theory is derived ths law raises an lq gain as the controlled error converges towards the origin it can be computed off line and it guarantees that input bounds are never exceeded without causing input saturation a succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest lq gain possible in the presence of the input bounds the formulation gives rise to an iteration function whose solution is a fixed point fp the fp defines the optimal gain and the invariant set which comprise ellipsoids fp existence uniqueness and convergence properties are proven we explain the algorithms for the generation and subsequent implementation of the plc law the notion of a switching function is introduced consideration is given to the computational aspects of the fp the plc scheme is demonstrated for the linearized dynamics of the simple pendulum and the puma 560 robot a discrete time plc law can also be derived using the discrete time algebraic riccati equation

optimal control of sampled data piecewise affine systems this paper discusses the optimal continuous time control problem of a class of piecewise affine pwa systems where the switching action of the discrete state is determined at each sampling time according to a condition on the continuous state such a system is called here the sampled data pwa sd pwa system first important remarks on the control design of this system via the continuous or discrete time pwa model are pointed out which motivate us to use the sd pwa model next based on the good properties of the proposed model an optimal continuous time controller of the sd pwa systems is proposed

a new first order algorithmic framework for optimization problems with orthogonality constraints in this paper we consider a class of optimization problems with orthogonality constraints the feasible region of which is called the stiefel manifold our new framework combines a function value reduction step with a correction step different from the existing approaches the function value reduction step of our algorithmic framework searches along the standard euclidean descent directions instead of the vectors in the tangent space of the stiefel manifold and the correction step further reduces the function value and guarantees a symmetric dual variable at the same time we construct two types of algorithms based on this new framework the first type is based on gradient reduction including the gradient reflection gr and the gradient projection gp algorithms the other one adopts a columnwise block coordinate descent cbcd scheme with a novel idea for solving the corresponding cbcd subproblem inexactly we prove that both gr gp with a fixed step size and cbcd belong to our algorithmic framework

bayesian optimization of the estimation of the age composition of a fish population abstract to estimate the age distribution of a fish population a two phase stratified sample design is proposed bayesian methods are employed to determine a optimal estimators of the desired population proportions and b the optimal choice of second phase sample sizes subject to a fixed budget a summary of the results of some numerical examples is presented these examples demonstrate typical expected gains obtained by using the bayesian optima as compared to alternative choices of second phase sample sizes suggested in the substantive literature

zames falb multipliers for quadratic programming in constrained linear model predictive control a quadratic program must be solved on line at each control step and this constitutes a nonlinearity if zero is a feasible point for this quadratic program then the resultant nonlinearity is sector bounded we show that if the nonlinearity is static then it is also monotone and slope restricted hence we show the existence of zames falb multipliers for such a nonlinearity the multipliers may be used in a general and versatile analysis of the robust stability of input constrained model predictive control

the explicit constrained min max model predictive control of a discrete time linear system with uncertain disturbances in this technical brief we develop an algorithm to determine the explicit solution of the constrained min max model predictive control problem for a discrete time linear system with bounded additive uncertain disturbance the control law is determined to be piecewise affine from a quadratic cost function and the state space is partitioned into corresponding polyhedral cones by moving the on line implementation to an off line explicit evaluation the computational burden is decreased and the applicability of min max optimization is broadened the results of this approach are shown via computer simulations