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

complexity of stability and controllability of elementary hybrid systems in this paper we consider simple classes of nonlinear systems and prove that basic questions related to their stability and controllability are either undecidable or computationally intractable np hard as a special case we consider a class of hybrid systems in which the state space is partitioned into two halfspaces and the dynamics in each halfspace correspond to a different linear system

brief computational complexity of randomized algorithms for solving parameter dependent linear matrix inequalities randomized algorithms are proposed for solving parameter dependent linear matrix inequalities and their computational complexity is analyzed the first proposed algorithm is an adaptation of the algorithms of polyak and tempo syst control lett 43 5 2001 343 and calafiore and polyak ieee trans autom control 46 11 2001 1755 for the present problem it is possible however to show that the expected number of iterations necessary to have a deterministic solution is infinite in order to make this number finite the improved algorithm is proposed the number of iterations necessary to have a probabilistic solution is also considered and is shown to be independent of the parameter dimension a numerical example is provided

survey paper research on probabilistic methods for control system design a novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty the main objective of this paper is to provide a broad perspective on this area of research known as probabilistic robust control and to address in a systematic manner recent advances the focal point is on design methods based on the interplay between uncertainty randomization and convex optimization and on the illustration of specific control applications

fast construction of robustness degradation function we develop a fast algorithm to construct the robustness degradation function which describes quantitatively the relationship between the proportion of systems guaranteeing the robustness requirement and the radius of the uncertainty set this function can be applied to predict whether a controller design based on an inexact mathematical model will perform satisfactorily when implemented on the true system

synthesis of optimal controllers for piecewise affine systems with sampled data switching this paper discusses the optimal control problem of the continuous time piecewise affine pwa systems with sampled data switching where the switching action is executed based upon a condition on the state at each sampling time first an algebraic characterization for the problem to be feasible is derived next an optimal continuous time controller is derived for a general class of pwa systems with sampled data switching for which the optimal control problem is feasible but whose subsystems in some modes may be uncontrollable in the usual sense finally as an application of the proposed approach the high speed and energy saving control problem of the cpu processing is formulated and the validity of the proposed methods is shown by numerical simulations

randomized algorithms for robust controller synthesis using statistical learning theory by now it is known that several problems in the robustness analysis and synthesis of control systems are np complete or np hard these negative results force us to modify our notion of solving a given problem an approach that is recently gaining popularity is that of using randomized algorithms which can be used to solve a problem approximately most of the time we begin with the premise that many problems in robustness analysis and synthesis can be formulated as the minimization of an objective function with respect to the controller parameters it is argued that in order to assess the performance of a controller as the plant varies over a prespecified family it is better to use the average performance of the controller as the objective function to be minimized rather than its worst case performance as the worst case objective function usually leads to rather conservative designs then it is shown that a property from statistical learning theory known as uniform convergence of empirical means ucem plays an important role in allowing us to construct efficient randomized algorithms for a wide variety of controller synthesis problems in particular whenever the ucem property holds there exists an efficient i e polynomial time randomized algorithm using very recent results in statistical learning theory it is shown that the ucem property holds in any problem in which the satisfaction of a performance constraint can be expressed in terms of a finite number of polynomial inequalities in particular several problems such as robust stabilization and weighted h 2 h norm minimization are amenable to the randomized approach

on the sample size of random convex programs with structured dependence on the uncertainty the scenario approach provides an intuitive method to address chance constrained problems arising in control design for uncertain systems it addresses these problems by replacing the chance constraint with a finite number of sampled constraints scenarios the sample size critically depends on helly s dimension a quantity always upper bounded by the number of decision variables however this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability we derive improved bounds of helly s dimension for problems where the chance constraint has certain structural properties the improved bounds lower the number of scenarios required for these problems leading both to improved objective value and reduced computational complexity our results are generally applicable to randomized model predictive control of chance constrained linear systems with additive uncertainty and affine disturbance feedback

randomized receding horizon navigation the note combines weak control lyapunov function based nonlinear receding horizon control with randomized optimization this approach is applied to the problem of robot navigation in the presence of state and input constraints it is shown that under certain conditions relaxing the definiteness requirements on the terminal cost function allows one to select control inputs through a monte carlo optimization scheme in a way that preserves the stability and convergence properties of the closed loop system while the particular randomized optimization scheme used here can be substituted for the nonlinear optimal control method of choice the introduction of randomization in receding horizon optimization is anticipated to offer additional trade offs between performance and computation speed compared to the fixed overhead nonlinear optimal control strategies typically employed

randomized algorithms for probabilistic robustness with real and complex structured uncertainty there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems unlike classical worst case methods these algorithms provide probabilistic estimates assessing for instance if a certain design specification is met with a given probability one of the advantages of this approach is that the robustness margins can be often increased by a considerable amount at the expense of a small risk in this sense randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information the applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres sampling in more general uncertainty sets is generally performed through overbounding at the expense of an exponential rejection rate in the paper randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general m spl delta configuration are studied in particular efficient polynomial time algorithms for uncertainty structures spl delta consisting of an arbitrary number of full complex blocks and uncertain parameters are developed

probabilistic design of lpv control systems this paper presents an alternative approach to design of linear parameter varying lpv control systems in contrast to previous methods which are focused on deterministic algorithms this paper is based on a probabilistic setting the proposed randomized algorithm provides a sequence of candidate solutions converging with probability one to a feasible solution in a finite number of steps the main features of this approach are as follows i the randomized algorithm gives a method for general lpv plants with state space matrices depending on scheduling parameters in a nonlinear manner that is the probabilistic setting does not need a gridding of the set of scheduling parameters or approximations such as a linear fractional transformation of the state space matrices ii the proposed algorithm is sequential and at each iteration it does not require heavy computational effort such as solving simultaneously a large number of linear matrix inequalities

soft decision demodulation and transform coding of images this paper describes a transform image coding system that uses soft decision demodulation to control channel errors in soft decision demodulation if certain received bits of a codeword representing a coefficient are unreliable then the codeword is rejected and the corresponding coefficient is replaced with an estimate by monitoring the three highest energy dct coefficients the reconstructed image quality can be improved for a channel with a bit error probability of 10 2

vapnik chervonenkis bounds for generalization the authors review the vapnik and chervonenkis theorem as applied to the problem of generalization by combining some of the technical modifications proposed in the literature they derive tighter bounds and a new version of the theorem bounding the accuracy in the estimation of generalization probabilities from finite samples a critical discussion and comparison with the results from statistical mechanics is given

some integrable hierarchies in 2 1 dimensions and their twistor description the twistor space construction due to mason and sparling for solutions of the nonlinear schrodinger and korteweg devries hierarchies is generalized resulting in families of integrable models in 2 1 dimensions soliton solutions to one such hierarchy associated with the gauge group su 2 are constructed by adapting the methods used to construct instanton and monopole solutions these integrable models have the feature that their solutions depend on a number of arbitrary functions as well as arbitrary constants in contrast to the closely related davey stewartson hierachy

stochastic optimization on continuous domains with finite time guarantees by markov chain monte carlo methods we introduce bounds on the finite time performance of markov chain monte carlo mcmc algorithms in solving global stochastic optimization problems defined over continuous domains it is shown that mcmc algorithms with finite time guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm this work is inspired by the concept of finite time learning with known accuracy and confidence developed in statistical learning theory

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

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

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

near optimal solutions to least squares problems with stochastic uncertainty in this chapter we consider least squares problems where the regression data is affected by stochastic uncertainty in this setting we study the problem of minimizing the expected value with respect to the uncertainty of the least squares residual for general nonlinear dependence of the data on the uncertain parameters determining an exact solution to this problem is known to be computationally prohibitive here we follow a probabilistic approach and determine a probable near optimal solution by minimizing the empirical mean of the residual finite sample convergence of the proposed method is assessed using statistical learning methods in particular we prove that if one constructs the empirical approximation of the mean using a finite number n of samples then the minimizer of this empirical approximation is with high probability an suboptimal solution for the original problem moreover this approximate solution can be efficiently determined numerically by a standard recursive algorithm comparisons with gradient algorithms for stochastic optimization are also discussed in this contribution and some numerical examples illustrate the proposed methodology

brief probabilistic solutions to some np hard matrix problems during recent years it has been shown that a number of problems in matrix theory are np hard including robust nonsingularity robust stability robust positive semidefiniteness robust bounded norm state feedback stabilization with structural and norm constraints etc in this paper we use standard bounds on empirical probabilities as well as recent results from statistical learning theory on the vc dimension of families of sets defined by a finite number of polynomial inequalities to show that for each of the above problems as well as for still more general and more difficult problems there exists a polynomial time randomized algorithm that can provide a yes or no answer to arbitrarily small levels of accuracy and confidence

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

a statistical learning theory approach for uncertain linear and bilinear matrix inequalities abstract in this paper we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities which depend in a possibly nonlinear way on a vector of uncertain parameters motivated by recent results in statistical learning theory we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms in particular we show that the vapnik chervonenkis dimension vc dimension of the two problems is finite and we compute upper bounds on it in turn these bounds allow us to derive explicitly the sample complexity of these problems using these bounds in the second part of the paper we derive a sequential scheme based on a sequence of optimization and validation steps the algorithm is on the same lines of recent schemes proposed for similar problems but improves both in terms of complexity and generality the effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters

probabilistic robustness analysis explicit bounds for the minimum number of samples in this paper we study robustness analysis of control systems affected by bounded uncertainty motivated by the difficulty to perform this analysis when the uncertainty enters into the plant coefficients in a nonlinear fashion we study a probabilistic approach in this setting the uncertain parameters q are random variables bounded in a set q and described by a multivariate density function f q we then pose the following question given a performance level what is the probability that this level is attained the main content of this paper is to derive explicit bounds for the number of samples required to estimate this probability with a certain accuracy and confidence a priori specified it is shown that the number obtained is inversely proportional to these thresholds and it is much smaller than that of classical results finally we remark that the same approach can be followed to study several problems in a control system context for example we can evaluate the worst case h norm of the sensitivity function of multi input multi output systems or compute when the robustness margin is of interest

design of near optimal decision rules in multistage adaptive mixed integer optimization in recent years decision rules have been established as the preferred solution method for addressing computationally demanding multistage adaptive optimization problems despite their success existing decision rules a are typically constrained by their a priori design and b do not incorporate in their modeling adaptive binary decisions to address these problems we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions respectively we then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed integer optimization we demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems we provide global lower bounds and show that our approach is i practically tractable and ii provides high quality solutions that outperform alternative methods

distributions of the two dimensional dct coefficients for images for a two dimensional discrete cosine transform dct image coding system there have been different assumptions concerning the distributions of the transform coefficients this paper presents results of distribution tests that indicate that for many images the statistics of the coefficients are best approximated by a gaussian distribution for the dc coefficient and a laplacian distribution for the other coefficients furthermore from a simulation of the dct coding system it is shown that the assumption that the coefficients are laplacian yields a higher actual output signal to noise ratio and a much better agreement between theory and simulation than the gaussian assumption

robust h control design for best mean performance over an uncertain parameters box abstract this paper deals with robust polytopic probabilistic h analysis and state feedback synthesis of linear systems and focuses on the performance distribution over the uncertainty region rather than on the performance bound the proposed approach allows different disturbance attenuation levels dals at the vertices of the uncertainty polytope it is shown that the mean disturbance attenuation level dal over an uncertain parameters box is the average of the dals at the vertices if each parameter has an independent symmetrical centered probability density function in such a most common case the mean dal over the uncertain parameters box can be optimized by minimizing the sum of the dals at the vertices the standard deviation of the dal over the uncertain parameters box is also addressed and a method to minimize this standard deviation is shown a new robust h state feedback synthesis theorem is given it is based on a recent most efficient analysis method and applies the proposed multiple vertex dals approach a state feedback design example utilizing the latter theorem shows that a control design which minimizes the sum of the vertex dals leads to a better actual closed loop performance than a similar design which minimizes only the bound of the dal over the uncertainty polytope the comparison is based on the statistics of a population of closed loop point wise h norms created by a monte carlo mechanism

brief some applications of randomized algorithms for control system design in this paper a few difficult problems related to simultaneous stabilization of three plants equivalent to a certain problem related to unit interpolation in h have been addressed through the framework of randomized algorithms these problems which were proposed by blondel simultaneous stabilization of linear systems springer berlin 1994 and blondel and gevers math control signals systems 6 1994 135 concern the existence of a controller

a differential difference system related to toroidal lie algebra we present a novel differential difference system in 2 1 dimensional space time one discrete two continuum arising from bogoyavlensky s 2 1 dimensional korteweg de vries hierarchy our method is based on the bilinear identity of the hierarchy which is related to the vertex operator representation of the toroidal lie algebra tor2

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

convexity and convex approximations of discrete time stochastic control problems with constraints we investigate constrained optimal control problems for linear stochastic dynamical systems evolving in discrete time we consider minimization of an expected value cost subject to probabilistic constraints we study the convexity of a finite horizon optimization problem in the case where the control policies are affine functions of the disturbance input we propose an expectation based method for the convex approximation of probabilistic constraints with polytopic constraint function and a linear matrix inequality lmi method for the convex approximation of probabilistic constraints with ellipsoidal constraint function finally we introduce a class of convex expectation type constraints that provide tractable approximations of the so called integrated chance constraints performance of these methods and of existing convex approximation methods for probabilistic constraints is compared on a numerical example

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

np hardness of some linear control design problems we show that some basic linear control design problems are np hard implying that unless p np they cannot be solved by polynomial time algorithms the problems that we consider include simultaneous stabilization by output feedback stabilization by state or output feedback in the presence of bounds on the elements of the gain matrix and decentralized control these results are obtained by first showing that checking the existence of a stable matrix in an interval family of matrices is np hard

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

a probabilistic framework for problems with real structured uncertainty in systems and control the objective of this paper is twofold first the problem of generation of real random matrix samples with uniform distribution in structured spectral norm bounded sets is studied this includes an analysis of the distribution of the singular values of uniformly distributed real matrices and an efficient i e polynomial time algorithm for their generation second it is shown how the developed techniques may be used to solve in a probabilistic setting several hard problems involving systems subject to real structured uncertainty

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

hamming coding of dct compressed images over noisy channels theoretical and simulation results of using hamming codes with the two dimensional discrete cosine transform 2d dct at a transmitted data rate of 1 bit pixel over a binary symmetric channel bsc are presented the design bit error rate ber of interest is 10 2 the 7 4 15 11 and 31 26 hamming codes are used to protect the most important bits in each 16 by 16 transformed block where the most important bits are determined by calculating the mean squared reconstruction error mse contributed by a channel error in each individual bit a theoretical expression is given which allows the number of protected bits to achieve minimum mse for each code rate to be computed by comparing these minima the best code and bit allocation can be found objective and subjective performance results indicate that using the 7 4 hamming code to protect the most important 2d dct coefficients can substantially improve reconstructed image quality at a ber of 10 2 furthermore the allocation of 33 out of the 256 bits per block to channel coding does not noticeably degrade reconstructed image quality in the absence of channel errors

convex dynamic programming for hybrid systems a classical linear programming approach to optimization of flow or transportation in a discrete graph is extended to hybrid systems the problem is finite dimensional if the state space is discrete and finite but becomes infinite dimensional for a continuous or hybrid state space it is shown how strict lower bounds on the optimal loss function can be computed by gridding the continuous state space and restricting the linear program to a finite dimensional subspace upper bounds can be obtained by evaluation of the corresponding control laws

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

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

reachability and controllability of switched linear discrete time systems this paper investigates the reachability and controllability issues for switched linear discrete time systems geometric characterization of controllability is presented for reversible systems the controllable sets and the reachable sets are identified in wonham s geometric approach and verifiable conditions for reachability and controllability are also presented

stochastic robustness of linear time invariant control systems a simple numerical procedure for estimating the stochastic robustness of a linear time invariant system is described monte carlo evaluation of the system s eigenvalues allows the probability of instability and the related stochastic root locus to be estimated this analysis approach treats not only gaussian parameter uncertainties but also nongaussian cases including uncertain but bound variations confidence intervals for the scalar probability of instability address computational issues inherent in monte carlo simulation trivial extensions of the procedure admit consideration of alternate discriminants thus the probabilities that stipulated degrees of instability will be exceeded or that closed loop roots will leave desirable regions can also be estimated results are particularly amenable to graphical presentation

existence and uniqueness of solutions of piecewise nonlinear systems abstract in this paper we consider the existence and the uniqueness of solutions of piecewise nonlinear systems we will present some necessary and sufficient conditions for the existence and the uniqueness of solutions of this class of systems

randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems in this paper we study two general semi infinite programming problems by means of a randomized strategy based on statistical learning theory the sample size results obtained with this approach are generally considered to be very conservative by the control community the first main contribution of this paper is to demonstrate that this is not necessarily the case utilizing as a starting point one sided results from statistical learning theory we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy in particular we show that the number of required samples grows with the accuracy parameter epsiv as 1 epsivln 1 epsiv and this is a significant improvement when compared to the existing bounds which depend on 1 epsiv2ln 1 epsiv2 secondly we present new results for optimization and feasibility problems involving boolean expressions consisting of polynomials in this case when the accuracy parameter is sufficiently small an explicit bound that only depends on the number of decision variables and on the confidence and accuracy parameters is presented for convex optimization problems we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence thirdly we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived

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

polynomial time probabilistic observability analysis of sampled data piecewise affine systems this paper proposes a polynomial time probabilistic approach to solve the observability problem of sampled data piecewise affine systems first an algebraic characterization for the system to be observable is derived next based on the characterization we propose a randomized algorithm that can determine if the system is observable in a probabilistic sense or the system is not observable in a deterministic sense finally it is shown with some examples for which it is hopeless to check the observability in a deterministic way that the proposed algorithm is very useful

an extension of the korteweg de vries hierarchy arising from a representation of a toroidal lie algebra abstract in this article we show how to construct hierarchies of partial differential equations from the vertex operator representations of toroidal lie algebras in the smallest example rank 2 toroidal cover of sl2 we obtain an extension of the kdv hierarchy we use the action of the corresponding infinite dimensional group to construct solutions for these non linear pdes

controllability analysis of biosystems based on piecewise affine systems approach this paper discusses the controllability problem of biosystems based on the piecewise affine system model representation first we consider what kind of controllability problems are useful for analyzing and controlling biosystems and then the controllable set problem that is a problem of finding a state set in which each state can be driven to a given target state is formulated it is shown that this problem will be very useful for many kinds of problems on control of biosystems such as the input allocation problem and the stabilization problem next based on our previous probabilistic controllability analysis technique for hybrid dynamical systems a more sophisticated method for solving the above complex problem in an approximated and suitable way for biosystem analysis is proposed finally the proposed framework is applied to the quorum sensing system of the pathogen pseudomonas aeruginosa for explaining how it is formulated and what solutions are obtained

computational complexity of spl mu calculation the structured singular value spl mu measures the robustness of uncertain systems numerous researchers over the last decade have worked on developing efficient methods for computing spl mu this paper considers the complexity of calculating spl mu with general mixed real complex uncertainty in the framework of combinatorial complexity theory in particular it is proved that the spl mu recognition problem with either pure real or mixed real complex uncertainty is np hard this strongly suggests that it is futile to pursue exact methods for calculating spl mu of general systems with pure real or mixed uncertainty for other than small problems

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

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

a primal dual probabilistic setting for quadratic stability of uncertain systems abstract this paper introduces a theoretical framework for the development of randomized algorithms for quadratic stability of uncertain systems affected by structured and unstructured uncertainty we propose a general duality setting which deals with a probabilistic version of the classical primal dual problems in particular the probabilistic primal problem is defined so that a linear matrix inequality has a solution with probability one and the probabilistic dual problem is formulated as a solvability problem for a certain matrix integral inequality on the class of positive semidefinite symmetric matrix valued functions the main result of the paper establishes a rigorous equivalence between infeasibility of the primal problem and existence of a solution of the dual problem in the second part of the paper we study various applications of this result in the context of uncertain systems

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

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

binary lattice vector quantization with linear block codes and affine index assignments we determine analytic expressions for the performance of some low complexity combined source channel coding systems the main tool used is the hadamard transform in particular we obtain formulas for the average distortion of binary lattice vector quantization with affine index assignments linear block channel coding and a binary symmetric channel the distortion formulas are specialized to nonredundant channel codes for a binary symmetric channel and then extended to affine index assignments on a binary asymmetric channel various structured index assignments are compared our analytic formulas provide a computationally efficient method for determining the performance of various coding schemes one interesting result shown is that for a uniform source and uniform quantizer the natural binary code is never optimal for a nonsymmetric channel even though it is known to be optimal for a symmetric channel

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

randomized methods for design of uncertain systems in this paper we study randomized methods for feedback design of uncertain systems the first contribution is to derive the sample complexity of various constrained control problems in particular we show the key role played by the binomial distribution and related tail inequalities and compute the sample complexity this contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm these results are then applied to the analysis of worst case performance and design with robust optimization the second contribution of the paper is to introduce a general class of sequential algorithms denoted as sequential probabilistic validation spv in these sequential algorithms at each iteration a candidate solution is probabilistically validated and corrected if necessary to meet the required specifications the results we derive provide the sample complexity which guarantees that the solutions obtained with spv algorithms meet some pre specified probabilistic accuracy and confidence the performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

n soliton solutions to the bogoyavlenskii schiff equation and a quest for the soliton solution in 3 1 dimensions we study the integrable systems in higher dimensions which can be written by the trilinear form instead of by the hirota s bilinear form we explicitly discuss the bogoyavlenskii schiff equation in 2 1 dimensions its analytical proof of multisoliton solution and a new feature are given being guided by the strong symmetry we also propose a new equation in 3 1 dimensions

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

on control system design using random samples of contractive block toeplitz matrices contractive lower triangular block toeplitz ltbt matrices have recently been parametrized in a closed form by a sequence of unstructured contractive matrices a problem naturally arises between uniform samplings over contractive ltbt matrices and their matrix parameters that is which one is more appropriate in control system analysis and synthesis this paper makes it clear that uniform sampling over matrix parameters results in a contractive ltbt matrix with a more rapid concentration speed towards the boundary of the uncertainty set this is not an attractive property in control related problems numerical comparisons also show that uniform sampling directly over contractive ltbt matrices outperforms that over the matrix parameter set in robustness analysis for a closed loop system as well as in strongly stable closed loop system design with complexity constraints on the controller as a sacrifice uniform sampling over contractive ltbt matrices is more mathematically complicated and more time cons

probability guaranteed robust h performance analysis and state feedback design abstract this paper addresses the common engineering practice of specifying a required probability of attaining some performance level the problem setup is that of a robust h performance analysis state feedback synthesis of an affinely parameter dependent linear system except that the parameter hyper rectangle box is allowed to shrink representing a probability less than one in order to accommodate a polytopic performance goal that is better than the one attainable for the original parameter box a new version of the bounded real lemma brl which assigns a different lyapunov function to each of the vertices of the uncertainty polytope and includes a scalar free parameter seems to reduce the overdesign associated with the polytopic problem to the bare minimum the shrinking of the parameter box leads to bi linear matrix inequalities blmis since the final vertices are also unknown these blmis are solved iteratively three steps have sufficed both in the analysis and in the state feedback design examples a uniform distribution is assumed for all the system parameters following the uniformity principle the probability requirement is expressed by a set of linear matrix inequalities lmis that is derived by extending an existing second order cone method these lmis are concurrently solved with the blmis of the brl the features of the proposed method are demonstrated via two examples

controllability and observability of hybrid systems abstract a special class of hybrid systems is considered whose state space is a cross product space of a euclidean space and a finite state space such models may be used to represent systems subject to known abrupt parameter variations such as commu tated networks they may also be used to approximate some types of time varying systems controllability observability and stability of hybrid systems are inves tigated in particular a necessary and sufficient algebraic condition a simple algebraic criterion and a computationally simple algebraic sufficient test for controllability and observability are derived moreover a simple sufficient stability condition is provided

probabilistic robust design with linear quadratic regulators in this paper we study robust design of uncertain systems in a probabilistic setting by means of linear quadratic regulators lqr we consider systems affected by random bounded nonlinear uncertainty so that classical optimization methods based on linear matrix inequalities cannot be used without conservatism the approach followed here is a blend of randomization techniques for the uncertainty together with convex optimization for the controller parameters in particular we propose an iterative algorithm for designing a controller which is based upon subgradient iterations at each step of the sequence we first generate a random sample and then we perform a subgradient step for a convex constraint defined by the lqr problem the main result of the paper is to prove that this iterative algorithm provides a controller which quadratically stabilizes the uncertain system with probability one in a finite number of steps in addition at a fixed step we compute a lower bound of the probability that a quadratically stabilizing controller is found