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

order statistics and probabilistic robust control abstract order statistics theory is applied in this paper to probabilistic robust control theory to compute the minimum sample size needed to come up with a reliable estimate of an uncertain quantity under continuity assumption of the related probability distribution also the concept of distribution free tolerance intervals is applied to estimate the range of an uncertain quantity and extract the information about its distribution to overcome the limitations imposed by the continuity assumption in the existing order statistics theory we have derived a cumulative distribution function of the order statistics without the continuity assumption and developed an inequality showing that this distribution has an upper bound which equals to the corresponding distribution when the continuity assumption is satisfied by applying this inequality we investigate the minimum computational effort needed to come up with an reliable estimate for the upper bound or the lower bound and the range of a quantity we also give conditions which are much weaker than the absolute continuity assumption for the existence of such minimum sample size furthermore the issue of making tradeoff between performance level and risk is addressed and a guideline for making this kind of tradeoff is established this guideline can be applied in general without continuity assumption

stochastic output feedback control convex lifting approach abstract this paper presents a method to design output feedback control for discrete time linear systems affected by bounded additive state output disturbances and subject to chance constraints on the state and hard constraints on the control input this method relies on a so called convex lifting which is a nonnegative convex piecewise affine function equal to 0 over a given stochastic positively invariant set and strictly positive outside this set accordingly it is shown that this function is strictly decreasing along the closed loop dynamics outside this invariant set and convergent to 0 as time tends to infinity consequently the state is convergent to the given invariant set while the method only requires solving a linear program at each sampling instant

brief paper polynomial time algorithms for probabilistic solutions of parameter dependent linear matrix inequalities a randomized approach is considered for a feasibility problem on a parameter dependent linear matrix inequality lmi in particular a gradient based and an ellipsoid based randomized algorithms are improved by introduction of a stopping rule the improved algorithms stop after a bounded number of iterations and this bound is of polynomial order in the problem size when the algorithms stop either of the following two events occurs i they find with high confidence a probabilistic solution which satisfies the given lmi for most of the parameter values ii they detect in an approximate sense the non existence of a deterministic solution which satisfies the given lmi for all the parameter values these results are important because the original randomized algorithms have issues to be settled on detection of convergence on the speed of convergence and on the assumption of feasibility the improved algorithms can be adapted for an optimization problem constrained by a parameter dependent lmi a numerical example shows the efficacy of the proposed algorithms

a probabilistic analytic center cutting plane method for feasibility of uncertain lmis many robust control problems can be formulated in abstract form as convex feasibility programs where one seeks a solution x that satisfies a set of inequalities of the form f f x d 0 d d this set typically contains an infinite and uncountable number of inequalities and it has been proved that the related robust feasibility problem is numerically hard to solve in general in this paper we discuss a family of cutting plane methods that solve efficiently a probabilistically relaxed version of the problem specifically under suitable hypotheses we show that an analytic center cutting plane scheme based on a probabilistic oracle returns in a finite and pre specified number of iterations a solution x which is feasible for most of the members of f except possibly for a subset having arbitrarily small probability measure

a randomized cutting plane method with probabilistic geometric convergence we propose a randomized method for general convex optimization problems namely the minimization of a linear function over a convex body the idea is to generate n random points inside the body choose the best one and cut the part of the body defined by the linear constraint we analyze the convergence properties of the algorithm from a theoretical viewpoint i e under the rather standard assumption that an algorithm for uniform generation of random points in the convex body is available under this assumption the expected rate of convergence for such method is proved to be geometric this analysis is based on new results on the statistical properties of the empirical minimum over a convex body that we obtained in this paper moreover explicit sample size results on convergence are derived in particular we compute the minimum number of random points that should be generated at each step in order to guarantee that in a probabilistic sense the method performs better than the deterministic center of gravity algorithm from a practical viewpoint we show how the method can be implemented using hit and run versions of markov chain monte carlo algorithms and exemplify the performance of this implementable modification via a number of illustrative problems a crucial notion for the hit and run implementation is that of boundary oracle which is available for most optimization problems including linear matrix inequalities and many other kinds of constraints preliminary numerical results for semidefinite programs are presented confirming that the randomized approach might be competitive to modern deterministic convex optimization methods

notes on the scenario design approach the scenario optimization method developed in is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimization problems arising in systems and control design that would otherwise be hard to tackle via standard deterministic techniques in this note we explore some further aspects of the scenario methodology and present two results pertaining to the tightness of the sample complexity bounds we also state a new theorem that enables the user to make a priori probabilistic claims on the scenario solution with one level of probability only

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

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

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

an ellipsoid algorithm for probabilistic robust controller design abstract in this paper a new iterative approach to probabilistic robust controller design is presented which is applicable to any robust controller filter design problem that can be represented as an lmi feasibility problem recently a probabilistic subgradient iteration algorithm was proposed for solving lmis it transforms the initial feasibility problem to an equivalent convex optimization problem which is subsequently solved by means of an iterative algorithm while this algorithm always converges to a feasible solution in a finite number of iterations it requires that the radius of a non empty ball contained into the solution set is known a priori this rather restrictive assumption is released in this paper while retaining the convergence property given an initial ellipsoid that contains the solution set the approach proposed here iteratively generates a sequence of ellipsoids with decreasing volumes all containing the solution set at each iteration a random uncertainty sample is generated with a specified probability density which parameterizes an lmi for this lmi the next minimum volume ellipsoid that contains the solution set is computed an upper bound on the maximum number of possible correction steps that can be performed by the algorithm before finding a feasible solution is derived a method for finding an initial ellipsoid containing the solution set which is necessary for initialization of the optimization is also given the proposed approach is illustrated on a real life diesel actuator benchmark model with real parametric uncertainty for which a h 2 robust state feedback controller is designed

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

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

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

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

stochastic algorithms for exact and approximate feasibility of robust lmis in this note we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities lmis of the form f x spl delta spl les 0 where x is the optimization variable and spl delta is the uncertainty which belongs to a given set spl delta the proposed algorithms are based on uncertainty randomization the first algorithm finds a robust solution in a finite number of iterations with probability one if a strong feasibility condition holds in case no robust solution exists the second algorithm computes an approximate solution which minimizes the expected value of a suitably selected feasibility indicator function the theory is illustrated by examples of application to uncertain linear inequalities and quadratic stability of interval matrices

brief recursive algorithms for inner ellipsoidal approximation of convex polytopes in this paper fast recursive algorithms for the approximation of an n dimensional convex polytope by means of an inscribed ellipsoid are presented these algorithms consider at each step a single inequality describing the polytope and under mild assumptions they are guaranteed to converge in a finite number of steps for their recursive nature the proposed algorithms are better suited to treat a quite large number of constraints than standard off line solutions and have their natural application to problems where the set of constraints is iteratively updated as on line estimation problems nonlinear convex optimization procedures and set membership identification

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

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

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

fast fast algorithm for the scenario technique the scenario approach is a recently introduced method to obtain feasible solutions to chance constrained optimization problems based on random sampling it has been noted that the sample complexity of the scenario approach rapidly increases with the number of optimization variables and this may pose a hurdle to its applicability to medium and large scale problems we here introduce the fast algorithm for the scenario technique a variant of the scenario optimization algorithm with reduced sample complexity subject classifications stochastic programming chance constrained optimization randomized algorithms sample based methods scenario approach area of review optimization

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

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

hit and run mixes fast it is shown that the hit and run algorithm for sampling from a convex body k introduced by r l smith mixes in time o n2r2 r2 where r and r are the radii of the inscribed and circumscribed balls of k thus after appropriate preprocessing hit and run produces an approximately uniformly distributed sample point in time o n3 which matches the best known bound for other sampling algorithms we show that the bound is best possible in terms of r r and n

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

repetitive scenario design repetitive scenario design rsd is a randomized approach to robust design based on iterating two phases a standard scenario design phase that uses n scenarios design samples followed by randomized feasibility phase that uses n o test samples on the scenario solution we give a full and exact probabilistic characterization of the number of iterations required by the rsd approach for returning a solution as a function of n n o and of the desired levels of probabilistic robustness in the solution this novel approach broadens the applicability of the scenario technology since the user is now presented with a clear tradeoff between the number n of design samples and the ensuing expected number of repetitions required by the rsd algorithm the plain one shot scenario design becomes just one of the possibilities sitting at one extreme of the tradeoff curve in which one insists in finding a solution in a single repetition this comes at the cost of possibly high n other possibilities along the tradeoff curve use lower n values but possibly require more than one repetition

randomized algorithms for synthesis of switching rules for multimodal systems in this paper we consider the design of globally asymptotically stabilizing state dependent switching rules for multimodal systems first restricting attention to linear time invariant lti systems with only two states for the switch and then generalizing the results to multimodal lti systems and to nonlinear systems in all cases the systems considered do not allow the construction of a single quadratic lyapunov function and hence fall in the class of problems that require multiple lyapunov functions and thus are nonconvex to address the challenge of nonconvexity we introduce probabilistic algorithms and prove their probability one convergence under a new notion of convergence then to reduce complexity we develop modified versions of the algorithm we also present a class of more general nonconvex problems to which this approach can be applied the results are illustrated using two and three dimensional systems with multiple switch states

technical communique policy set iteration for markov decision processes this communique presents an algorithm called policy set iteration psi for solving infinite horizon discounted markov decision processes with finite state and action spaces as a simple generalization of policy iteration pi psi generates a monotonically improving sequence of stationary markovian policies p k based on a set manipulation as opposed to pi s single policy manipulation at each iteration k when the set involved with psi at k contains n independently generated sample policies from a given distribution d the probability that the expected value of any sampled policy from d with respect to an initial state distribution is greater than that of p k converges to zero with o n k rate moreover psi converges to an optimal policy no slower than pi in terms of the number of iterations for any d

uniform sample generations from contractive block toeplitz matrices this note deals with generating a series of random matrices uniformly and independently from contractive lower triangular block toeplitz ltbt matrices all the contractive ltbt matrices are parameterized nonlinearly but recursively in a closed form by a sequence of contractive unstructured matrices moreover the conditional probability density function pdf is obtained in a closed form for all the matrix parameters as well as the squares of their singular values it has been made clear that the conditional pdfs of the matrix parameters are unitarily invariant a procedure is developed to accurately produce the required random matrix samples

probabilistic enhancement of classical robustness margins the unirectangularity concept abstract the focal point of this paper is a control system subjected to parametric uncertainty motivated by the newly emerging theory of probabilistic robustness the risk of performance violation is assessed with uncertainty bounds which exceed classical deterministic margins for a wide class of problems the uniformity principle up developed by barmish and lagoa math control signals systems 10 1997 203 222 makes it possible to estimate the probability of performance satisfaction with almost no a priori statistical information about the uncertainty the application of the up is however limited to problems satisfying certain convexity and symmetricity conditions since such conditions are violated in many practical problems the objective in this paper is to extend the application of the up to this end by working with a so called unirectangularity condition a procedure is implemented for computing probabilities of performance and the associated improvements of deterministic robustness margins that is given any robustness radius r 0 which is computable via deterministic methods a probabilistic enhancement of this margin r 0 e r 0 with pre specified level of risk e 0 is provided the radius r 0 e is called a risk adjusted robustness margin

an algorithm for sampling subsets of h sub spl infin with applications to risk adjusted performance analysis and model in validation in spite of their potential to reduce computational complexity the use of probabilistic methods in robust control has been mostly limited to parametric uncertainty since the problem of sampling causal bounded operators is largely open in this note we take steps toward removing this limitation by proposing a computationally efficient algorithm aimed at uniformly sampling suitably chosen subsets of h sub spl infin as we show in the note samples taken from these sets can be used to carry out model in validation and robust performance analysis in the presence of structured dynamic linear time invariant uncertainty problems known to be np hard in the number of uncertainty blocks

probabilistic optimal estimation with uniformly distributed noise the classical approach to system identification is based on stochastic assumptions about the measurement error and provides estimates that have random nature worst case identification on the other hand only assumes the knowledge of deterministic error bounds and establishes guaranteed estimates thus being in principle better suited for the use in control design however a main limitation of such deterministic bounds lies in their potential conservatism thus leading to estimates of restricted use in this paper we propose a rapprochement between the stochastic and worst case system identification viewpoints which is based on the probabilistic setting of information based complexity the main idea in this line of research is to discard sets of measure at most where is a probabilistic accuracy from the set of deterministic estimates therefore we are decreasing the so called worst case radius of information at the expense of a given probabilistic risk in the case of uniformly distributed noise we derive new computational results and in particular we compute a trade off curve called violation function which shows how the radius of information decreases as a function of the accuracy to this end we construct randomized and deterministic algorithms which provide approximations of this function we report extensive simulations showing numerical comparisons between the stochastic worst case and probabilistic approaches thus demonstrating the efficacy of the methods proposed in this paper

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

statistical learning control of uncertain systems theory and algorithms it has recently become clear that many control problems are too difficult to admit analytic solutions new results have also emerged to show that the computational complexity of some solved control problems is prohibitive many of these control problems can be reduced to decidability problems or to optimization questions even though such questions may be too difficult to answer analytically or may not be answered exactly given a reasonable amount of computational resources researchers have shown that we can approximately answer these questions most of the time and have high confidence in the correctness of the answers

brief paper guaranteed cost regulator design a probabilistic solution and a randomized algorithm this paper presents a gradient based randomized algorithm to design a guaranteed cost regulator for a plant with general parametric uncertainties the algorithm either provides with high confidence a probabilistic solution that satisfies the design specification with high probability for a randomly sampled uncertainty or claims that the feasible set of the design parameters is too small to contain a ball with a given radius in both cases the number of iterations executed in the algorithm is of polynomial order of the problem size and is independent of the dimension of the uncertainty

the exact feasibility of randomized solutions of uncertain convex programs many optimization problems are naturally delivered in an uncertain framework and one would like to exercise prudence against the uncertainty elements present in the problem in previous contributions it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization in this paper we establish new feasibility results for randomized algorithms specifically the exact feasibility for the class of the so called fully supported problems is obtained it turns out that all fully supported problems share the same feasibility properties revealing a deep kinship among problems of this class it is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully supported problems the feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully supported problems

brief paper probabilistic sorting and stabilization of switched systems in this paper we consider lyapunov stability of switched linear systems whose switching signal is constrained to a subset of indices we propose a switching rule that chooses the most stable subsystem among those belonging to the subset this rule is based on an ordering of the subsystems using a common lyapunov function we develop randomized algorithms for finding the ordering as well as for finding a subset of systems for which a common lyapunov function exists it is shown that the class of randomized algorithms known as the las vegas type is useful in the design procedure a third order example illustrating the efficacy of the approach is presented

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

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

improved sample complexity estimates for statistical learning control of uncertain systems probabilistic methods and statistical learning theory have been shown to provide approximate solutions to difficult control problems unfortunately the number of samples required in order to guarantee stringent performance levels may be prohibitively large this paper introduces bootstrap learning methods and the concept of stopping times to drastically reduce the bound on the number of samples required to achieve a performance level we then apply these results to obtain more efficient algorithms which probabilistically guarantee stability and robustness levels when designing controllers for uncertain systems

zeta functions for curves and log canonical models the topological zeta function and igusa s local zeta function are respectively a geometrical invariant associated to a complex polynomial f and an arithmetical invariant associated to a polynomial f over a p adic field when f is a polynomial in two variables we prove a formula for both zeta functions in terms of the so called log canonical model of f 1 0 in bbb a 2 this result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions also in the formula for igusa s local zeta function appears a remarkable non symmetric q deformation of the intersection matrix of the minimal resolution of a hirzebruch jung singularity 1991 mathematics subject classification 32s50 11s80 14e30 14g20

on avoiding vertexization of robustness problems the approximate feasibility concept for a large class of robustness problems with uncertain parameter vector q confined to a box q there are many papers providing results along the following lines the desired performance specification is robustly satisfied for all q spl isin q if and only if it is satisfied at each vertex q sup i of q since the number of vertices of q explodes combinatorically with the dimension of q the computation associated with the implementation of such results is often intractable the main point of this paper is to introduce a new approach to such problems aimed at alleviation of this computational complexity problem to this end the notion of approximate feasibility is introduced and the theory which follows from this definition is vertex free

distributed randomized algorithms for probabilistic performance analysis randomized algorithms are a useful tool for analyzing the performance of complex uncertain systems their implementation requires the generation of a large number n of random samples representing the uncertainty scenarios and the corresponding evaluation of system performance when n is very large and or performance evaluation is costly or time consuming it can be necessary to distribute the computational burden of such algorithms among many cooperating computing units this paper studies distributed versions of randomized algorithms for expected value probability and extrema estimation over a network of computing nodes with possibly time varying communication links explicit a priori bounds are provided for the sample and communication complexity of these algorithms in terms of number of local samples number of computing nodes and communication iterations

simulated annealing for convex optimization we apply the method known as simulated annealing to the following problem in convex optimization minimize a linear function over an arbitrary convex set where the convex set is specified only by a membership oracle using distributions from the boltzmann gibbs family leads to an algorithm that needs only o n phases for instances in rn this gives an optimization algorithm that makes o n4 5 calls to the membership oracle in the worst case compared to the previous best guarantee of o n5 the benefits of using annealing here are surprising because such problems have no local minima that are not also global minima hence we conclude that one of the advantages of simulated annealing in addition to avoiding poor local minima is that in these problems it converges faster to the minima that it finds we also give a proof that under certain general conditions the boltzmann gibbs distributions are optimal for annealing on these convex problems

efficient monte carlo procedures for generating points uniformly distributed over bounded regions we consider the monte carlo problem of generating points uniformly distributed within an arbitrary bounded measurable region the class of markovian methods considered generate points asymptotically uniformly distributed within the region computational experience suggests the methods are potentially superior to conventional rejection techniques for large dimensional regions

interval predictor models identification and reliability this paper addresses the problem of constructing reliable interval predictors directly from observed data differently from standard predictor models interval predictors return a prediction interval as opposed to a single prediction value we show that in a stationary and independent observations framework the reliability of the model that is the probability that the future system output falls in the predicted interval is guaranteed a priori by an explicit and non asymptotic formula with no further assumptions on the structure of the unknown mechanism that generates the data this fact stems from a key result derived in this paper which relates at a fundamental level the reliability of the model to its complexity and to the amount of available information number of observed data

reliability estimation for dynamical systems subject to stochastic excitation using subset simulation with splitting a new subset simulation approach is proposed for reliability estimation for dynamical systems subject to stochastic excitation the basic idea of subset simulation is to factor a small failure probability into a product of larger failure probabilities conditional on intermediate failure events the new method proposed in this work does not require markov chain monte carlo simulation in contrast to the original method to estimate the conditional probabilities instead only direct monte carlo simulation is needed the method employs splitting of a trajectory that reaches an intermediate failure level into multiple trajectories subsequent to the corresponding first passage time the new approach still enjoys most of the advantages of the original subset simulation e g it is applicable to general causal dynamical systems and it is robust with respect to the dimension of the uncertain input variables the statistical properties of the failure probability estimators are presented where it is shown that they are unbiased and formulas are derived to assess the error of estimation including the coefficient of variation we also discuss the selection of intermediate failure events and the number of samples for each failure level the resulting algorithm is simple and easy to implement two examples are presented to demonstrate the effectiveness of the new approach and the results are compared with the original subset simulation and with direct monte carlo simulation

probabilistic reachability and safety for controlled discrete time stochastic hybrid systems in this work probabilistic reachability over a finite horizon is investigated for a class of discrete time stochastic hybrid systems with control inputs a suitable embedding of the reachability problem in a stochastic control framework reveals that it is amenable to two complementary interpretations leading to dual algorithms for reachability computations in particular the set of initial conditions providing a certain probabilistic guarantee that the system will keep evolving within a desired safe region of the state space is characterized in terms of a value function and maximally safe markov policies are determined via dynamic programming these results are of interest not only for safety analysis and design but also for solving those regulation and stabilization problems that can be reinterpreted as safety problems the temperature regulation problem presented in the paper as a case study is one such case

contribution of jumping numbers by exceptional divisors abstract we investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension inspired by the results for curves on surfaces by smith and thompson 17 and tucker 18 in particular we construct an example of an exceptional divisor that is not contracted in the log canonical model and does not contribute any jumping numbers

a scenario approach for non convex control design randomized optimization is an established tool for control design with modulated robustness while for uncertain convex programs there exist efficient randomized approaches this is not the case for non convex problems methods based on statistical learning theory are applicable to non convex problems but they usually are conservative in achieving the desired probabilistic guarantees in this paper we derive a novel scenario approach for a wide class of random non convex programs with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program but for all feasible solutions inside a set of a priori chosen complexity we also address measure theoretic issues for uncertain convex and non convex programs among the family of non convex control design problems that can be addressed via randomization we apply our scenario approach to stochastic model predictive control for chance constrained nonlinear control affine systems

mixed deterministic randomized methods for fixed order controller design in this paper we propose a general methodology for designing fixed order controllers for single input single output plants the controller parameters are classified into two classes randomized and deterministically designed for the first class we study randomized algorithms in particular we present two low complexity algorithms based on the chernoff bound and on a related bound often called ldquolog over logrdquo bound which is generally used for optimization problems secondly for the deterministically designed parameters we reformulate the original problem as a set of linear equations then we develop a technique which efficiently solves it using a combination of matrix inversions and sensitivity methods a detailed complexity analysis of this technique is carried on showing its superiority from the computational point of view to existing algorithms based on linear programming in the second part of the paper these results are extended to h infin performance one of the contributions is to prove that the deterministically designed parameters enjoy a special convex characterization this characterization is then exploited in order to design fixed order controllers efficiently we then show further extensions of these methods for stabilization of interval plants in particular we derive a simple one parameter formula for computing the so called critical frequencies which are required by the algorithms

probabilistic enhancement of classical robustness margins a class of nonsymmetric distributions in this note we address the problem of risk assessment when the robustness margin is exceeded without a priori knowledge of the distribution of the uncertainty the only assumption is that the distribution belongs to a given class in contrast to previous work this class contains both symmetric and nonsymmetric distributions we prove that the assessment of risk can be done using only a subset of the admissible distributions also if the set of uncertainties that verify the specifications is convex it is proven that risk assessment can be done using only a finite subset of the class finally a way of estimating risk is provided for the nonconvex case

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

stochastic model predictive control of lpv systems via scenario optimization a stochastic receding horizon control approach for constrained linear parameter varying discrete time systems is proposed in this paper it is assumed that the time varying parameters have stochastic nature and that the system s matrices are bounded but otherwise arbitrary nonlinear functions of these parameters no specific assumption on the statistics of the parameters is required by using a randomization approach a scenario based finite horizon optimal control problem is formulated where only a finite number m of sampled predicted parameter trajectories scenarios are considered this problem is convex and its solution is a priori guaranteed to be probabilistically robust up to a user defined probability level p the p level is linked to m by an analytic relationship which establishes a tradeoff between computational complexity and robustness of the solution then a receding horizon strategy is presented involving the iterated solution of a scenario based finite horizon control problem at each time step our key result is to show that the state trajectories of the controlled system reach a terminal positively invariant set in finite time either deterministically or with probability no smaller than p the features of the approach are illustrated by a numerical example

on incremental approximate saddle point computation in zero sum matrix games we consider the problem of approximately computing saddle point of a zero sum matrix game when either the columns of the matrix are revealed incrementally in time or the matrix is too large to apply traditional methods we leverage the established adaptive multiplicative weights algorithm but introduce a novel simple criterion to determine whether the approximately computed minimizer s best strategy needs to be re computed when a new column of the matrix is introduced our main results are two fold first we show that our proposed incremental approach achieves the same accuracy as applying the adaptive multiplicative weights algorithm on the entire matrix if known a priori second when the columns of the matrix are generated independently and from the same distribution we show that the expected number of times the approximate strategy is re computed grows at most logarithmically with the number of columns of the matrix thereby being computationally efficient

random convex programs random convex programs rcps are convex optimization problems subject to a finite number n of random constraints the optimal objective value j of an rcp is thus a random variable we study the probability with which j is no longer optimal if a further random constraint is added to the problem violation probability v it turns out that this probability rapidly concentrates near zero as n increases we first develop a theory for rcps leading to explicit bounds on the upper tail probability of v then we extend the setup to the case of rcps with r a posteriori violated constraints rcpvs a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control explicit and nonasymptotic bounds are derived also in this case the upper tail probability of v is upper bounded by a multiple of a beta distribution irrespective of the distribution on the random constraints all results are derived under no feasibility assumptions on the problem further the relation between rcpvs and chance constrained problems ccp is explored showing that the optimal objective j of an rcpv with the generic constraint removal rule provides with arbitrarily high probability an upper bound on the optimal objective of a corresponding ccp moreover whenever an optimal constraint removal rule is used in the rcpvs then appropriate choices of n and r exist such that j approximates arbitrarily well the objective of the ccp

nonfragile filtering for discrete time linear systems in finite frequency domain this article investigates the problem of nonfragile filter design for discrete time linear systems subject to noises with known frequency ranges additive interval uncertainty reflecting imprecision in filter implementation is considered by the aid of generalised kyp lemma both deterministic and randomised filtering algorithms are proposed to deal with noises in low middle and high frequency domain respectively the proposed nonfragile finite frequency filters can get a better noise attenuation performance when frequency ranges of noises are known beforehand an example about f 18 aircraft model is given to illustrate the effectiveness of the proposed algorithms

learning noisy functions via interval models this paper considers the problem of identification of an interval model for an unknown static function using a finite batch of stochastic input output data u i y i u i y i i 1 ni 1 n the criterion used for identification is that the width of the interval output of the model should be minimized while containing a given fraction of the observed outputs y i y i we show that for suitable finite nn the resulting model will be reliable that is it will explain any other unseen output up to a given and arbitrary high probability

efficient optimal design of uncertain discrete time dynamical systems in this paper we consider the problem of optimal design of an uncertain discrete time nonlinear dynamical system the problem is formulated using an a posterori design criterion which can account for uncertainties generated by the dynamics of the system itself as well as parametric uncertainties in general for most uncertain complex dynamical systems this type of method is difficult to solve analytically a numerical scheme is developed for the optimal design that involves two steps first in order to obtain a numerical algorithm for the optimal solution we apply randomized algorithms for average performance synthesis to approximate the optimal solution second using the properties of the perron frobenius operator we develop an efficient computation approach to calculate the stationary distribution for the uncertain dynamical systems and the average performance criteria

on the road between robust optimization and the scenario approach for chance constrained optimization problems we propose a new method for solving chance constrained optimization problems that lies between robust optimization and scenario based methods our method does not require prior knowledge of the underlying probability distribution as in robust optimization methods nor is it based entirely on randomization as in the scenario approach it instead involves solving a robust optimization problem with bounded uncertainty where the uncertainty bounds are randomized and are computed using the scenario approach to guarantee that the resulting robust problem is solvable we impose certain assumptions on the dependency of the constraint functions with respect to the uncertainty and show that tractability is ensured for a wide class of systems our results lead immediately to guidelines under which the proposed methodology or the scenario approach is preferable in terms of providing less conservative guarantees or reducing the computational cost

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