witten s lectures on crumpling abstract crumpling is a distortion brought about by a strong compression of a surface in which energy condenses in extremely small regions rather than being stored uniformly scaling laws describe how much energy goes into how little space as crumpling of a sheet proceeds a network of ridges and vertices develops the precise nature of these geometrical singularities depends on the stretching and bending characteristics it is shown that for two dimensional surfaces in three dimensional space energy mainly condenses in stretching ridges the scaling arguments are put on a firm basis by using the mathematical description of an elastic membrane through the von karman equations and the universal properties of ridges are discussed from this theory one can understand that crumpling confers strength finally it is sketched how energy condensation and its scaling laws depend on the dimensions of the confined sheet and embedding space

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

stochastic mpc framework for controlling the average constraint violation this technical note considers linear discrete time systems with additive bounded disturbances subject to hard control input bounds and a stochastic constraint on the amount of state constraint violation averaged over time the amount of violations is quantified by a loss function and the averaging can be weighted corresponding to exponential forgetting of past violations the freedom in the choice of the loss function makes this formulation highly flexible for instance probabilistic constraints or integrated chance constraints can be enforced by an appropriate choice of the loss function for the type of constraint considered we develop a recursively feasible receding horizon control scheme exploiting the averaged over time nature by explicitly taking into account the amount of past constraint violations when determining the current control input this leads to a significant reduction in conservatism as a simple extension of the proposed approach we show how time varying state constraints can be handled within our framework the computational complexity online as well as offline is comparable to existing model predictive control schemes the effectiveness of the proposed methodology is demonstrated by means of a numerical example from building climate control

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

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

a numerically robust state space approach to stable predictive control strategies recent work with predictive control strategies has shown that the stable gpc sgpc approach has significant computational and numerical advantages however sgpc is cast in the transfer function framework which limits its application here we develop a means of extending the results to state space models and show that improved numerical conditioning can be obtained for many stable predictive control strategies

on the computational complexity and generalization properties of multi stage and stage wise coupled scenario programs abstract we discuss the computational complexity and feasibility properties of scenario sampling techniques for uncertain optimization programs we propose an alternative way of dealing with a special class of stage wise coupled programs and compare it with existing methods in the literature in terms of feasibility and computational complexity we identify trade offs between different methods depending on the problem structure and the desired probability of constraint satisfaction to illustrate our results an example from the area of approximate dynamic programming is considered

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

model predictive control of linear systems with multiplicative unbounded uncertainty and chance constraints this paper presents a novel stochastic model predictive control algorithm for linear systems characterized by multiplicative and possibly unbounded model uncertainty probabilistic constraints on the states and inputs are considered and a quadratic cost function is minimized the stochastic control problem and in particular the probabilistic constraints are reformulated in deterministic terms by means of the cantelli inequality so that the on line computational burden of the algorithm is similar to the one of a standard mpc method the properties of the algorithm namely the recursive feasibility and the pointwise convergence of the state are proven by suitably selecting the terminal cost and the constraints on the mean and the variance of the state at the end of the prediction horizon and by considering as additional optimization variables also the mean and the covariance of the state at the beginning of the prediction horizon an extension to deal with the case of expectation rather than probabilistic constraints is reported the numerical issues related to the off line selection of the algorithm s parameters and its on line implementation are discussed simulation results referred to a system with unbounded uncertainty are shown to compare the performances achievable with probabilistic and expectation constraints

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

lyapunov based model predictive control of stochastic nonlinear systems in this work we design a lyapunov based model predictive controller lmpc for nonlinear systems subject to stochastic uncertainty the lmpc design provides an explicitly characterized region from where stability can be probabilistically obtained the key idea is to use stochastic lyapunov based feedback controllers with well characterized stabilization in probability to design constraints in the lmpc that allow the inheritance of the stability properties by the lmpc the application of the proposed lmpc method is illustrated using a nonlinear chemical process system example

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

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

robust model predictive control via scenario optimization this paper discusses a novel probabilistic approach for the design of robust model predictive control mpc laws for discrete time linear systems affected by parametric uncertainty and additive disturbances the proposed technique is based on the iterated solution at each step of a finite horizon optimal control problem fhocp that takes into account a suitable number of randomly extracted scenarios of uncertainty and disturbances followed by a specific command selection rule implemented in a receding horizon fashion the scenario fhocp is always convex also when the uncertain parameters and disturbance belong to nonconvex sets and irrespective of how the model uncertainty influences the system s matrices moreover the computational complexity of the proposed approach does not depend on the uncertainty disturbance dimensions and scales quadratically with the control horizon the main result in this work is related to the analysis of the closed loop system under receding horizon implementation of the scenario fhocp and essentially states that the devised control law guarantees constraint satisfaction at each step with some a priori assigned probability p while the system s state reaches the target set either asymptotically or in finite time with probability at least p the proposed method may be a valid alternative when other existing techniques either deterministic or stochastic are not directly usable due to excessive conservatism or to numerical intractability caused by lack of convexity of the robust or chance constrained optimization problem

nonlinear q design for convex stochastic control in this note we describe a version of the q design method that can be used to design nonlinear dynamic controllers for a discrete time linear time varying plant with convex cost and constraint functions and arbitrary disturbance distribution choosing a basis for the nonlinear q parameter yields a convex stochastic optimization problem which can be solved by standard methods such as sampling in principle for a large enough basis and enough sampling this method can solve the controller design problem to any degree of accuracy in any case it can be used to find a suboptimal controller using convex optimization methods we illustrate the method with a numerical example comparing a nonlinear controller found using our method with the optimal linear controller the certainty equivalent model predictive controller and a lower bound on achievable performance obtained by ignoring the causality constraint

sequential randomized algorithms for robust convex optimization sequential randomized algorithms are considered for robust convex optimization which minimizes a linear objective function subject to a parameter dependent convex constraint employing convex optimization and random sampling of parameter these algorithms enable us to obtain a suboptimal solution within reasonable computational time the suboptimal solution is feasible in a probabilistic sense and the suboptimal value belongs to an interval which contains the optimal value the maximum of the interval is the optimal value of the robust convex optimization plus a specified tolerance on the other hand its minimum is the optimal value of the chance constrained optimization which is a probabilistic relaxation of the robust convex optimization with high probability

stable receding horizon scenario predictive control for markov jump linear systems abstract this paper discusses predictive control for constrained discrete time markov jump linear systems mjls which jump between a finite set of modes according to a markov probabilistic transition observation model minimising an average cost due to the exponential explosion of the number of possible realisations as horizon grows scenario approaches consider only a subset of them prior works cast the problem as a tree based optimisation one but enforce stability and feasibility via artificial lyapunov related constraints the proposed approach avoids this route proposing instead terminal ingredients and tree properties trim contained strictly complete properly generalising the stability feasibility ideas in linear and mjls literature

a constraint sampling approach for multi stage robust optimization we propose a tractable approximation scheme for convex not necessarily linear multi stage robust optimization problems we approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization we obtain a priori probabilistic guarantees on the feasibility properties of the optimal decision rule by applying existing constraint sampling techniques to the semi infinite problem arising from the decision rule approximation we demonstrate that for a suitable choice of basis functions the approximation converges as the size of the basis and the number of sampled constraints tend to infinity the approach yields an algorithm parameterized in the basis size the probability of constraint violation and the confidence that this probability will not be exceeded these three parameters serve to tune the trade off between optimality and feasibility of the decision rules and the computational cost of the algorithm we assess the convergence and scalability properties of our approach in the context of two inventory management problems

stochastic receding horizon control of constrained linear systems with state and control multiplicative noise we develop a receding horizon control approach to stochastic linear systems with control and state multiplicative noise that also contain constraints our receding horizon formulation is based upon an on line optimization that utilizes open loop plus linear feedback and is solved as a semi definite programming problem we also provide a characterization of stability performance and constraint satisfaction properties of the receding horizon controlled system under a specific choice of terminal weight and terminal constraint a simple numerical example is used to illustrate the approach

compensation of gaussian curvature in developable cones is local we use the angular deficit scheme v borrelli f cazals and j m morvan comput aided geom des 20 319 2003 to determine the distribution of gaussian curvature in developable cones d cones e cerda s chaieb f melo and l mahadevan nature london 401 46 1999 numerically these d cones are formed by pushing a thin elastic sheet into a circular container negative gaussian curvatures are identified at the rim where the sheet touches the container around the rim there are two narrow bands with positive gaussian curvatures the integral of the negative gaussian curvature near the rim is almost completely compensated by that of the two adjacent bands this suggests that the gauss bonnet theorem which constrains the integral of gaussian curvature globally does not explain the spontaneous curvature cancellation phenomenon t liang and t a witten phys rev e 73 046604 2006 the locality of the compensation seems to increase for decreasing d cone thickness the angular deficit scheme also provides a way to confirm the curvature cancellation phenomenon

stochastic mpc with inequality stability constraints stochastic uncertainty is a common feature of many control engineering problems and is also present in a wider class of applications e g finance and sustainable development recent work proposed a constrained mpc approach that took explicit account of the distributions of uncertain model parameters but used terminal equality constraints to ensure stability the present paper reformulates the problem in order to relax the stability constraints by invoking appropriate terminal inequalities the application of the proposed strategy and its advantages over earlier work are illustrated by means of a numerical example

nonquadratic stochastic model predictive control a tractable approach this paper deals with the finite horizon stochastic optimal control problem with the expectation of the p norm as the objective function and jointly gaussian although not necessarily independent additive disturbance process we develop an approximation strategy that solves the problem in a certain class of nonlinear feedback policies while ensuring satisfaction of hard input constraints a bound on suboptimality of the proposed strategy in this class of nonlinear feedback policies is given for the special case of p 1 we also develop a recursively feasible receding horizon policy with respect to state chance constraints and or hard control input constraints in the presence of bounded disturbances the performance of the proposed policies is examined in two numerical examples

on constraint sampling in the linear programming approach to approximate dynamic programming in the linear programming approach to approximate dynamic programming one tries to solve a certain linear program the alp that has a relatively small numberk of variables but an intractable numberm of constraints in this paper we study a scheme that samples and imposes a subset ofm

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

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

continuous time probabilistic ultimate bounds and invariant sets the concepts of ultimate bounds and invariant sets play a key role in several control theory problems as they replace the notion of asymptotic stability in the presence of unknown disturbances however when the disturbances are unbounded as in the case of gaussian white noise no ultimate bounds nor invariant sets can in general be found to overcome this limitation we introduced in previous work the notions of probabilistic ultimate bound pub and probabilistic invariant set pis for discrete time systems this article extends the notions of pub and pis to continuous time systems studying their main properties and providing tools for their calculation in addition the use of these concepts in robust control design by covariance assignment is presented

an approach to output feedback mpc of stochastic linear discrete time systems in this paper we propose an output feedback model predictive control mpc algorithm for linear discrete time systems affected by a possibly unbounded additive noise and subject to probabilistic constraints in case the noise distribution is unknown the probabilistic constraints on the input and state variables are reformulated by means of the chebyshev cantelli inequality the recursive feasibility is guaranteed the convergence of the state to a suitable neighbor of the origin is proved under mild assumptions and the implementation issues are thoroughly addressed two examples are discussed in detail with the aim of providing an insight into the performance achievable by the proposed control scheme

the scenario approach for stochastic model predictive control with bounds on closed loop constraint violations many practical applications in control require that constraints on the inputs and states of the system are respected while some performance criterion is optimized in the presence of model uncertainties or disturbances it is often sufficient to satisfy the state constraints for at least a prescribed share of the time such as in building climate control or load mitigation for wind turbines for such systems this paper presents a new method of scenario based model predictive control scmpc the basic idea is to optimize the control inputs over a finite horizon subject to robust constraint satisfaction under a finite number of random scenarios of the uncertainty and or disturbances previous scmpc approaches have suffered from a substantial gap between the rate of constraint violations specified in the optimal control problem and that actually observed in closed loop operation of the controlled system this paper identifies the two theoretical explanations for this gap first accounting for the special structure of the optimal control problem leads to a substantial reduction of the problem dimension second the probabilistic constraints have to be interpreted as average in time rather than pointwise in time based on these insights a novel scmpc method can be devised for general linear systems with additive and multiplicative disturbances for which the number of scenarios is significantly reduced the presented method retains the essential advantages of the general scmpc approach namely a low computational complexity and the ability to handle arbitrary probability distributions moreover the computational complexity can be adjusted by a sample and remove strategy

brief paper explicit use of probabilistic distributions in linear predictive control the guarantee of feasibility given feasibility at initial time is an issue that has been overlooked by many of the recent papers on stochastic model predictive control effective solutions have recently been proposed but these carry considerable online computational load and a degree of conservativism for the case that the elements of the random additive disturbance vector are independent the current paper ensures that probabilistic constraints are met and that a quadratic stability condition is satisfied a numerical example illustrates the efficacy of the proposed algorithm which achieves tight satisfaction of constraints and thereby attains near optimal performance

stochastic tubes in model predictive control with probabilistic constraints stochastic model predictive control mpc strategies can provide guarantees of stability and constraint satisfaction but their online computation can be formidable this difficulty is avoided in the current technical note through the use of tubes of fixed cross section and variable scaling a model describing the evolution of predicted tube scalings facilitates the computation of stochastic tubes furthermore this procedure can be performed offline the resulting mpc scheme has a low online computational load even for long prediction horizons thus allowing for performance improvements the efficacy of the approach is illustrated by numerical examples

brief a probabilistically constrained model predictive controller we propose a novel control algorithm probabilistically constrained predictive control to deal with the uncertainties of system disturbances the output is to be controlled in the constrained range with a desired probability under the assumption of a linear system the formulated joint probabilistically constrained problem is convex thus it can be solved with a nonlinear programming solver the probabilities and gradients of the constraints composed of disturbance sequences with multivariate normal distribution are computed using an efficient simulation approach the results of a test problem show the effectiveness of the proposed algorithm

probabilistic tubes in linear stochastic model predictive control this paper considers constrained control of linear systems with additive and multiplicative stochastic uncertainty and linear input state constraints both hard and soft constraints are considered and bounds are imposed on the probability of soft constraint violation assuming the plant parameters to be finitely supported a method of constraint handling is proposed in which a sequence of tubes corresponding to a sequence of confidence levels on the predicted future plant state is constructed online around nominal state trajectories a set of linear constraints is derived by imposing bounds on the probability of constraint violation at each point on an infinite prediction horizon through constraints on one step ahead predictions a guarantee of the recursive feasibility of the online optimization ensures that the closed loop system trajectories satisfy both the hard and probabilistic soft constraints the approach is illustrated by a numerical example

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

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

nearly tight bounds on the contiguity and linearity of cographs in this paper we show that the contiguity and linearity of cographs on n vertices are both o logn moreover we show that this bound is tight for contiguity as there exists a family of cographs on n vertices whose contiguity is w logn we also provide an w logn loglogn lower bound on the maximum linearity of cographs on n vertices as a by product of our proofs we obtain a min max theorem which is worth of interest in itself stating equality between the rank of a tree and the minimum height of one of its path partitions

a necessary and sufficient rhc stabilizability condition for stochastic control with delayed input abstract we study the mean square stabilizability of ito stochastic systems with delayed input in contrast to most previous results where only sufficient condition was derived this paper proposes a necessary and sufficient stabilization condition by exploring the receding horizon control rhc and the coupled riccati equations it is shown that the system controlled by rhc is stabilizable if and only if the two coupled lyapunov type inequalities are satisfied the key is to design a novel cost function based on the optimal cost of finite horizon optimization problem

model predictive control for systems with stochastic multiplicative uncertainty and probabilistic constraints abstract robust predictive control handles constrained systems that are subject to stochastic uncertainty but propagating the effects of uncertainty over a prediction horizon can be computationally expensive and conservative this paper overcomes these issues through an augmented autonomous prediction formulation and provides a method of handling probabilistic constraints and ensuring closed loop stability through the use of an extension of the concept of invariance namely invariance with probability p

robust filtering for discrete time systems with bounded noise and parametric uncertainty this note presents a new approach to finite horizon guaranteed state prediction for discrete time systems affected by bounded noise and unknown but bounded parameter uncertainty our framework handles possibly nonlinear dependence of the state space matrices on the uncertain parameters the main result is that a minimal confidence ellipsoid for the state consistent with the measured output and the uncertainty description may be recursively computed in polynomial time using interior point methods for convex optimization with n states l uncertain parameters appearing linearly in the state space matrices with rank one matrix coefficients the worst case complexity grows as o l n l sup 3 5 with unstructured uncertainty in all system matrices the worst case complexity reduces to o n sup 3 5

scaling properties of randomly folded plastic sheets we study the scaling properties of randomly folded aluminum sheets of different thicknesses h and widths l we found that the fractal dimension d 2 30 0 01 and the force scaling exponent delta 0 21 0 02 are independent of the sheet thickness and close to those obtained in numerical simulations with a coarse grained model of triangulated self avoiding surfaces with bending and stretching rigidity so our findings suggest that finite bending rigidity and self avoidance play the predominant roles in the scaling behavior of randomly folded plastic sheets

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

chance constrained h control for a class of time varying systems with stochastic nonlinearities the finite horizon case in this paper a new finite horizon h control problem is considered for a class of time varying systems with stochastic nonlinearities measurements degradation and chance constraints the purpose of the addressed problem is to design the time varying controller such that the closed loop system satisfies the prespecified h disturbance attenuation requirement and certain chance constraints on the controlled output vector zk i e the probability of the controlled output zk belonging to a given set is larger than a prescribed value a modified maximum volume inscribed ellipsoid mvie method is employed to convert the chance constraint into some tractable inequalities which could be conveniently handled by the recursive linear matrix inequality rlmi approach then by using stochastic control analysis method sufficient conditions are derived for the existence of the desired multi objective controller and furthermore the gains of the desired controllers are characterized by means of rlmis finally two illustrative examples and a practical system are proposed to highlight the effectiveness and applicability of the presented controller design technology

on the stability of receding horizon control for continuous time stochastic systems abstract we study the stability of receding horizon control for continuous time non linear stochastic differential equations we illustrate the results with a simulation example in which we employ receding horizon control to design an investment strategy to repay a debt

risk and complexity in scenario optimization scenario optimization is a broad methodology to perform optimization based on empirical knowledge one collects previous cases called scenarios for the set up in which optimization is being performed and makes a decision that is optimal for the cases that have been collected for convex optimization a solid theory has been developed that provides guarantees of performance and constraint satisfaction of the scenario solution in this paper we open a new direction of investigation the risk that a performance is not achieved or that constraints are violated is studied jointly with the complexity as precisely defined in the paper of the solution it is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk this result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios in particular no extra knowledge on the distribution by which scenarios are generated is assumed so that the result is broadly applicable this deep seated result unveils a fundamental and general structure of data driven optimization and suggests practical approaches for risk assessment

on the connection between compression learning and scenario based single stage and cascading optimization problems we investigate the connections between compression learning and scenario based optimization we first show how to strengthen or relax the consistency assumption at the basis of compression learning and provide novel learnability conditions for the underlying algorithms we then consider different constrained optimization problems affected by uncertainty represented by means of scenarios we show that the compression learning perspective provides a unifying framework for scenario based optimization since the issue of providing guarantees on the probability of constraint violation reduces to a learning problem for an appropriately chosen algorithm that satisfies some consistency assumption to illustrate this we revisit the scenario approach within the developed context moreover using the compression learning machinery we provide novel results on the probability of constraint violation for the class of cascading optimization problems

randomized solutions to convex programs with multiple chance constraints the scenario based optimization approach scenario approach provides an intuitive way of approximating the solution to chance constrained optimization programs based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty scenarios a key merit of this approach is that it neither requires explicit knowledge of the uncertainty set as in robust optimization nor of its probability distribution as in stochastic optimization the scenario approach is also computationally efficient because it only requires the solution to a convex optimization program even if the original chance constrained problem is nonconvex recent research has obtained a rigorous foundation for the scenario approach by establishing a direct link between the number of scenarios and bounds on the constraint violation probability these bounds are tight in the general case of an uncertain optimization problem with a single chance constraint this paper shows that the bounds can be improved

stochastic receding horizon control with output feedback and bounded controls we study the problem of receding horizon control for stochastic discrete time systems with bounded control inputs and incomplete state information given a suitable choice of causal control policies we first present a slight extension of the kalman filter to estimate the state optimally in mean square sense we then show how to augment the underlying optimization problem with a negative drift like constraint yielding a second order cone program to be solved periodically online we prove that the receding horizon implementation of the resulting control policies renders the state of the overall system mean square bounded under mild assumptions we also discuss how some quantities required by the finite horizon optimization problem can be computed off line thus reducing the on line computation

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

convergence of stochastic nonlinear systems and implications for stochastic model predictive control the stability of stochastic model predictive control mpc subject to additive disturbances is often demonstrated in the literature by constructing lyapunov like inequalities that ensure closed loop performance bounds and boundedness of the state but tight ultimate bounds for the state and non conservative performance bounds are typically not determined in this work we use an input to state stability property to find conditions that imply convergence with probability 1 of a disturbed nonlinear system to a minimal robust positively invariant set we discuss implications for the convergence of the state and control laws of stochastic mpc formulations and we prove convergence results for several existing stochastic mpc formulations for linear and nonlinear systems

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

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

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

effect of ridge ridge interactions in crumpled thin sheets we study whether and how the energy scaling based on the single ridge approximation is revised in an actual crumpled sheet namely in the presence of ridge ridge interactions molecular dynamics simulation is employed for this purpose in order to improve the data quality modifications are introduced to the common protocol as crumpling proceeds we find that the average storing energy changes from being proportional to one third of the ridge length to a linear relation while the ratio of bending and stretching energies decreases from 5 to 2 the discrepancy between previous simulations and experiments on the material dependence for the power law exponent is resolved we further determine the average ridge length to scale as 1 d 1 3 the ridge number as d 2 3 and the average storing energy per unit ridge length as d 0 881 where d denotes the volume density of the crumpled ball these results are accompanied by experimental proofs and are consistent with mean field predictions finally we extend the existent simulations to the high pressure region and verify the existence of a scaling relation that is more general than the familiar power law at covering the whole density range

receding horizon control for multiplicative noise stochastic systems with input delay abstract this paper is concerned with finite horizon stabilization control for a class of discrete time stochastic systems subject to multiplicative noise and input delay by constructing a new cost function a complete solution to the problem of finite horizon stabilization is given for the first time based on previous work zhang et al 2015 it is shown that the system can be stabilized in the mean square sense with the receding horizon control rhc if and only if two new inequalities on the terminal weighting matrices are satisfied moreover the two inequalities can be solved by using iterative algorithm the explicit stabilizing controller is derived by solving a finite horizon optimal control problem simulations demonstrate the effectiveness of the proposed method

learning decision trees from random examples needed for learning abstract we define the rank of a decision tree and show that for any fixed r the class of all decision trees of rank at most r on n boolean variables is learnable from random examples in time polynomial in n and linear in 1 and log 1 where is the accuracy parameter and is the confidence parameter using a suitable encoding of variables rivest s polynomial learnability result for decision lists can be interpreted as a special case of this result for rank 1 as another corollary we show that decision trees on n boolean variables of size polynomial in n are learnable from random examples in time linear in n o log n 1 and log 1 as a third corollary we show that boolean functions that have polynomial size dnf expressions for both their positive and their negative instances are learnable from random examples in time linear in n o log n 2 1 and log 1

model predictive control for constrained systems with serially correlated stochastic parameters and portfolio optimization in this paper we consider mpc for constrained discrete time systems with stochastic parameters which are assumed to be a set of serially correlated time series a generalized performance criterion is composed of a weighted sum of a linear combination of the a expected value of quadratic forms of state and control vectors b quadratic forms of the expected value of the state vector and c the linear component of the expected value of the state vector the purpose of the present paper is to design optimal control strategies that are independent of distributional assumptions on the stochastic parameters and subject to hard constraints on the input manipulated variables and to provide a numerically tractable algorithm for practical applications all expressions are presented in terms of the first and second order conditional moments the results are applied to a problem of investment portfolio optimization with serially correlated returns we present the numerical modelling results based on stocks traded on the russian stock exchanges micex

ambiguous chance constrained problems and robust optimization in this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain we focus primarily on the special case where the uncertainty set of the distributions is of the form where p denotes the prohorov metric the ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure the main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability this result is established using the strassen dudley representation theorem that states that when the distributions of two random variables are close in the prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability we also show that the robust sampled problem can be solved efficiently both in theory and in practice

optimal feedback control strategies for state space systems with stochastic parameters two different optimal feedback laws are derived for state space systems parametrized through an independent identically distributed vector sequence both feedback laws are obtained by minimizing the expectation of a multistep quadratic loss function at each time step they differ on the assumptions made about the future inputs the properties and implementability of the feedback laws are discussed for the infinite horizon case