characterization of zero locations of polytopes of real polynomials given a stability region of a polytope of polynomials the edge theorem can be used to verify certain root clustering properties of the polytope the authors consider the inverse problem starting with a polytope of real polynomials they present a result that allows the construction of the smallest set of regions in the complex plane which characterizes its zero location

a penalty method for the vorticity velocity formulation we present a new vorticity velocity formulation and implementation for the unsteady three dimensional navier stokes equations based on a penalty method it relies on an equivalence theorem that employs exact boundary conditions and the vorticity definition on the domain boundary this approach is particularly attractive for high order methods for which the often used influence matrix method fails to converge for t 0 the accuracy and the robustness of the new method is demonstrated in the context of several spectral element simulations of unsteady two and three dimensional internal and external flows in particular the flow past a finite span cylinder attached to end plates is studied in some detail in order to evaluate the effects of the aspect ratio on the formation length

an edge theorem for polytopes of polynomials which can drop in degree abstract a new version of the edge theorem for robust stability of polynomials that can drop in degree is given the proof is of elementary nature and a counterexample is given to show that the original edge theorem of bartlett hollot and lin does not hold for the general case of arbitrary degree dropping

robust finite settling time stabilization of discrete time system with structured uncertainty abstract the robust finite settling time fst stabilization problem is considered for an uncertain discrete time system fst stability is a generalization of the dead beat response 9 which requires that all the external signals as well as internal signals of a feedback system settle to new steady states after a finite time for any step inputs the uncertain discrete time system under consideration is described by a causal real rational function which includes uncertain parameters where the uncertain parameters are restricted to prescribed bounding intervals and enter affine linearly into the coefficients of the transfer functions a necessary and sufficient condition for robust fst stability is presented in terms of a finite of transfer functions the condition leads to a computationally verifiable necessary and sufficient condition for robust fst stabilizability and a systematic procedure for designing a robust fst feedback control system

method of robust stability analysis with highly structured uncertainties this paper presents a method of analyzing the stability of linear time invariant interconnected systems with uncertainties each subsystem is single input single output and its vector locus lies inside a polygon at each frequency it is shown that the convex closure of the image of the cartesian product of these polygons under the mapping phi det i fh agrees with the convex closure of the image of the vertices of these polygons under the mapping from this result the image can be estimated by a finite dumber of calculations a sufficient stability condition is obtained by applying this result to the multivariable nyquist stability criterion

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

a few results concerning the hurwitz stability of polytopes of complex polynomials abstract let c f 1 f m be a polytope generated by complex polynomials f 1 f m whose degrees differ at most by one the main goal of this note is to provide a tool for verifying whether a polynomial family c f 1 f m is stable the note extend a few important results of the robust stability theory the edge theorem given by bartlett et al 1988 5 its generalizations proposed by sideris and barmish 1989 7 and fu and barmish 1989 6 and the eigenvalue criterions of bialas 1985 2004 4 11 to more general cases concerning complex polynomial families without degree invariant assumptions numerical examples are presented to complete and illustrate the results

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

supermodularity and affine policies in dynamic robust optimization this paper considers a particular class of dynamic robust optimization problems where a large number of decisions must be made in the first stage which consequently fix the constraints and cost structure underlying a one dimensional linear dynamical system we seek to bridge two classical paradigms for solving such problems namely 1 dynamic programming dp and 2 policies parameterized in model uncertainties also known as decision rules obtained by solving tractable convex optimization problems we show that if the uncertainty sets are integer sublattices of the unit hypercube the dp value functions are convex and supermodular in the uncertain parameters and a certain technical condition is satisfied then decision rules that are affine in the uncertain parameters are optimal we also derive conditions under which such rules can be obtained by optimizing simple i e linear objective functions over the uncertainty sets our results suggest new modeling paradigms for dynamic robust optimiz

ambiguous joint chance constraints under mean and dispersion information we study joint chance constraints where the distribution of the uncertain parameters is only known to belong to an ambiguity set characterized by the mean and support of the uncertainties and by an upper bound on their dispersion this setting gives rise to pessimistic optimistic ambiguous chance constraints which require the corresponding classical chance constraints to be satisfied for every for at least one distribution in the ambiguity set we demonstrate that the pessimistic joint chance constraints are conic representable if i the constraint coefficients of the decisions are deterministic ii the support set of the uncertain parameters is a cone and iii the dispersion function is of first order that is it is positively homogeneous we also show that pessimistic joint chance constrained programs become intractable as soon as any of the conditions i ii or iii is relaxed in the mildest possible way we further prove that the optimistic joint chance constraints are conic representab

the post processing approach in the finite element method part 1 calculation of displacements stresses and other higher derivatives of the displacements this is the first in a series of three papers in which we discuss a method for post processing a finite element solution to obtain high accuracy approximations for displacements stresses stress intensity factors etc rather than take the values of these quantities directly from the finite element solution we evaluate certain weighted averages of the solution over the entire region these yield approximations are of the same order of accuracy as the strain energy we obtain error estimates and also present some numerical examples to illustrate the practical effectiveness of the technique in the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation

linear controller design for chance constrained systems this paper is concerned with the design of a linear control law for a linear system with stationary additive disturbances the objective is to find a state feedback gain that minimizes a quadratic stage cost function while observing chance constraints on the input and or the state unlike most of the previous literature the chance constraints and the stage cost are not considered on each input state of the transient response instead they refer to the input state of the closed loop system in its stationary mode of operation hence the control is optimized for the long run rather than for finite horizon operation the controller synthesis problem can be cast as a convex semi definite program sdp the chance constraints appear as linear matrix inequalities both single chance constraints sccs and joint chance constraints jccs on the input and or the state can be included if the disturbance is gaussian this information can be used to improve the controller design the presented approach can also be extended to the case of output feedback the entire design procedure is flexible and easy to implement as demonstrated on a short illustrative example

some properties of zeros of polynomials with vanishing coefficients abstract it is well known that when a polynomial whose coefficients are continuous functions of a parameter loses its degree then some of its zeros must vanish at infinity in this paper we consider such a situation we examine how roots of a complex polynomial tend to infinity as some of its coefficients including the leading one tend to zero we show among other things that in such a situation the unbounded paths traced by the roots of the polynomial have asymptotes we also obtain their formulas some examples are presented to complete and illustrate the results

generalized gauss inequalities via semidefinite programming a sharp upper bound on the probability of a random vector falling outside a polytope based solely on the first and second moments of its distribution can be computed efficiently using semidefinite programming however this chebyshev type bound tends to be overly conservative since it is determined by a discrete worst case distribution in this paper we obtain a less pessimistic gauss type bound by imposing the additional requirement that the random vector s distribution must be unimodal we prove that this generalized gauss bound still admits an exact and tractable semidefinite representation moreover we demonstrate that both the chebyshev and gauss bounds can be obtained within a unified framework using a generalized notion of unimodality we also offer new perspectives on the computational solution of generalized moment problems since we use concepts from choquet theory instead of traditional duality arguments to derive semidefinite representations for worst case probability bounds

a dynamic game approach to distributionally robust safety specifications for stochastic systems abstract this paper presents a new safety specification method that is robust against errors in the probability distribution of disturbances our proposed distributionally robust safe policy maximizes the probability of a system remaining in a desired set for all times subject to the worst possible disturbance distribution in an ambiguity set we propose a dynamic game formulation of constructing such policies and identify conditions under which a non randomized markov policy is optimal based on this existence result we develop a practical design approach to safety oriented stochastic controllers with limited information about disturbance distributions however an associated bellman equation involves infinite dimensional minimax optimization problems since the disturbance distribution may have a continuous density to alleviate computational issues we propose a duality based reformulation method that converts the infinite dimensional minimax problem into a semi infinite program that can be solved using existing convergent algorithms we prove that there is no duality gap and that this approach thus preserves optimality the results of numerical tests confirm that the proposed method is robust against distributional errors in disturbances while a standard stochastic safety verification tool is not

robust root clustering for linear uncertain systems using generalized lyapunov theory in this paper the problem of matrix root clustering in sub regions of complex plane for linear state space models with real parameter uncertainty is considered the nominal matrix root clustering theory of gutman and jury 1981 ieee trans aut control ac 26 403 using generalized lyapunov equation is extended to the perturbed matrix case and bounds are derived on the perturbation to maintain root clustering inside a given region the theory allows us to get an explicit relationship between the parameters of the root clustering region and the uncertainty range of the parameter space the current literature available on perturbation bounds for robust stability becomes a special case of this unified theory

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

monte carlo turbulence simulation using rational approximations to von karman spectra turbulence simulation is computational ly much simpler using rational spectra but turbulence falls off as y s 3 jn frequency ranges of interest to aircraft response and as predicted by von karman s model rational approximations to von karman spectra should satisfy three requirements 1 the rational spectra should provide a good approximation to the von karman spectra in the frequency range of interest 2 for stability the resulting rational transfer function should have all its poles in the left half plane and 3 at high frequencies the rational spectra must fall off as an integer power of frequency and since the 2 power is closest to the 5 3 power the rational approximation should roll off as the 2 power at high frequencies rational approximations to von karman spectra that satisfy these three criteria are presented along with spectra from simulated turbulence agreement between the spectra of the simulated turbulence and von karman spectra is excellent

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

numerical analysis of singularities in two dimensions part 2 computation of generalized flux stress intensity factors summary a numerical method for the computation of the generalized flux stress intensity factors gfifs gsifs for the asymptotic solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points is described special attention is given to heat transfer and elasticity problems the singularities may be caused by re entrant corners and abrupt changes in material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs computed in a companion paper characterize the straining modes and their amplitudes the gfifs gsifs quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the gfifs gsifs this paper addresses a general method based on the complementary weak formulation for determining the gfifs gsifs numerically as a post solution operation on the finite element solution vector importantly the method is applicable to anisotropic materials multi material interfaces and cases where the singularities are characterized by complex eigenpairs an error analysis is sketched and numerical examples are presented to illustrate the effectiveness of the technique

from cvar to uncertainty set implications in joint chance constrained optimization we review and develop different tractable approximations to individual chance constrained problems in robust optimization on a variety of uncertainty sets and show their interesting connections with bounds on the conditional value at risk cvar measure we extend the idea to joint chance constrained problems and provide a new formulation that improves upon the standard approach our approach builds on a classical worst case bound for order statistics problems and is applicable even if the constraints are correlated we provide an application of the model on a network resource allocation problem with uncertain demand

on kharitonov type theorems for real polynomials when is degree drop admissible a root locus approach is adopted to show that extension of kharitonov type theorem for real interval polynomial with degree drop is not only applicable to the open left half plane but also other regions such as the left shifted open left half plane and the left sector

distributionally robust joint chance constraints with second order moment information we develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints assuming that only the first and second order moments as well as the support of the uncertain parameters are given it is known that robust chance constraints can be conservatively approximated by worst case conditional value at risk cvar constraints we first prove that this approximation is exact for robust individual chance constraints with concave or not necessarily concave quadratic constraint functions and we demonstrate that the worst case cvar can be computed efficiently for these classes of constraint functions next we study the worst case cvar approximation for joint chance constraints this approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations the tightness depends on a set of scaling parameters which can be tuned via a sequential convex optimization algorithm we show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the worst case cvar can be evaluated efficiently if the scaling parameters are kept constant we evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations

infinite horizon performance bounds for uncertain constrained systems we present a new method to bound the performance of controllers for uncertain linear systems with mixed state and input constraints and bounded disturbances we take as a performance metric either an expected value or minimax discounted cost over an infinite horizon and provide a method for computing a lower bound on the achievable performance of any causal control policy in either case our lower bound is compared to an upper performance bound provided by restricting the choice of controller to one that is affine in the observed disturbances and we show that the two bounds are closely related in particular the lower bounds have a natural interpretation in terms of affine control policies that are optimal for a problem with a restricted disturbance set we show that our performance bounds can be computed via solution of a finite dimensional convex optimization problem and provide numerical examples to illustrate the efficacy of our method

expansion of det a b and robustness analysis of uncertain state space systems in this note an expansion for the determinant of the sum of two matrices det a b is given a robustness analysis approach for state space systems with uncertain parameters is investigated by using kharitonov s theorem some useful results are obtained several illustrative examples are given

optimal convergence analysis for the extended finite element method we establish some optimal a priori error estimate on some variants of the extended finite element method xfem namely the xfem with a cut off function and the standard xfem with a fixed enrichment area the results are established for the lame system homogeneous isotropic elasticity and the laplace problem the convergence of the numerical stress intensity factors is also investigated we show some numerical experiments which corroborate the theoretical results

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

output feedback receding horizon control of constrained systems this paper considers output feedback control of linear discrete time systems with convex state and input constraints which are subject to bounded state disturbances and output measurement errors we show that the non convex problem of finding a constraint admissible affine output feedback policy over a finite horizon to be used in conjunction with a fixed linear state observer can be converted to an equivalent convex problem when used in the design of a time varying robust receding horizon control law we derive conditions under which the resulting closed loop system is guaranteed to satisfy the system constraints for all time given an initial state estimate and bound on the state estimation error when the state estimation error bound matches the minimal robust positively invariant mrpi set for the system error dynamics we show that this control law is time invariant but its calculation generally requires solution of an infinite dimensional optimization problem finally using an invariant outer

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

optimization over state feedback policies for robust control with constraints this paper is concerned with the optimal control of linear discrete time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input it is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence this implies that a broad class of constrained finite horizon robust and optimal control problems where the optimization is over affine state feedback policies can be solved in a computationally efficient fashion using convex optimization methods this equivalence result is used to design a robust receding horizon control rhc state feedback policy such that the closed loop system is input to state stable iss and the constraints are satisfied for all time and all allowable disturbance sequences the cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance free state and input sequences the value of the receding horizon control law can be calculated at each sample instant using a single tractable and convex quadratic program qp if the disturbance set is polytopic or a tractable second order cone program socp if the disturbance set is given by a 2 norm bound

kharitonov s theorem extension to interval polynomials which can drop in degree a nyquist approach this paper studies the hurwitz stability of interval polynomials which can drop in degree from a nyquist point of view these families make it possible to deal with system families taking into account different dynamic behaviors in the modeling of a plant the behavior of the interval polynomials is studied and it is proven that kharitonov s theorem can be extended

polytopes of polynomials with zeros in a prescribed set in the publication by a c bartlett c v holot and h lin proc amer contr conf minneapolis mn 1987 a fundamental result is established on the zero locations of a family of polynomials it is shown that the zeros of a polytope p of nth order real polynomials are contained in a simply connected set d if and only if the zeros of all polynomials along the edges of p are contained in d the present authors are motivated by the fact that the requirement of simple connectedness of d may be too restrictive and applications such as dominant pole assignment and filter design where the separation of zeros is required they extend the edge criterion of bartlett et al to handle any set d whose complement d sup c has the following property every point d in d sup c lies on some continuous path which remains within d sup c and is unbounded this requirement is typically verified by inspection and allows for a large class of disconnected sets

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

a posteriori error estimation based on the superconvergent recovery by compatibility in patches the present work deals with an a posteriori error estimator for linear finite element analysis based on a stress recovery procedure called recovery by compatibility in patches the key idea of this procedure is to recover improved stresses by minimizing the complementary energy over patches of elements displacements computed by the finite element analysis are prescribed on the boundary of the patch here a new form of this recovery procedure is presented adopting a different patch configuration centred upon an element instead of a node allows to drastically simplify the recovery process thus improving efficiency and making the implementation in finite element codes much easier the robustness tests demonstrate that the error estimator associated to the new form of the recovery procedure retains the very good properties of the original one such as superconvergence the numerical results on two common benchmark problems confirm the effectiveness of the proposed error estimator which appears to be competitive with those currently available copyright 2006 john wiley sons ltd

state space measures of robustness of pole locations for structured and unstructured perturbations abstract measures of pole location robustness for linear feedback systems are derived from a state space model of the system the robustness tests ensure that the eigenvalues of the perturbed systems matrix a e remain in a desired region d of the complex plane containing the eigenvalues of the nominal system matrix a the region d may be any open set of the complex plane whatsoever the results are expressed in terms of induced matrix norms and apply to structured perturbations of the form e b c where b and c define the structure of e and may be nonsquare matrices rank one perturbations e of minimal norm and with the given structure that will cause a e to have an eigenvalue outside of d are constructed for the cases when the matrix norm is induced by the vector 1 norm or the vector norm the advantages of having robustness measures for several matrix norms that can be computed are illustrated with a simple example that demonstrates how the conservatism of single tests can be reduced using several tests i e several matrix norms a method for computing numerically the robustness measures for particular norms is presented it can be used to compute with a guaranteed degree of accuracy the maximum of the norm of the frequency response of a system

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

worst case conditional value at risk with application to robust portfolio management this paper considers the worst case conditional value at risk cvar in the situation where only partial information on the underlying probability distribution is available the minimization of the worst case cvar under mixture distribution uncertainty box uncertainty and ellipsoidal uncertainty are investigated the application of the worst case cvar to robust portfolio optimization is proposed and the corresponding problems are cast as linear programs and second order cone programs that can be solved efficiently market data simulation and monte carlo simulation examples are presented to illustrate the proposed approach

generalized edge theorem abstract in this paper we shall show that a family of analytic functions constructed from the convex hull of a finite number of vertex functions will have no zeros within a simply connected region in the complex plane if and only if all its edge functions have no zeros within the simply connected region the result is in fact a generalization of the edge theorem which has been derived for polynomial functions 1 2 7

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

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

a convergent algorithm for the output covariance constraint control problem this paper considers the optimal control problem of minimizing control effort subject to multiple performance constraints on output covariance matrices y i of the form y i leq overline y i where overline y i is given the contributions of this paper are a set of conditions that characterize global optimality and an iterative algorithm for finding a solution to the optimality conditions this iterative algorithm is completely described up to a user specified parameter we show that under suitable assumptions on problem data the iterative algorithm converges to a solution of the optimality conditions provided that this parameter is properly chosen both discrete and continuous time problems are considered

distributionally robust control of constrained stochastic systems we investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process i e when only the first two moments of the disturbance distribution are known we consider two types of distributionally robust constraints in the first case we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments these constraints are commonly referred to as distributionally robust chance constraints in the second case we impose conditional value at risk cvar constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information such constraints are referred to as distributionally robust cvar constraints with second order moment specifications we propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems we prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust cvar and chance constrained problems the proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives

superconvergent extraction of flux intensity factors and first derivatives from finite element solutions a superconvergent method for the computation of the derivatives of the solution and the coefficients of the asymptotic expansion at singular points is presented for the laplace problem in two dimensions the algorithm utilizes the complementary weak form on a localized small domain mathematical analysis demonstrates the superconvergent behavior and numerical experiments support our analysis this method is well suited for anisotropic multi material singular interface problems