stabilization of uncertain systems via linear control this note considers the problem of stabilizing a linear dynamical system whose state equation includes a time varying uncertain parameter vector q cdot given the dynamics dot x t a q t x t b q t u t and a bounding set q for the values q t the objective is to choose a control law u t p x t guaranteeing uniform asymptotic stability for all admissible variations of q cdot our results differ from previous work in one fundamental way that is we show that when working with linear controllers it is possible to dispense with all assumptions on b cdot which have been made by previous authors e g see 1 9 this elimination of hypotheses on b cdot is accomplished roughly as follows the system sigma underline underline delta a q b q is shown to be equivalent to another system sigma underline underline delta a q b as far as stabilization is concerned since b is a constant matrix independent of q the desired result is readily obtained

comparing bayes model averaging and stacking when model approximation error cannot be ignored we compare bayes model averaging bma to a non bayes form of model averaging called stacking in stacking the weights are no longer posterior probabilities of models they are obtained by a technique based on cross validation when the correct data generating model dgm is on the list of models under consideration bma is never worse than stacking and often is demonstrably better provided that the noise level is of order commensurate with the coefficients and explanatory variables here however we focus on the case that the correct dgm is not on the model list and may not be well approximated by the elements on the model list we give a sequence of computed examples by choosing model lists and dgm s to contrast the risk performance of stacking and bma in the first examples the model lists are chosen to reflect geometric principles that should give good performance in these cases stacking typically outperforms bma sometimes by a wide margin in the second set of examples we examine how stacking and bma perform when the model list includes all subsets of a set of potential predictors when we standardize the size of terms and coefficients in this setting we find that bma outperforms stacking when the deviant terms in the dgm point in directions accommodated by the model list but that when the deviant term points outside the model list stacking seems to do better overall our results suggest the stacking has better robustness properties than bma in the most important settings

robust stabilization of linear systems with norm bounded time varying uncertainty abstract in this paper robust stabilization of a class of linear systems with norm bounded time varying uncertainties is considered it is shown that for this class of uncertain systems quadratic stabilizability via linear control is equivalent to the existence of a positive definite symmetric matrix solution to a parameter dependent riccati equation also a construction for the stabilizing feedback law is given in terms of the solution to the riccati equation

an algorithm for minimizing the norm of state feedback controllers in eigenvalue assignment abstract minimization of the norm of parametric feedback controllers which assign desired eigenvalues to the closed loop system is achieved by implementation of vector companion forms described in an earlier paper the simple algorithm which emerges gives insight into the mechanism by which parametric eigenvalue assignment is affected a couple of numerical examples are presented to illustrate advantages of this algorithm which easily computes the controller gain matrix with minimum norm

combining predictions in pairwise classification an optimal adaptive voting strategy and its relation to weighted voting weighted voting is the commonly used strategy for combining predictions in pairwise classification even though it shows good classification performance in practice it is often criticized for lacking a sound theoretical justification in this paper we study the problem of combining predictions within a formal framework of label ranking and under some model assumptions derive a generalized voting strategy in which predictions are properly adapted according to the strengths of the corresponding base classifiers we call this strategy adaptive voting and show that it is optimal in the sense of yielding a map prediction of the class label of a test instance moreover we offer a theoretical justification for weighted voting by showing that it yields a good approximation of the optimal adaptive voting prediction this result is further corroborated by empirical evidence from experiments with real and synthetic data sets showing that even though adaptive voting is sometimes able to achieve consistent improvements weighted voting is in general quite competitive all the more in cases where the aforementioned model assumptions underlying adaptive voting are not met in this sense weighted voting appears to be a more robust aggregation strategy

constrained stabilization problem and transient mismatch phenomenon in singularly perturbed systems we initially study the control of linear systems by state feedback and simultaneously require the feedback gain matrix to satisfy a certain constraint equation we provide two fundamentally different methods to solve the constrained stabilization problem the first method parametrizes the class of stabilizing gain matrices and converts the problem to an equivalent nonlinear programming problem the second method on the other hand converts the problem to an equivalent stabilization problem by a static output controller where the static gain is required to take a special structure the motivation behind introducing this problem and its solution is the application of the results to the design of observer based controllers for the singularly perturbed systems it is shown that the transient mismatch phenomenon which occurs in this type of design may be alleviated if the observer gain of the fast observer is chosen based on the solution of a certain constraint stabilization problem which involves the outpu

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

optimal pole shifting for continuous multivariable linear systems a method for shifting the real parts of the open loop poles to any desired positions while preserving the imaginary parts is presented the method is based on the mirror image property which has been reported by molinari in other words molinari s results are extended and then a recursive approach is developed in each step of this approach it is required to solve a first order or a second order linear matrix lyapunov equation for shifting one real pole or two complex conjugate poles respectively the presented method yields a solution which is optimal with respect to a quadratic performance index the attractive feature of this method is that it enables solutions to complex problems to be easily found without solving any non linear algebraic riccati equation

classification by pairwise coupling we discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes and then coupling the estimates together the coupling model is similar to the bradley terry method for paired comparisons we study the nature of the class probability estimates that arise and examine the performance of the procedure in real and simulated data sets classifiers used include linear discriminants nearest neighbors adaptive nonlinear methods and the support vector machine

pole assignment by state transition graph the pole assignment problem is considered using the graph representation of a matrix the parametrization of controllers which have a specified characteristic polynomial is given a simple algorithm based on graphs is presented and two examples are given

quadratic stabilizability of a new class of linear systems with structural independent time varying uncertainty this paper investigates the problem of designing a linear state feedback control to stabilize a new class of single input uncertain linear dynamical systems uncertain parameters in the system matrices are time varying and bounded in given compact sets we first provide a concept called new standard system where some of the entries are required to be negative sign invariant and sign invariant and each entry varies independently in an arbitrarily large range then for a class of new standard systems we derive a necessary and sufficient condition under which a system can be quadratically stabilized by a linear control for all admissible variations of uncertainties the result extends the main result in wei 1990 ieee transactions on automatic control 35 3 268 277

pole placement by performance criterion modification a procedure is presented that modifies the state weighting matrix q and introduces a degree of relative stability into the original performance criterion to shape the resulting optimal dynamics by positioning the closed loop eigenvalues along the real axis of the optimal system in the linear quadratic regulator problem it is based on the results of the algebraic riccati equation are which establish the invariance of certain eigenspaces of the associated hamiltonian matrix with respect to certain perturbations of q and the degree of relative stability

parametric time optimal control of linear discrete time systems by state feedback part 2 irregular kronecker invariants in an earlier paper part 1 a parametric state feedback time optimal control law was devised for the regular kronecker invariants on the basis of parametrization of the vector companion form the present paper describes a straight forward technique for obtaining the parametric feedback matrix by exploiting the parametric vector companion forms in the case of irregular kronecker invariants hidden properties of vector companion forms are also given which result in entirely different and interesting techniques for parametrization of the control law an illustrative example is presented

a non parametric approach to extending generic binary classifiers for multi classification ensemble methods which combine generic binary classifier scores to generate a multi classification output are commonly used in state of the art computer vision and pattern recognition systems that rely on multi classification in particular we consider the one vs one decomposition of the multi class problem where binary classifier models are trained to discriminate every class pair we describe a robust multi classification pipeline which at a high level involves projecting binary classifier scores into compact orthogonal subspaces followed by a non linear probabilistic multi classification step using kernel density estimation kde we compare our approach against state of the art ensemble methods dcs drcw on 16 multi class datasets we also compare against the most commonly used ensemble methods vote nest on 6 real world computer vision datasets finally we measure the statistical significance of our approach using non parametric tests experimental results show that our approach gives a statistically significant improvement in multi classification performance over state of the art highlightsensemble methods combine binary classifiers to yield a multi classification output one vs one ensemble binary classifiers trained to discriminate each class pair we propose a robust non parametric probabilistic one vs one ensemble method kdemrp kdemrp improves classification performance over state of the art dcs drcw kdemrp improvements are statistically significant

parameterization of the class of deadbeat controllers via the theory of decoupling a novel approach is presented for parameterizing the class of minimum time deadbeat controllers mtdc through the minimum number of parameters the approach is based on the theory of decoupling and the properties of square decouplable systems the main result is a compact parametric form for the class of mtdc of a discrete time system contrary to many existing techniques no special assumption on the invertibility of the transition matrix is required

in defense of one vs all classification we consider the problem of multiclass classification our main thesis is that a simple one vs all scheme is as accurate as any other approach assuming that the underlying binary classifiers are well tuned regularized classifiers such as support vector machines this thesis is interesting in that it disagrees with a large body of recent published work on multiclass classification we support our position by means of a critical review of the existing literature a substantial collection of carefully controlled experimental work and theoretical arguments

a two stage linear discriminant analysis via qr decomposition linear discriminant analysis lda is a well known method for feature extraction and dimension reduction it has been used widely in many applications involving high dimensional data such as image and text classification an intrinsic limitation of classical lda is the so called singularity problems that is it fails when all scatter matrices are singular many lda extensions were proposed in the past to overcome the singularity problems among these extensions pca lda a two stage method received relatively more attention in pca lda the lda stage is preceded by an intermediate dimension reduction stage using principal component analysis pca most previous lda extensions are computationally expensive and not scalable due to the use of singular value decomposition or generalized singular value decomposition in this paper we propose a two stage lda method namely lda qr which aims to overcome the singularity problems of classical lda while achieving efficiency and scalability simultaneously the key difference between lda qr and pca lda lies in the first stage where lda qr applies qr decomposition to a small matrix involving the class centroids while pca lda applies pca to the total scatter matrix involving all training data points we further justify the proposed algorithm by showing the relationship among lda qr and previous lda methods extensive experiments on face images and text documents are presented to show the effectiveness of the proposed algorithm

quadratic stabilizability of linear systems with structural independent time varying uncertainties the author investigates the problem of designing a linear state feedback control to stabilize a class of single input uncertain linear dynamical systems the systems under consideration contain time varying uncertain parameters whose values are unknown but bounded in given compact sets the method used to establish asymptotical stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the author first shows that to ensure a stabilizable system some entries of the system matrices must be sign invariant he then derives necessary and sufficient conditions under which a system can be quadratically stabilized by a linear control for all admissible variations of uncertainties the conditions show that all uncertainties can only enter the system matrices in such a way as to form a particular geometrical pattern called an antisymmetric stepwise configuration for the systems satisfying the stabilizability conditions a computational control design procedure is also provided and illustrated via an example

robust stabilization of nonlinear cascaded systems this paper considers the problem of robust stabilization for an uncertain nonlinear system which is a cascaded interconnection of two subsystems both of the subsystems are allowed to be nonlinear multi variable and containing uncertain parameters we present a new approach to designing stabilizing controllers which assure both robust global asymptotic stability and local quadratic stability compared with existing results the assumptions required for such robust stabilizing controllers to exist are significantly simplified

paper linear ultimate boundedness control of uncertain dynamical systems given a state equation whose description includes uncertain parameters it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded when facing such a problem one often has available or assumes an upper bound for the excursions of the uncertain parameters within this framework the objective is to guarantee no matter how nature selects the uncertainty that the state will eventually end up and remain within some pre specified region when this region is a small neighborhood about the origin the concept of uniform ultimate boundedness is tantamount to practical asymptotic stability the objectives of this paper are twofold first and foremost our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control can in point of fact be solved via linear control this is the main result and is given in theorem 4 1 secondly we show that some of the so called matching assumptions which have been required in the earlier literature can be weakened somewhat this enables one to design controllers for a larger class of dynamical systems than previously considered

extreme lower probabilities we consider lower probabilities on finite possibility spaces as models for the uncertainty about the state these generalizations of classical probabilities can have some interesting properties for example k monotonicity avoiding sure loss coherence permutation invariance the sets formed by all the lower probabilities satisfying zero or more of these properties are convex we show how the extreme points and rays of these sets the extreme lower probabilities can be calculated and we give an illustration of our results

classification by pairwise coupling of imprecise probabilities we address combining imprecise classifiers in the framework of imprecise probabilities classifier outputs are probability intervals which are large for irrelevant classifiers the combination consists in computing probabilities consistent with the intervals this approach is particularly interesting when combining few classifiers in this paper we are interested in making decisions by combining classifiers providing uncertain outputs in the form of sets of probability distributions more precisely each classifier provides lower and upper bounds on the conditional probabilities of the associated classes the classifiers are combined by computing the set of unconditional probability distributions compatible with these bounds by solving linear optimization problems when the classifier outputs are inconsistent we propose a correcting step that restores this consistency the experiments show the interest of our approach for solving multi class classification problems particularly when information is scarce i e a limited number of classifiers is available in this case modeling the lack of information associated with classifier outputs gives good results even when they are poorly regularized or overfit the data

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

properties of min max controllers in uncertain dynamical systems we discuss properties of min max controllers previously obtained for uncertain dynamical systems in particular we present a new model for switching action and re examine asymptotic stability we demonstrate the attractiveness of switching surfaces and comment on the insensitivity of solutions in those surfaces to the uncertainty

stabilization of linear time invariant interval systems via constant state feedback control this paper investigates the stabilization problem of linear uncertain systems via constant state feedback control the systems under consideration contain time invariant uncertain parameters whose values are unknown but bounded in given compact sets and are thus called interval systems the criterion for the asymptotic stability of the closed loop system obtained when a state feedback control is applied is that all the eigenvalues of the resulting system matrix are in the strict left half of the complex plane first the author shows that to insure an interval system stabilizable some entries of the system matrices must be sign invariant more precisely the number of the least required sign invariant entries in system matrices is equal to the system order then the author studies the stabilizability of a set of interval systems called standard systems which contain sufficient numbers of sign invariant entries in proper locations after dividing all standard systems into some subsets by the uncertainty locations the author then derives necessary and sufficient conditions under which every system in a subset is stabilizable regardless of its parameter varying bounds the conditions show that all uncertain entries in system matrices should form a particular geometrical pattern called a generalized antisymmetric stepwise configuration for an interval system satisfying the stabilizability conditions a computational control design procedure is also provided and illustrated via an example the result is further generalized for nonstandard systems via linear transformation

analysis on robust stability for interval descriptor systems this paper considers the robust stability problem for linear continuous time descriptor systems with interval system matrices we 4rst derive some necessary conditions for the considered interval descriptor system to be regular impulse free and stable then under some constraints a necessary and su5cient condition is given by using the kronecker product and analysis c 2001 elsevier science b v all rights reserved

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

probability estimates for multi class classification by pairwise coupling pairwise coupling is a popular multi class classification method that combines all comparisons for each pair of classes this paper presents two approaches for obtaining class probabilities both methods can be reduced to linear systems and are easy to implement we show conceptually and experimentally that the proposed approaches are more stable than the two existing popular methods voting and the method by hastie and tibshirani 1998

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

on stabilization of a new class of linear time invariant interval systems via constant state feedback control the paper is concerned with stabilization problem of a new class of linear time invariant uncertain systems via linear state feedback control we derive a necessary and sufficient condition that extends a previous result in terms of system form

consistency of random forests and other averaging classifiers in the last years of his life leo breiman promoted random forests for use in classification he suggested using averaging as a means of obtaining good discrimination rules the base classifiers used for averaging are simple and randomized often based on random samples from the data he left a few questions unanswered regarding the consistency of such rules in this paper we give a number of theorems that establish the universal consistency of averaging rules we also show that some popular classifiers including one suggested by breiman are not universally consistent

structural stabilization of uncertain systems necessity of the matching condition this paper investigates one aspect of the problem of stabilizing an uncertain linear system that is the systems under consideration contain uncertain parameters which are unknown but bounded the question arises as to whether such a system can be stabilized via feedback control in some of the previous papers in this area the system is assumed to satisfy a so called matching condition this type of assumption is used to assure that the uncertain system can be stabilized it is known however that the matching condition is not a necessary condition for stabilizability this paper introduces a strengthened notion of stabilizability referred to as structural stabilizability via a nominally determined quadratic lyapunov function it is then shown that if a system is to have this stronger property the matching condition must necessarily be satisfied

on ultimate boundedness control of uncertain systems in the absence of matching assumptions this paper describes an approach to ultimate boundedness control under hypotheses less stringent than heretofore required instead of imposing the so called matching conditions of previous authors a certain decomposition of the dynamics is performed the resulting decomposed system has two parts a matched portion and a mismatched portion a certain measure of mismatch m is defined and it is shown that effective control is possible as long as m does not exceed some critical mismatch threshold m it is seen that this threshold depends on the choice of feedback hence it becomes possible to investigate the tradeoffs between the size of the mismatch threshold and the sizes of the feedback gains which achieve the desired ultimate boundedness of solutions

multiclass from binary expanding one versus all one versus one and ecoc based approaches recently there has been a lot of success in the development of effective binary classifiers although many statistical classification techniques have natural multiclass extensions some such as the support vector machines do not the existing techniques for mapping multiclass problems onto a set of simpler binary classification problems run into serious efficiency problems when there are hundreds or even thousands of classes and these are the scenarios where this paper s contributions shine we introduce the concept of correlation and joint probability of base binary learners we learn these properties during the training stage group the binary leaner s based on their independence and with a bayesian approach combine the results to predict the class of a new instance finally we also discuss two additional strategies one to reduce the number of required base learners in the multiclass classification and another to find new base learners that might best complement the existing set we use these two new procedures iteratively to complement the initial solution and improve the overall performance this paper has two goals finding the most discriminative binary classifiers to solve a multiclass problem and keeping up the efficiency i e small number of base learners we validate and compare the method with a diverse set of methods of the literature in several public available datasets that range from small 10 to 26 classes to large multiclass problems 1000 classes always using simple reproducible scenarios