new results on composite control of singularly perturbed uncertain linear systems abstract we consider the problem of stabilizing uncertain singularly perturbed linear time varying systems whose fast dynamics can be unstable we propose a class of nonlinear composite controllers which assure global uniform ultimate boundedness of the trajectories of the closed loop system provided the singular perturbation parameter is sufficiently small these controllers consist of a fast controller which stabilizes the fast dynamics and a slow controller which yields the desired stability properties for the slow dynamics we consider the structure of the fast controller to be simpler than structures previously proposed in the literature to obtain these controllers we first develop some new results for singularly perturbed systems under output feedback in particular it is shown that the matching assumption which deals with the manner in which the uncertainties enter the system and is made on the reduced order system is invariant under linear feedback of the fast variable

constructing descriptive and discriminative nonlinear features rayleigh coefficients in kernel feature spaces we incorporate prior knowledge to construct nonlinear algorithms for invariant feature extraction and discrimination employing a unified framework in terms of a nonlinearized variant of the rayleigh coefficient we propose nonlinear generalizations of fisher s discriminant and oriented pca using support vector kernel functions extensive simulations show the utility of our approach

a sequential algorithm for sparse support vector classifiers support vector machines svms though accurate are not preferred in applications requiring high classification speed or when deployed in systems of limited computational resources due to the large number of support vectors involved in the model to overcome this problem we have devised a primal svm method with the following properties 1 it solves for the svm representation without the need to invoke the representer theorem 2 forward and backward selections are combined to approach the final globally optimal solution and 3 a criterion is introduced for identification of support vectors leading to a much reduced support vector set in addition to introducing this method the paper analyzes the complexity of the algorithm and presents test results on three public benchmark problems and a human activity recognition application these applications demonstrate the effectiveness and efficiency of the proposed algorithm

robustness of dynamic output feedback control for singular perturbation systems abstract the problem of the robustness of dynamic output feedback control for singular perturbation systems is investigated the solution to this problem is reduced to the simultaneous design of static output feedback controllers for the fast subsystem and the so called auxiliary system of the slow subsystem some conditions are proposed to ensure the robustness of the actual closed loop system the exact upper bound of the parasitic parameter for the controlled system is also determined finally an actual model which failed in dynamic output feedback control in 4 is reexamined successfully here

a further note on the robustness of output feedback control methods to modeling errors in an earlier note it was shown that designing a nonstrictly proper output feedback compensator using a model that neglects parasitic elements might lead to an unstable closed loop system even when the neglected modes are asymptotically stable the purpose of this note is to show that this instability phenomenon might take place even when the reduced order design model is strictly proper

robust stabilization of singularly perturbed systems abstract we discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models in 1 it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the true system containing both fast and slow modes this phenomenon was then expanded to include the case where the simplified system is strictly proper in 2 the objectives of this note are threefold i to show that given any proper system and any stabilizing controller for it that is proper but not strictly proper there exists a singular perturbation of the system that is destabilized by that controller ii to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter and iii to provide a characterization in the same spirit as 3 4 of the set of all strictly proper controllers that stabilize a given proper plant by combining these results it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system

a sampled data control scheme for stabilization of systems with unmodeled high frequency dynamics the output feedback stabilization problem is considered for linear sampled data control systems in the presence of unmodeled high frequency dynamics sufficient conditions for stabilization of such systems were given by m vidyasagar 1985 for strictly proper systems this paper extends the class of systems in order to include the proper systems case and proposes a conceptually novel digital control scheme in this scheme the control law based on the reduced order system is applied to the full system only during a short fraction of the sampling period during the second part of the sampling period the control is taken equal to zero such that without control the neglected fast states have sufficient time to converge to their equilibrium point

a new stable compensator design for exact and approximate loop transfer recovery abstract in this paper a new compensator structure for loop transfer recovery ltr is proposed the proposed compensator a is open loop stable b guarantees closed loop stability and above all c requires much smaller values of gain than the conventional observer based controller for the same degree of loop transfer recovery the fact that the new compensator requires much smaller values of gain than the conventional controller results in several practical advantages the most important among them being the reduction in controller band width and freedom from the woes of saturation the trade off between the value of gain and the degree of loop transfer recovery as well as the bounds on singular values of sensitivity and complementary sensitivity functions is shown clearly both full and reduced order compensators for ltr when the design specifications are reflected either at the input or at the output point of the given plant are considered numerical examples illustrate the advantages of the new compensator structure the new compensator structure is inspired by a careful and clear understanding of how loop transfer recovery occurs when conventional observer based controllers are used to motivate and deduce our new compensator structure a unified treatment of observer theory for ltr is presented in the context of such a unification some new results are also given

near optimal output feedback regulation of ill conditioned linear systems a two time scale approximation for the linear quadratic optimal output feedback regulator program is examined necessary conditions for optimality as well as an algorithm for computing locally near optimal gains are derived if it is assumed that the slow and fast subsystem initial conditions are uniformly distributed optimal gains for the two time scale problem provide a second order approximation to optimal closed loop performance in the unperturbed system this is verified with a numerical example

hierarchical mixing linear support vector machines for nonlinear classification support vector machines svms play a very dominant role in data classification because of their good generalization performance however they suffer from the high computational complexity in the classification stage when there are a considerable number of support vectors svs it is desirable to design efficient algorithms in the classification stage to deal with datasets obtained from real time pattern recognition systems to this end we propose a novel classifier called hmlsvms hierarchical mixing linear support vector machines which has a hierarchical structure with a mixing linear svm classifier at each node it predicts the label of a sample using only a few hyperplanes we also give a generalization error bound for the class of locally linear svms llsvms based on the rademacher theory which ensures that overfitting can be effectively avoided experimental evaluations show that the proposed classifier achieves a high efficiency in the classification stage while the classification performance approaches that of kernel svms highlightsh mlsvms does not require a large memory to store kernel values in the training process because h mlsvms is composed of lsvms and there are very efficient algorithms for training lsvms 48 49 the hierarchical structure makes h mlsvms predict the labels of new arrived samples via a few of lsvms and it is much faster than nonlinear svms classifiers we quantify the generalization error bound for the class of llsvms based on the rademacher complexity and the stop criterion based on minimizing this bound ensures that h mlsvms can effectively avoid overfitting and have a good classification performance

robust redesign of adaptive control effects of unmodeled high frequency dynamics on stability and performance of adaptive control schemes are analyzed in the regulation problem global stability properties are no longer guaranteed but a region of attraction exists for exact adaptive regulation the dependence of the region of attraction on unmodeled parasitics is examined first then the general case of model reference adaptive control is considered in which parasitics can destroy stability and boundedness properties a modified adaptive law is proposed guaranteeing the existence of a region of attraction from which all signals converge to a residual set which contains the equilibrium for exact tracking the size of this set depends on design parameters the frequency range of parasitics and the reference input signal characteristics

sparseness of support vector machines support vector machines svms construct decision functions that are linear combinations of kernel evaluations on the training set the samples with non vanishing coefficients are called support vectors in this work we establish lower asymptotical bounds on the number of support vectors on our way we prove several results which are of great importance for the understanding of svms in particular we describe to which limit svm decision functions tend discuss the corresponding notion of convergence and provide some results on the stability of svms using subdifferential calculus in the associated reproducing kernel hilbert space

on the robustness of output feedback control methods to modeling errors singular perturbation analysis is employed to study the robustness of output feedback control schemes for linear time invariant systems the effect of designing the control strategy using a model that neglects unknown parasitic elements is considered it is shown that such a strategy may in general be destabilizing even when the neglected modes are asymptotically stable special structures that will not permit this situation and will guarantee asymptotic stability are illustrated

a study on reduced support vector machines recently the reduced support vector machine rsvm was proposed as an alternate of the standard svm motivated by resolving the difficulty on handling large data sets using svm with nonlinear kernels it preselects a subset of data as support vectors and solves a smaller optimization problem however several issues of its practical use have not been fully discussed yet for example we do not know if it possesses comparable generalization ability as the standard svm in addition we would like to see for how large problems rsvm outperforms svm on training time in this paper we show that the rsvm formulation is already in a form of linear svm and discuss four rsvm implementations experiments indicate that in general the test accuracy of rsvm are a little lower than that of the standard svm in addition for problems with up to tens of thousands of data if the percentage of support vectors is not high existing implementations for svm is quite competitive on the training time thus from this empirical study rsvm will be mainly useful for either larger problems or those with many support vectors experiments in this paper also serve as comparisons of 1 different implementations for linear svm and 2 standard svm using linear and quadratic cost functions

simple and robust methods for support vector expansions most support vector sv methods proposed in the recent literature can be viewed in a unified framework with great flexibility in terms of the choice of the kernel functions and their constraints we show that all these problems can be solved within a unique approach if we are equipped with a robust method for finding a sparse solution of a linear system moreover for such a purpose we propose an iterative algorithm that can be simply implemented finally we compare the classical sv approach with other recently proposed cross correlation based alternative methods the simplicity of their implementation and the possibility of exactly calculating their computational complexity constitute important advantages in a real time signal processing scenario

an equivalence between sparse approximation and support vector machines in the first part of this paper we show a similarity between the principle of structural risk minimization principle srm vapnik 1982 and the idea of sparse approximation as defined in chen donoho and saunders 1995 and olshausen and field 1996 then we focus on two specific approximate implementations of srm and sparse approximation which have been used to solve the problem of function approximation for srm we consider the support vector machine technique proposed by v vapnik and his team at at t bell labs and for sparse approximation we consider a modification of the basis pursuit de noising algorithm proposed by chen donoho and saunders 1995 we show that under certain conditions these two techniques are equivalent they give the same solution and they require the solution of the same quadratic programming problem

output feedback control of linear two time scale systems output feedback control of linear time invariant singularly perturbed systems is studied the set of all compensators that stabilize a singularly perturbed system while preserving its two time scale structure is parameterized the parameterization is used to show that any two frequency scale stabilizing compensator can be asymptotically approximated by a compensator designed via a sequential procedure in this procedure a fast high frequency compensator is designed first to stabilize the fast model of the system then a strictly proper slow low frequency compensator is designed to stabilize a modified slow model the parallel connection of the two compensators forms a two frequency scale stabilizing compensator for the singularly perturbed system

building sparse multiple kernel svm classifiers the support vector machines svms have been very successful in many machine learning problems however they can be slow during testing because of the possibly large number of support vectors obtained recently wu 2005 proposed a sparse formulation that restricts the svm to use a small number of expansion vectors in this paper we further extend this idea by integrating with techniques from multiple kernel learning mkl the kernel function in this sparse svm formulation no longer needs to be fixed but can be automatically learned as a linear combination of kernels two formulations of such sparse multiple kernel classifiers are proposed the first one is based on a convex combination of the given base kernels while the second one uses a convex combination of the so called ldquoequivalentrdquo kernels empirically the second formulation is particularly competitive experiments on a large number of toy and real world data sets show that the resultant classifier is compact and accurate and can also be easily trained by simply alternating linear program and standard svm solver

multiple kernel learning algorithms in recent years several methods have been proposed to combine multiple kernels instead of using a single one these different kernels may correspond to using different notions of similarity or may be using information coming from multiple sources different representations or different feature subsets in trying to organize and highlight the similarities and differences between them we give a taxonomy of and review several multiple kernel learning algorithms we perform experiments on real data sets for better illustration and comparison of existing algorithms we see that though there may not be large differences in terms of accuracy there is difference between them in complexity as given by the number of stored support vectors the sparsity of the solution as given by the number of used kernels and training time complexity we see that overall using multiple kernels instead of a single one is useful and believe that combining kernels in a nonlinear or data dependent way seems more promising than linear combination in fusing information provided by simple linear kernels whereas linear methods are more reasonable when combining complex gaussian kernels

a sparsity driven kernel machine based on minimizing a generalization error bound a new sparsity driven kernel classifier is presented based on the minimization of a recently derived data dependent generalization error bound the objective function consists of the usual hinge loss function penalizing training errors and a concave penalty function of the expansion coefficients the problem of minimizing the non convex bound is addressed by a successive linearization approach whereby the problem is transformed into a sequence of linear programs the algorithm produced comparable error rates to the standard support vector machine but significantly reduced the number of support vectors and the concomitant classification time

on output feedback control of singularly perturbed systems we treat the problem of robustness of output feedback controllers with respect to singular perturbations given a singularly perturbed control system whose boundary layer system is exponentially stable and whose reduced order system is exponentially stabilizable via a possibly dynamical output feedback controller we present a sufficient condition which ensures that the system obtained by applying the same controller to the original full order singularly perturbed control system is exponentially stable for sufficiently small values of the perturbation parameter this condition which is less restrictive than those previously given in the literature is shown to be always satisfied when the singular perturbation is due to the presence of fast actuators and or sensors furthermore we show explicitly that in the linear time invariant case if this condition is not satisfied then there exists an output feedback controller which stabilizes the reduced order system but destabilizes the full order system

numerical fixed point solution for near optimum regulators of linear quadratic gaussian control problems for singularly perturbed systems the fixed point method is developed for obtaining an efficient numerical solution of the linear quadratic gaussian problem for singularly perturbed systems it is shown that each iteration step improves the accuracy by an order of magnitude that is the accuracy 0 ek can be obtained by performing only k 1 iterations in addition only low order systems are involved in algebraic manipulations and no analyticity requirements are imposed on the system coefficients

exact decomposition of the algebraic riccati equation of deterministic multimodeling optimal control problems in this paper we show how to exactly decompose the algebraic riccati equations of deterministic multimodeling in terms of one pure slow and two pure fast algebraic riccati equations the algebraic riccati equations obtained are of reduced order and nonsymmetric however their o spl epsiv perturbations where spl epsiv spl par sub spl epsiv 2 sup spl epsiv 1 spl par and spl epsiv sub 1 spl epsiv sub 2 are small positive singular perturbation parameters are symmetric the newton method is perfectly suited for solving the nonsymmetric reduced order pure slow and pure fast algebraic riccati equations since excellent initial guesses are available from their o spl epsiv perturbed reduced order symmetric algebraic riccati equations that can be solved rather easily the proposed decomposition scheme might facilitates new approaches to multimodeling control problems that are conceptually simpler and numerically more efficient than the ones previously used

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

extended polynomial growth transforms for design and training of generalized support vector machines growth transformations constitute a class of fixed point multiplicative update algorithms that were originally proposed for optimizing polynomial and rational functions over a domain of probability measures in this paper we extend this framework to the domain of bounded real variables which can be applied towards optimizing the dual cost function of a generic support vector machine svm the approach can therefore not only be used to train traditional soft margin binary svms one class svms and probabilistic svms but can also be used to design novel variants of svms with different types of convex and quasi convex loss functions in this paper we propose an efficient training algorithm based on polynomial growth transforms and compare and contrast the properties of different svm variants using several synthetic and benchmark data sets the preliminary experiments show that the proposed multiplicative update algorithm is more scalable and yields better convergence compared to standard quadratic and nonlinear programming solvers while the formulation and the underlying algorithms have been validated in this paper only for svm based learning the proposed approach is general and can be applied to a wide variety of optimization problems and statistical learning models

on the robustness of sampled data control to unmodeled high frequency dynamics robustness of sampled data control designs to unmodeled high frequency dynamics is studied using singular perturbation theory it is argued that when the plant is preceded by a zero order hold a direct transmission term of the reduced order model which results from neglecting high frequency dynamics should be modeled as a delay element in order to ensure robustness

bounds on the optimum quadratic cost of structure constrained controllers in this note lower and upper bounds are derived on the quadratic cost of optimal output feedback controllers subject to constraints on the controller structure the use of these bounds in forming a comparative assessment of alternative controller confugurations is described and an illustrative example is given the principal advantage of these bounds arises from the fact that they can save considerable computational effort

reduced support vector machines a statistical theory in dealing with large data sets the reduced support vector machine rsvm was proposed for the practical objective to overcome some computational difficulties as well as to reduce the model complexity in this paper we study the rsvm from the viewpoint of sampling design its robustness and the spectral analysis of the reduced kernel we consider the nonlinear separating surface as a mixture of kernels instead of a full model the rsvm uses a reduced mixture with kernels sampled from certain candidate set our main results center on two major themes one is the robustness of the random subset mixture model the other is the spectral analysis of the reduced kernel the robustness is judged by a few criteria as follows 1 model variation measure 2 model bias deviation between the reduced model and the full model and 3 test power in distinguishing the reduced model from the full one for the spectral analysis we compare the eigenstructures of the full kernel matrix and the approximation kernel matrix the approximation kernels are generated by uniform random subsets the small discrepancies between them indicate that the approximation kernels can retain most of the relevant information for learning tasks in the full kernel we focus on some statistical theory of the reduced set method mainly in the context of the rsvm the use of a uniform random subset is not limited to the rsvm this approach can act as a supplemental algorithm on top of a basic optimization algorithm wherein the actual optimization takes place on the subset approximated data the statistical properties discussed in this paper are still valid

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

parallel reduced order controllers for stochastic linear singularly perturbed discrete systems an approach to the decomposition and approximation of linear quadratic gaussian control problems for singularly perturbed discrete systems at steady state is presented the global kalman filter is decomposed into separate reduced order local filters through the use of a decoupling transformation a near optimal control law is derived by approximating coefficients of the optimal control law the proposed method allows parallel processing of information and reduces offline and online computational requirements a real world example demonstrates the efficiency of the proposed method

the recursive algorithm for the optimal static output feedback control problem of linear singularity perturbed systems a recursive algorithm is developed for solving the algebraic equations comprising the solution of the optimal static output feedback control problem of singularly perturbed linear systems the algorithm is very efficient from the numerical point of view since only low order systems are involved in algebraic calculations and the required solution can be easily obtained up to an arbitrary order of accuracy that is o epsilon sup k where epsilon is a small perturbation parameter the real world example demonstrates the failure of o epsilon theory used so far in the study of this problem and the necessity for the existence of o epsilon sup k theory

a new approach to robust guaranteed cost control for uncertain multimodeling systems the guaranteed cost control problem for multimodeling systems with norm bounded uncertainty is investigated the main contribution in this paper is that a new independent controller is derived by solving the reduced order slow and fast algebraic riccati equations ares whose dimension is smaller than the dimension of full order multiparameter algebraic riccati equation mare it is shown that if these ares have a positive definite stabilizing solution then the closed loop system is quadratically stable and has the cost bound

a robust minimax approach to classification when constructing a classifier the probability of correct classification of future data points should be maximized we consider a binary classification problem where the mean and covariance matrix of each class are assumed to be known no further assumptions are made with respect to the class conditional distributions misclassification probabilities are then controlled in a worst case setting that is under all possible choices of class conditional densities with given mean and covariance matrix we minimize the worst case maximum probability of misclassification of future data points for a linear decision boundary this desideratum is translated in a very direct way into a convex second order cone optimization problem with complexity similar to a support vector machine problem the minimax problem can be interpreted geometrically as minimizing the maximum of the mahalanobis distances to the two classes we address the issue of robustness with respect to estimation errors in the means and covariances of the classes via a simple modification of the input data we also show how to exploit mercer kernels in this setting to obtain nonlinear decision boundaries yielding a classifier which proves to be competitive with current methods including support vector machines an important feature of this method is that a worst case bound on the probability of misclassification of future data is always obtained explicitly

convergence of a numerical algorithm for calculating optimal output feedback gains a sequential numerical algorithm is described which obtains gains minimizing a broad class of performance indexes including the standard lq case the primary contribution is a proof that the algorithm converges to a local minimum under nonrestrictive assumptions numerical examples illustrate the theory the second example demonstrates an important lq design technique which permits the designer to prespecify the feedback structure subject to the requirement of output feedback stabilizability

regularized vector field learning with sparse approximation for mismatch removal in vector field learning regularized kernel methods such as regularized least squares require the number of basis functions to be equivalent to the training sample size n the learning process thus has o n 3 and o n 2 in the time and space complexity respectively this poses significant burden on the vector learning problem for large datasets in this paper we propose a sparse approximation to a robust vector field learning method sparse vector field consensus sparsevfc and derive a statistical learning bound on the speed of the convergence we apply sparsevfc to the mismatch removal problem the quantitative results on benchmark datasets demonstrate the significant speed advantage of sparsevfc over the original vfc algorithm two orders of magnitude faster without much performance degradation we also demonstrate the large improvement by sparsevfc over traditional methods like ransac moreover the proposed method is general and it can be applied to other applications in vector field learning

learning sparse kernel classifiers for multi instance classification we propose a direct approach to learning sparse kernel classifiers for multi instance mi classification to improve efficiency while maintaining predictive accuracy the proposed method builds on a convex formulation for mi classification by considering the average score of individual instances for bag level prediction in contrast existing formulations used the maximum score of individual instances in each bag which leads to nonconvex optimization problems based on the convex mi framework we formulate a sparse kernel learning algorithm by imposing additional constraints on the objective function to enforce the maximum number of expansions allowed in the prediction function the formulated sparse learning problem for the mi classification is convex with respect to the classifier weights therefore we can employ an effective optimization strategy to solve the optimization problem that involves the joint learning of both the classifier and the expansion vectors in addition the proposed formulation can explicitly control the complexity of the prediction model while still maintaining competitive predictive performance experimental results on benchmark data sets demonstrate that our proposed approach is effective in building very sparse kernel classifiers while achieving comparable performance to the state of the art mi classifiers

relevance units latent variable model and nonlinear dimensionality reduction a new dimensionality reduction method called relevance units latent variable model rulvm is proposed in this paper rulvm has a close link with the framework of gaussian process latent variable model gplvm and it originates from a recently developed sparse kernel model called relevance units machine rum rum follows the idea of relevance vector machine rvm under the bayesian framework but releases the constraint that relevance vectors rvs have to be selected from the input vectors rum treats relevance units rus as part of the parameters to be learned from the data as a result a rum maintains all the advantages of rvm and offers superior sparsity rulvm inherits the advantages of sparseness offered by the rum and the experimental result shows that rulvm algorithm possesses considerable computational advantages over gplvm algorithm

frequency domain results for systems with slow and fast dynamics the mathematical treatment of systems with slow and fast dynamics has traditionally involved singular perturbation theory for differential equations this paper suggests a set of conditions to be placed on a frequency domain description of a system to guarantee two time scale behavior some basic stability and approximation results are presented

technical communique on the klimushchev krasovskii theorem the klimushchev krasovskii theorem describes a sufficient condition for the asymptotic stability of singularly perturbed systems motivated by some subtle examples this note presents a new necessary and sufficient condition by imposing an additional assumption on the statement of the klimushchev krasovskii theorem this note also extends the main inertia theorem to singularly perturbed systems

robust stabilization for composite observer based control of discrete systems abstract design criteria of robust observer based controllers for linear discrete time singularly perturbed systems with slow and fast modes are considered the developed stability measure complements the quantitative information for unmodeled fast modes that cannot be provided by the usual qualitative robustness analyses a numerical example is provided to show the analytical results obtained

paper optimal output feedback design of systems with ill conditioned dynamics singular perturbation concepts are exploited to develop a procedure for designing a constant gain output feedback control system it is assumed that the original system is ill conditioned in the sense that the plant contains widely separated dynamics and that an accurate description for the high frequency behavior may not be available the design procedure attempts to stabilize the system by minimizing a quadratic cost function made up of the control and states associated with a reduced order low frequency model for the plant and a measure of stability for the neglected fast dynamics the resulting design procedure does not require knowledge of the fast dynamics

linear compensator designs based exclusively on input output information are never robust with respect to unmodeled dynamics the effects of unmodeled higher order dynamics or parasitics on the stability of linear control systems are investigated a description is first given of a class of perturbations of a given state equation that cannot be distinguished from the original on the basis of input output measurements alone then it is shown that given any plant compensator pair perturbations of each system of this type can always be found that destabilize the closed loop configuration the effect of destabilizing perturbations on output behavior is explored

applications of singular perturbation techniques to control problems this paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years the first three sections are devoted to the standard model and its time scale stability and controllability properties the next two sections deal with linear quadratic optimal control and one with cheap near singular control then the composite control and trajectory optimization are considered in two sections and stochastic control in one section the last section returns to the problem of modeling this time in the context of large scale systems the bibliography contains more than 250 titles

recursive algorithm for the optimal static output feedback control problem of linear weakly coupled systems a recursive algorithm is developed for solving the algebraic equations comprising the solution of the optimal static output feedback control problem of linear weakly coupled systems the proposed algorithm is very efficient from the numerical point of view since only low order systems are involved in algebraic calculations and the required solution can easily be obtained up to an arbitrary order of accuracy that is 0 e2k where e is a small weak coupling parameter the real world example demonstrates the effectiveness of the proposed reduced order algorithm and its advantages over the global full order algorithm

paper instability analysis and improvement of robustness of adaptive control the effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed five possible types of instability mechanisms parameter drift linear instability fast adaptation instability high frequency instability and throughput instability are analyzed using simple examples a procedure is used to construct lyapunov like functions for a modified adaptive controller applied to a dominant plant of relative degree one in the presence of parasitics and disturbances and obtain sufficient conditions under which none of the five types of instability can occur the modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set the size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast the region of attraction becomes the whole space

online kernel principal component analysis a reduced order model kernel principal component analysis kernel pca is an elegant nonlinear extension of one of the most used data analysis and dimensionality reduction techniques the principal component analysis in this paper we propose an online algorithm for kernel pca to this end we examine a kernel based version of oja s rule initially put forward to extract a linear principal axe as with most kernel based machines the model order equals the number of available observations to provide an online scheme we propose to control the model order we discuss theoretical results such as an upper bound on the error of approximating the principal functions with the reduced order model we derive a recursive algorithm to discover the first principal axis and extend it to multiple axes experimental results demonstrate the effectiveness of the proposed approach both on synthetic data set and on images of handwritten digits with comparison to classical kernel pca and iterative kernel pca

robustness of linear feedback control systems to unmodelled high frequency dynamics a basic issue in the control of linear time invariant plants is the effect of neglected high frequency dynamics on the performance in particular on the closed loop stability of the control system in this paper the robustness of various output feedback control designs based on a reduced order model with neglected high frequency dynamics is investigated using singular perturbation techniques a general robust design rule is to avoid static output feedback for systems with unmodelled high frequency dynamics from a frequency domain standpoint the robust design rule is to avoid closing high frequency plant loops by using strictly proper controllers or controllers with a low pass filtering property

optimal output feedback control of discrete linear singularly perturbed stochastic systems the static output feedback control problem for discrete linear singularly perturbed stochastic systems is studied a recursive algorithm is presented to solve the corresponding coupled non linear algebraic equations the algorithm removes the ill conditioning by decomposing the higher order equations into lower order equations and develops a technique which enables us to obtain an arbitrary accuracy that is 0 ej j 1 2 3 a approximation for this problem a numerical example is presented to support the theoretical results communicated by professor a a ioannou

singular perturbation of strictly positive realness abstract the robustness of the strictly positive real spr condition of linear time invariant multi input multi output lti mimo singularly perturbed systems is examined a necessary and sufficient condition under which the full system is spr is proved this robustness result is an extension of the analysis for the popov kalman yakubovich lemma by v r saksena and p v kokotovic 7

fast suboptimal solution to the static output control problem of linear singularly perturbed systems abstract the reduced order suboptimal solution to the static output control problem of linear singularly perturbed system is obtained in terms of the fast variables only assuming the special structure of initial conditions for the slow and fast variables the problem of big initial conditions of the fast variables is considered

suboptimal constant output feedback and its application to modern flight control system design this paper describes an applications oriented approach to the generation of optimal output feedback gains for linear time invariant systems which is independent of the open loop stability the standard requirement for the provision of initial stabilizing output feedback gains for priming the computational process is circumvented in lieu of initial stabilizing gains the proposed algorithm employs the full state feedback solution which guarantees stability under the mild condition of stabilizabitity in this paper the problem of the generation of sub optimal output feedback is cast in the setting of a constrained parameter optimization problem the solution of this constrained optimization employs hestenes method of multipliers with some modifications a primal dual problem is considered where the primal minimization employs a davidon fletcher powell method and the dual maximization is accomplished via a quasi newton procedure this approach provides the designer with a means of suppressing to zero sele

feedback systems and multiple time scales we study feedback design for systems with multiple time scale structure uncovering an intrinsic time scale structure for a state feedback system with full state output from this we construct a multiple time scale asymptotic observer and consider a system with state feedback via a two time scale asymptotic observer finally we obtain more general results for a restricted class of systems

the discrete time h sub spl infin control problem with strictly proper measurement feedback this paper is concerned with the discrete time h sub spl infin control problem with strictly proper measurement feedback we derive necessary and sufficient conditions for the existence of a strictly proper compensator which achieves a given h sub spl infin norm bound note that contrary to the continuous time in discrete time there might exist a suitable proper compensator but no suitable strictly proper compensator finally we give an explicit formula for one controller which achieves this bound

on the design of stabilizing controllers for singularly perturbed systems the robustness of output feedback control designs for singularly perturbed systems based on the reduced systems is discussed a frequency domain approach is presented to determine the conditions for the stability of linear time invariant systems subject to neglected high frequency dynamics in contrast with the qualitative analyses for robust stability that have appeared in the literature this approach gives an explicit computable bound on unmodeled dynamics which does not destabilize the systems stability conditions for various feedback control schemes are presented it is noted that the approach extended to systems with mixed singular regular perturbations can be used to derive the robust stability condition

a direct method for building sparse kernel learning algorithms many kernel learning algorithms including support vector machines result in a kernel machine such as a kernel classifier whose key component is a weight vector in a feature space implicitly introduced by a positive definite kernel function this weight vector is usually obtained by solving a convex optimization problem based on this fact we present a direct method to build sparse kernel learning algorithms by adding one more constraint to the original convex optimization problem such that the sparseness of the resulting kernel machine is explicitly controlled while at the same time performance is kept as high as possible a gradient based approach is provided to solve this modified optimization problem applying this method to the support vectom machine results in a concrete algorithm for building sparse large margin classifiers these classifiers essentially find a discriminating subspace that can be spanned by a small number of vectors and in this subspace the different classes of data are linearly well separated experimental results over several classification benchmarks demonstrate the effectiveness of our approach

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods

study of the discrete singularly perturbed linear quadratic control problem by a bilinear transformation this paper presents a new approach in the study of the linear quadratic control problem of singularly perturbed discrete systems by applying a bilinear transformation the algebraic discrete riccati equation is converted into a continuous one which can be solved by using the reduced order recursive method already documented in the control literature this method produces the reduced order near optimal solution up to an arbitrary order of accuracy and reduces the size of required computations the method is very suitable for parallel programming a real world example an f 8 aircraft demonstrates the efficiency of the proposed method