a stability result for linear parameter varying systems abstract a negative real part eigenvalue assumption on a linear parameter varying system implies exponential stability under a condition that includes either slowly varying parameters or slowly accelerating parameters

singularity conditions for the non existence of a common quadratic lyapunov function for pairs of third order linear time invariant dynamic systems abstract the condition that a finite collection of stable matrices a 1 a m has no common quadratic lyapunov function cqlf is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters these conditions are applied to the case of two stable 3 3 matrices where they are used to derive necessary and sufficient conditions for the non existence of a cqlf

a linear programming oriented procedure for quadratic stabilization of uncertain systems abstract this paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty a constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain some experiments are presented

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

hyperelliptic curves and their invariants geometric arithmetic and algorithmic aspects we apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and conversely propose algorithms for reconstructing hyperelliptic models from given invariants we focus on genus 3 hyperelliptic curves both geometric and arithmetic aspects are considered

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

properties of predictors for autoregressive time series abstract the prediction of the n s th observation of the pth order autoregressive process is investigated the mean square of the predictor error through terms of order n 1 conditional on yn y n 1 y n p 1 is obtained for the stationary normal process the mean squared error expression is similar to the usual regression formula for the variance of the predictor error the usual regression formula for the estimated variance of a predictor error and its generalization to s period prediction is shown to provide a consistent estimator of the mean squared error of the least squares predictor for both stationary and non stationary processes

a noniterative method for the design of linear robust controllers a procedure for determining a linear control law which guarantees asymptotic stability for an uncertain system is derived using an elementary matrix identity an alternative proof of the existence of stabilizing control law for the matched case is provided bounds for the rates of decay are established and by using these bounds it is shown that every trajectory of the closed loop system can be made to decay at a prespecified exponential rate the technique is simpler to use than existing techniques since it does not require a numerical procedure but allows the control law to be obtained from some simple formulas

inverse moment bounds for sample autocovariance matrices based on detrended time series and their applications abstract in this paper we assume that observations are generated by a linear regression model with short or long memory dependent errors we establish inverse moment bounds for k n dimensional sample autocovariance matrices based on the least squares residuals also known as the detrended time series where k n n k n and n is the sample size these results are then used to derive the mean square error bounds for the finite predictor coefficients of the underlying error process based on the detrended time series we further estimate the inverse of the n dimensional autocovariance matrix r n 1 of the error process using the banded cholesky factorization by making use of the aforementioned inverse moment bounds we obtain the convergence of moments of the difference between the proposed estimator and r n 1 under spectral norm

necessary and sufficient conditions for regional stabilisability of generic switched linear systems with a pair of planar subsystems in this article the regional stabilisability issues of a pair of planar lti systems are investigated through the geometrical approach and easily verifiable necessary and sufficient conditions are derived the main idea of the article is to characterise the best case switching signals based upon the variations of the constants of the integration of the subsystems the conditions are generic as all possible combinations of the subsystem dynamics are considered

design of controllers for linear parameter varying systems by the gain scheduling technique abstract design of controllers for linear multi input multi output systems whose dynamics depend on a time varying parameter is studied design algorithms for the state feedback control law based on the gain scheduling technique are proposed state feedback control laws are designed so that one of the following goals is achieved i the poles of the closed loop system for all fixed parameter values are placed in a small neighborhood of a set of desired closed loop poles ii the stability of the closed loop system for all fixed parameter values guarantees that the of the system with the time varying parameter independent of the rate of change of the parameter rigorously justified and computationally appealing design algorithms are developed

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

robust performance of linear parametrically varying systems using parametrically dependent linear feedback abstract in this paper a parameter dependent control problem for linear parametrically varying lpv systems is presented sufficient conditions are given that guarantee an lpv system is exponentially stable and achieves an induced l 2 norm performance objective from the disturbance to error signals the usefulness of parameter dependent controllers is motivated from a gain scheduling viewpoint the resulting synthesis problem is reformulated into a convex optimization problem which can be solved using efficient new algorithms

common polynomial lyapunov functions for linear switched systems in this paper we consider linear switched systems dot x t a u t x t x in r n u in u a u u in u compact and the problem of asymptotic stability for arbitrary switching functions uniform with respect to switching uas given a uas system it is always possible to build a common polynomial lyapunov function our main result is that the degree of that common polynomial lyapunov function is not uniformly bounded over all the uas systems this result answers a question raised by dayawansa and martin a generalization to a class of piecewise polynomial lyapunov functions is given

on the performance of nonlinear dynamical systems under parameter perturbation we present a method for analysing the deviation in transient behaviour between two parameterised families of nonlinear odes as initial conditions and parameters are varied within compact sets over which stability is guaranteed this deviation is taken to be the integral over time of a user specified positive definite function of the difference between the trajectories for instance the l 2 norm we use sum of squares programming to obtain two polynomials which take as inputs the possibly differing initial conditions and parameters of the two families of odes and output upper and lower bounds to this transient deviation equality can be achieved using symbolic methods in a special case involving linear time invariant parameter dependent systems we demonstrate the utility of the proposed methods in the problems of model discrimination and location of worst case parameter perturbation for a single parameterised family of ode models

hierarchically supervised control of linear plants with unpredictable mode switch dynamics a mode switch process is defined allowing a linear piecewise time invariant plant to switch from time to time from the actual configuration to another one of a given family of linear time invariant plants when and where switching occurs is not a priori known it is required to find a stabilizing controller yielding an output response with some desired transient specifications the solution proposed is given by the connection of multiple families of linear controllers with a hierarchically supervised switching scheme

a riccati equation approach to the design of linear robust controllers abstract in this paper by the lyapunov stability criterion and the riccati equation we derive a new procedure for determining a linear control law to stabilize an uncertain system the main features of this approach are that no precompensator is needed the required feedback gains are small and the uncertain system can be exponentially stabilized with any prescribed exponential rate in addition the robust control law can be made linear quadratic optimal with respect to an index function for the nominal system two examples are given to illustrate the results

effects of not knowing the order of an autoregressive process on the mean squared error of prediction i abstract given a realization of t consecutive observations of an autoregressive process x of unknown order m a process of an arbitrary order p is assumed to be fitted for predicting the future values of another process xtilde t where xtilde t is independent of xt but otherwise has identical probabilistic structure the mean squared error of prediction when the autoregressive coefficients are estimated by least squares is obtained to terms of order t 1 the bias in estimating the autoregressive coefficients is examined the results are illustrated for three processes with m 2 by considering fitting of a process with p 1

properties of predictors in misspecified autoregressive time series models abstract this article investigates major effects of misspecification in stationary linear time series models when we fit a pth order autoregressive model the true model can be an autoregressive moving average model we derive the formulas of bias and mean squared error mse of the least squares estimator and the hth period ahead prediction mse in the time domain contrary to previous studies the process in estimation is not necessarily independent of the process in prediction and the distribution of process is not necessarily gaussian we examine the effects of this dependence and nonnormality on prediction in misspecified models

sufficient conditions for stability of linear time varying systems abstract in this paper we consider sufficient conditions for the exponential stability of linear time varying systems of the form dot x t a t x t t 0 stability guaranteeing upper bounds for different measures of parameter variations are derived

convergence of gaussian quasi likelihood random fields for ergodic levy driven sde observed at high frequency this paper investigates the gaussian quasi likelihood estimation of an exponentially ergodic multidimensional markov process which is expressed as a solution to a levy driven stochastic differential equations whose coefficients are supposed to be known except for the finite dimensional parameters to be estimated we suppose that the process is observed under the condition for the rapidly increasing experimental design by means of the polynomial type large deviation inequality the mighty convergence of the corresponding statistical random fields is derived which especially leads to the asymptotic normality at rate p nhn for all the target parameters and also to the convergence of their moments in our results the diffusion coefficient may be degenerate or even null although the resulting estimator is not asymptotically efficient in the presence of jumps we do not require any specific form of the driving levy measure rendering that the proposed estimation procedure is practical and somewhat robust to underlying model specification

on common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form in this note the problem of determining necessary and sufficient conditions for the existence of a common quadratic lyapunov function for a pair of stable linear time invariant systems whose system matrices a sub 1 and a sub 2 are in companion form is considered it is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product a sub 1 a sub 2 does not have an eigenvalue that is real and negative examples are presented to illustrate the result

on unified model selection for stationary and nonstationary short and long memory autoregressive processes summary the question of model choice for the class of stationary and nonstationary fractional and nonfractional autoregressive processes is considered this class is defined by the property that the dth difference for 2 d oo is a stationary autoregressive process of order p0 00 a version of the akaike information criterion aic for determining an appropriate autoregressive order when d and the autoregressive parameters are estimated simultaneously by a maximum likelihood procedure beran 1995 is derived and shown to be of the same general form as for a stationary autoregressive process but with d treated as an additional estimated parameter moreover as in the stationary case this criterion is shown not to provide a consistent estimator of p0 the corresponding versions of the bic of schwarz 1978 and the hic of hannan quinn 1979 are shown to yield consistent estimators of po the results provide a unified treatment of fractional and nonfractional stationary and integrated nonstationary autoregressive models

brief stability preserving interpolation methods for the synthesis of gain scheduled controllers synthesis of gain scheduled controllers for nonlinear plants often requires that a parameter varying controller be generated from a finite set of linear time invariant controllers we propose interpolation methods for this task with the property that stability of the linearized closed loop system is preserved for all values of the scheduling parameter in addition slow variation arguments are presented that establish local stability of the nonlinear closed loop system with gain scheduled controller

structure and control of piecewise linear systems elementary algebraic topology is used in the study of the structure and optimal control of piecewise linear and non linear systems

on the control of linear systems whose coefficients are functions of parameters the stabilization and regulation of linear discrete time systems whose coefficients depend on one or more parameters is studied for linear systems whose coefficients are continuous functions of real or complex parameters respectively analytic or rational functions of real parameters it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous analytic rational functions of the parameters closed form expressions for a collection of stabilizing gains are given in terms of the reachability gramian for systems which are stabilizable for all values of the parameters it is shown that continuous analytic rational stabilizing gains can be computed from a finite time solution to a riccati difference equation whose coefficients are functions of the parameters these results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters

on the dynamic instability of a class of switching system in this note we use floquet theory to establish that a sufficient condition for the existence of a periodic destabilizing switching sequence for the system x a t x a t a1 a2 am where the ai are hurwitz matrices is given by the existence of non negative real constants 1 2 m i 0 i 0 such that the matrix pencil has at least one eigenvalue with a positive real part we use this result to provide insights into the relationship between the non existence of a common quadratic lyapunov function and the existence of a destabilizing switching sequence for low order systems as well as the robustness of a class of switching system that is known to be exponentially stable

computing twists of hyperelliptic curves abstract we give an efficient algorithm to compute equations of twists of hyperelliptic curves c of arbitrary genus over any perfect field k of characteristic different from 2 starting with a cocycle in h 1 gal k k aut c we also discuss some interesting examples

the bias of autoregressive coefficient estimators abstract this article presents simple expressions for the bias of estimators of the coefficients of an autoregressive model of arbitrary but known finite order the results include models both with and without a constant term the effects of overspecification of the model order on the bias are described the emphasis is on least squares and yule walker estimators but the methods extend to other estimators of similar design although only the order t 1 component of the bias is captured where t is the series length this asymptotic approximation is shown to be very accurate for least squares estimators through some numerical simulations the simulations examine fourth order autoregressions chosen to resemble some data series from the literature the order t 1 bias approximations for yule walker estimators need not be accurate especially if the zeros of the associated polynomial have moduli near 1 examples are given where the approximation is accurate and where it is useless the bias expressions are

order selection for same realization predictions in autoregressive processes assume that observations are generated from an infinite order autoregressive ar process shibata ann statist 8 1980 147 164 considered the problem of choosing a finite order ar model allowing the order to become infinite as the number of observations does in order to obtain a better approximation he showed that for the purpose of predicting the future of an independent replicate akaike s information criterion aic and its variants are asymptotically efficient although shibata s concept of asymptotic efficiency has been widely accepted in the literature it is not a natural property for time series analysis this is because when new observations of a time series become available they are not independent of the previous data to overcome this difficulty in this paper we focus on order selection for forecasting the future of an observed time series referred to as same realization prediction we present the first theoretical verification that aic and its variants are still asymptotically elficient in the sense defined in section 4 for same realization predictions to obtain this result a technical condition easily met in common practice is introduced to simplify the complicated dependent structures among the selected orders estimated parameters and future observations in addition a simulation study is conducted to illustrate the practical implications of aic this study shows that aic also yields a satisfactory saute realization prediction in finite samples on the other hand a limitation of aic in same realization settings is pointed out it is interesting to note that this limitation of aic does not exist for corresponding independent cases

multiple model adaptive control with mixing despite the remarkable theoretical accomplishments and successful applications of adaptive control the field is not sufficiently mature to solve challenging control problems where strict performance and robustness guarantees are required critical to the design of practical control systems for these challenging applications and currently lacking in parameter estimation based adaptive control schemes is an approach that explicitly accounts for robust performance and stability specifications towards this goal this paper describes a robust adaptive control approach called adaptive mixing control that makes available the full suite of powerful design tools from lti theory e g mixed synthesis the stability and robustness properties of adaptive mixing control are analyzed it is shown that the mean square regulation error is of the order of the modeling error provided the unmodeled dynamics satisfy a norm bound condition and when the parameter estimate converges to its true value which is guaranteed if a persistence of excitation condition is satisfied the adaptive closed loop system converges exponentially fast to a closed loop system comprising the plant and some lti controller that satisfies the control objective a benchmark example is presented which is used to compare the adaptive mixing controller with other adaptive schemes

the gain scheduling and the robust state feedback stabilization problems we consider dynamical uncertain systems of the form x spl dot a x w b x w u where w t spl isin w is an unknown but bounded time varying parameter we consider the robust state feedback stabilization problem in which we consider a control of the form u spl phi x and the gain scheduling stabilization problem in which a control of the form u spl phi x w full information control is admitted we show that for convex processes those systems in which for fixed x the set of all a x w b x w is convex the two problems are equivalent i e if there exists a locally lipschitz gain scheduling stabilizing control then there exists a robustly stabilizing control continuous everywhere except possibly at the origin thus for convex process stabilization knowledge of w t is not an advantage for the compensator then we consider the special class of polytopic lpv systems and show that there is no loss of regularity as in the general case if we pass from a gain scheduling controller to a state feedback controller in particular no discontinuity at the origin may occur we show that the existence of a dynamic controller always implies the existence of a static one finally we show that we can always find a linear gain scheduling controller for a stabilizable system this means that a possible advantage of the online measurement of the parameter w t is that this always allows for linear compensators whose implementation can be easier than that of nonlinear ones

parameter dependent state observer design for affine lpv systems in this paper the design of a parameter dependent state observer that allows to estimate the state of an affine linear parameter varying lpv system is investigated the observer has the property of being parameter dependent since the corresponding state space matrices are scheduled using an interpolation method moreover the stability of the estimation error is based on the existence of an affine parameter dependent lyapunov function the main contribution of this paper is that the problem of the observer design and the existence of such a lyapunov function is interpreted as a flexible lmi feasibility condition some illustrative examples are presented at the end of this paper

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

interpolation of observer state feedback controllers for gain scheduling the authors propose a method of interpolating linear time invariant controllers with observer state feedback structure in order to generate a continuously varying family of controllers that stabilizes a family of linear plants gain scheduling is a motivation for this work and the interpolation method yields guidelines for the design of gain scheduled controllers the method is illustrated with the design of a missile autopilot using loop shaping h infinity controllers

gain scheduling dynamic linear controllers for a nonlinear plant abstract for a general type of nonlinear tracking problem we consider gain scheduling based on a family of linear dynamic controllers designed for a family of plant linearizations about constant operating points a necessary and sufficient condition for the existence of gain scheduled controllers satisfying appropriate requirements is presented using this condition the impact of linear controller configuration on the existence of a gain scheduled controller can be assessed and ad hoc approaches to scheduling can be analyzed

a cyclic switching strategy for parameter adaptive control this paper introduces a new strategy called cyclic switching to deal with the well known certainty equivalence control synthesis problem which arises in the design of identifier based adaptive controllers because of the existence of points in parameter space where the design model spl sigma sub d upon which certainty equivalence synthesis is based loses stabilizability unlike most previously suggested methods for handling this problem the technique proposed here can be employed with or without process excitation for the technique to work it is not necessary for there to be a mechanism for moving tuned parameters away from values at which spl sigma sub d loses stabilizability and no such mechanism is used

on the stability of switched positive linear systems it was recently conjectured that the hurwitz stability of the convex hull of a set of metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching in this note we show that 1 this conjecture is true for systems constructed from a pair of second order metzler matrices 2 the conjecture is true for systems constructed from an arbitrary finite number of second order metzler matrices and 3 the conjecture is in general false for higher order systems the implications of our results both for the design of switched positive linear systems and for research directions that arise as a result of our work are discussed toward the end of the note