discrete time lq optimal control problems for infinite markov jump parameter systems optimal control problems for discrete time linear systems subject to markovian jumps in the parameters are considered for the case in which the markov chain takes values in a countably infinite set two situations are considered the noiseless case and the case in which an additive noise is appended to the model the solution for these problems relies in part on the study of a countably infinite set of coupled algebraic riccati equations icare conditions for existence and uniqueness of a positive semidefinite solution to the icare are obtained via the extended concepts of stochastic stabilizability ss and stochastic detectability sd which turn out to be equivalent to the spectral radius of certain infinite dimensional linear operators in a banach space being less than one for the long run average cost ss and sd guarantee existence and uniqueness of a stationary measure and consequently existence of an optimal stationary control policy furthermore an extension of a lyapunov equation result is derived for the countably infinite markov state space case

adaptive control of linear systems with markov perturbations the stochastic model considered is a linear jump diffusion process x for which the coefficients and the jump processes depend on a markov chain z with finite state space first we study the optimal filtering and control problem for these systems with non gaussian initial conditions given noisy observations of the state x and perfect measurements of z we derive a new sufficient condition which ensures the existence and the uniqueness of the solution of the nonlinear stochastic differential equations satisfied by the output of the filter we study a quadratic control problem and show that the separation principle holds next we investigate an adaptive control problem for a state process x defined by a linear diffusion for which the coefficients depend on a markov chain the processes x and z being observed in independent white noises suboptimal estimates for the process x z and approximate control law are investigated for a large class of probability distributions of the initial state asymptotic properties of these filters and this control law are obtained upper bounds for the corresponding error are given

on the existence of maximal solution for generalized algebraic riccati equations arising in stochastic control abstract existence of maximal solution is proved for a generalized version of the well known standard algebraic riccati equations which arise in certain stochastic optimal control problems

some remarks about stochastic controllability for linear composite systems sufficient conditions of e controllability for a class of continuous parameter stochastic linear composite systems are established in this case the objective is to analyze composite systems in their lower order subsystems and in terms of their interconnecting structure an example is given to illustrate the criterion

riccati differential equation in optimal filtering of periodic non stabilizable systems this paper discusses the periodic solutions of the matrix riccati differential equation in the optimal filtering of periodic systems special emphasis is given to non stabilizable systems and the question addressed is the existence and uniqueness of a steady state periodic non negative definite solution of the periodic riccati differential equation which gives rise to an asymptotically stable steady state filter the results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution the convergence of the general solution of the periodic riccati differential equation to a periodic equilibrium solution is also investigated the results are extensions of existing time invariant systems results to the case of periodic systems

generalized riccati difference and differential equations abstract in this paper we investigate generalized riccati differential and difference equations obtained from standard riccati equations by adding a semidefinite perturbation term for such equations we give results on the monotonic dependence of the solutions on the coefficients and initial values as well as results on convergence of solutions

stabilizability and positiveness of solutions of the jump linear quadratic problem and the coupled algebraic riccati equation this note addresses the jump linear quadratic problem of markov jump linear systems and the associated algebraic riccati equation necessary and sufficient conditions for stability of the optimal control and positiveness of riccati solutions are developed we show that the concept of weak detectability is not only a sufficient condition for the finiteness of cost functional to imply stability of the associated trajectory but also a necessary one this together with a characterization developed here for the kernel of the riccati solution allows us to show that the control solution stabilizes the system if and only if the system is weakly detectable and that the riccati solution is positive definite if and only if the system is weakly observable the connection between the algebraic riccati equation and the control problem is made as far as the minimal positive semidefinite solution for the algebraic riccati equation is identified with the optimal solution of the linear quadratic problem illustrative numerical examples and comparisons are included

brief paper the extended periodic lyapunov lemma in this paper the so called lyapunov lemma for linear systems with periodically time varying coefficients is dealt with by this lemma the asymptotic stability of a given periodic system is linked with the existence of a periodic positive definite solution of the associated lyapunov equation under the assumption on the system being observable by relaxing the observability assumption to detectability an extended version of the lemma is obtained where the system stability is linked with the existence of a periodic positive semidefinite solution of the lyapunov equation

the filtering problem for continuous time linear systems with markovian switching coefficients abstract the stochastic model under consideration is a markovian jump process with finite state space feeding the parameters of a linear diffusion process x the processes y and z observe linearly and separately x and in independent white noises some properties of the finite optimal filter for the x and processes given the history of measurements z are investigated apart from their theoretical interest these results have an interesting practical bearing on the general filtering problem by providing a natural finite suboptimal solution preliminary experimental results show the effectiveness of our approach to estimate the state trajectory even with a relatively low signal to noise ratio on the measurement processes

maximal versus strong solution to algebraic riccati equations arising in infinite markov jump linear systems abstract we deal with a perturbed algebraic riccati equation in an infinite dimensional banach space which appears for instance in the optimal control problem for infinite markov jump linear systems from now on imjls infinite or finite here has to do with the state space of the markov chain being infinite countable or finite see e g m d fragoso j baczynski optimal control for continuous time lq problems with infinite markov jump parameters siam j control optim 40 1 2001 270 297 by using a certain concept of stochastic stability a sort of l 2 stability we have proved in j baczynski m d fragoso maximal solution to algebraic riccati equations linked to infinite markov jump linear systems internal report lncc no 6 2006 existence and uniqueness of maximal solution for this class of equations as it is noticed in this paper unlike the finite case including the linear case we cannot guarantee anymore that maximal solution is a strong solution in this setting via a discussion on the main mathematical hindrance behind this issue we devise some mild conditions for this implication to hold specifically our main result here is that under stochastic stability along with a condition related with convergence in the infinite dimensional scenario and another one related to spectrum weaker than spectral continuity we ensure the maximal solution to be also a strong solution these conditions hold trivially in the finite case allowing us to recover the result of strong solution of c e de souza m d fragoso on the existence of maximal solution for generalized algebraic riccati equations arising in stochastic control systems control lett 14 1990 233 239 set for mjls the issue of whether the convergence condition is restrictive or not is brought to light and together with some counterexamples unveil further differences between the finite and the infinite countable case

existence and comparison theorems for algebraic riccati equations for continuous and discrete time systems abstract we discuss two comparison theorems for algebraic riccati equations of the form xbr 1 b x x a br 1 c a br 1 c x q c r 1 c 0 simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a riccati equation from the existence of a hermitian solution of a second riccati equation further similar results are given for discrete algebraic riccati equations

the interacting multiple model algorithm for systems with markovian switching coefficients an important problem in filtering for linear systems with markovian switching coefficients dynamic multiple model systems is the management of hypotheses which is necessary to limit the computational requirements a novel approach to hypotheses merging is presented for this problem the novelty lies in the timing of hypotheses merging when applied to the problem of filtering for a linear system with markovian coefficients the method is an elegant way to derive the interacting multiple model imm algorithm evaluation of the imm algorithm shows that it performs well at a relatively low computational load these results imply a significant change in the state of the art of approximate bayesian filtering for systems with markovian coefficients

analysis of the periodic lyapunov and riccati equations via canonical decomposition a new proof is given of the following result a periodic riccati equation admits a unique positive semidefinite periodic solution and the solution is stabilizing if and only if the underlying system is stabilizable and detectable the proof hinges on the decomposition of the riccati equation induced by the system canonical decomposition the solution structure is therefore naturally pointed out

optimality analysis of adaptive sampled control of hybrid systems with quadratic index this note is concerned with the sampled data based linear quadratic lq adaptive control of continuous time systems with unknown markov jump parameters a parameter estimator and a control design method are given it is shown that when the sample step size is small the sampled data based adaptive control is suboptimal under lq index the result is illustrated by a simulation example

convergence properties of the riccati difference equation in optimal filtering of nonstabilizable systems this paper studies the convergence and properties of the solutions of the riccati difference equation special emphasis is given to systems which are not necessarily stabilizable in the filtering sense particularly those having uncontrollable roots on the unit circle besides generalizing and unifying previous work the results have application to a number of important problems including filtering and control of systems with purely deterministic disturbances such as sinusoids and drift components

robust stability of a class of uncertain markov jump nonlinear systems this note addresses the problem of robust stability analysis for a class of markov jump nonlinear systems subject to polytopic type parameter uncertainty a condition for robust local exponential mean square stability in terms of linear matrix inequalities is developed an estimate of a robust domain of attraction of the origin is also provided the approach is based on a stochastic lyapunov function with polynomial dependence on the system state and uncertain parameters a numerical example illustrates the proposed result

a separation principle for the h2 control of continuous time infinite markov jump linear systems with partial observations abstract in this paper we devise a separation principle for the h 2 optimal control problem of continuous time markov jump linear systems with partial observations and the markov process taking values in an infinite countable set s we consider that only an output and the jump parameters are available to the controller it is desired to design a dynamic markov jump controller such that the closed loop system is stochastically stable and minimizes the h 2 norm of the system as in the case with no jumps we show that an optimal controller can be obtained from two sets of infinite coupled algebraic riccati equations one associated with the optimal control problem when the state variable is available and the other one associated with the optimal filtering problem an important feature of our approach not previously found in the literature is to introduce an adjoint operator of the continuous time markov jump linear system to derive our results

statistical behavior of joint least square estimation in the phase diversity context the images recorded by optical telescopes are often degraded by aberrations that induce phase variations in the pupil plane several wavefront sensing techniques have been proposed to estimate aberrated phases one of them is phase diversity for which the joint least square approach introduced by gonsalves et al is a reference method to estimate phase coefficients from the recorded images in this paper we rely on the asymptotic theory of toeplitz matrices to show that gonsalves technique provides a consistent phase estimator as the size of the images grows no comparable result is yielded by the classical joint maximum likelihood interpretation e g as found in the work by paxman et al finally our theoretical analysis is illustrated through simulated problems

mode independent cal h _ infty filters for markovian jump linear systems this note addresses the problem of hinfin filtering for continuous time linear systems with markovian jumping parameters the main contribution of the note is to provide a method for designing an asymptotically stable linear time invariant hinfin filter for systems where the jumping parameter is not accessible the cases where the transition rate matrix of the markov process is either exactly known or unknown but belongs to a given polytope are treated the robust hinfin filtering problem for systems with polytopic uncertain matrices is also considered and a filter design method based on a lyapunov function that depends on the uncertain parameters is developed the proposed filter designs are given in terms of linear matrix inequalities

borel quantum kinematics of rank k on smooth manifolds the authors propose a model for quantizing a nonrelativistic physical system that admits a finite number of internal degrees of freedom they assume that the configuration space of the given system carries the structure of a smooth manifold m the kinematics of the given system is assumed to be describable by a flow model under certain technical restrictions momentum and position operators are seen to act on a hilbert space of square integrable sections of a hermitian vector bundle over m they determine their necessary shape up to isometric isomorphism and distinguish two types in particular they show that the number of inequivalent quantizations depends on the topology of the underlying configuration space

comments on robust jump linear quadratic control a mode stabilizing solution in the above paper an optimal control problem with discounting was proposed to achieve closed loop mode stabilization for linear systems with jump parameters the paper used the unstated hypothesis that the diagonal entries of the generator of the jump process are equal this correspondence discusses the situation in the general case

optimal control for infinite dimensional stochastic differential equations with infinite markov jumps and multiplicative noise abstract in this paper we solve an infinite horizon linear quadratic control problem for a class of differential equations with countably infinite markov jumps and multiplicative noise the global solvability of the associated differential riccati type equations is studied under detectability hypotheses a nonstochastic operatorial approach is used some properties of the linear stochastic systems such as stability stabilizability and detectability are also discussed on the basis of a new solution representation result a generalized ito s formula which applies to infinite dimensional stochastic differential equations with countably infinite markov jumps is also provided

optimal quadratic control of jump linear systems with separately controlled transition probabilities abstract the optimal control of discrete time jump systems that would be linear but for randomly jumping parameters are considered these parameter jumps which are indexed by a finite state form process can be used to model systems subject to component failures or structural changes the control of such systems when the jump probabilities can be controlled by choice of a finite valued input is considered this input form control can be thought of as corresponding to a maintenance or repair policy the optimal control problem with quadratic costs whose weighting matrices depend on the jumping parameters and form control is formulated for such systems the optimal solution for finite and infinite time horizon problems is developed in addition sufficient conditions for the existence of a steady state stabilizing solution to the infinite time problem are given in terms of appropriately defined controllability and observability properties

almost sure and moments stability of jump linear systems abstract the stability of piecewise deterministic linear systems driven by an underlying finite markov chain is analyzed necessary and sufficient conditions for moment stability are obtained by means of an explicit formula for the corresponding liapunov exponent the relationship between almost sure and moment stability is elucidated revealing the possible occurrence of high order moment instability and large deviations from a stable sample path behavior

asymptotic eigenvalue distribution of block toeplitz matrices and application to blind simo channel identification szego s 1984 theorem states that the asymptotic behavior of the eigenvalues of a hermitian toeplitz matrix is linked to the fourier transform of its entries this result was later extended to block toeplitz matrices i e covariance matrices of multivariate stationary processes the present work gives a new proof of szego s theorem applied to block toeplitz matrices we focus on a particular class of toeplitz matrices those corresponding to covariance matrices of single input multiple output simo channels they satisfy some factorization properties that lead to a simpler form of szego s theorem and allow one to deduce results on the asymptotic behavior of the lowest nonzero eigenvalue for which an upper bound is developed and expressed in terms of the subchannels frequency responses this bound is interpreted in the context of blind channel identification using second order algorithms and more particularly in the case of band limited channels

stability analysis for stochastic hybrid systems this survey addresses stability analysis for stochastic hybrid systems shs which are dynamical systems that combine continuous change and instantaneous change and that also include random effects we re emphasize the common features found in most of the models that have appeared in the literature which include stochastic switched systems markov jump systems impulsive stochastic systems switching diffusions stochastic impulsive systems driven by renewal processes diffusions driven by levy processes piecewise deterministic markov processes general stochastic hybrid systems and stochastic hybrid inclusions then we review many of the stability concepts that have been studied including lyapunov stability lagrange stability asymptotic stability and recurrence next we detail lyapunov based sufficient conditions for these properties and additional relaxations of lyapunov conditions many other aspects of stability theory for shs like converse lyapunov theorems and robustness theory are not fully developed hence we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces

mean square stochastic stability of linear time delay system with markovian jumping parameters this paper deals with the mean square stochastic stability of the class of linear time delay systems with markovian jumping parameters a delay independent sufficient condition for checking the stochastic stability of this class of systems is established a numerical example is given to show the usefulness of the proposed theoretical results

on the phase portrait of the matrix riccati equation arising from the periodic control problem a comprehensive description is provided of the properties of the matrix riccati differential equation in which the coefficient matrices are t periodic for some t 0 the main results include 1 a classification of all the periodic equilibria t periodic solutions 2 necessary and sufficient conditions for convergence to the uniformly asymptotically stable completely unstable periodic equilibrium as t to infty t to infty 3 necessary and sufficient conditions for a solution to escape in finite forward or backward time 4 a description of every almost periodic solution 5 a proof that every solution which does not escape in finite time is asymptotically almost periodic and an explicit formula for the limiting almost periodic solution

circulant preconditioners for toeplitz matrices with piecewise continuous generating functions the authors consider the solution of n by n toeplitz systems t sub n x b by preconditioned conjugate gradient methods the preconditioner c sub n is the t chan circulant preconditioner which is defined to be the circulant matrix that minimizes parallel b sub n t sub n parallel sub f over all circulant matrices b sub n for toeplitz matrices generated by positive 2 pi periodic continuous functions they have shown earlier that the spectrum of the preconditioned system c sup minus 1 sub n t sub n is clustered around 1 and hence the convergence rate of the preconditioned system is superlinear however in this paper they show that if instead the generating function is only piecewise continuous then for all epsilon sufficiently small there are o log n eigenvalues of c sup minus 1 sub n t sub n that lie outside the interval 1 epsilon 1 epsilon in particular the spectrum of c sup minus 1 sub n t sub n cannot be clustered around 1 numerical examples are given to verify that the convergence rate of the method is no longer superlinear in general 20 refs

some remarks on the riccati equation arising in an optimal control problem with state and control dependent noise this paper solves a quadratic optimal control for a linear stochastic evolution equation with unbounded coefficients it is assumed that the stochastic noise depends both on the state and on the control the dynamic programming approach is used and attention is focused on the riccati equation in 5 and 6 some attractivity and maximality properties of the solutions of the algebraic riccati equation are proved and it is shown that in some special cases there exists a maximal solution

controllability observability and discrete time markovian jump linear quadratic control this paper is concerned with the controllability and observability of discrete time linear systems that possess randomly jumping parameters described by finite stale markov processes and the relationship between these properties and the solution of the infinite time jump linear quadratic jlq optimal control problem the solution of the markovian jlq problem with finite or infinite time horizons is known necessary and sufficient conditions for the existence of optimal constant control laws that lead to finite optimal expected costs as the time horizon becomes infinite are also known sufficient conditions for these steady state control laws to stabilize the controlled system are also available chizeck et al 1986 these conditions are not easy to test however various definitions of controllability and observability for stochastic systems exist in the literature these definitions are unfortunately not related to the steady state jlq control problem in a manner that is analogous to the role of deter

lyapunov coupled equations for continuous time infinite markov jump linear systems this paper deals with lyapunov equations for continuous time markov jump linear systems mjls out of the bent which wends most of the literature on mjls we focus here on the case in which the markov chain has a countably infinite state space it is shown that the infinite mjls is stochastically stabilizable ss if and only if the associated countably infinite coupled lyapunov equations have a unique norm bounded strictly positive solution it is worth mentioning here that this result do not hold for mean square stabilizability mss since ss and mss are no longer equivalent in our set up see e g j baczynski optimal control for continuous time lq problems with infinite markov jump parameters ph d thesis federal university of rio de janeiro ufrj coppe 2000 to some extent a decomplexification technique and tools from operator theory in banach space and in particular from semigroup theory are the very technical underpinning of the paper

controllability of linear stochastic systems abstract we discuss several concepts of controllability for stochastic systems and derive sufficient and in most cases also necessary algebraic conditions for the controllability of the state and the output of white and colored noise linear systems these criteria reduce to the well known deterministic ones in the absence of noise

infinite markov jump bounded real lemma abstract a bounded real lemma is established providing an equivalent condition to stochastic stability ss of a continuous time infinite markovian jump linear system mjls with a prescribed l 2 stochastic disturbance attenuation level in terms of existence of solutions to an infinite set of coupled linear matrix inequalities lmis besides the interest in its own right the main result provides a fundamental tool for the development of an h like theory devoted to this class of systems

on controlability of linear systems with stochastic jump parameters this note studies the notion of stochastic e controllability in probability for linear systems subject to sudden jumps in parameter values by using a lyapunov like approach sufficient conditions are obtained in terms of a set of coupled bernoulli equations the potential use of the results is illustrated by an example

on an infinite dimensional perturbed riccati differential equation arising in stochastic control existence and uniqueness of solutions for a class of infinite dimensional perturbed riccati differential equations in a certain banach space is our main concern in this paper besides the interest in its own right this class arises in control problems involving linear systems with countably infinite markov jump parameters

a new perspective on the robustness of markov jump linear systems this paper is concerned with the robust analysis and control of continuous time markov jump linear systems the centerpiece of the paper is an alternative to the small gain theorem which hinges on the robustification of a certain adjoint lyapunov operator by means of this technique it is proven that the small gain theorem of markov jump linear systems may sometimes be arbitrarily conservative even when nonlinear lipschitz disturbances are taken into account the adjoint approach on the other hand provides the maximal degree of robustness for this particular setup in addition we prove that the adjoint design of controllers is solved more efficiently than the design based on small gain analysis bearing these new facts in mind we derive an iterative algorithm for the design of robust controllers based on linear matrix inequalities by means of numerical examples regarding the robust control of an underactuated manipulator arm and of a simplified power systems model it is shown that the adjoint design methodology can be much more advantageous than its small gain counterpart

solution and asymptotic behavior of coupled riccati equations in jump linear systems a new necessary and sufficient condition for the existence of a positive semidefinite solution of coupled riccati equations occurring in jump linear systems is derived by verifying a riccati inequality it is shown that such a solution exists in addition two numerical algorithms are given to compute it an example is given to illustrate the proposed method

asymptotic eigenvalue distribution of block toeplitz matrices the need for approximating block toeplitz with toeplitz block matrices by means of block circulant with circulant block matrices with the objective of transforming an inherently ill posed image deconvolution problem to a well posed one motivated a surge of papers on the analysis of the quality as well as the speed of convergence of the algorithms that produce such approximants to serve as preconditioners for conjugate gradient methods this correspondence contributes to that surge by giving a simple proof of the fact that the sequence of eigenvalues of the hermitian block toeplitz with toeplitz block matrices are asymptotically equidistributed to do this weyl s 1916 results on the distribution properties of multidimensional sequences are exploited inferences to related results are made

stochastic stability properties of jump linear systems jump linear systems are defined as a family of linear systems with randomly jumping parameters usually governed by a markov jump process and are used to model systems subject to failures or changes in structure the authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties it is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability and a testable necessary and sufficient condition for second moment stability is derived the lyapunov exponent method for the study of almost sure sample stability is discussed and a theorem which characterizes the lyapunov exponents of jump linear systems is presented finally for one dimensional jump linear system it is proved that the region for delta moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0 sup

electric load model synthesis by diffusion approximation of a high order hybrid state stochastic system a statistical approach is used to model the dynamics of the electric demand of large aggregates of electric space heaters or air conditioners the importance of such loads is twofold first they account for a significant portion of power system dynamics following a power outage second because they are associated with energy storage they are often selected for load shedding within a load management program the derivation of the aggregate electrical dynamics is considered first for a homogeneous group of devices subsequently a perturbation analysis yields the dynamics for a nonhomogeneous group the homogeneons group aggregrate load model is a system of coupled ordinary and partial differential equations fokker planck equations it is obtained by writing evolution equations for the probability density of a hybrid state markov system used to model the switching dynamics of individual devices this result is new and could give a clue to the analysis of a broad class of hybrid state stochastic systems in turn this could provide a new impetus not only in the area of electric load modeling but other areas such as power system reliability and the design of relay control systems where stochastic hybrid state models occur frequently simulation results which illustrate the dynamical properties of the model are presented

functions of banded hermitian block toeplitz matrices in signal processing the paper deals with the entries of functions of large banded hermitian block toeplitz matrices and their perturbations for general continuous functions convergence results are established and for analytic functions these results are accompanied by estimates of the convergence speed the applications to signal processing include among others the direct derivation of a closed form solution of the minimum mean square error mmse for a double sided infinite length linear equalizer for mimo channels from the finite length mmse expression and the derivation of a formula for the contribution of each channel input to the overall mmse

a unified approach for stochastic and mean square stability of continuous time linear systems with markovian jumping parameters and additive disturbances necessary and sufficient conditions for stochastic stability ss and mean square stability mss of continuous time linear systems subject to markovian jumps in the parameters and additive disturbances are established we consider two scenarios regarding the additive disturbances one in which the system is driven by a wiener process and one characterized by functions in l 2 m omega cal f mathbb p which is the usual scenario for the h infty approach the markov process is assumed to take values in an infinite countable set mathcal s it is shown that ss is equivalent to the spectrum of an augmented matrix lying in the open left half plane to the existence of a solution for a certain lyapunov equation and implies is equivalent for mathcal s finite asymptotic wide sense stationarity awss it is also shown that ss is equivalent to the state x t belonging to l 2 n omega cal f mathbb p whenever the disturbances are in l 2 m omega cal f mathbb p for the case in which mathcal s is finite ss and mss are equivalent and the lyapunov equation can be written down in two equivalent forms with each one providing an easier to check sufficient condition

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

stability of nonlinear discrete repetitive processes with markovian switching repetitive processes are a class of 2d systems that operate over a subset of the upper right quadrant of the 2d plane applications include iterative learning control where experimental verification has been reported based on a linear time invariant model approximation of the dynamics this paper considers discrete nonlinear repetitive processes with markovian switching and applies as one application the resulting stability theory to iterative learning control for a class of networked systems where time varying dynamics arise

a miscellany of results of an equation of count j f riccati a collection of results on the riccati equation is presented the questions addressed are the existence of strong solutions of the algebraic riccati equation and the convergence of solutions of the riccati difference equation to those of the algebraic equation the results derived utilize detestability conditions only

mixed h sub 2 h sub spl infin control of discrete time markovian jump linear systems in this paper we consider the mixed h sub 2 h sub spl infin control problem for the class of discrete time linear systems with parameters subject to markovian jump linear systems it is assumed that both the state variable and the jump variable are available to the controller the transition probability matrix may not be exactly known but belongs to an appropriate convex set for this controlled discrete time markovian jump linear system the problem of interest can be stated in the following way find a robust with respect to the uncertainty on the transition markov probability matrix mean square stabilizing state and jump feedback controller that minimizes an upper bound for the h sub spl infin norm under the restriction that the h sub spl infin norm is less than a prespecified value spl delta the problem of the determination of the smallest h sub spl infin norm is also addressed we present an approximate version of these problems via linear matrix inequality optimization

array algorithm for filtering of discrete time markovian jump linear systems this note develops an array algorithm for optimal filtering of discrete time markovian jump linear systems dtmjlss the known advantages of this kind of algorithm which was originally developed for normal state space systems remain valid when it is applied to linear systems subject to markovian jumps it is numerically more stable in the sense that it presents better conditioning and reduced dynamical range a numerical example based on fixed point implementations is presented in order to demonstrate the advantage of this algorithm

optimal control for continuous time linear quadratic problems with infinite markov jump parameters the subject matter of this paper is the optimal control problem for continuous time linear systems subject to markovian jumps in the parameters and the usual infinite time horizon quadratic cost what essentially distinguishes our problem from previous ones inter alia is that the markov chain takes values on a countably infinite set to tackle our problem we make use of powerful tools from semigroup theory in banach space and a decomplexification technique the solution for the problem relies in part on the study of a countably infinite set of coupled algebraic riccati equations icare conditions for existence and uniqueness of a positive semidefinite solution of the icare are obtained via the extended concepts of stochastic stabilizability ss and stochastic detectability sd these concepts are couched into the theory of operators in banach space and parallel to the classical linear quadratic lq case bound up with the spectrum of a certain infinite dimensional linear operator

asymptotically equivalent sequences of matrices and multivariate arma processes the present paper considers a special class of vector random processes that we call multivariate asymptotically wide sense stationary wss processes a multivariate random process is said to be asymptotically wss if it has constant mean and the sequence of its autocorrelation matrices is asymptotically equivalent a e to the sequence of autocorrelation matrices of some multivariate wss process it is shown that this class of processes contains meaningful processes other than multivariate wss processes in particular we give sufficient conditions for multivariate moving average ma processes multivariate autoregressive ar processes and multivariate autoregressive moving average arma processes to be asymptotically wss furthermore in order to solve multiple input multiple output mimo problems in communications and signal processing involving this kind of processes we extend the gray definition of a e sequences of matrices and his main results on these sequences to non square matrices as an example the derived results on a e sequences of non square matrices are applied to compute the differential entropy rate and the minimum mean square error mmse for a linear predictor of a multivariate asymptotically wss process

fft based preconditioners for toeplitz block least squares problems discretized two dimensional deconvolution problems arising e g in image restoration and seismic tomography can be formulated as least squares computations min b tx 2 where t is often a large scale rectangular toeplitz block matrix the authors consider solving such block least squares problems by the preconditioned conjugate gradient algorithm using square nonsingular circulant block and related preconditioners constructed from the blocks of the rectangular matrix t preconditioning with such matrices allows efficient implementation using the one dimensional or two dimensional fast fourier transform fft two block preconditioners related to those proposed by t chan and j olkin for square nonsingular toeplitz block systems are derived and analyzed it is shown that for important classes of t the singular values of the preconditioned matrix are clustered around one this extends the authors earlier work on preconditioners for toeplitz least squares iterations for one dimensiona

a note on the maximal solution of the periodic riccati equation the periodic symmetric solutions of the periodic riccati differential equation associated with the filtering problem are considered by the authors it is proven that under the sole assumption of detectability there exists a maximal solution moreover such a solution turns out to be strong i e the characteristic multipliers of the associated closed loop system belong to the closed unit disk the proof relies on an iterative linearization technique which calls for a sequence of periodic lyapunov equations similar results are given for the minimal solution

on hermitian solutions of the symmetric algebraic riccati equation the structure of the set of hermitian solutions of the matrix quadratic equation xdx xa a x c 0 is studied under the conditions that c c d is positive semidefinite and a d is stabilizable new features e g nonexistence of the minimal solution appear in contrast with the known case when a d is controllable

the stability of stochastic coupled systems with time varying coupling and general topology structure we introduce a class of novel stochastic coupled systems in which the coupling structure is time varying and the topology structure is not strongly connected and first establish the system on a digraph with a time varying weight matrix motivated by du and li 2014 we give a hierarchical method to deal with digraphs without strong connectivity and establish the corresponding hierarchical algorithm to realize this approach also an example is given to illustrate our hierarchical algorithm and its feasibility in the sequel based on the theory of asymptotically autonomous systems kirchhoff s matrix tree theorem and lyapunov method several moment exponential stability criteria are presented including a lyapunov type theorem and a coefficient type criterion furthermore theoretical results are applied to stochastic coupled oscillators with time varying coupling structure sctcs and the stability criterion of sctcs is obtained finally the effectiveness of theoretical results is illustrated by two numerical examples

on exponential almost sure stability of random jump systems this paper is concerned with a class of random jump systems represented by transition operators which includes switched linear systems with strictly stationary switching signals in infinite modes space as its special case a series of necessary and sufficient conditions are established for almost sure stability of this class of random jump systems under different scenarios the stability criteria obtained are further extended to markov jump linear systems with infinite states and hence a unified approach to describing the almost sure stability of mjlss is addressed under this context all the results in the work are developed for both the continuous and discrete time systems

maximal solution of a certain class of periodic riccati differential equations this paper deals with a generalized version of the well known periodic riccati differential equation which arises in certain optimal filtering problems for linear periodic systems a monotonicity property for symmetric periodic equilibrium solutions is established and it is proved that there exists a maximal symmetric periodic equilibrium solution

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

controllability stabilizability and continuous time markovian jump linear quadratic control consideration is given to the control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes the relationship between appropriately defined controllability stabilizability properties and the solution of the infinite time jump linear quadratic jlq optimal control problems is also examined although the solution of the continuous time markov jlq problem with finite or infinite time horizons is known only sufficient conditions for the existence of finite cost constant stabilizing controls for the infinite time problem appear in the literature in this paper necessary and sufficient conditions are established these conditions are based on new definitions of controllability observability stabilizability and detectability that are appropriate for continuous time markovian jump linear systems these definitions play the same role for the jlq problem as the deterministic properties do for the linear quadratic regulator lqr problem

properties of nonlinear schrodinger equations associated with diffeomorphism group representations the authors recently derived a family of nonlinear schrodinger equations on r3 from fundamental considerations of generalized symmetry ih cross delta t psi h cross 2 2m del 2 psi f psi psi psi ih cross d del 2 psi mod del psi mod 2 mod psi mod 2 psi where f is an arbitrary real functional and d a real continuous quantum number these equations descriptive of a quantum mechanical current that includes a diffusive term correspond to unitary representations of the group diff m parametrized by d where m r3 is the physical space in the present paper we explore the most natural ansatz for f which is labelled by five real coefficients we discuss the invariance properties describe the stationary states and some non stationary solutions and determine the extra dissipative terms that occur in the ehrenfest theorem we identify an interesting galilean invariant subfamily whose properties we investigate including the case where the dissipative terms vanish

stabilizability and detectability of linear periodic systems abstract in this paper four characterizations of stabilizability and detectability of linear periodic systems are considered two of them look as natural extensions of the classical definitions given for time invariant systems the remaining two are modal characterizations which turn out to be useful in the analysis of the periodic lyapunov and riccati equations it is shown that all these notions of stabilizability and detectability are in fact equivalent to each other one of the various definitions calls for the existence of the kalman canonical decomposition of periodic systems this issue is addressed in the appendix

stability results for discrete time linear systems with markovian jumping parameters stability results for discrete time linear systems subject to random jumps in the parameters are obtained first necessary and sufficient conditions for mean square stability mss including the case in which the system is driven by an independent wide sense stationary random sequence are derived it is shown that mss is equivalent to the spectral radius of an augmented matrix being less than one or to the existence of a solution of a certain lyapunov equation in addition it is shown that the lyapunov equation can be derived in four equivalent forms and as a by product each one gives risc to an easier to check sufficient condition these results give inter alia a unified and rather complete picture for mss of markovian jump linear systems next we derive sufficient conditions for almost sure stability of the noiseless ease finally we conclude by presenting an illustrative application to adaptive filtering where a mild condition for almost sure convergence is provided

jump linear quadratic gaussian control in continuous time the optimal quadratic control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes is addressed the systems are also subject to gaussian input and measurement noise the optimal solution for the jump linear quadratic gaussian jlqc problem is given this solution is based on a separation theorem the optimal state estimator is sample path dependent if the plant parameters are constant in each value of the underlying jumping process then the controller portion of the compensator converges to a time invariant control law however the filter portion of the optimal infinite time horizon jlqc compensator is not time invariant thus a suboptimal filter which does converge to a steady state solution under certain conditions is derived and a time invariant compensator is obtained

asymptotically equivalent sequences of matrices and hermitian block toeplitz matrices with continuous symbols applications to mimo systems for the engineering community gray s tutorial monograph on toeplitz and circulant matrices has been and remains the best elementary introduction to the szego theory on large toeplitz matrices in this paper the most important results of the cited monograph are generalized to block toeplitz bt matrices by maintaining the same mathematical tools used by gray that is by using asymptotically equivalent sequences of matrices as applications of these results the geometric minimum mean square error mmse for both an infinite length multivariate linear predictor and an infinite length decision feedback equalizer dfe for multiple input multiple output mimo channels are obtained as a limit of the corresponding finite length cases similarly a short derivation of the well known capacity of a time invariant mimo gaussian channel with intersymbol interference isi and fixed input covariance matrix is also presented

information filtering and array algorithms for discrete time markovian jump linear systems this technical note develops information filter and array algorithms for a linear minimum mean square error estimator of discrete time markovian jump linear systems a numerical example for a two mode markovian jump linear system to show the advantage of using array algorithms to filter this class of systems is provided

discrete time markovian jump linear quadratic optimal control this paper is concerned with the optimal control of discrete time linear systems that possess randomly jumping parameters described by finite state markov processes for problems having quadratic costs and perfect observations the optimal control laws and expected costs to go can be precomputed from a set of coupled riccati like matrix difference equations necessary and sufficient conditions are derived for the existence of optimal constant control laws which stabilize the controlled system as the time horizon becomes infinite with finite optimal expected cost

a q schrodinger equation based on a hopf q deformation of the witt algebra in an earlier paper a q schrodinger equation was obtained based on a particular quantization procedure called borel quantization and a related q deformation of the witt algebra this q deformation is a deformation in the category of lie algebras and hence the corresponding q witt algebra has a trivial hopf algebra structure in this paper we extend the above algebra by the addition of a set of shift type generators which appear in the expression for the quantum mechanical position operator and hence lead to a new type of quantum kinematics the latter gives rise to a new kind of evolution equation and it is shown that in the limit q1 a specific class of schrodinger equations is obtained from it this specification of a particular class is a new phenomenon because in earlier references where a different q deformation has been implemented or no deformation has been used at all such a class could not be determined uniquely the extended algebra used here has a nontrivial hopf structure the appearance of the shift type generator in the q deformed picture hence leads to a selection of a particular type of dynamics and delivers in the limit q1 new information for the characterization of the corresponding dynamics in the undeformed situation

controllability of stochastic linear systems in the paper necessary and sufficient conditions for various types of stochastic controllability of the linear stochastic system of the form d x ax d t bu d t c d w r x 0 x 0 are given some possible extensions are indicated as well

output feedback for a class of linear systems with stochastic jump parameters a class of linear systems subject to sudden jumps in parameter values is studied for a quadratic performance index the optimal controller has already been proposed it consists of a full state feedback that requires the synthesis of an observer for the plant state and mode to avoid this difficulty the present note develops new results that lead to the selection of the best feedback control law depending only on measurable outputs necessary optimality conditions are derived and two computationnal algorithms proposed an example is included

lyapunov and riccati equations periodic inertia theorems the lyapunov and riccati differential equations with periodically time varying coefficients are considered under the assumption of detectability of the underlying periodic system two inertia theorems are provided linking the inertia of the solution to the one of the so called monodromy matrix

on a partially observable lqg problem for systems with markovian jumping parameters in this paper a variant of the lqg problem for systems with markovian jumping parameters is studied it is assumed that the system is partially observable in the sense that we do not have access to the jumping parameter via nonlinear filtering the problem is recast as one with complete information it is then shown that a certain averaged control is an optimal policy vis a vis a modified quadratic cost apart from its intrinsic theoretical interest this result has an important bearing on a number of different analyses related with the original quadratic cost