from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

isospectral flows related to frobenius stickelberger thiele polynomials the isospectral deformations of the frobenius stickelberger thiele fst polynomials introduced in spiridonov et al commun math phys 272 139 165 2007 are studied for a specific choice of the deformation of the spectral measure one is led to an integrable lattice fst lattice which is indeed an isospectral flow connected with a generalized eigenvalue problem in the second part of the paper the spectral problem used previously in the study of the modified camassa holm mch peakon lattice is interpreted in terms of the fst polynomials together with the associated fst polynomials resulting in a map from the mch peakon lattice to a negative flow of the finite fst lattice furthermore it is pointed out that the degenerate case of the finite fst lattice unexpectedly maps to the interlacing peakon ode system associated with the two component mch equation studied in chang et al adv math 299 1 35 2016

brief paper guaranteed cost control of stochastic uncertain systems with slope bounded nonlinearities via the use of dynamic multipliers this paper presents a new approach to constructive output feedback robust nonlinear guaranteed cost controller design the approach involves a class of controllers which include copies of the slope bounded nonlinearities occurring in the plant dynamic multipliers are introduced to exploit these repeated nonlinearities the linear part of the controller is synthesized using minimax lqg control theory

recurrence coefficients for discrete orthonormal polynomials and the painleve equations we investigate semi classical generalizations of the charlier and meixner polynomials which are discrete orthogonal polynomials that satisfy three term recurrence relations it is shown that the coefficients in these recurrence relations can be expressed in terms of wronskians of modified bessel functions and confluent hypergeometric functions respectively for the generalized charlier and generalized meixner polynomials these wronskians arise in the description of special function solutions of the third and fifth painleve equations

robust mean field linear quadratic gaussian games with unknown l 2 disturbance this paper considers a class of mean field linear quadratic gaussian games with model uncertainty the drift term in the dynamics of the agents contains a common unknown function we take a robust

toda type differential equations for the recurrence coefficients of orthogonal polynomials and freud transformation transformations of the measure of orthogonality for orthogonal polynomials namely freud transformations are considered jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations dynamics of the coefficients are described by generalized toda equations the classical toda lattice equations are the simplest special case of dynamics of the coefficients under the freud transformation of the measure of orthogonality also dynamics of hankel determinants its minors and other notions corresponding to the orthogonal polynomials are studied

minimality and observability of group systems abstract group systems are a generalization of willems type linear systems that are useful in error control coding it is shown that the basic ideas of willems s treatment of linear systems are easily generalized to linear systems over arbitrary rings and to group systems the interplay between systems behaviors and trellises evolution laws is discussed with respect to completeness minimality controllability and observability it is pointed out that for trellises of group systems and willems type linear systems minimality is essentially the same as observability the development is universal algebraic in nature and holds unconditionally for linear systems over the real numbers

dynamical systems and convolutional codes over finite abelian groups polynomial algebraic techniques have always played a central role in linear systems theory and also in the theory of convolutional codes we show how such techniques can be generalized to study systems and codes defined over abelian groups the systems are considered from the behavioral point of view as developed by willems in the 1980s and some of our results can be seen as extensions of willems results to group systems we also address a certain number of coding oriented questions and we propose concrete methods based on these algebraic techniques for the synthesis of encoders inverters and syndrome formers for codes over finite abelian groups

robust control of nonlinear jump parameter systems governed by uncertain chains we consider an infinite horizon minimax optimal control problem for stochastic uncertain systems governed by a discrete state uncertain continuous time chain using existing risk sensitive control results a robust suboptimal absolutely stabilizing guaranteed cost controller is constructed conditions are presented under which this suboptimal controller is minimax optimal we then present a numeric algorithm for calculating a robust sub optimal controller using a markov chain approximation technique

calculating laplace transforms in terms of the toda molecule a method for calculating a continued fraction expansion of laplace transforms is presented in terms of the tau function of the continuous time toda molecule equation and its integrable discretization the z transform and a discrete analogue of the laplace transform of a set of finite samples are also formulated

state distributions and minimum relative entropy noise sequences in uncertain stochastic systems the discrete time case this paper is concerned with dissipativity theory and robust performance analysis and design of discrete time stochastic systems driven by statistically uncertain random noise the uncertainty is quantified by the conditional relative entropy of the actual probability law of the noise with respect to a nominal product measure corresponding to a white noise sequence we discuss a balance equation dissipation inequality and superadditivity property for the corresponding conditional relative entropy supply as a function of time the problem of minimizing the supply required to drive the system between given state distributions over a specified time horizon is considered such variational problems involving entropy and probabilistic boundary conditions are known in the literature as schrodinger bridge problems in application to control systems the minimum required conditional relative entropy supply characterizes the robustness of the system with respect to a statistically uncertain random noise we o

the equivalence between infinite horizon optimal control of stochastic systems with exponential of integral performance index and stochastic differential games a new method based on the theory of large deviations from the invariant measure is introduced for the analysis of stochastic systems with an infinite horizon exponential of integral performance index it is shown that the infinite horizon optimal exponential of integral stochastic control problem is equivalent to a stationary stochastic differential game for an auxiliary system as an application of the developed technique the infinite horizon risk sensitive lqg problem is analyzed for both the completely observed and partially observed case

model simplification and optimal control of stochastic singularly perturbed systems under exponentiated quadratic cost we study the optimal control of a general class of stochastic singularly perturbed linear systems with perfect and noisy state measurements under positively and negatively exponentiated quadratic cost the expected cost function to be minimized is actually taken as the long term time average of the logarithm of the expected value of an exponentiated quadratic loss we identify appropriate slow and fast subproblems obtain their optimum solutions compatible with the corresponding measurement structure and subsequently study the performances they achieve on the full order system as the singular perturbation parameter epsilon becomes sufficiently small with the expressions given in all cases being exact to within o sqrt epsilon it is shown that the composite controller obtained by appropriately combining the optimum slow and fast controllers achieves a performance level close to the optimal one whenever the full order problem has a solution the slow controller on the other hand achieves asymptotically as epsilon to 0 only a finite performance level but not necessarily optimal provided that the fast subsystem is open loop stable if the intensity of the noise in the system dynamics decreases to zero however the slow controller also achieves a performance level close to the optimal one the paper also presents a more direct derivation than heretofore available of the solution to the linear exponential quadratic gaussian leqg problem under noisy state measurements which allows for a general quadratic cost with cross terms in the exponent and correlation between system and measurement noises and obtains both necessary and sufficient conditions for existence of an optimal solution such a general leqg problem is encountered in the slow fast decomposition of the full order problem even if the original problem does not feature correlated noises in this general context the paper also establishes a complete equivalence between the leqg problem and the hi optimal control problem with measurement feedback though this equivalence does not extend to the slow and fast subproblems arrived at after time scale separation

the dynamics of group codes dual abelian group codes and systems fundamental results concerning the dynamics of abelian group codes behaviors and their duals are developed duals of sequence spaces over locally compact abelian lca groups may be defined via pontryagin duality dual group codes are orthogonal subgroups of dual sequence spaces the dual of a complete code or system is finite and the dual of a laurent code or system is anti laurent if c and c sup spl perp are dual codes then the state spaces of c act as the character groups of the state spaces of c sup spl perp the controllability properties of c are the observability properties of c sup spl perp in particular c is strongly controllable if and only if c sup spl perp is strongly observable and the controller memory of c is the observer memory of c sup spl perp the controller granules of c act as the character groups of the observer granules of c sup spl perp examples of minimal observer form encoder and syndrome former constructions are given finally every observer granule of c is an end around controller granule of c

robust finite horizon minimax filtering for discrete time stochastic uncertain systems we study a finite horizon robust minimax filtering problem for time varying discrete time stochastic uncertain systems the uncertainty in the system is characterized by a set of probability measures under which the stochastic noises driving the system are defined the optimal minimax filter has been found by applying techniques of risk sensitive lqg control the structure and properties of resulting filter are analyzed and compared to h and kalman filters

brief paper risk sensitivity conditions for stochastic uncertain model validation the paper presents sufficient and necessary conditions that verify the relevance of an assumed linear stochastic system model for problems in which probabilistic characteristics of the plant are not known exactly the approach is to establish the existence of an admissible probability space on which the output of the candidate stochastic system model is consistent in a stochastic sense with the noisy output of the plant

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

convolutional codes over groups the basic algebraic structure theory of convolutional codes and their trellises is developed simultaneously for codes over groups rings and fields the first part which covers fundamental notions such as minimality and observability is semi tutorial in that most definitions are already standard within the modern behavioral theory as are some of the formally stated results however some of the pivotal results emphasizing the role of observability as the basic well behavedness condition for codes are new and several previous results are given simplified proofs the usefulness of the behavioral approach even for convolutional codes over fields is demonstrated by a new minimality test for encoders as well as by the straightforward derivation of some known minimality criteria for generator matrices from the basic minimality criteria for group trellises the second part of the paper deals with issues that are specific to codes over rings and groups the main result is a concise characterization the first such of those groups that can appear as the branch group of any group trellis it is further shown how such groups are presented by shift registers a new large class of noncommutative convolutional codes is also given

robust stability and performance of stochastic uncertain systems on an infinite time interval abstract in this paper we consider a robust stability problem for continuous time stochastic uncertain systems the uncertainty in the system is characterized in terms of an uncertain probability distribution on the noise input this uncertainty is assumed to satisfy a certain relative entropy constraint the solution to a specially parametrized risk sensitive performance analysis problem is used to estimate the level of guaranteed performance for the stochastic uncertain system under consideration this solution is obtained by solving an algebraic riccati equation the corresponding performance bound holds for all admissible uncertainties and is nonconservative

robust output feedback stabilization via risk sensitive control we consider a problem of robust linear quadratic gaussian lqg control for discrete time stochastic uncertain systems with partial state measurements for a finite horizon case the problem was recently introduced by petersen et al ieee trans automat control 45 2000 398 in this paper an infinite horizon extension of the results of petersen et al ieee trans automat control 45 2000 398 is discussed we show that for a broad class of uncertain systems under consideration a controller constructed in terms of the solution to a specially parameterized risk sensitive stochastic control problem absolutely stabilizes the stochastic uncertain system

minimax lqg control of stochastic partially observed uncertain systems we consider an infinite horizon linear quadratic minimax optimal control problem for stochastic uncertain systems with output measurement a new description of stochastic uncertainty is introduced using a relative entropy constraint for the stochastic uncertain system under consideration a connection between the worst case control design problem and a specially parametrized risk sensitive stochastic control problem is established using this connection a minimax optimal lqg controller is constructed which is based on a pair of algebraic matrix riccati equations arising in risk sensitive control it is shown that this minimax optimal controller absolutely stabilizes the stochastic uncertain system

observability of linear stochastic uncertain systems a concept of observability of uncertain systems has received some attention in the recent literature in this note we extend this concept to stochastic uncertain systems we show that the observability of a stochastic uncertain system can be characterized using a relative entropy functional which plays the same role as the observability gramian plays in linear system analysis also we establish a connection between the observability of a linear stochastic uncertain system under consideration and risk sensitive performance of an associated reference system

dichotomy and absolute stability of nonlinear systems with periodically nonstationary linear part abstract systems with periodically nonstationary linear part are considered inputs and outputs of nonlinearities satisfy a certain integral quadratic constraint the conditions necessary and sufficient of dichotomy and absolute stability are obtained

robust guaranteed cost state estimation for nonlinear stochastic uncertain systems via an iqc approach abstract this paper presents an approach to robust nonlinear state estimation based on the use of integral quadratic constraints and minimax lqg control the approach involves a class of state estimators which includes copies of the system nonlinearities in the state estimator the nonlinearities being considered are those which satisfy a certain global lipschitz condition the linear part of the state estimator is synthesized using minimax lqg control theory which is closely related to h control theory and this leads to a nonlinear state estimator which gives an upper bound on the estimation error cost

guaranteed cost lqg control for uncertain systems with a normalized coprime factor uncertainty structure the paper extends the robust minimax lqg control design methodology to stochastic uncertain systems with a general uncertainty structure which includes normalized coprime factor uncertainty passive uncertainty and sector norm bounded uncertainty as special cases the derivation of the result uses a special parameter dependent girsanov measure transformation the efficacy of the proposed methodology is illustrated using the problem of frequency locking of an optical cavity which occurs in the area of experimental quantum optics

hierarchical dobinski type relations via substitution and the moment problem we consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson a a 1 monomials of the form exp a ra r 1 2 under the composition of their exponential generating functions they turn out to be of sheffer type we demonstrate that two key properties of these sequences remain preserved under substitutional composition a the property of being the solution of the stieltjes moment problem and b the representation of these sequences through infinite series dobinski type relations we present a number of examples of such composition satisfying properties a and b we obtain new dobinski type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences

minimal realizations of linear systems the shortest basis approach given a discrete time linear system c a shortest basis for c is a set of linearly independent generators for c with the least possible lengths a basis b is a shortest basis if and only if it has the predictable span property i e has the predictable delay and degree properties and is non catastrophic or alternatively if and only if it has the subsystem basis property for any interval j the generators in b whose span is in j is a basis for the subsystem cj the dimensions of the minimal state spaces and minimal transition spaces of c are simply the numbers of generators in a shortest basis b that are active at any given state or symbol time respectively a minimal linear realization for c in controller canonical form follows directly from a shortest basis for c and a minimal linear realization for c in observer canonical form follows directly from a shortest basis for the orthogonal system c this approach seems conceptually simpler than that of classical minimal realization theory

absolute stabilization and minimax optimal control of uncertain systems with stochastic uncertainty this paper is concerned with existence and optimality properties of so called guaranteed cost controllers for an uncertain system subject to structured uncertainty the uncertainty in the system is assumed to have a stochastic character and to satisfy certain stochastic integral constraints it is shown that a minimax optimal guaranteed cost state feedback controller for a stochastic system can be synthesized as a state feedback controller absolutely stabilizing this system for each initial state of the system this controller can be found by parametric optimization of solutions of a parameter dependent generalized matrix riccati equation arising in stochastic h infty theory

robust controllability of linear stochastic uncertain systems the paper extends the concept of robust controllability via linear state feedback to stochastic uncertain systems we show that the controllability of a stochastic uncertain system can be characterized using solutions to a game type differential riccati equation

nonlinear versus linear control in the absolute stabilizability of uncertain systems with structured uncertainty this note considers a stabilization problem for a class of uncertain linear systems containing structured uncertainty described by a certain integral quadratic constraint the notion of stabilizability considered is that of absolute stabilizability the main result gives a necessary and sufficient condition for the absolute stabilizability of this class of uncertain systems in terms of the existence of a solution to a corresponding diagonally scaled h sup spl infin control problem it follows from this result that absolute stabilizability via nonlinear control implies absolute stabilizability via linear control

partial skew orthogonal polynomials and related integrable lattices with pfaffian tau functions skew orthogonal polynomials sops arise in the study of the n point distribution function for orthogonal and symplectic random matrix ensembles motivated by the average of characteristic polynomials of the bures random matrix ensemble studied in forrester and kieburg commun math phys 342 1 151 187 2016 we propose the concept of partial skew orthogonal polynomials psops as a modification of the sops and then the psops with a variety of special skew symmetric kernels and weight functions are addressed by considering appropriate deformations of the weight functions we derive nine integrable lattices in different dimensions as a consequence the tau functions for these systems are shown to be expressed in terms of pfaffians and the wave vectors psops in fact the tau functions also admit the multiple integral representations among these integrable lattices some of them are known while the others are novel to the best of our knowledge in particular one integrable lattice is related to the partition function of the bures ensemble besides we derive a discrete integrable lattice which can be used to compute certain vector pade approximants this yields the first example regarding the connection between integrable lattices and generalised inverse vector valued pade approximants about which hietarinta joshi and nijhoff pointed out that this field remains largely to be explored in the recent monograph hietarinta et al in discrete systems and integrability vol 54 cambridge university press cambridge 2016 section 4 4

controllability and observability indices for linear time invariant systems a new approach abstract this paper is devoted to an investigation of the controllability and observability indices for linear time invariant discrete time systems these indices are here introduced at an intrinsic level in the context of the behavioral approach to systems theory for the class of the complete controllable systems it is proven that the controllability indices can be interpreted as a minimal and complete set of invariants with respect to a very natural systems equivalence autonomous phenomena are also discussed

on toda lattices and orthogonal polynomials first we derive a simple connection between toda and langmuir lattices and give a characterization of toda lattices with the help of stieltjes functions then it is shown how to generate by orthogonal polynomials in an elementary way periodic and almost periodic toda lattices the particles of the toda lattice are not even restricted as usual to move on the real line they may also move in the complex plane with the help of this result for special cases explicit solutions are obtained in terms of elliptic functions

robust filtering of stochastic uncertain systems on an infinite time horizon in this paper we consider a robust filtering problem for continuous time stochastic uncertain systems the uncertainty in the system is characterized in terms of an uncertain probability distribution on the noise input this uncertainty is assumed to satisfy a certain relative entropy constraint the solution to a specially parameterized risk sensitive stochastic filtering problem is used to construct a filter for the original uncertain system which guarantees an optimal worst case filtering error the corresponding minimax optimal filter is obtained by solving a pair of algebraic riccati equations

classification problems for shifts on modules over a principal ideal domain in this paper we study symbolic dynamics over alphabets which are modules over a principal ideal domain considering topological shifts which are also submodules our main result is the classification up to algebraic and topological conjugacy of the torsion free transitive finite memory shifts

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

minimax optimal control of stochastic uncertain systems with relative entropy constraints this paper considers a new class of discrete time stochastic uncertain systems in which the uncertainty is described by a constraint on the relative entropy between a nominal noise distribution and the perturbed noise distribution this uncertainty description is a natural extension to the case of stochastic uncertain systems of the sum quadratic constraint uncertainty description this paper solves problems of worst case robust performance analysis and output feedback minimax optimal controller synthesis in a general nonlinear setting specializing these results to the linear case leads to a minimax linear quadratic gaussian lqg optimal controller this controller is defined by riccati difference equations and a kalman filter like state equation the paper also shows that the minimax lqg problem will have a solution if and only if a corresponding h sup spl infin control problem has a solution a linear example is presented to illustrate the minimax lqg methodology

special solutions of the toda chain and combinatorial numbers we classify all solutions of the restricted toda chain from the ansatz that all moments are expressed in terms of polynomials of some variable we show that each such solution corresponds to the sheffer class orthogonal polynomials it is shown that corresponding hankel determinants are related to some well known combinatorial numbers

the lanczos algorithm for extensive many body systems in the thermodynamic limit we establish rigorously the scaling properties of the lanczos process applied to an arbitrary extensive many body system which is carried to convergence n and the thermodynamic limit n taken in this limit the solution for the limiting lanczos coefficients are found exactly and generally through two equivalent sets of equations given initial knowledge of the exact cumulant generating function the measure and the orthogonal polynomial system associated with the lanczos process in this regime are also given explicitly some important representations of these lanczos functions are given including taylor series expansions and theorems controling their general properties are proven

addition formulas for polynomials built on classical combinatorial sequences abstract we prove addition formulas for some polynomials built on combinatorial sequences catalan numbers bell numbers or related to classical polynomials hermite these relations give also an explicit computation of several hankel determinants

elliptic solutions of the restricted toda chain lame polynomials and generalization of the elliptic stieltjes polynomials we construct new families of elliptic solutions of the restricted toda chain the main tool is a special so called stieltjes ansatz for the moments of corresponding orthogonal polynomials we show that the moments thus obtained are related to three types of lam polynomials the corresponding orthogonal polynomials can be considered as a generalization of the stieltjes carlitz elliptic polynomials

shifts on compact and discrete lie groups algebraic topological invariants and classification problems abstract this paper is devoted to an investigation of various dynamical concepts for group shift systems which are invariant by algebraic conjugacy i e topological conjugacy preserving the group structure the concept of controllability which is stronger than topological transitivity and the concept of limit dimension which is analogous to topological entropy are discussed at length particular attention is dedicated to the abelian case for which duality results are also established