recovering the initial state of an infinite dimensional system using observers let a be the generator of a strongly continuous semigroup t on the hilbert space x and let c be a linear operator from d a to another hilbert space y possibly unbounded with respect to x not necessarily admissible we consider the problem of estimating the initial state z 0 d a with respect to the norm of x from the output function y t ct tz 0 given for all t in a bounded interval 0 t we introduce the concepts of estimatability and backward estimatability for a c in a more general way than currently available in the literature we introduce forward and backward observers and we provide an iterative algorithm for estimating z 0 from y this algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the gramian our results lead also to a very general formulation of russell s principle i e estimatability and backward estimatability imply exact observability this general formulation of the principle does not require t to be invertible we illustrate our estimation algorithms on systems described by wave and schrodinger equations and we provide results from numerical simulations

an observer for infinite dimensional dissipative bilinear systems we consider bilinear systems of the form t a t u t b t y t t c on an infinite dimensional hilbert space h where a is the generator of a semigroup of contraction b is a bounded dissipative operator and c h the input signal u l r such that u t 0 for almost every t r we present a simple observer for this class of systems with the estimation error converging weakly to zero in h for every sufficiently rich input inputs that we call regularly persistent our result is a generalization of the previous results in 1 2

a behavioral approach to linear exact modeling the behavioral approach to system theory provides a parameter free framework for the study of the general problem of linear exact modeling and recursive modeling the authors present the solution of the continuous time polynomial exponential time series modeling problem both recursive and nonrecursive solutions are provided and classified according to properties like complexity and controllability it is shown in particular that recursive modeling corresponds to updating by means of a cascade inter connection of systems as a special case the solution of several other problems such as rational interpolation realization and modeling of arbitrary discrete time time series is obtained

state and output feedback boundary control for a coupled pde ode system abstract this note is devoted to stabilizing a coupled pde ode system with interaction at the interface first a state feedback boundary controller is designed and the system is transformed into an exponentially stable pde ode cascade with an invertible integral transformation where pde backstepping is employed moreover the solution to the resulting closed loop system is derived explicitly second an observer is proposed which is proved to exhibit good performance in estimating the original coupled system and then an output feedback boundary controller is obtained for both the state and output feedback boundary controllers exponential stability analyses in the sense of the corresponding norms for the resulting closed loop systems are provided the boundary controller and observer for a scalar coupled pde ode system as well as the solutions to the closed loop systems are given explicitly

structured matrices and unconstrained rational interpolation problems we describe a fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept we use the interpolation data to implicitly define a convenient non hermitian structured matrix and then apply a computationally efficient procedure for its triangular factorization this leads to a transmission line interpretation that makes evident the interpolation properties we further discuss connections with the lagrange interpolating polynomial as well as questions regarding the minimality and the admissible degrees of complexity of the solutions

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

matrix polynomial solutions of tangential lagrange sylvester interpolation conditions of low mcmillan degree abstract a matrix polynomial of mcmillan degree n 1 which satisfies n tangential lagrange interpolation conditions is constructed explicitly in realization form analogous results are given for more general bitangential lagrange sylvester interpolation conditions a corollary is a type of euclidean algorithm for matrix polynomials and mcmillan degree the main tool is the recent parametrization of all matrix polynomial solutions of a set of lagrange sylvester interpolation conditions due to ball gohberg and rodman

sampled data observers for 1 d parabolic pdes with non local outputs abstract the present work provides a systematic approach for the design of sampled data observers to a wide class of 1 d parabolic pdes with non local outputs the studied class of parabolic pdes allows the presence of globally lipschitz nonlinear and non local terms in the pde two different sampled data observers are presented one with an inter sample predictor for the unavailable continuous measurement signal and one without an inter sample predictor explicit conditions on the upper diameter of the uncertain sampling schedule for both designs are derived for exponential convergence of the observer error to zero in the absence of measurement noise and modeling errors moreover explicit estimates of the convergence rate can be deduced based on the knowledge of the upper diameter of the sampling schedule when measurement noise and or modeling errors are present input to output stability ios estimates of the observer error hold for both designs with respect to noise and modeling errors the main results are illustrated by two examples which show how the proposed methodology can be extended to other cases e g boundary point measurements

compensating actuator and sensor dynamics governed by diffusion pdes for possibly unstable ode systems with actuator delay predictor based infinite dimensional feedback can compensate for actuator delay of arbitrary length and achieve stabilization we extend this concept to another class of pde ode cascades where the infinite dimensional part of the plant is of diffusive rather than convective type we derive predictor like feedback laws and observers with explicit gain kernels the gain kernels involve second order matrix exponentials of the system matrix of the ode plant which is the result of the second order in space character of the actuator sensor dynamics the construction of the kernel functions is performed using the continuum version of the backstepping method robustness to small perturbations in the diffusion coefficient is proved

differential methods in inverse scattering this paper discusses a new set of differential methods for solving the inverse scattering problem associated to the propagation of waves in an inhomogeneous medium by writing the medium equations in the form of a two component system describing the interaction of rightward and leftward propagating waves the causality of the propagation phenomena is exploited in order to identify the medium layer by layer the recursive procedure that we obtain constitutes a continuous version of an algorithm first derived by schur in order to test for the boundedness of functions analytic inside the unit circle it recovers the local reflectivity function of the medium using similar ideas some other differential methods can also be derived to reconstruct alternative parametrizations of the layered medium in terms of the local impedance or of the potential function the differential inverse scattering methods turn out to be very efficient since in some sense they let the medium perform the inversion by itself and thus

output feedback stabilization of an unstable wave equation we consider the problem of stabilization of a one dimensional wave equation that contains instability at its free end and control on the opposite end in contrast to classical collocated boundary damper feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition the controllers and the associated observers designed in the paper are more complex due to the open loop instability of the plant the controller and observer gains are designed using the method of backstepping which results in explicit formulae for the gain functions we prove exponential stability and the existence and uniqueness of classical solutions for the closed loop system we also derive the explicit compensators in frequency domain the results are illustrated with simulations

backstepping boundary control for first order hyperbolic pdes and application to systems with actuator and sensor delays abstract we consider a problem of boundary feedback stabilization of first order hyperbolic partial differential equations pdes these equations serve as a model for physical phenomena such as traffic flows chemical reactors and heat exchangers we design controllers using a backstepping method which has been recently developed for parabolic pdes with the integral transformation and boundary feedback the unstable pde is converted into a delay line system which converges to zero in finite time we then apply this procedure to finite dimensional systems with actuator and sensor delays to recover a well known infinite dimensional controller analog of the smith predictor for unstable plants we also show that the proposed method can be used for the boundary control of a korteweg de vries like third order pde the designs are illustrated with simulations

new stability and exact observability conditions for semilinear wave equations the problem of estimating the initial state of 1 d wave equations with globally lipschitz nonlinearities from boundary measurements on a finite interval was solved recently by using the sequence of forward and backward observers and deriving the upper bound for exact observability time in terms of linear matrix inequalities lmis fridman 2013 in the present paper we generalize this result to n d wave equations on a hypercube this extension includes new lmi based exponential stability conditions for n d wave equations as well as an upper bound on the minimum exact observability time in terms of lmis for 1 d wave equations with locally lipschitz nonlinearities we find an estimate on the region of initial conditions that are guaranteed to be uniquely recovered from the measurements the efficiency of the results is illustrated by numerical examples

column reduced rational matrix functions with given null pole data in the complex plane abstract explicit formulas are given for rational matrix functions which have a prescribed null pole structure in the complex plane and are column reduced at infinity a full parametrization of such functions is obtained the results are specified and developed further for matrix polynomials

fast algorithms for generalized displacement structures and lossless systems abstract we derive an efficient recursive procedure for the triangular factorization of strongly regular matrices with generalized displacement structure that includes as special cases a variety of previously studied classes such as toeplitz like and hankel like matrices the derivation is based on combining a simple gaussian elimination procedure with displacement structure and leads to a transmission like interpretation in terms of two cascades of first order sections we further derive state space realizations for each section and for the entire cascades and show that these realizations satisfy a generalized embedding result and a generalized notion of j losslessness the cascades turn out to have intrinsic blocking properties which can be shown to be equivalent to interpolation constrains

natural second order observers for second order distributed parameter systems the aim of this manuscript is to present an alternative method for designing state observers for second order distributed parameter systems without resorting to a first order formulation this method has the advantage of utilizing the algebraic structure that second order systems enjoy with the obvious computational savings in observer gain calculations the proposed scheme ensures that the derivative of the estimated position is indeed the estimate of the velocity component and to achieve such a result a parameter dependent lyapunov function was utilized to ensure the asymptotic convergence of the state estimation error

global exponential sampled data observers for nonlinear systems with delayed measurements abstract this paper presents new results concerning the observer design for certain classes of nonlinear systems with both sampled and delayed measurements by using a small gain approach we provide sufficient conditions which involve both the delay and the sampling period ensuring exponential convergence of the observer system error the proposed observer is robust with respect to measurement errors and perturbations of the sampling schedule moreover new results on the robust global exponential state predictor design problem are provided for wide classes of nonlinear systems

observers and initial state recovering for a class of hyperbolic systems via lyapunov method a b s t r a c t recently the problem of estimating the initial state of some linear infinite dimensional systems from measurements on a finite interval was solved by using the sequence of forward and backward observers ramdani tucsnak and weiss 2010 in the present paper we introduce a direct lyapunov approach to the problem and extend the results to the class of semilinear systems governed by wave and beam equations with boundary measurements from a finite interval we first design forward observers and derive linear matrix inequalities lmis for the exponential stability of the estimation errors further we obtain simple finite dimensional conditions in terms of lmis for an upper bound t on the minimal time that guarantees the convergence of the sequence of forward and backward observers on 0 t for the initial state recovering this t represents also an upper bound on the observability time for observation times bigger than t these lmis give upper bounds on the convergence rate of the iterative algorithm in the norm defined by the lyapunov functions in our approach t is found as the minimal dwelling time for the switched exponentially stable forward and backward estimation error systems with the different lyapunov functions liberzon 2003 the efficiency of the results is illustrated by numerical examples 2013 elsevier ltd all rights reserved

coupling operators wedderburn forney spaces and generalized inverses abstract the pole zero structure of a transfer function has been analyzed by two distinct approaches module theoretic coordinate free and analytic basis dependent the former the wyman sain conte perdon school emphasizes pole modules zero modules exact sequences of maps and abstract realization theory applications developed include the model matching problem zero structure of one sided inverses and connections with geometric systems theory the latter the gohberg school works with pole chains zero chains matrix equations and concrete factorization theory minimal wiener hopf inner outer and interpolation zero pole lagrange sylvester nevanlinna pick in this paper we make explicit the connection between the two approaches in particular we describe the zero pole exact sequence of a transfer function without a pole or zero at infinity in the coordinate system provided by a global left null pole triple an important tool for us and an object of interest in its own right is a generalized inverse g of a transfer function g we relate the poles of g with the zeros of g we also show that if the pole module of g is isomorphic to the zero module of g then left right pole pairs of g are left right null pairs of g

observer design for a class of nonlinear ode pde cascade systems abstract the problem of state observation is addressed for nonlinear systems that can be modeled by an ode pde series association the ode subsystem assumes a triangular structure while the pde element is of heat diffusion type the aim is to accurately estimate online the state vector of the ode subsystem and the distributed state of the pde element one major difficulty is that the state observation must only rely on the global system output i e the pde state at the terminal boundary in particular the connection point between the ode and the pde blocks is not accessible to measurements the observation problem is dealt with by designing a high gain type observer sufficient conditions involving the pde domain length are formally established that ensure the observer exponential convergence

on the extended luenberger type observer for semilinear distributed parameter systems the design of an extended luenberger observer is considered to solve the state observation problem for semilinear distributed parameter systems for this a backstepping based technique is proposed for the design of the output injection weights by making use of the extended linearization of the semilinear observer error system with respect to the observer state stability of both the linearized and the semilinear observer error dynamics is analyzed theoretically moreover an efficient sample and hold implementation is considered to improve the computational efficiency of the observer design simulation examples are provided for a bistable semilinear partial differential equation and the simplified model of a bioreactor with monod kinetics

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

the stabilization of a one dimensional wave equation by boundary feedback with noncollocated observation this note addresses the stabilization of a one dimensional wave equation with control at one end and noncollocated observation at another end a simple exponentially convergent observer is constructed the dynamical stabilizing boundary output feedback is designed via the observed state while the closed loop system is nondissipative we show its exponential stability using riesz basis approach

a recursive method for solving unconstrained tangential interpolation problems an efficient recursive solution is presented for the one sided unconstrained tangential interpolation problem the method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data the recursive procedure admits a physical interpretation in terms of discretized transmission lines in this framework the generating system is constructed as a cascade of first order sections singular steps occur only when the input data is contradictory i e only when the interpolation problem does not have a solution various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants the algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general hermite matrix interpolation problem a recursive method is proposed to compute a column reduced generating system that can be used to solve the minimal tangential interpolation problem

exact dynamic stiffness matrix of non symmetric thin walled curved beams subjected to initial axial force abstract an improved numerical method to exactly evaluate the dynamic element stiffness matrix is proposed for the spatially coupled free vibration analysis of non symmetric thin walled curved beams subjected to uniform axial force for this purpose firstly equations of motion boundary conditions and force deformation relations are rigorously derived from the total potential energy for a curved beam element next systems of linear algebraic equations with non symmetric matrices are constructed by introducing 14 displacement parameters and transforming the fourth order simultaneous differential equations into the first order simultaneous equations and then explicit expressions for displacement parameters are numerically evaluated via eigensolutions and the exact 14 14 element stiffness matrix is determined using force deformation relations in order to demonstrate the validity and the accuracy of this study the spatially coupled natural frequencies of non symmetric thin walled curved beams subjected to uniform compressive and tensile forces are evaluated and compared with analytical and finite element solutions using hermitian curved beam elements or abaqus s shell element in addition some results by the parametric study are reported

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

from continuous time design to sampled data design of observers in this work a sampled data nonlinear observer is designed using a continuous time design coupled with an inter sample output predictor the proposed sampled data observer is a hybrid system it is shown that under certain conditions the robustness properties of the continuous time design are inherited by the sampled data design as long as the sampling period is not too large the approach is applied to linear systems and to triangular globally lipschitz systems