Finite-dimensional State Space Research Articles

Markov Processes Associated With Positivity Preserving Coercive Forms

AbstractCoercive closed forms on L2-spaces are studied whose associated L2-semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.

The optimal projection equations for finite-dimensional fixed-order ℋ2-optimal control of infinite-dimensional discrete-time systems

In this paper we are concerned with the fixed, finite-order ℋ2-optimal control of linear, infinite-dimensional discrete-time systems. The set of necessary conditions for the existence of the ℋ2-optimal controller is expressed in terms of four operator equations (two modified Riccati equations and two modified Lyapunov equations) coupled by a projector having the rank given by the dimension of the compensator state. It is shown that, when the order of the compensator is equal to the order of the plant, the four operator equations are decoupled and we obtain the infinite-dimensional generalization of the finite-dimensional state-space solution to the ℋ2-optimal control problem for time-varying systems, as given by lonescu and Weiss (1992).

Identification of the deterministic part of MIMO state space models given in innovations form from input-output data

In this paper we describe two algorithms to identify a linear, time-invariant, finite dimensional state space model from input-output data. The system to be identified is assumed to be excited by a measurable input and an unknown process noise and the measurements are disturbed by unknown measurement noise. Both noise sequences are discrete zero-mean white noise. The first algorithm gives consistent estimates only for the case where the input also is zero-mean white noise, while the same result is obtained with the second algorithm without this constraint. For the special case where the input signal is discrete zero-mean white noise, it is explicitly shown that this second algorithm is a special case of the recently developed Multivariable Output-Error State Space ( moesp) class of algorithms based on instrumental variables. The usefulness of the presented schemes is highlighted in a realistic simulation study.

Geometric representation of ignition and extinction concepts in microheterogeneous combustion

For microheterogeneous reaction-diffusion systems, the question as to the character of their stationary states has been investigated. For this, the two limiting cases of quasi-homogeneous and of essentially heterogeneous combustion have been introduced. In both cases, initiation of wave transition could be related to singularities in an appropriate state surface in a finite-dimensional state space

Continuity of dynamical systems: The continuous-time case

The purpose of this paper is to study continuity of the parametrization of continuous-time linear time-invariant differential systems having a finite-dimensional state space. We show that convergence of the behavior of such systems corresponds to convergence of the coefficients of a set of associated differential equations. For this to hold, both the behavior and the convergence need to be appropriately defined.

Tensor products in generalized measure theory

Klay, Randall, and Foulis established that the signed weight space of the tensor product of two quasimanuals each having a positive, finite-dimensional state space is isomorphic to the algebraic tensor product of the signed-weight spaces of the factors. We obtain a generalization of this result for arbitrary quasimanuals. A compactness condition due to Cook—here calleddiscreteness—is discussed and shown to be preserved under the formation of tensor products. It is shown that the predual of the signed weight space of a tensor product of discrete manuals is the projective (ordered) tensor product of the preduals of the signed weight spaces of the factors.

Symbolic dynamics of successive quantum-mechanical measurements.

We consider successive measurements on quantum-mechnical systems and investigate the way in which sequences of measurements produce information. The eigenvalues of suitable projection operators form symbolic sequences that characterize the quantum system under consideration. For several model systems with finite-dimensional state space, we explicitly calculate the probabilities to observe certain symbol sequences and determine the corresponding R\'enyi entropies K(\ensuremath{\beta}) with the help of the transfer-matrix method. A nontrivial dependence on \ensuremath{\beta} is observed. We show that the R\'enyi entropies as well as the symbol-sequence probabilities of the quantum-mechanical measurement process coincide with those of appropriate classes of one-dimensional chaotic maps.

A general framework for linear periodic systems with applications to H/sup infinity / sampled-data control

The authors present a framework for dealing with continuous-time periodic systems. The main tool is a lifting technique which provides a strong correspondence between continuous-time periodic systems and certain types of discrete-time time-invariant systems with infinite-dimensional input and output spaces. Despite the infinite dimensionality of the input and output spaces, a lifted system has a finite-dimensional state space if the original system does. This fact permits rather constructive methods for analyzing these systems. As a demonstration of the utility of this framework, the authors use it to describe the continuous-time (i.e., intersample) behavior of sampled-data systems, and to obtain a complete solution to the problem of parameterizing all controllers that constrain the L/sup 2/-induced norm of a sampled-data system to within a certain bound.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Non-finite-dimensional computability of filters in the presence of entrance boundaries

Let X be a one-dimensional diffusion with a constant diffusion coefficient. Suppose that we observe X in additive white noise. Bene[sbreve] showed that the conditional density of X(t) given the noisy observations up to time t can be computed explicitly by a stochastic system with a finite-dimensional state space if the drift of X is a differentiable, everywhere defined solution of a one dimensional Ricatti equation with a quadratic potential. Suppose now that the drift satisfies the same Ricatti equation, but only on a bounded or semi-bounded interval I at the endpoints of which it becomes singular. In this caseX will have entrance boundaries at the finite endpoints of I We show that unnormalized conditional estimates will no longer be finite-dimensionally computable in this situation. Our proof uses the following principle, which is of independent interest: the property of finite dimensional computability of a statistic is invariant under the application of the Markov semigroup of X

From simple to complicated: noncausal bounded input, bounded output stability in linear discrete time models in the deterministic and stochastic cases

We consider the question of bounded input, bounded output stability for time-invariant linear systems with a finite dimensional state space and with time axis Z, i.e., all the integers. Our approach also includes the stochastic ARMA models. We do not assume that the inputs are necessarily nonanticipating, and in this respect our results differ from most existing ones. Similar results have been given by Hannan and Deistler and by Brockwell and Davis. Our approach is polynomial-algebra-oriented, and does not use strictly rational functions.

Weak global controllability of nonlinear systems

The problem of the controllability of nonlinear systems in finite-dimensional state spaces is considered in a measure-theoretical framework, in which we deal with a set of measured defined by the boundary conditions and the differential equations of the problem. The property of weak controllability is then equivalent to the existence of a positive measure satisfying a set of linear equalities and inequalities. This problem is solved by considering the extension of an associated linear functional, defined in a suitable subspace of the space of functions being used. Necessary and sufficient conditions for weak controllability are obtained.

Aircraft Parameter Estimation with Non-Rational Turbulence Model

The maximum likelihood technique is applied to estimate aircraft stability and control parameters from flight data allowing for gust (turbulence) disturbance to be modeled by the von Karman spectrum rather than the more commonly used Dryden spectrum. The von Karman spectrum is a more accurate model of atmospheric turbulence, but unfortunately it is a non-rational function. This leads to complications in the theory because the finite dimensional state space representations can no longer be used. The identification and the associated filtering problems are solved by posing them in an artificially defined infinite dimensional state space. Numerical method of computing the optimal filter weighting function is presented. This algorithm is demonstrated by using actual flight test data.

Hilbert—Schmidt Systems with Finite-Dimensional State Spaces

The problem of characterizing the input-output behaviour of those linear systems which admit internal representations with finite-dimensional state spaces is studied in this paper. The systems are assumed to be Hilbert-Schmidt systems, and the main result is derived using well-known properties of the kernels of the associated Hankel operators. A necessary and sufficient condition is derived for a system to admit a completely observable and completely accessible internal representation with a finite-dimensional state space.

On the Problems of Constructing Minimal Realizations for Two-Dimensional Filters

The input-output behavior of a two-dimensional linear filter is defined by a formal power series in two variables. If the power series is rational the dynamics of the filter is described by updating equations on finite dimensional local state spaces. The class of realizations considered in this paper is constituted by doubly indexed dynamical systems of reduced structure. The construction of the class of minimal realizations is based on matrix representation techniques of noncommutative power series.

Computation of reachable and observable realizations of spatial filters

The input-output behaviour of a two-dimensional linear filter is defined by a formal power series in two variables- If the power series is rational, the dynamics of the filter is described by updating equations on finite dimensional local state spaces. The notions of local reachability and observability are defined in a natural way and an algorithm for obtaining a reachable and observable realization is given. In general reachability and observability do not imply the minimality of the realization. Nevertheless the dimension of a minimal realization is the least rank in a family of Hankel matrices.

State-space realization theory of two-dimensional filters

The realization problem of two-dimensional linear filters is approached from a system theoretic point of view. The input-output behavior of such a system is defined by formal power series in two variables, and a Nerode state space is constructed. This state space is, in general, infinite dimensional. If the power series is rational, the dynamics of the filter is described by updating equations on finite-dimensional local state space. The notions of local reachability and observability are defined in a natural way and an algorithm for obtaining a reachable and observable realization is given. In general, local reachability and observability do not imply the minimality of the realization.

Liapunov-Razumikhin conditions for asymptotic stability in functional differential equations of Volterra type

In [l], B.S. Razumikhin gave conditions for the stability and asymptotic stability of the zero state of systems of ordinary differential equations involving a fixed finite time delay. His conditions make use of Liapunov functions defined on the finite-dimensional state space rather than on the space of functions continuous on the interval of delay; for methods involving functions of the latter type, cf., for example, Yoshizawa 121. In a recent paper [3], the author has used Liapunov functions of Razumikhin type to give conditions sufficient for the stability of the zero state of a system of ordinary differential equations involving an interval of delay which becomes unbounded as t + + co; an example of such a system is an integrodifferential equation of Volterra type:

System identification via state characterization

Arbib and Zeiger's generalization of Ho's algorithm for system identification is presented from an alternative viewpoint called “state characterization”. This is an essentially nonlinear, nonparametric method for designing experiments which will directly determine a state space representation for an unknown black box. This method assumes (1) a finite dimensional state space is possible, (2) the unknown black box can be reset to its initial state at will, so that an arbitrary set of experiments can be performed, and (3) there is no plant noise, so that applying given inputs to the initial state will always yield the same state, so various experiments can be performed on it. In the initial, informal discussion the properties of this method of system identification are compared to the properties of the method of parameter estimation via loss function minimization, e.g. least square error or maximum likelihood, which is the only general approach to system identification presently available. Loss function minimization will utilize any data, but it is computationally difficult and essentially parametric. State characterization eliminates the computational requirement at the expense of requiring specific data, and it is essentially nonparametric. It is proposed that state characterization may have practical application in determining an approximate, low order description of a complex system about which we have little prior information, for instance in the social sciences and medicine. The formal presentation is limited to finite state automata and discrete time linear systems. The situation is considered in which the order of the unknown black box, the minimum size state space required to represent it, is not known a priori; so it is necessary to continue experimenting indefinitely, using the results to obtain successively better descriptions of the unknown black box. Detailed algorithms are presented for obtaining either a Moore model or a Mealy model description. Branching rules for using past data to choose subsequent experiments in order to hasten convergence are presented. Known theorems are used to prove that after some finite time these algorithms will yield a correct description, and new theorems show that the subsequent representations will be invariant after a correct description is obtained—although there is no way for the experimenter to know when he has obtained a correct description.

Reliability Prediction Studies of Complex Systems Having Many Failed States

In this paper, the theory of discrete Markov processes is used to develop methods for predicting the reliability and moments of the first time to failure of complex systems having many failed states. It is assumed that these complex systems operate in a repair environment and are composed of subsystems that have known constant failure and repair rates. Specifically, complex systems composed of any finite number of subsystems are considered. The complex system at any time is in an acceptable state or in a failed state. The methods presented for the reliability modeling of such complex systems assume a state behavior that is characterizable by a stationary Markov process (also called Markov chain) with finite-dimensional state space and a discrete time set. It is shown that once the matrix of the constant failure and repair rates of the subsystems is known, and the state assignment is made, then it is a straightforward matter to obtain the probabilistic description of the complex system.