Stochastic theory of minimal realization

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.

After Crouch [2], we clarify the relation between minimal realizations Sussmann [7]-[11] and weakly minimal realizations Hermann and Krener [3] of a given input-output mapping in the analytic case. For a given input-output mapping, all the weakly minimal analytic realizations are classified. Our result, applied to realization theory, is exactly Crouch's one, and can be derived from Sussmann's paper [8]. Otherwise, it is closely related to other works of Krener [4] and Hermann and Krener [3]. Our contribution is essentially a presentation of the above-mentioned results in a general unified manner.

Geometric framework and minimal realizations of nonlinear systems on fibre bundle

On parametrizations for the minimal partial realization problem

We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou (1987) in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length.

Several results exposing the interrelations between state-space and frequency-domain descriptions of multivariable linear systems are presented. Three canonical forms for constant parameter autoregressive-moving average (ARMA) models for input-output relations are described and shown to corrrespond to three particular canonical forms for the state variable realization of the model. Invariant parameters for the partial realization problem are characterized. For stochastic processes, it is shown how to construct an ARMA model, driven by white noise, whose output has a specified covariance. A two-step procedure is given, based on minimal realization and Cholesky-factorization algorithms. Though the goal is an ARMA model, it proves useful to introduce an artificial state model and to employ the recently developed Chandrasekhar-type equations for state estimation. The important case of autoregressive processes is studied and it is shown how the Chandrasekhar-type equations can be used to obtain and generalize the well known Levinson-Wiggins-Robinson (LWR) recursion for estimation of stationary autoregressive processes.

Whereas optimal prediction of Gaussian sequences requires the employment of a linear filter with consistently identifiable parameters and with Gaussian white noise input, the optimal predictor of non-Gaussian sequences is n nonlinear filter, having an independent noise input. Since the latter cannot be identified directly without prior knowledge of the non-linearity, the optimal linear predictor is usually identified where a non-Gaussian white noise input is considered and which is fully optimal only when that input turns out to be independent in all moments. However, if the non-Gaussian sequence is the outcome of a Gaussian sequence passed through a zero memory non-linearity or through non-linear measurement elements, a transformation of the non-Gaussian sequence into a Gaussian one is possible, such that optimal non-linear prediction may be approximated to any required degree, as is shown by the analysis of the present work. Furthermore, the parameters of that predictor may be consistently identified in the absence of any parameter information.

An application of the Fokker-Planck equation in stochastic reservoir theory

The partial realization problem for 2D discrete linear shift-invariant systems is discussed. The authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2D linear system characterized by a given 2D impulse response array. The method used is based on the 2D Berlekamp-Massey algorithm which has a close connection with the 2D Hankel matrix derived from the 2D array. While it compares in efficiency with other methods for identification and approximation of 2D linear systems, it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree. >

Smoothing algorithms of various kinds have been around for several decades. However, some basic issues regarding the dynamical structure and the minimal dimension of the steady-state algorithm are still poorly understood. In this paper, we derive a realization of minimal dimension of the optimal smoother for a signal admitting a state-space description of dimension n. It is shown that the dimension of the smoothing algorithm can vary from n to 2n, depending on the zero structure of the signal model. The dynamics (pole structure) of the steady-state smoother is also characterized explicitly and is related to the zero structure of the model. We use several recent ideas from stochastic realization theory. In particular, a minimal Markovian representation of the smoother is derived, which requires solving a nonsymmetric Wiener-Hopf factorization problem. In this way, the smoother is naturally expressed as the cascade of a whitening filter and a linear filter of least possible dimension, whose state space is a minimal Markovian subspace containing the smoothed estimate x/spl I.cap/. This, among other aspects, affords a very simple calculation of the error covariance matrix of the smoother. A reduced-order two-filter implementation of the type due to Mayne (1966) and Fraser (1967) is obtained by solving a Riccati equation of reduced dimension, which is in general smaller than the dimension of the Riccati equations considered in the literature.

The problem of finding minimal realizations of linear constant systems from finite order input-output Markov matrix sequences is considered. The paper identifies from the sequences sets of independent structural and numerical quantities which are invariants of equivalent state space representations and completely characterize any minimal realization of the sequence. These sets, termed bases of invariants, acquire a “nesting” property by which a subsequent basis of a higher order finite sequence is obtained from the previous basis by addition of some new invariants. Two canonical state space representations of special forms that reflect the input and output structural properties of the underlying systems are presented and readily derived from these bases by a simple algorithm which is provided. Necessary and sufficient conditions for the existence of a unique minimal partial realization to a given finite Markov sequence are given in terms of the invariants of its nested basis. The set of all minimal partial realizations $S_{n_r }^r $ that, in the case of existence of more than one solution, corresponds to many distinct systems, is thoroughly investigated. A minimal set of undetermined quantities that parametrize $S_{n_r }^r $ is obtained. These parameters are used to characterize $S_{n_r }^r $ either in the form of bases of invariants or in the form of the canonical representations, and it is also shown that an arbitrary assignment of values to these parameters leads to a minimal realization of the given finite sequence. Additional properties of these parameters that may be desirable in certain identification problems are also discussed.

Digital averaging technique is widely used in weak signal detection. The weak signal buried in strong noise is normally digitized with a multi-bit analog-to-digital converter (ADC) before the averaging. When using large number of averaging, the digitization can be accomplished by a 2-level comparator, instead of a multi-bit ADC. In this paper, we studied the statistical properties of ADCs with additive white Gaussian noise (AWGN) input. Our analysis shows that, with low signal-to-noise ratio (SNR) input, a comparator introduces 20% amplitude SNR loss compared with a fine multi-bit ADC, which can be mostly recovered with a post filter. It also shows that the comparator is linear and insensitive to input offset with low SNR input and AWGN.

The realization of linear time-varying systems specified by an analytic weighting pattern is approached in a novel manner using an algebraic framework defined over the ring of analytic functions. Realizations are given by a state representation consisting of a first-order vector differential equation and an output equation, both with analytic coefficients. Various new criteria for realizability are derived, including conditions given in terms of the finiteness of modules over the ring of analytic functions generated by the elementary rows or columns of a (generalized) Hankel matrix. These results are related to local criteria for realizability specified in terms of the rank of matrix functions, as developed in the work of Silverman and Meadows [5], [8], [9] and Kalman [7]. It is shown that the construction of minimal realizations reduces to the problem of computing a basis for a finite free module defined over the ring of analytic functions. A minimal realization algorithm is then derived using a constructive procedure for computing bases for finite free modules over a Bezout domain. The Silverman-Meadows realization algorithm [5] is a special case of the procedure given here. In the last part of the paper, the realization algorithm is applied to the problem of system reduction.

A directional partial realization problem

Based on a generalization of the Markov parameters, a generalized method for the construction of minimal realizations is presented. The method can give a family of minimal realizations, which may be useful, for example, in the problem of minimal partial realization. In contrast to the Markov parameters, special forms of the generalized Markov parameters may readily be obtained from experimental data. In particular, it is shown that the Laguerre expansion coefficients of a system provide a special form of the generalized Markov parameters of an associated system, and by using the proposed method, an exact minimal state variable model of the system can be restored from its Laguerre truncation.