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New results are presented concerning the state isomorphism approach to global identifiability analysis of parameterized classes of nonlinear state space systems with specified initial states. In particular we study the class of homogeneous systems, for which, under certain conditions, the local state isomorphism for a pair of indistinguishable parameter vectors is shown to be homogeneous of degree one. For homogeneous polynomial systems, conditions are given under which the local state isomorphism becomes linear. Here, the issue of whether or not the observability rank condition holds at the origin is shown to be of key importance. The scope of the results, which extend to the multivariable case, is discussed and illustrated by a number of worked examples. This demonstrates how the developed theory can be put to use to investigate the global identifiability properties of parameterized model classes.

In this paper we introduce two variants of a new parametrization for state–space systems which we will both call separable least squares data driven local co-ordinates (slsDDLC). SlsDDLC is obtained by modifying the parametrization by data driven local co-ordinates (DDLC). These modifications lead to analogous parametrizations, and we show how they can be used for a suitably concentrated likelihood criterion function. The concentration step can be done by an ordinary or generalized least squares step. An obvious consequence is the reduced number of parameters in the iterative search algorithm. The application of the parametrizations to maximum likelihood identification is exemplified. Simulations indicate that the usage of slsDDLC for concentrated likelihood functions has numerical advantages as compared to the usage of the more commonly used echelon canonical form or conventional DDLC for the likelihood function. Copyright © 2004 John Wiley & Sons, Ltd.

State space methods have proved to be powerful theoretical and computational tools in a number of areas of applications, in particular filtering and control theory. In this paper we advocate the use of state space methods for the study of discrete probability densities on the set {0,1,2,…}. The fundamental approach is to consider the class D of all discrete probability densities that can be represented as the impulse response/convolution kernel of a finite dimensional discrete time state space system. We show that all standard operations such as the calculation of moments, convolution, scaling, translation, product, etc. can be carried out using system representations.

In this paper we present a simulation study as well as two applications of a newly proposed approach towards I(1) cointegration using so-called state-space error correction models (SSECMs). SSECMs have been introduced in Ribarits and Hanzon (2014) where questions of model selection, parametrization and maximum likelihood estimation are treated in detail. The present paper shows the potential of this new approach to I (1) cointegration analysis.

In this paper we consider cointegrated I (1) processes in the state-space framework. We introduce the state-space error correction model (SSECM) and discuss in detail how to estimate SSECMs by (pseudo) maximum likelihood methods, including reduced rank regression techniques which allow for a successive reduction of the number of parameters in the original constrained likelihood optimization problem. In doing so, we follow very closely the Johansen approach for the VAR case; see Johansen (1995). Finally, the remaining free parameters will be represented using a local parametrization technique and we show how efficient gradient-type algorithms can be employed for the numerical optimization in the resulting lower dimensional unconstrained problem. Simulation studies and applications are presented in a separate 'companion paper'; see Ribarits and Hanzon (2014).

This paper deals with the problem of computing a best stable rational L/sup 2/ approximation of specified order to a given multivariable transfer function. The problem is equivalently formulated as a minimization problem over the manifold of stable all-pass (or lossless) transfer functions of fixed order. Some special Schur parameters are used to describe this manifold. Such a description presents numerous advantages: it takes into account the stability constraint, possesses a good numerical behavior and provides a model in state-space form, which is very useful in practice. A rigorous and convergent algorithm is proposed to compute local minima which has been implemented using standard MATLAB subroutines. The effectiveness of our approach to solve model reduction problems as well as identification problems in frequency domain is demonstrated through several examples, including real-data simulations.

A new approach to the question of parametrizing state-space systems is considered. The approach consists of using input-normal state-space representations. These representations are unique up to an isometric state isomorphism. By decomposing the tangent space of the set of normalized controllable matrix pairs into the tangent space of the equivalence class and the orthogonal complement one obtains an interesting set of local coordinates with a number of remarkable properties. Here the approach is presented in the context of a separable least squares approach to maximum likelihood estimation of a linear system.

A state-space approach to H 2 model reduction of stable systems is pursued, using techniques from global rational optimization, and aimed at finding global optima. First, a concentrated version of the H 2 criterion is derived, by optimizing analytically with respect to a subset of parameters. Next, a bound is developed, which is shown to be sharp for reduction to models of order 1. The choice of parameterization is such as to keep all these criteria rational. To illustrate the approach, a nontrivial example of a multivariable stable system of order 4 was designed, together with an approximation of order 2. Detailed analysis shows the approximation to be the global optimum among all stable system of order 2, constituting the first such system described in the literature. The fact that it also yields a global optimum to the bound, which happens to be sharp at the optimum, leads to a couple of questions for further research.

The asymptotic Fisher information matrix (FIM) has several applications in linear systems theory and statistical parameter estimation. It occurs in relation to the Cramer-Rao lower bound for the covariance of unbiased estimators. It is explicitly used in the method of scoring and it determines the asymptotic convergence properties of various system identification methods. It defines the Fisher metric on manifolds of systems and it can be used to analyze questions on identifiability of parametrized model classes. For many of these applications, exact symbolic computation of the FIM can be of great use. In this paper two different methods are described for the symbolic computation of the asymptotic FIM. The first method applies to parametrized MIMO state-space systems driven by stationary Gaussian white noise and proceeds via the solution of discrete-time Lyapunov and Sylvester equations, for which a method based on Faddeev sequences is used. This approach also leads to new short proofs of certain well-known results on the structure of the FIM for SISO ARMA systems. The second method applies to parametrized SISO state-space systems and uses an extended Faddeev algorithm. For both algorithms the concept of a Faddeev reachability matrix and the solution of discrete-time Sylvester equations in controller companion form are central issues. The methods are illustrated by two worked examples. The first concerns three different parametrizations of the class of stable AR(n) systems. The second concerns a model for an industrial mixing process, in which the value of exact computation to answer identifiability questions becomes particularly clear.

In this paper the boundaries of several families of (time-invariant) ARMA models and corresponding linear state space models are described. The topology of pointwise convergence of the Markov parameters is used.

In this paper we are concerned with linear (stochastic) systems like dxt = (Axt+B1ut)dt + B2dwt, dyt = Cxt dt + dvt or (more or less equivalent) ARMAX models and the problem of identifying A, B1, B2, C on the basis of observations of the inputs ut and outputs yt. In particular we are interested in the problem of whether there exists a machine (a system) driven by the instant neous observations (ut, yt) which as output produces a "best" estimate of the unknown system (recursive estimation). And even more particularly we are interested on how big (in state space dimension) such a machine must be. Introducing additional state space parameters a ij , b k l 1 , b k l 2 , c r, s and equations da ij = db k l 1 = db k l 2 = dc rs = 0 converts the original problem into a nonlinear filtering problem. For such problems the socalled estimation Lie algebra contains a good deal of information (on how hard the problem is), and this is what we try to exploit in this paper.