Linear Time-invariant State-space Models Research Articles

Subspace model identification Part 1. The output-error state-space model identification class of algorithms

In this paper, we present two novel algorithms to realize a finite dimensional, linear time-invariant state-space model from input-output data. The algorithms have a number of common features. They are classified as one of the subspace model identification schemes, in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input-output data. This structure is then exploited in the calculation of a realization. Another common feature is their algorithmic organization: an RQ factorization followed by a singular value decomposition and the solution of an overdetermined set (or sets) of equations. The schemes assume that the underlying system has an output-error structure and that a measurable input sequence is available. The latter characteristic indicates that both schemes are versions of the MIMO Output-Error State Space model identification (MOESP) approach. The first algorithm is denoted in particular as the (elementary MOESP scheme). The subspace approximation step requires, in addition to input-output data, knowledge of a restricted set of Markov parameters. The second algorithm, referred to as the (ordinary MOESP scheme), solely relies on input-output data. A compact implementation is presented of both schemes. Although we restrict our presentation here to error-free input-output data, a framework is set up in an identification context. The identification aspects of the presented realization schemes are treated in the forthcoming Parts 2 and 3.

Time moment and Padé approximation methods applied to the order reduction of MIMO linear systems

The combined use of the time moment and Padé approximation methods (in time space and frequency domain description), coupled with three powerful dominant pole selection criteria, is introduced for the purpose of application to high-order MIMO linear time-invariant state-space original system models, in order to obtain corresponding adequate reduced order model(s). The proposed approach has been applied successfully to a 10th-order two-input two-output time-invariant linear model of a practical power system.

The stability robustness determination of state space models with real unstructured perturbations

This paper considers the robust stability of a linear time-invariant state space model subject to real parameter perturbations. The problem is to find the distance of a given stable matrix from the set of unstable matrices. A new method, based on the properties of the Kronecker sum and two other composite matrices, is developed to study this problem; this new method makes it possible to distinguish real perturbations from complex ones. Although a procedure to find the exact value of the distance is still not available, some explicit lower bounds on the distance are obtained. The bounds are applicable only for the case of real plant perturbations, and are easy to compute numerically; if the matrix is large in size, an iterative procedure is given to compute the bounds. Various examples including a 46th-order spacecraft system are given to illustrate the results obtained. The examples show that the new bounds obtained can have an arbitrary degree of improvement over previously reported ones.

On time-invariant realizations of discrete random processes

A class of discrete second-order random processes, described by linear time-invariant state-space models, is investigated. Equivalent realizations with reduced number of noise inputs are presented. In contrast to the innovations approach these realizations have time-invariant parameters.