Michel Verhaegen

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

The problem of Linear Multivariable State Space model identification from input-output data can under the presence of process- and measurement noise be solved in a non-iterative way when incorporating instrumental variables constructed from both input and output sequences in the recently developed class of multivariable output-error state space model class of subspace model identification schemes. Abstract--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 tloise 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.

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)...

Identifying MIMO Wiener systems using subspace model identification methods

Abstract In this paper we show that the MOESP (Multivariable Output-Error State sPace) class of subspace model identification (SMI) schemes can be extended to identify Wiener systems, a series connection of a linear dynamic system followed by a static nonlinearity. Methods are developed for two cases: firstly when the nonlinearity is not an even function, and secondly for systems which include even nonlinearities. Simulations are used to demonstrate the efficacy of these algorithms, and to compare them to existing, correlation based techniques.

Subspace model identification Part 2. Analysis of the elementary output-error state-space model identification algorithm

The elementary MOESP algorithm presented in the first part of this series of papers is analysed in this paper. This is done in three different ways. First, we study the asymptotic properties of the estimated state-space model when only considering zero-mean white noise perturbations on the output sequence. It is shown that, in this case, the MOESPl implementation yields asymptotically unbiased estimates. An important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the Hankel matrices, constructed from the input and output data at the beginning of the computations, depends on the number of non-zero Markov parameters. This analysis, however, leads to a second implementation of the elementary MOESP scheme, namely MOESP2. The latter implementation has the same asymptotic properties without the finite impulse response constraint. Secondly, we compare the MOESP2 algorithm with a classical state space model identification scheme. The latter...

Distributed Control for Identical Dynamically Coupled Systems: A Decomposition Approach

We consider the problem of designing distributed controllers for a class of systems which can be obtained from the interconnection of a number of identical subsystems. If the state space matrices of these systems satisfy a certain structural property, then it is possible to derive a procedure for designing a distributed controller which has the same interconnection pattern as the plant. This procedure is basically a multiobjective optimization under linear matrix inequality constraints, with system norms as performance indices. The explicit expressions for computing these controllers are given for both H infin or H 2 performance, and both for static state feedback and dynamic output feedback (in discrete time). At the end of the paper, two application examples illustrate the effectiveness of the approach.

Subspace algorithms for the identification of multivarible dynamic errors-in-variables models

Abstract We consider the problem of identifying multivariable finite dimensional linear time-invariant systems from noisy input/output measurements. Apart from the fact that both the measured input and output are corrupted by additive white noise, the output may also be contaminated by a term which is caused by a white input process noise; furthermore, all these noise processes are allowed to be correlated with each other. We shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open- or closed-loop. Two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms.

Recursive subspace identification of linear and non-linear Wiener state-space models

The MOESP class of identification algorithms are made recursive on the basis of various updating schemes for subspace tracking

Subspace identification of multivariable linear parameter-varying systems

A subspace identification method is discussed that deals with multivariable linear parameter-varying state-space systems with affine parameter dependence. It is shown that a major problem with subspace methods for this kind of system is the enormous dimension of the data matrices involved. To overcome the curse of dimensionality, we suggest using only the most dominant rows of the data matrices in estimating the model. An efficient selection algorithm is discussed that does not require the formation of the complete data matrices, but processes them row by row.

Subspace identification of Bilinear and LPV systems for open- and closed-loop data

In this paper we present a novel algorithm to identify LPV systems with affine parameter dependence operating under open- and closed-loop conditions. A factorization is introduced which makes it possible to form a predictor that predicts the output, which is based on past inputs, outputs, and scheduling data. The predictor contains the LPV equivalent of the Markov parameters. Using this predictor, ideas from closed-loop LTI identification are developed to estimate the state sequence from which the LPV system matrices can be constructed. A numerically efficient implementation is presented using the kernel method. It turns out that if structure is present in the scheduling sequence the computational complexity reduces even more.

Identifying MIMO Hammerstein systems in the context of subspace model identification methods

In this paper, we outline the extension of the MOESP family of subspace model identification schemes to the Hammerstein-type of nonlinear system. Two types of identification problem are considered. The first type assumes the (polynomial) structure of the static nonlinearity to be given and the task is to identify both the linear system dynamics and the unknown proportional constants in the para-metrization of the static nonlinearity. The second type addresses the identification of both the linear dynamic part and the static nonlinearity, where only limited a priori information regarding the structure of the nonlinearity is available. The improved robustness properties of the algorithms developed for this second type of Hammerstein identification problem over existing correlation-based schemes is illustrated by a numerical example.