Handling Certain Structure Information in Subspace Identification

Abstract

The prediction-error approach to parameter estimation of linear models often involves solving a non-convex optimization problem. In some cases, it is therefore difficult to guarantee that the global optimum will be found. A common way to handle this problem is to find an initial estimate, hopefully lying in the region of attraction of the global optimum, using some other method. The prediction-error estimate can then be obtained by a local search starting at the initial estimate. In this paper, a new approach for finding an initial estimate of certain linear models utilizing structure and the subspace method is presented. The polynomial models are first written on the observer canonical state-space form, where the specific structure is later utilized, rendering least-squares estimation problems with linear equality constraints.Keywords: System identification, Subspace method, Black-box models. Handling Certain Structure Information in Subspace IdentificationC. Lyzell * M. Enqvist * L. Ljung * * Division of Automatic Control, Department of Electrical Engineering, Linköpings Universitet, SE -581 83, Linköping, Sweden.(E-mail: {lyzell,maren,ljung}@isy.liu.se)Abstract: The prediction-error approach to parameter estimation of linear models often involves solving a non-convex optimization problem. In some cases, it is therefore difficult to guarantee that the global optimum will be found. A common way to handle this problem is to find an initial estimate, hopefully lying in the region of attraction of the global optimum, using some other method. The prediction-error estimate can then be obtained by a local search starting at the initial estimate. In this paper, a new approach for finding an initial estimate of certain linear models utilizing structure and the subspace method is presented. The polynomial models are first written on the observer canonical state-space form, where the specific structure is later utilized, rendering least-squares estimation problems with linear equality constraints.

On subspace identification of cascade structured systems

“…The problem of imposing some structure into the subspace method could be found in [6] where they try to identify OE and ARMAX models using subspace methods. In [7] and [8] they want to guarantee that the identified model with a subspace method is stable when the true linear system is known to be stable.…”

“…The proposed method here solves this problem by finding a similarity transform that takes the system to cascade form while minimizing the mean square error between the estimated output and the measured output. The method introduced here is related to the method proposed in [6] where OE and ARMAX models are estimated with subspace methods by finding suitable transformations, to get the system to the desired form. The method denoted method 1 has three steps:…”

Continuous-time gray-box identification of mechanical systems using subspace-based identification methods

“…Methods for estimation of gray-box models for industrial robots have been considered in [19], and in [20] with a method based on a least-squares approach. Moreover, structural reformulations in subspace identification, similar to the one used in this paper, for general dynamic systems have been investigated in [21], [22]. The main contribution of this paper is the development and validation of a time-domain (in contrast to previously suggested frequency-domain and leastsquares methods addressing the same problem) subspacebased gray-box identification method for continuous-time mechanical system models.…”

Subspace identification with prior steady-state information

“…Prior information on locations of poles was found beneficial in subspace identification [17], [18]. Modified subspace methods have been proposed for systems with known model structures including large-scale circulant systems [19], cascade systems [20], the Output Error (OE) and AutoRegressive Moving Average with eXternal input (AR-MAX) models [21]. Pre-specification of zero transfer functions in identified state-space models were also studied [22].…”