N2SID: Nuclear norm subspace identification of innovation models

Michel Verhaegen,Anders Hansson

doi:10.1016/j.automatica.2016.05.021

Michel Verhaegen, Anders Hansson

Open Access

https://doi.org/10.1016/j.automatica.2016.05.021

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Abstract

The identification of multivariable state space models in innovation form is solved in a subspace identification framework using convex nuclear norm optimization. The convex optimization approach allows to include constraints on the unknown matrices in the data-equation characterizing subspace identification methods, such as the lower triangular block-Toeplitz of weighting matrices constructed from the Markov parameters of the unknown observer. The classical use of instrumental variables to remove the influence of the innovation term on the data equation in subspace identification is avoided. The avoidance of the instrumental variable projection step has the potential to improve the accuracy of the estimated model predictions, especially for short data length sequences.

Highlights

Subspace IDentification (SID) methods for the identification of Linear Time-Invariant (LTI) state space models as developed originally in [20, 10, 21] derive approximate models rather than models that are “optimal” with respect to a goodness of fit criterion defined in terms of the weighted norm of the difference between the measured output and the model predicted output
In this paper we present a new SID method for identifying multivariable state space models in innovation form within the framework of nuclear norm optimization
Subspace identification of multivariable state space innovation models is revisited in this paper in the scope of nuclear norm optimization methods and using the observer form

Summary

Introduction

Subspace IDentification (SID) methods for the identification of Linear Time-Invariant (LTI) state space models as developed originally in [20, 10, 21] derive approximate models rather than models that are “optimal” with respect to a goodness of fit criterion defined in terms of the weighted norm of the difference between the measured output and the model predicted output. A number of recent developments have been made to integrate the low rank approximation step in SID with a goodness of fit into a single multi-criteria convex optimization problem. These contributions were inspired by the work in [3] to approximate a constraint on the rank of a matrix by minimizing its nuclear norm. In this paper we present a new SID method for identifying multivariable state space models in innovation form within the framework of nuclear norm optimization.

The Subspace Identification Problem

Pareto optimal Subspace Identification

A convex relaxation A convex relaxation of the NP hard problem formulation

Concluding Remarks