On-line structured subspace identification with application to switched linear systems

Abstract

This article is concerned with the identification of switched linear multiple-inputs–multiple-outputs state-space systems in a recursive way. First, a structured subspace identification scheme for linear systems is presented which turns out to have many attractive features. More precisely, it does not require any singular value decomposition but is derived using orthogonal projection techniques; it allows a computationally appealing implementation and it is closely related to input–output models identification. Second, it is shown that this method can be implemented on-line to track both the range space of the extended observability matrix and its dimension and thereby, the system matrices. Third, by making use of an on-line switching times detection strategy, this method is applied to blindly identify switched systems and to label the obtained submodels. Simulation results on noisy data illustrate the abilities and the benefits of the proposed approach.

Keywords:

• subspace methods
• multivariable systems identification
• recursive methods
• linear systems
• switched systems
• state space models

Acknowledgements

The authors would like to thank the anonymous reviewers, whose constructive comments greatly helped to improve the quality of this manuscript.

Notes

Notes

1. That is, a matrix whose characteristic polynomial is equal to its minimal polynomial. Note that A is non-derogatory is a necessary condition for a MISO system to be observable.

2. gcd refers to the greatest common divisor, taken here to be a monic polynomial.

3. Recall that, as a permutation matrix, S satisfies S ^⊤ S = SS ^⊤ = I.

4. One can indeed show that when the input is PE(f + n _i ), this relation holds.

5. We recall that τ₁ − 1 is the last time index of the data involved in (43).

6. Even if it is not necessarily followed by an order change.