Alternative Form of Predictor Based Identification of LPV-SS Models with Innovation Noise

Pepijn B. Cox, Roland Tóth

In this paper, we present an approach to identify linear parameter-varying (LPV) systems with a state-space (SS) model structure in an innovation form where the coefficient functions have static and affine dependency on the scheduling signal. With this scheme, the curse of dimensionality problem is reduced, compared to existing predictor based LPV subspace identification schemes. The investigated LPV-SS model is reformulated into an equivalent impulse response form, which turns out to be a moving average with exogenous inputs (MAX) system. The Markov coefficient functions of the LPV-MAX representation are multi-linear in the scheduling signal and its time-shifts, contrary to the predictor based schemes where the corresponding LPV auto-regressive with exogenous inputs system is multi-quadratic in the scheduling signal and its time-shifts. In this paper, we will prove that under certain conditions on the input and scheduling signals, the $\ell_2$ loss function of the one-step-ahead prediction error for the LPV-MAX model has only one unique minimum, corresponding to the original underlying system. Hence, identifying the LPV-MAX model in the prediction error minimization framework will be consistent and unbiased. The LPV-SS model is realized by applying an efficient basis reduced Ho-Kalman realization on the identified LPV-MAX model. The performance of the proposed scheme is assessed on a Monte Carlo simulation study.

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