# Periodic input estimation for linear periodic systems: Automotive engine applications

Auteurs: Jonathan Chauvin, Gilles Corde, Nicolas Petit et Pierre Rouchon

Automatica. Vol. 43, No. 6, pp 971--980. 2007, DOI: 10.1016/j.automatica.2006.12.012

In this paper, we consider periodic linear systems driven by T0 -periodic signals that we desire to reconstruct. The systems under consideration are of the form

where A(t ), A0 (t ), and C(t ) are T0 -periodic matrices. The period T0 is known. The T0 -periodic input signal w(t ) is unknown but is assumed to admit a ﬁnite dimensional Fourier decomposition.

Our contribution is a technique to estimate w from the measurements y. In both full state measurement and partial state measurement cases, we propose an efﬁcient observer for the coefﬁcients of the Fourier decomposition of w(t ). The proposed techniques are particularly attractive for automotive engine applications where sampling time is short. In this situation, standard estimation techniques based on Kalman ﬁlters are often discarded (because of their relative high computational burden). Relevance of our approach is supported by two practical cases of application.

Detailed convergence analysis is also provided. Under standard observability conditions, we prove asymptotic convergence when the tuning PeriodicInputparameters are chosen sufﬁciently small.

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Automatica. Vol. 43, No. 6, pp 971--980. 2007, DOI: 10.1016/j.automatica.2006.12.012

In this paper, we consider periodic linear systems driven by T0 -periodic signals that we desire to reconstruct. The systems under consideration are of the form

where A(t ), A0 (t ), and C(t ) are T0 -periodic matrices. The period T0 is known. The T0 -periodic input signal w(t ) is unknown but is assumed to admit a ﬁnite dimensional Fourier decomposition.

Our contribution is a technique to estimate w from the measurements y. In both full state measurement and partial state measurement cases, we propose an efﬁcient observer for the coefﬁcients of the Fourier decomposition of w(t ). The proposed techniques are particularly attractive for automotive engine applications where sampling time is short. In this situation, standard estimation techniques based on Kalman ﬁlters are often discarded (because of their relative high computational burden). Relevance of our approach is supported by two practical cases of application.

Detailed convergence analysis is also provided. Under standard observability conditions, we prove asymptotic convergence when the tuning PeriodicInputparameters are chosen sufﬁciently small.

Download PDF