Adaptive state estimation of state-affine systems with unknown time-varying parameters

ABSTRACT

In this paper we provide two significant extensions to the recently developed parameter estimation-based observer design technique for state-affine systems. First, we consider the case when the full state of the system is reconstructed in spite of the presence of unknown, time-varying parameters entering into the system dynamics. Second, we address the problem of reduced order observers with finite convergence time. For the first problem, we propose a simple gradient-based adaptive observer that converges asymptotically under the assumption of generalised persistent excitation. For the reduced order observer we invoke the advanced dynamic regressor extension and mixing parameter estimator technique to show that we can achieve finite convergence time under the weak interval excitation assumption. Simulation results that illustrate the performance of the proposed adaptive observers are given. This include, an unobservable system, an example reported in the literature and the widely popular, and difficult to control, single-ended primary inductor converter.

KEYWORDS:

• Linear time-varying systems
• state-affine systems
• adaptive state observer design

Acknowledgements

This paper is partially supported by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 See Appendix 1 for the definition of the PMS.

2 We recall that a bounded vector signal $m:{\mathbb{R}}_{+}\to {\mathbb{R}}^q$ is IE if there exists a time interval $[t_0,t_0+t_a]\subset [0,\mathrm{\infty })$ such that ${\int }_{t_0}^{t_0+t_a}m\left(t\right)m^{\mathrm{\top }}\left(t\right)\ge ϵI_q$, for some $t_a>0$ and $ϵ>0$.

3 Some special scenarios where it is possible to satisfy this assumption are given in Mazenc et al. (2020, Subsection 2.3).

4 To generate the regressor extension we use the construction proposed by Kreisselmeier (1977).

Additional information

Funding

This work was supported by Ministry of Science and Higher Education of Russian Federation [goszadanie no. 2019-0898].