A singular perturbation approach to nonlinear observer design with an application to electromagnetic actuators

ABSTRACT

A constructive method for the design of nonlinear observers is discussed. To formulate conditions for the construction of the observer gains, stability results for nonlinear singularly perturbed systems are utilised. The nonlinear observer is designed directly in the given coordinates, where the error dynamics between the plant and the observer becomes singularly perturbed by a high-gain part of the observer injection, and the information of the slow manifold is exploited to construct the observer gains of the reduced-order dynamics. This is in contrast to typical high-gain observer approaches, where the observer gains are chosen such that the nonlinearities are dominated by a linear system. It will be demonstrated that the considered approach is particularly suited for self-sensing electromechanical systems. Two variants of the proposed observer design are illustrated for a nonlinear electromagnetic actuator, where the mechanical quantities, i.e. the position and the velocity, are not measured.

KEYWORDS:

• Nonlinear observers
• singular perturbations
• stability analysis
• electromagnetic actuators
• self-sensing

Acknowledgements

The authors thank the anonymous reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

T. Braun http://orcid.org/0000-0003-4866-1089

J. Reuter http://orcid.org/0000-0003-2069-5682

J. Rudolph http://orcid.org/0000-0001-8402-2458

Notes

1. In the notation, the variables that are considered as additional design parameters are separated by semicolons.

2. The consideration of the invariant set $\mathcal{A}$ is based on Shim et al. (2003).

3. For the definition of comparison functions see Hahn (1967) or Khalil (2002).

4. Roughly speaking, a nonlinear nonautonomous dynamical system is called detectable if two different solutions of the system, that yield indistinguishable outputs, are convergent (see e.g. Praly, 2015).

5. Compare with the related discussion in the context of sliding observers in Slotine et al. (1987).

6. Compare with Kokotović et al. (1986), where a similar value of ϑ is obtained in the consideration of the region of attraction for singular perturbed systems.