Shaping state and time-dependent convergence rates in non-linear control and observer design

TL;DR

This paper presents a method for designing controllers and observers for non-linear, time-varying systems to achieve specified, state- and time-dependent convergence rates with global stability guarantees, extending contraction theory.

Contribution

It introduces a novel gain-scheduling approach for non-linear control and a dual observer design for broad classes of systems, generalizing classical linear techniques.

Findings

Achieves specified convergence rates in non-linear, time-varying systems.

Provides a dual observer design for complex non-linear systems.

Demonstrates applications in aerospace, chemical plants, and navigation.

Abstract

This paper derives for non-linear, time-varying and feedback linearizable systems simple controller designs to achieve specified state-and timedependent complex convergence rates. This approach can be regarded as a general gain-scheduling technique with global exponential stability guarantee. Typical applications include the transonic control of an aircraft with strongly Mach or time-dependent eigenvalues or the state-dependent complex eigenvalue placement of the inverted pendulum. As a generalization of the LTI Luenberger observer a dual observer design technique is derived for a broad set of non-linear and time-varying systems, where so far straightforward observer techniques were not known. The resulting observer design is illustrated for non-linear chemical plants, the Van-der-Pol oscillator, the discrete logarithmic map series prediction and the lighthouse navigation problem. These results [23] allow one to shape globally the state- and time-dependent convergence behaviour ideally suited to the non-linear or time-varying system. The technique can also be used to provide analytic robustness guarantees against modelling uncertainties. The derivations are based on non-linear contraction theory [18], a comparatively recent dynamic system analysis tool whose results will be reviewed and extended.