Dynamic gain-scheduled control and extended linearisation: extensions, explicit formulae and stability

Abstract

A controller designed for linearisations at various trim/operating points of a nonlinear system using linear approaches is not necessarily performing well or stable once scheduled with a state under dynamic conditions; the key idea of using this scheduled control law design is to retain states close to the current, usually dynamically varying, operating point. Dynamic gain scheduling (DGS) is a technique aimed to resolve this controller scheduling issue for rapidly changing dynamics and states. It entails scheduling the control law gains with a fast-varying state variable rather than with a slowly varying state. It has been successfully applied to the aircraft system models, allowing also for nested loops. The aim of this paper is to extend DGS to a more general setting allowing for more complex multi-input dynamics incorporating the more simple static scheduling within the same controller. Hence, given a linear design, suitable transformations are provided which allow fast scheduling of multi-variable controllers. Important parallels to the approach of extended linearisation are drawn. Theoretical results are shown, providing explicit formulae related to nonlinear dynamic inversion control. Several examples of varying nonlinear complexity are presented in order to emphasise the characteristics of the approach.

Keywords:

• dynamic gain scheduling
• nonlinear systems
• extended linearisation

Acknowledgements

The authors would like to acknowledge private conversations on the use of gain scheduling with several engineers with longstanding work background in the aerospace industry.

Notes

1. The signal w ∈ ℜ^m is an external signal and generally not constant.

2. The demand trajectory has been limited in terms of its slope over time. This is not due to the limitation of the gain scheduling approach but due to the generic inability of a control law to track a demand of significant temporal change. A generic option to deal with that is to feed forward the information of to allow faster controller action. To retain simplicity, this is not done here, which implies that the magnitude may cause a controller error.

Additional information

Funding

This work was supported in part by the Chinese Scholarship Council (CSC) [grant number 2008611087].