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

A controller designed for linearizations 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 aircraft system models, allowing also for nested loops. The aim of this paper is to extend dynamic gain scheduling 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 linearization are drawn. Theoretical results are shown, providing explicit formulae related to nonlinear dynamic inversion (NDI) control. Several examples of varying nonlinear complexity are presented in order to emphasize the characteristics of the approach.