A Design of Robust Global Stabilizing Controllers for A Class of Pure-Feedback Nonlinear Systems
Satoru Tao, March 2002 (in Japanese)

This paper focuses on a class of pure-feedback nonlinear systems. The class of systems is broader than the strict-feedback form in which control theory has experienced major advance in the last decade. This thesis does not require affine assumptions which are imposed by the strict-feedback form on systems. Instead of an `affine' lower-triangular form we assume for the strict-feedback systems, we allow the systems to be in a `power' lower-triangular form. It is known that the power lower-triangular form sometimes arises in designing nonholonomic systems. Nonholonomic systems are systems which have attracted a lot of attention recently because of their unique motions, their curious mechanisms and their practical potentials. However, ordinary approaches which have been developed mostly for holonomic systems are not applicable to the nonholonomic systems at all. It has been reported that a class of nonholonomic systems can be transformed into the power lower-triangular form. Therefore, the power lower-triangular form can be considered as a very important class of systems we need to tackle. Although many systematic approaches are available for strict-feedback systems, only few attention has been paid to power lower-triangular systems. We have yet to develop a systematic approaches to the power lower-triangular systems.

There are many cases where a linear controller works well in a small neighborhood of an equilibrium point even if the system we need to control is nonlinear. However, when the system operates far away from the equilibrium, nonlinearities often cause divergent responses as well as violent responses. To obtain stable responses in a larger region by a linear controller, the controller should resort to extremely high gain. That is why appropriate nonlinear compensation is crucial for controlling nonlinear systems, and the local gain of the nonlinear compensation can remain reasonably small.

For the purpose of designing such appropriate nonlinear compensation with reasonably small local gain, a design method which is based a state-dependent quadratic form and state-dependent scaling has been proposed in the nonlinear control literature recently. If the systems are in the strict-feedback form, the method provides us with a systematic procedure, and various uncertainties affecting the systems can be treated in a unified way. However, the method is not directly applicable to systems in the power lower-triangular form. There is also another method which has been developed exclusively for the power lower-triangular systems, the method sometimes leads to a controller of unreasonably high gain. Uncertainties arising in various places may not be treated in a unified way.

The objective of this thesis is to propose a new design procedure for designing nonlinear controllers which require smaller local gain. The development is a natural extension of the abovementioned approach based on the state-dependent quadratic form and state-dependent scaling. Due to the result of this thesis, we are able to deal with uncertainties in the power lower-triangular systems systematically in a unified manner.