A new method to design discontinuous stabilizing controllers for cascade systems

In this paper, we propose a new method to design discontinuous stabilizing controllers for a class of cascade systems including various kind of nonholonomic systems such as chained systems and symmetric affine systems. The cascade system consists of two systems: One system is a linear system with constant coefficient matrices, and another system is a linear system with matrices depending on the variables of the first system. We replace the independent variable of the second system with the variable that corresponds to the slowest mode of the first system. The result is a differential equation with regular singularity at the origin. If this system is not "controllable", it can be transformed into a "controllable" system by "similar transformation". This transformation results in a discontinuous state feedback. Our approach can be seen as a generalization of the method based on σ process that is developed by Astolfi and Laiou.