A quotient method for designing nonlinear controllers

An algorithmic method is proposed to design stabilizing control laws for a class of nonlinear systems that comprises single-input feedback-linearizable systems and a particular set of single-input non-feedback-linearizable systems. The method proceeds iteratively and consists of two stages: (i) the forward stage converts the system into a sequence of cascade forms while reducing the dimension at every step by creating quotient manifolds and (ii) the backward stage constructs the feedback law iteratively. The paper shows the construction of these quotient manifolds and provides algorithms for both stages. A strong feature of the approach is the possibility to use residual degrees of freedom to overcome singularities. A Lyapunov function is introduced to prove global asymptotic stability. Furthermore, the relation of the quotient method to backstepping is discussed. The approach is illustrated via the simulation of a field-controlled DC motor.