Application of Backstepping Techniques to the Time-Varying Exponential Stabilisation of Chained Form Systems

It is known that the kinematic model of several non-holonomic systems can be converted into a chained form control system. Asymptotic stabilisation of any equilibrium point of this system cannot be achieved by means of a continuous pure state feedback, but can be obtained by using a time-varying continuous feedback [26]. In the present paper, a backstepping technique is used to derive explicit time-varying feedbacks that ensure exponential stability of the closed-loop system. Two classes of control laws are proposed, with one of them involving a dynamic extension of the original chained system. As in other recent studies on the same topic, exponential convergence is obtained by using the properties associated with homogeneous systems. The control laws so obtained are continuous in both the state and time variables. A complementary and novel feature of the proposed control design technique lies in the estimation of a lowerbound of the asymptotic rate of convergence as a function of a reduced set of control parameters which is independent of the system's dimension. Moreover, the fact that this lowerbound may take any positive value indicates that any pre-specified exponential rate of convergence can be achieved via a suitable choice of the control parameters.