Backstepping control of linear time-varying systems with known and unknown parameters

The backstepping control design procedure has been used to develop stabilizing controllers for time invariant plants that are linear or belong to some class of nonlinear systems. The use of such a procedure to design stabilizing controllers for plants with time varying parameters has been an open problem. In this paper we consider the backstepping design procedure for linear time varying (LTV) plants with known and unknown parameters. We first show that a backstepping controller can be designed for an LTV plant by following the same steps as in the linear time-invariant (LTI) case and treating the plant parameters as constants at each time t. Its stability properties however cannot be established by using the same Lyapunov function and techniques as in the LTI case. We then introduce a new parametrization and filter structure that takes into account the plant parameter variations leading to a new backstepping controller. The new control design guarantees exponential convergence of the tracking error to zero if the plant parameters are exactly known. If the parameters are not precisely known but the time variations of the parameters associated with the system zeros are known, the appropriate choice of certain design parameters, without using any adaptive law, leads to closed-loop stability and perfect regulation. This new control design is modified and supplemented with an update law to be applicable to LTV plants with unknown parameters. In the adaptive control design, the notion of structured parameter variations is used in order to include possible a priori information about the plant parameter variations. With this formulation, only the unstructured plant parameters are estimated and are required to be slowly time varying, and the structured plant parameters are allowed to have any finite speed of variation. The adaptive controller is shown to be robust with respect to the unknown but slow parameter variations in the global sense. We derive performance bounds which can be used to select certain design parameters for performance improvement. The properties of the proposed control scheme are demonstrated using simulation results.