Adaptive robust control schemes of uncertain time-delay systems and its applications to water pollution control systems

The problem of adaptive robust stabilization is considered for a class of uncertain nonlinear time-delay dynamical systems. It is assumed that the upper bound of the nonlinear delayed state perturbations is a linear function of some parameters which are assumed to be unknown. It is also assumed that the time delays are time-varying, and can be any nonnegative continuous and bounded functions. In this paper, it is not required that the derivatives of the time-varying delays have to be less than one. For such a class of uncertain nonlinear time-delay systems, a new method is presented whereby a class of memoryless continuous adaptive robust state feedback controllers with a rather simpler structure is proposed. That is, being completely different from the related works reported in the control literature, the nonlinear perturbations are not included in the proposed control schemes. By employing a quasi-Lyapunov function, it is shown that the solutions of uncertain nonlinear time-delay systems can be guaranteed to be uniformly exponentially convergent towards a ball which can be as small as desired. Finally, as an application of the results, the problem of water pollution control is considered for uncertain river time-delay systems due to industrial waste treatment facility, and the corresponding simulation is given.