Nonlinear control under delays that depend on delayed states

We recently introduced a design methodology for the stabilization of nonlinear systems with input and state delays that depend on the current state of the plant. In the present paper we consider nonlinear systems with delays that depend on the delayed state of the plant, that is, the delay is defined implicitly as a nonlinear function of the state at a past time which depends on the delay itself. Since the prediction horizon and the delay depend on the state of the plant, the key design challenges are how to compute the predictor state and the delay (since the delay needs to be available in order to compute the predictor). We resolve these challenges and we establish closed-loop stability with the aid of a strict Lyapunov functional that we construct. We also design a predictor feedback law for systems with state delays that depend on delayed states. We present an example of a strict-feedforward nonlinear system with input delay.