Compensation of state-dependent input delay for nonlinear systems

We introduce and solve stabilization problems for linear and nonlinear systems with state-dependent input delay. Since the state-dependence of the delay makes the prediction horizon dependent on the future value of the state, which means that it is impossible to know a priori how far in the future the prediction is needed, the key design challenge is how to determine the predictor state. We resolve this challenge and establish closed-loop stability of the resulting infinite-dimensional nonlinear system for general nonnegative-valued delay functions of the state. Due to an inherent limitation on the allowable delay rate in stabilization of systems with time-varying delays, in the case of state-dependent delay, where the delay rate becomes dependent on the gradient of the delay function and on the state and control input, only regional stability results are achievable. For forward-complete nonlinear systems we establish global asymptotic stability results and for linear systems we prove exponential stability. Global stability is established under a restrictive but a priori verifiable Lyapunov-like condition that the delay rate be bounded by unity irrespective of the values of the state and input. Several illustrative examples are provided, including unicycle stabilization subject to input delay that grows with the distance from the reference position.