State Estimation and Control of Nonlinear Systems with Large and Variable Measurement Delays

This chapter deals with the problem of output feedback control of nonlinear systems affected by time-varying measurement delay. A control law is presented, which is made of an observer-controller cascade where the controller is a classic state-linearizing scheme, and the observer is a high-gain observer of chaintype. It is shown that under suitable conditions on the system, the observer is globally exponentially convergent, and the replacement of the true state with the observer state in the control law results in an exponentially stabilizing feedback scheme. The main limitation with a single observer is the presence of a delay bound that depends on the Lipschitz constant of the nonlinear system. To overcome this limitation it is possible to resort to a chain of observers that, at the cost of a growing realization space and convergence time, can in principle allow to compensate any delay. This design is straightforward when the delay is known and constant but its extension to time-varying delays requires special attention, in particular when the delay is not continuous with respect to time, as it frequently happens in the applications. We therefore introduce a classification of delay functions with respect to the available output information and illustrate how to design the cascade of elementary observers to solve the state reconstruction problem. We also characterize the class of delay functions for which this approach fails to provide a viable implementation.