Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data

In this paper, we consider estimation problems involving a class of nonlinear systems characterized by two non-standard attributes: (i) such systems evolve over multiple stages; and (ii) the dynamics in each stage involve unknown time-delays and unknown system parameters. These unknown quantities are to be estimated such that a least-squares error function between the system output and a set of noisy measurement data from a real plant is minimized. We first present the classical parameter estimation formulation, where the expectation of the error function is regarded as the cost function. However, in practice, there exists uncertainty in the distribution of the measurement data. The optimal parameter estimate should be able to withstand this uncertainty. Accordingly, we propose a new parameter estimation formulation, in which the cost function is the variance of the error function and the constraint indicates an allowable sacrifice from the optimal expectation value of the classical parameter estimation problem. For these two estimation problems, we show that the gradients of their cost functions and the constraint function with respect to the time-delays and system parameters can be computed through solving a set of auxiliary time-delay systems in conjunction with the governing multistage time-delay system, simultaneously. On this basis, we develop gradient-based optimization algorithms to determine the unknown time-delays and system parameters. Finally, we consider two example problems to illustrate the effectiveness and applicability of our proposed algorithms.