A general estimation framework for nonlinear singularly perturbed systems

Abstract

Estimation of unmeasured variables is a central objective in a broad range of applications. However, the estimation process turns into a challenging task when the underlying model is nonlinear and even more so when additionally it exhibits multiple time-scales. The current results on estimation for systems with two time-scales apply to linear systems, and limited classes of nonlinear plants and specific observers. Therefore, a new and robust estimation framework for nonlinear systems with variables evolving in different time-scales is needed. This work focuses on developing a rigorous theoretical body for the state estimation of a general class of nonlinear singularly perturbed systems by assuming that the input and output are measured.

In the first part of the thesis, we consider the estimation of the slow state of globally Lipschitz nonlinear singularly perturbed systems by using a full-order observer synthesised for the reduced order (slow) model. We deal with the case when the measured output is disturbed by bounded measurement noise. We prove a global exponential input-to-state (ISS) practical stability property for the estimation error with ISS gain from the measurement noise. Moreover, we show that our assumptions are such that they also imply practical L2 stability of the error dynamics. Our findings apply to a general class of nonlinear globally Lipschitz singularly perturbed systems, and to a number of full-order observers. In order to prove the robustness results to singular perturbations and to measurement noise of the observer, we first show that the plant has bounded solutions under an appropriate set of assumptions on the corresponding boundary layer and reduced order models. We demonstrate the applicability of our findings by showing that the stated assumptions hold for at least four classes of plants and nonlinear observers. Moreover, we present simulation results for numerical examples.

In the second part of the thesis, we generalise current results in the literature and the results of the first part by considering broader classes of plants and estimators of general dimension to cover reduced-order, full-order and higher-order observers. Similarly to the first part of the thesis, we first prove a boundedness result for the plant based on a set of assumptions imposed on the reduced order and boundary layer systems. We then exploit this boundedness property of the plant to show that the error dynamics of the observer designed for the reduced system are semi-globally input-to-state practically (ISpS) stable when the observer is implemented on the original plant. Furthermore, we conclude L stability results when the measurement noise simultaneously belongs to the L2 and L infinity spaces. In the absence of measurement noise, we state results on semi-global practical asymptotical (SPA) stability for the error dynamics. We illustrate the generality of our main results through four classes of systems with corresponding observers.

In the third part of this thesis, we address the parameter and state estimation problem of nonlinear systems with unknown slowly time-varying parameters where the unknown parameter is assumed to belong to a compact set. We tackle this problem by using a multi-observer approach under the supervisory framework. This estimation technique requires a finite number of sample points taken from the compact set where the unknown parameter belongs to. Then, by using these samples as potential parameter estimates, a state observer is designed for each sample to construct a bank of observers to generate potential state estimates. The selection of the parameter and state estimates is performed under the supervisory framework by using a set of monitoring signals and a selection criterion. The monitoring signals characterise the quality of the output estimation errors so that the selection criterion chooses the estimate that gives the smallest difference between the measured and the estimated output.

In this thesis, we propose a novel dynamic sampling policy to generate the parameter samples. This new policy allows the application of the multi-observer technique on systems with slowly time-varying parameters so that our proposed approach is a non-trivial generalisation of the multi-observer technique for parameter and state estimation for systems with constant parameters. We prove that our proposed technique generates parameter and state estimates that are ultimately bounded where the ultimate bounds can be made arbitrarily small if the parameter is sufficiently slow, and if there is a sufficiently large number of observers. We have addressed the parameter and state estimation problem since the slow state of a singularly perturbed system can be regarded as a slowly time-varying parameter to the fast dynamics. Hence, the multi-observer technique for parameter and state estimation of nonlinear systems with unknown slowly time-varying parameters is natural to the singular perturbations framework.