Observers and observability for uncertain nonlinear systems

Uncertainties bring adverse effects on the system’s general performance and stability. Therefore, control theories have been developed to ensure normal operation of systems under various uncertainties. Up to now, many available strategies for control of uncertain systems rely largely on observer design to recover and compensate the uncertainty to force the system to meet the desired performance. Despite the development of different observers for practical estimation of uncertainties, several limitations pose a great threat to their theoretical analysis. For instance, the structure of uncertain nonlinear systems is limited to a special form especially the cascade form subject to uncertainty matched with the control input. Additionally, the assumptions on large-time behavior of uncertainties are conservative such as the one to require the uncertainty to converge to a constant as time goes to infinity. These are taking into consideration that uncertainties may affect the system from each channel and be composed of the time-varying external disturbances and linear/nonlinear unknown dynamics depending on system states.

Recently, researchers at the Chinese Academy of Sciences: Dr. Sen Chen, Professor Yi Huang and Professor Bao-Zhu Guo together with Dr. Wenyan Bai from Beijing Aerospace Automatic Control Institute and Dr. Ze-Hao Wu from Foshan University investigated observers and observability for general uncertain nonlinear systems. Fundamentally, they proposed a structural condition for ensuring the convergence of the observer and observability for a large class of uncertain nonlinear systems. The work is currently published in International Journal of Robust and Nonlinear Control.

The authors cross-examined the performance of the observers for a large class of uncertain nonlinear systems to address their convergence. Secondly, the nature of the system observability was analyzed to determine the observer properties and essential conditions for ensuring their convergence. Whereas an algebraic criterion of the observability was used for uncertain nonlinear systems, a biased estimation error was employed for unobservable uncertain nonlinear systems.

For a large class of uncertain nonlinear systems, the convergence of observers comprising of augmenting state for uncertainty estimation required both the observability condition for the augment matrix pair and the proposed structural condition. Specifically, the combination of the two conditions was a necessary and sufficient condition for the convergence of the observer and observability for the uncertain nonlinear systems. On the other hand, unobservable uncertain nonlinear systems do not satisfy this necessary and sufficient condition. However, the biased estimation error proved an effective tool for evaluating the estimation performance of designed observers.

The theoretical analysis was validated based on three sets of numerical simulation which produced satisfactory results. For instance, a benchmark two-mass-spring system was considered to demonstrate that the observability for the system is a necessary and sufficient condition for the designed observer to achieve unbiased estimations. In a statement to Advances in Engineering, Dr. Ze-Hao Wu, the corresponding author said “It is worth noting that the study was designed to bridge the existing gaps in the previous studies. For example, it first proposed a structural condition for a large class of uncertain nonlinear systems which was previously ignored, to determine the necessary conditions for general uncertain nonlinear systems to be observable and the convergence of the designed observers ”. The innovative study, therefore, made a breakthrough in the observability and observers design problems for general uncertain nonlinear systems, and the results are very important to the design and theoretical analysis in the extended state observer (ESO) and active disturbance rejection control (ADRC).