(246h) Functional Observers for Discrete-Time Nonlinear Systems with Applications to Fault Detection and Estimation

In control theory, a functional observer is an auxiliary system that is driven by the available system outputs and mirrors the dynamics of a physical process in order to estimate one or more functions of the system states (Luenberger, 1966, 1971). Besides being of theoretical importance, the use of functional observers arises in many applications. For example, functional estimates are useful in feedback control system design because the control signal is often a linear combination of the states, and it is possible to utilize a functional observer to directly estimate the feedback control signal (Kravaris, 2016; Luenberger, 1966, 1971).

Over the past fifty years, considerable research has been carried out on estimating functions of the state vector for linear systems ever since Luenberger introduced the concept of functional observers in 1966 (Luenberger, 1966) and proved that it is feasible to construct a functional observer with number of states equal to observability index minus one. Subsequent research has focused on lower order functional observers where necessary and sufficient conditions for their existence and stability have been derived (Darouach, 2000).

For continuous-time nonlinear systems, the problem of designing functional observers for estimating a single nonlinear functional has been tackled for general nonlinear systems from the point of view of observer error linearization (Kravaris, 2016; Kravaris and Venkateswaran, 2021) and the approach has been extended to a disturbance decoupled fault detection and isolation (Venkateswaran et al., 2020). For discrete-time nonlinear systems, however, results have been limited. The goal of the present work is to develop a direct generalization of Luenberger’s functional observers to discrete time nonlinear systems. The concept of functional observers for discrete-time nonlinear systems is defined and the observer design problem is considered from the point of view of observer error linearization and is analogous to the methods in (Kravaris, 2016; Venkateswaran et al., 2020). It will be shown that, with the proposed formulation, easy-to-check necessary and sufficient conditions for the existence of such a functional observer can be derived, leading to simple formulas for observer design with eigenvalue assignment. Furthermore, the formulation also lends itself to fault detection and estimation in discrete-time nonlinear systems and this will also be investigated. The functional observer design scheme and the fault detection and estimation capabilities will be tested on a non-isothermal CSTR case study.

References

Darouach, M., 2000. Existence and design of functional observers for linear systems. IEEE Transactions on Automatic Control 45, 940-943.

Kravaris, C., 2016. Functional observers for nonlinear systems. IFAC-PapersOnLine 49, 505-510.

Kravaris, C., Venkateswaran, S., 2021. Functional Observers with Linear Error Dynamics for Nonlinear Systems. arXiv preprint arXiv:2101.11148.

Luenberger, D., 1966. Observers for multivariable systems. IEEE Transactions on Automatic Control 11, 190-197.

Luenberger, D., 1971. An introduction to observers. IEEE Transactions on automatic control 16, 596-602.

Venkateswaran, S., Liu, Q., Wilhite, B.A., Kravaris, C., 2020. Design of linear residual generators for fault detection and isolation in nonlinear systems. International Journal of Control, 1-17.