(687f) Functional Observers for Nonlinear Processes

The problem of estimating a function of the state vector, without the need of estimating the entire state vector, arises in many monitoring and control applications. Properties like 95% boiling point, fuel cloud point, polymer melt viscosity, etc., can play a critical role in the operation of industrial processes, and these properties need to be estimated on line, for the purpose of monitoring and fault detection. Moreover, such an estimator will be the heart of an inferential control system for the purpose of regulating the estimated properties to set point. Finally, in control theory, in the state-space synthesis of output feedback controllers, it is only the state feedback function that needs to be estimated and not the entire state vector.

The foregoing applications motivate the development of functional observers, the aim being to build an observer specifically for the needed function of the states, of low enough order, in order to lower computational effort in real-time implementation. For linear systems, basic theory of functional observers (see e.g. [1]) is based on Luenberger theory for linear multivariable systems ([2]). On the other hand, for nonlinear systems, there is well-developed theory only for full-state observers, including the direct extension of full-state Luenberger observers to nonlinear systems in the context of exact observer linearization ([3]). The goal of the present work is to develop a direct generalization of Luenberger functional observers to nonlinear systems

This paper will first develop a theoretical formulation of the problem of designing functional observers for general nonlinear systems. Notions of functional observer linearization will also be formulated, the objective being to achieve exactly linear error dynamics in transformed coordinates, with prescribed rate of decay of the error. Necessary and sufficient conditions for the existence of a lower-order functional observer with linear dynamics and linear output map will be derived.

Theoretical results will be applied to the design of nonlinear functional observers in two monitoring case studies: (i) a nonisothermal CSTR where reactant conversion needs to be monitored from on line temperature measurements; (ii) an anaerobic bioreactor, where organic substrate conversion needs to be monitored from on line measurement of biogas production rate.

References:

[1] C.T. Chen, Linear system theory and design, Holt, Rinehart & Winston, New York, 1984.

[2] D. G. Luenberger, “Observers for linear multivariable systems,” IEEE Trans. Automat. Contr., Vol. AC-11, pp.190-197, 1966.

[3] N. Kazantzis and C. Kravaris, Nonlinear observer design using Lyapunov's auxiliary theorem, Systems & Control Lett., Vol. 34, pp. 241-247, 1998.