(212b) Nonlinear Observer Based Robust and Offset Free Nonlinear Predictive Control of Air Separation Units in Power Plants

Nonlinear model predictive control (NMPC) is becoming an accepted tool for controlling nonlinear multi-variable systems in the presence of constraints. Numerous successful applications have been reported, especially for energy intensive applications that require frequent changes in operations. Important large-scale examples of these include grade changes for polymer processes [1] and dynamic demand satisfaction in air separation units [2]. However, in practical applications, a state observer is usually operating in parallel to reconstruct the estimated states feeding back to the NMPC. Moreover, disturbances and modeling errors are usually present and often not predictable or measurable, causing plant-model mismatches. Consequently, a natural concern of NMPC-observer pairs in the presence of plant-model mismatches is the robust stability.

For unconstrained linear systems, the well-known separation principle guarantees that the controller-observer pair is nominally stable if both the controller and the observer are stable. For a class of nonlinear systems, a similar separation principle has been proposed which ensures the nominal stability [3]. Recently, Findeisen et al. [4] obtained practically stable systems by synthesizing a sufficiently fast high-gain observer with a robust NMPC controller. These results focused on the nominal closed-loop stability or stability in the presence of vanishing perturbations. In practice, however, as time progresses or operating conditions change drastically, plant-model mismatch develops into non-vanishing perturbations. Thus it becomes necessary to extend the above analysis to a more general case where the NMPC-observer pair operates under sustained changes in model parameters.

On the other hand, from the performance's point of view, offset free regulatory behavior is an important requirement in control practice. For linear systems, several strategies relying on augmented states have been proposed [5,6,7]. For nonlinear systems, the most common method in NMPC is to compare the measured output of the process to the model prediction at current time step to generate an output disturbance estimate. Then this output disturbance is used to compute a new target state in NMPC objective function [8]. Moreover, the authors also showed that NMPC with the new target state guarantees offset free regulator behavior. The main limitation of this approach is that it is restricted to open loop stable systems where model simulation can be used for state estimation.

Recently, Deshpande et al. [9] have proposed an NMPC scheme for achieving offset free tracking based on the direct use of prior and posterior state estimation errors, computed using extended Kalman filter. While they demonstrate applicability of their scheme to open loop unstable systems, they do not provide any stability proofs for their formulation. In this work, we study the robust stability of this NMPC-observer pair. Moreover we propose a robust NMPC formulation based on general extended recursive observers, by using the observer error feedback in NMPC formulation. In addition, we show that target state concept proposed in [8] can be extended to NMPC-observer pair. The proposed target setting scheme considers both state and output disturbances while computing the target state. We show that the proposed NMPC with the calculated target state guarantees offset free regulator behavior. The proposed schemes are demonstrated through simulation studies on a large-scale air separation unit [2].

Reference:

[1] V. M. Zavala, and L.T.Biegler. Optimization-Based Strategies for the Operation of Low-Density Polyethylene Tubular Reactors: Nonlinear Model Predictive Control. Computers and Chemical Engineering, to appear, 2009.

[2] R. Huang, V. M. Zavala and L. T. Biegler. Advanced Step Nonlinear Model Predictive Control for Air Separation Units. Journal of Process Control, 2009, 19: 678-685.

[3] A. Atassi and H. Khalil. A separation principle for the stabilization of a class of nonlinear systems. IEEE Trans. On Auto. Contr. 1999, 44:1672.

[4] R. Findeisen, L. Imsland, F. Allgöwer and B. Foss. Output feedback stabilization of constrained systems with nonlinear predictive control. Inter. J. of Robust and Nonlinear Contr. 2003, 13:211.

[5] G. Pannocchia. Robust model predictive control with guaranteed setpoint tracking. J of Process Control. 2004, 14:927.

[6] G. Pannocchia and E. Kerrigan. Offset-free receding horizon control of constrained linear systems. AIChE J. 2005, 51:3134.

[7] G. Pannocchia and A. Bemporad. Combined design of disturbance model and observer for offset-free model predictive control. IEEE Trans. On Auto. Contr. 2007, 52: 1048.

[8] S. Meadows and J. Rawlings, Nonlinear Model Predictive Control, in Nonlinear Process Control, D. Seborg and M. Henson. (Eds.). Prentice Hall, 1997.

[9] A. Deshpande, S. Patwardhan and S. Narasimhan. Intelligent state estimation for fault tolerant nonlinear predictive control. J. of Process Control. 2009, 19: 187.