(712f) Combining Self-Optimizing Control and Extremum-Seeking Control – Applied to Ammonia Reactor Case Study

In this work, we consider self-optimizing control and extremum seeking control in the context of Real time optimization. Often real-time optimization (RTO) uses a process model to determine the optimal setpoints by solving an optimization problem online. Online computations are often expensive and is susceptible to plant-model mismatch. Alternatively, there are methods that avoid repeated numerical optimization by simply transforming the optimization problem into a feedback control problem. Extremum seeking control, Necessary conditions of optimality (NCO) tracking and self-optimizing control belong to such feedback-based methods.

Self-optimizing control uses a system model to solve an offline optimization problem for expected disturbances. This translates the economic objectives into control objectives by finding an appropriate set of controlled variables, which when kept at a constant setpoint gives acceptable loss [1]. This eliminates the need for an expensive online optimizer to re-optimize when disturbances occur and converts the slow optimization problem into a fast control problem. However, the optimal setpoint may change for large deviations from the nominal optimal point because of nonlinearity in the process. One of the main advantages of self-optimizing control is that it can handle unmeasured disturbances, but cannot handle unmodelled disturbances.

Extremum seeking control on the other hand is a class of model free methods that optimizes the system based on measurements. However, due to the time scale separation requirements, extremum-seeking control causes the convergence to be very slow [2]. Extremum seeking control is also often affected by disturbances. The effect of disturbance is clearly motivated in [3], with a suggested modification for measured disturbances. The effect of unmeasured disturbances, however, remains unsolved.

The self-optimizing control and extremum seeking control methods have been developed relatively independently since 2000. Self-optimizing control combined with NCO tracking was proposed in [4]. However, the simple finite difference method used in to obtain the gradient made it difficult to make the NCO tracking work in [4]. In this work, we propose the combination of extremum seeking control and self-optimizing control in a hierarchical fashion to exploit the synergy between the two methods. The self-optimizing control layer reacts fast to changes in unmeasured but modelled disturbances thereby keeping the process in the near optimal region, and extremum-seeking control accounts for the unmodelled disturbances and fine-tunes the optimal setpoint to further minimize the loss from the self-optimizing control. By combining self-optimizing control and extremum seeking control, we can handle a wider class of disturbances.

The proposed method is tested on an ammonia reactor case study from [5]. The self-optimal variables are identified using the exact local method [6] and the corresponding setpoints were provided by an extremum-seeking scheme as proposed in [7]. Simulation results show the performance improvements gained from the proposed method compared with just a self-optimizing control or extremum-seeking control. References

[1] Skogestad, S. (2000). Plantwide control: the search for the self-optimizing control structure. Journal of Process control, 10(5), 487{507.

[2] Nesic, D. (2009). Extremum seeking control: Convergence analysis. In Control Conference (ECC), 2009 European, 1702-1715.

[3] Krishnamoorthy, D., Pavlov, A., and Li, Q. (2016). Robust extremum seeking control with application to gas lifted oil wells. IFAC-PapersOnLine, 49(13), 205-210.

[4] Jäschke J, Skogestad S. NCO tracking and self-optimizing control in the context of real-time optimization. Journal of Process Control. 2011 Dec 31;21(10):1407-16.

[5] Morud, J. and Skogestad, S. (1998). Analysis of Instability in an Industrial Ammonia Reactor. AICHe Journal, 44(4), 888-895.

[6] Halvorsen, I.J., Skogestad, S., Morud, J.C., and Alstad, V. (2003). Optimal selection of controlled variables. Industrial & Engineering Chemistry Research, 42(14), 3273-3284.

[7] Hunnekens, B., Haring, M., van de Wouw, N., and Nijmeijer,H. (2014). A dither-free extremum-seeking control approach using 1st-order least squares fits for gradient estimation. In 53rd IEEE Conference on Decision and Control, 2679-2684.