(627e) Sensitivity-Based Output Feedback in Dynamic Optimization

We consider batch process optimization and robust implementation of optimal control policies. The dynamic optimization of such processes is in most cases model based, and therefore subject to uncertainties. This may lead to sub-optimal control trajectories with significant economic losses.

 Our goal is the development of a simple strategy which guarantees near-optimal operation under all conditions. Here, 'under all conditions' means for the defined disturbances, plant changes and implementation errors.

 The main idea is to apply sensitivity-based output feedback to update the control inputs using the deviation of the measurements (outputs) from the nominal optimal trajectory. In this approach, fast updates can be computed online without the need of reoptimization when disturbances occur. Similar idea has been presented in [1]. However, no systematic guidelines are given for the computation of such controller.

 The use of sensitivity information to update nominal input has been recently used in the context of nonlinear model predictive control (see [2]) and dynamic real time optimization (see [3]). In both cases, however, the disturbances need to be measured or estimated.

  We show here that an output feedback controller can be easily obtained from the optimal sensitivity of outputs and inputs provided there are enough independent measurements. This controller uses directly the process outputs and no state or disturbance estimation is required. The resulting closed-loop satisfies the necessary conditions of optimality to first order and is exact in the linear case. Interestingly, if the measurements outnumber the disturbances, there will be extra degrees of freedom that can be used to mitigate the effect of noise.

  The proposed method was tested in two case studies with nonlinear dynamics, namely, a fed-batch reactor with uncertain kinetic parameters and a high-order continuous stirred-tank reactor where the initial states are disturbed. In both cases the controller was able to recover almost completely the economical loss due to small perturbations when compared to a nominal open-loop approach.

 The method is simple and intuitive and yet has shown surprisingly good performance for relevant industrial processes.

References

 [1]  Z. K. Nagy and R. D. Braatz. ‘Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis”. Journal of Process Control, vol.14, 411-422, 2004.

 [2] V. M. Zavala and L. T. Biegler, “The advanced-step nmpc controller: Optimality, stability and robustness," Automatica, vol. 45, pp. 86-93, 2009.

 [3] L. Wrth, R. Hannemann, and W. Marquardt, “Neighboring-extremal updates for nonlinear model-predictive control and dynamic real-time optimization," Journal of Process Control, vol. 19, pp. 1277 -1288, 2009.