(713b) Self-Optimizing Control Using the Sensitivity Features of SIpopt

Self-optimizing control is a systematic framework for designing a control structure. The idea is to systematically find sets of controlled variables, which when controlled at constant setpoints, give optimal or near optimal operation in economic terms [1]. Thus, a self-optimizing control structure is required to have the two main properties:

1. The control structure optimizes process operation in terms of an economic cost

2. The control structure should be easy to implement (constant setpoints for the controlled variables)

Over the last decade, a framework has been developed to find and evaluate candidate sets of controlled variables. The methods are based on a linearization around the nominal optimal operating point, and yield sets of controlled variables, which minimize the economic loss in a vicinity of the nominal operating point [2].

However, one of the challenges when designing a self-optimizing control structure for large-scale problems is to find the optimal solution and the required sensitivity information. In particular, the sensitivity with respect to disturbances, the gain matrices and the reduced Hessian at the nominal solution are required for applying the tools from self-optimizing control theory. To the authors' knowledge, in all published case studies, the optimal sensitivity, the gain matrices and the reduced Hessian where estimated “manually”, by re-solving the optimization problem and using a finite difference approximation. This is quite tedious and prone to errors.

In a related research area there has been progress on this topic. In nonlinear model predictive control and dynamic real-time optimization, the sensitivity of the optimal solution has been used to find fast approximations to the optimal solution, e.g. [3].  Some modern nonlinear programming algorithms contain features, which can calculate the required sensitivities at very small additional cost [4].

The combination of these NLP sensitivity calculation features and the well established framework of self-optimizing control is the topic of this paper. We show how the data which is needed for the methods from self-optimizing control, can be obtained with the powerful NLP sensitivity calculation features included in the software sIPOPT [4]. This software reuses the matrix factorizations from the NLP solver to obtain the sensitivities of the KKT solution at minimal additional computational cost. Moreover, reduced Hessian information can be obtained efficiently.

Using sIPOPT, the required sensitivity information is obtained with a very high degree of accuracy, and can be readily used for developing a self-optimizing control structure. This facilitates the task of designing a good control structure significantly, and avoids the cumbersome “manual” sensitivity calculations. We present some case studies including optimization and controlled variable selection of distillation columns and a network of heat exchangers for preheating the feed of a crude oil unit in a refinery.

[1] S. Skogestad “Plantwide control: the search for the self-optimizing control structure”, J. Proc. Control, 10, 487-507 (2000)

[2] V. Alstad, S. Skogestad and E.S. Hori, “Optimal measurement combinations as controlled variables”, J. Proc. Control, 19, 138-148 (2009)

[3] R. Huang, V. M. Zavala, L.T. Biegler, “Advanced step nonlinear model predictive control for air separation units” J. Proc. Control, 19, 678-685 (2009)

[4] H. Pirnay, R. Lopez-Negrete, and L. T. Biegler. “Optimal sensitivity based on IPOPT”. submitted for publication, (2011).