Johannes Jäschke Invariants for Optimal Operation of Process Systems

State of the art strategies to achieve optimal process operation typically employ a hierarchical control structure, where different tasks are designated to different control layers. In the simplest case there is an optimization and a control layer. The optimization layer computes the optimal setpoints for the controlled variables, which are then implemented by the control layer. While the control layer is designed to keep the controlled variables at given setpoints, the optimization layer changes these setpoints to adapt operation optimally to varying conditions. For simple implementation, we want to change the setpoints only occasionally while still obtaining acceptable performance under varying disturbances. The focus of this thesis is to study how to find good controlled variables, whose optimal value is invariant or near invariant to disturbances. These invariants are called selfoptimizing variables, and keeping them constant will result in an acceptable, or in the ideal case, zero loss from optimality. In the first part of this thesis, we consider controlled variables, which are linear combinations of measurements. The loss is used as the criterion for selecting the best set of controlled variables. Applying the inverse Choleski factor of the Hessian with respect to the inputs as a weighting factor, we derive a first order accurate expression of the loss in terms of the weighted square norm of the gradient of the optimization problem. Next, we present a method for finding controlled variables by analyzing past optimal measurement data. Selecting combinations of measurements which correspond to directions of small singular values in the data, leads to controlled variables which mimic the original disturbance rejection. Furthermore, the relationship between self-optimizing control and necessary conditions of optimality (NCO) tracking1 is studied. We find the methods to be complementary, and propose to apply NCO tracking in the optimization layer, and self-optimizing control in the control layer. This will reject expected disturbances by self-optimizing control on a fast time scale, while unexpected disturbances are rejected by the setpoint updates from NCO tracking. In the second part of the thesis, we extend the concept of self-optimizing control to polynomial systems with constraints. By virtue of the sparse resultant, we use the model equations to eliminate the unknown variables from the optimality conditions. This yields invariants which are polynomials in the measurements; controlling these invariants is equivalent to controlling the optimality conditions. This procedure is not limited to steady state optimization, and therefore, we demonstrate that it can be used for finding invariants for polynomial input affine optimal control problems. Manipulating the inputs to control the invariant to zero gives optimal operation. 1François, G., Srinivasan, B., Bonvin, D. 2005. “Use of measurements for enforcing the necessary conditions of optimality in presence of constraints and uncertainty”. Journal of Process Control 15 (6). 701-712