Invariants for Optimal Operation of Process Systems
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State of the art strategies to achieve optimal process operation typically employ a hierarchicalcontrol structure, where different tasks are designated to different control layers.In the simplest case there is an optimization and a control layer. The optimization layercomputes the optimal setpoints for the controlled variables, which are then implementedby the control layer. While the control layer is designed to keep the controlled variables atgiven setpoints, the optimization layer changes these setpoints to adapt operation optimallyto varying conditions. For simple implementation, we want to change the setpoints onlyoccasionally while still obtaining acceptable performance under varying disturbances. The focus of this thesis is to study how to find good controlled variables, whose optimalvalue is invariant or near invariant to disturbances. These invariants are called self-optimizingvariables, and keeping them constant will result in an acceptable, or in the idealcase, zero loss from optimality. In the first part of this thesis, we consider controlled variables, which are linear combinationsof measurements. The loss is used as the criterion for selecting the best set ofcontrolled variables. Applying the inverse Choleski factor of the Hessian with respect tothe inputs as a weighting factor, we derive a first order accurate expression of the loss interms of the weighted square norm of the gradient of the optimization problem. Next, we present a method for finding controlled variables by analyzing past optimalmeasurement data. Selecting combinations of measurements which correspond to directionsof small singular values in the data, leads to controlled variables which mimic theoriginal disturbance rejection. Furthermore, the relationship between self-optimizing control and necessary conditionsof 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 controlin the control layer. This will reject expected disturbances by self-optimizing control ona fast time scale, while unexpected disturbances are rejected by the setpoint updates fromNCO tracking. In the second part of the thesis, we extend the concept of self-optimizing controlto polynomial systems with constraints. By virtue of the sparse resultant, we use themodel equations to eliminate the unknown variables from the optimality conditions. Thisyields invariants which are polynomials in the measurements; controlling these invariantsis equivalent to controlling the optimality conditions. This procedure is not limited to steady state optimization, and therefore, we demonstratethat it can be used for finding invariants for polynomial input affine optimal controlproblems. Manipulating the inputs to control the invariant to zero gives optimal operation.