(15e) Interior Point Solution of Integrated Plant and Control Design Problems with Embedded MPC

The potentially significant impact that the design of a process plant can have on its ability to be satisfactorily controlled has led to the development of integrated formulations in which dynamic performance requirements are included as constraints within an optimal design framework. Failure to account for the dynamic operation of a process could lead to reduced profits or even safety and environmental constraint violations. This motivates the need to design the process and its associated control system in an integrated manner.

Prior work in simultaneous process and control system design has focused primarily on the use of linear controllers such as proportional-integral control (Mohideen et al, 1996; Bahri et al., 1997; Schweiger and Floudas, 1998; Kookos and Perkins, 2001; Pistikopoulos and Sakizlis, 2002). Baker and Swartz (2004a) considered actuator saturation effects in integrated design and control. This results in model discontinuities, and to avoid potential difficulties with a sequential solution approach in which the integration of the model differential-algebraic equation system is separated from the optimization, they follow a simultaneous solution approach in which the actuator saturation is handled through mixed-integer constraints. They demonstrated that failure to account for actuator saturation could lead to suboptimal designs. The growth in solution time with problem size led to the development of an alternative approach in which saturation was modeled using complementarity constraints, and the resulting problem solved using an interior-point approach (Baker and Swartz, 2004b).

In this paper we consider integrated design and control with constrained model predictive control (MPC) as the regulatory control system. MPC has become the advanced control method of choice in the chemical process industry, thus direct accommodation of MPC within the integrated design framework is important. The resulting problem shares the characteristic of model discontinuity with the actuator saturation problem above, but is more complex, since the control calculation involves the solution of a quadratic programming (QP) problem at every sampling period. When embedded within a design optimization problem, this results in a multi-level optimization problem. We follow a simultaneous solution approach in which the MPC optimization sub-problems are replaced by their Karush-Kuhn-Tucker (KKT) optimality conditions. In recent work we have considered this formulation for the constraint back-off calculation problem with constrained MPC as the regulatory control system. This results in a single-level optimization problem, but with complementarity constraints. IPOPT-C (Raghunathan and Biegler, 2003), an interior point algorithm designed for mathematical programs with complementarity constraints, was found to solve these problems reliably and significantly faster than a mixed-integer quadratic programming formulation. Here, we extend the formulation to integrated design and control. Our approach differs from that of Sakizlis et al. (2003) who use a parametric formulation of the predictive controller in an integrated plant and control system design strategy. They use a sequential solution approach in which the Boolean algebraic conditions defining the regions of applicability of the explicit control laws are replaced with steep hyperbolic tangent functions.

We consider here linear constrained MPC applied to a nonlinear dynamic plant. This is a commonly occurring situation in practice, and is reasonable for operation close to a desired steady-state and/or for processes that are not strongly nonlinear. The internal dynamic model used by the predictive controller is based on a linearization around the nominal operating point; thus it is a function of both the design variables and the operating point. The nonlinear dynamic plant model is discretized using orthogonal collocation on finite elements, and the resulting equations included as constraints in the overall optimization problem. The controller equations comprise the KKT conditions of the MPC open-loop optimization problems (QP problems) corresponding to each sampling period. Together with the discretized plant model, these equations define the closed-loop dynamic response of the system. The complementarity constraints resulting from the KKT conditions of the QP sub-problems are handled through the application of an interior point solution approach.

Details of the problem formulation and solution strategy are provided, and the application to two case studies presented. This includes a multi-input multi-output two-reactor system based on the case study presented in Bahri et al. (1997). The design variables include the fraction of the feed stream routed to each reactor and the MPC tuning parameters. The performance of the solution strategy for this challenging class of integrated design and control problems is discussed.

References

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Baker, R. and Swartz, C.L.E. (2004a). Rigorous handling of input saturation in the design of dynamically operable plants. Ind. Eng. Chem. Res., 43(18), 5880-5887.

Baker, R. and Swartz, C.L.E. (2004b). Inclusion of actuator saturation as complementarity constraints in integrated design and control. Proceedings of the 7th International Conference on Dynamics and Control of Process Systems (DYCOPS 7), Boston.

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