Solution Model

NCO tracking relies on the notion of a solution model, which encapsulates useful qualitative knowledge regarding the optimal solution. To be useful for the design of optimizing control schemes, the solution model needs to include the input elements that are affected by uncertainty (the manipulated variables, MVs) and the corresponding NCO elements (the controlled variables, CVs) that will be regulated to zero to enforce optimality. To illustrate the NCO-based input parameterization, consider the input u(t) in a given time interval. With CVP parameterization, this input is parameterized using a (large) number of constant values, the values of which are computed numerically. If the analysis of the nominal optimal solution shows that the path constraint S is active during that interval and u is the input that pushes S to its bound, a much more parsimonious parameterization is u (t) = Gc (Sref – S(t)), where Gc is an appropriate controller that keeps S active at its reference value Sref. The parameterization of the optimal solution looks for MVs that can be used to track the CVs so as to satisfy the NCO. The solution model is not unique since the NCO depend on the parameterization that is used. The diversity in solution models can be exploited to ease up implementation.
The development of a solution model involves three steps:
(1) The optimal solution is first characterized in terms of the types and sequence of arcs. This step typically uses the available plant model and numerical optimization to compute the input profiles using CVP.
(2) A finite set of parameters is selected as MVs to represent the input profiles. The corresponding NCO are formulated, from which the CVs are obtained.
(3) A robustness analysis is performed to ensure that the optimal solution in the presence of uncertainty and the nominal optimal solution are structurally the same.
To ease implementation, it is often possible to approximate the optimal inputs with simpler profiles, which in fact is the strength of the approach, as the approximations introduced at the solution level are easier to assess in terms of optimality loss. The optimizing control problem can be solved as a centralized (multivariable) or decentralized (multi-loop) control problem. Note that the decentralized approach requires appropriate pairing of MVs and CVs.