The Application of Optimal Linear Regulator Theory to a Problem in Water Pollution

At present, many metropolitan sewer systems do not meet existing and proposed standards on water pollution. Existing systems were designed to overflow at prescribed locations in order to protect the sewage treatment plants whenever severe overload conditions exist (usually during storms). This discharge of untreated overflows into natural receiving waters is of growing concern to water pollution control authorities. The model considered in this paper is representative of the combined storm-sewer systems in cities such as Minneapolis-St. Paul, Minn., Seattle, Wash., and San Francisco, Calif. The objective is to utilize the total storage capacity available in the system in such a manner as to minimize the water pollution resulting from overflows at individual points within the system. In addition, it is required that no abrupt changes in control be admitted, as this is likely to lead to undesirable surges. The nonlinear model is shown to fit within the framework of an optimal regulator problem with derivative constraints. The optimal feedback control law is derived and compared with the optimal bang-bang controller. The solution technique that is presented may be applied to many combined storm-sewer systems in which the flows through the systems to the treatment plants may be controlled. It may be used by city engineers to determine necessary modifications to existing systems in order to meet the new standards regarding water pollution.