Date of Award

Summer 1976

Document Type


Degree Name

Doctor of Engineering Science in Electrical Engineering


Electrical Engineering

First Advisor

Harlan J. Perlis

Second Advisor

Marshall Chuan Yung Kuo

Third Advisor

Andrew Ulrich Meyer

Fourth Advisor

Robert Dresnack


This study concerns the application of a combination of multilevel hierarchical systems analysis techniques and Pontryagin's minimum principle (multilevel control) to the problem of controlling optimally two classes of dynamic distributed parameter plants representing concentrations balances in streams, rivers and estuaries. The concentrations treated in this study are those deemed the most effective indicators of water quality, dissolved oxygen (DO) and biochemical oxygen demand (BOD).

One class of plants treated in this study consists of linear continuous distributed parameter plants represented mathematically by sets of simultaneous partial differential equations. Optimal control of a plant of this class is initiated by applying spatial discretization followed by a combination of multilevel techniques and Pontryagin's minimum principle for lumped parameter systems. This approach reduces the original problem of optimally controlling a distributed parameter plant to a hierarchy of subproblems comprised of ordinary differential and algebraic equations that can be solved iteratively.

A general two-dimensional plant representative of a class of two-step discrete dynamic distributed parameter plants is derived from mass balances at the faces of a model of a volume element of a waterway. The resulting set of simultaneous finite-difference equations represents dynamic balances of concentrations at a finite number of spatial points in a reach of a waterway at selected time instants. Application of Pontryagin's minimum principle for discrete systems in conjunction with multilevel hierarchical systems analysis techniques reduces the problem of controlling such a plant optimally to a hierarchy of subproblems to be solved iteratively.

Implicit in the application of optimal control to a plant is the selection of a suitable performance index functional with which to measure the relative optimality of each solution iteration. A variety of performance indices based upon physical considerations is utilized in conjunction with several different control modes for a number of plants representative of the two classes treated in this study.

Subproblem hierarchies corresponding to both continuous and discrete distributed parameter plants representing concentrations balances in waterway reaches subject to multilevel optimal control are aggregated into super hierarchies. These super hierarchies possess at least one more level than those corresponding to the single reaches and represent, in this context, the concentrations balances in multireach or regional portions of waterways.

Sufficient boundary, initial and final conditions are presented for numerical solution of the subproblem hierarchies developed in this study. Flow charts for the corresponding digital computer programs also are depicted.

A proof of consistency between the ordinary differential equations of the spatially discretized plant and the partial differential equations of the continuous distributed parameter plant that it approximates is developed for a representative plant. A proof of convergence of the solutions of the equations of the same spatially discretized plant also is developed.

Stability analyses are conducted for representative continuous and discrete distributed parameter plants. The optimal control of the spatially discretized continuous distributed parameter plant is formulated as a linear regulator problem and the associated performance index is utilized as a Liapunov function. The optimal control of the discrete distributed parameter plant with time-varying mean volume flow rate is formulated as the problem of optimal control of a nonstationary system which is treated by transforming the nonstationary system to an equivalent stationary system. The z-transform is applied to the finite-difference equations of the plant to facilitate evaluation of the effect of the presence of transport lags.

The relationship between structural characteristics and computational efficiency of subproblem hierarchies is analyzed.

Multilevel hierarchical systems analysis techniques are applied to the sensitivity analysis of a spatially discretized distributed parameter plant subject to multilevel optimal control. The combination of discretization and multilevel techniques is shown to reduce the generation of trajectory sensitivity coefficients for an optimally controlled distributed parameter plant to generation of trajectory sensitivity coefficients for a series of lumped parameter plants under optimal control. A normalized performance index sensitivity function also is developed for the same plant.

Numerical results of multilevel optimization are presented for various control modes and configurations applied to plants representing: single reaches of a tidal river, four contiguous reaches of a tidal river, six contiguous reaches of a tidal river with taper and waste dischargers, and single reaches of an estuary.

The study culminates with the application of one of the single reach subproblem hierarchies for a discrete distributed parameter plant under multilevel optimal control and multilevel hierarchical systems analysis techniques to the problem of minimizing total treatment cost for a multireach portion of a tidal river. This demonstrates the feasibility and efficiency of the multilevel approach to the solution of dynamic systems optimization problems of regional scope.