Optimization of a Class of River Aeration Problems by the Use of Multivariable Distributed Parameter Control Theory

Self Cite

Optimal control theory based on Pontryagin's minimum principle has been applied to the problem of specifying the steady state distribution of artificial aeration in polluted rivers. The optimization was developed for a general integral type cost functional with weighted energy constraints. The system equations used were of the classical Streeter‐Phelps genre with an induced aeration term added as the control function. The formulation of the problem yielded the famous Riccati‐Kalman structure and an analytical solution. Several optimal and suboptimal aeration designs at different temperature levels were obtained using a digital computer and an automatic plotting routine. An analog computer was used in the critical stages of the numerical solutions. Results from the analog computer pointed the way to the ultimate use of Laplace transform techniques in conjunction with a digital computer for maximum accuracy.

Section: Objective Functionmentioning

confidence: 99%

Section: The Ruhr River Association In Essen Westmentioning

confidence: 99%

See 1 more Smart Citation

A Steady State Optimal Design of Artificial Induced Aeration in Polluted Streams by the Use of Pontryagin's Minimum Principle

Davidson¹,

Bradshaw²

1970

Self Cite

“…Many of these investigations have focused on DO/BOD dynamics and have employed either distributed [Tarassov et al, 1969] or lumped parameter models of one sort or another [Kendrick et al, 1970;Young and Beck, 1974;Ozunger and Perkins, 1979]. The control methodologies applied to these models have included dynamic programing [Naito et al, 1972], duality theory [Varaiya, 1972], differential game theory [Ozunger and Perkins, 1979], procedures based on pole placement [Young and Beck, 1974;Gourishankar and Raman, 1977], forms of hierarchical control [Tarnura, 1974] and Monte Carlo methods [Whitehead and Young, 1979]. Singh [1975] pointed out that many of these approaches are characterized by the considerable computational burden required to implement the control scheme.…”

Section: Introductionmentioning

confidence: 99%

Control of DO level in a river under uncertainty

Spear¹,

Hornberger²

1983

A previously developed regionalized sensitivity analysis for exposing critical uncertainties in models of environmental systems is extended to study control of systems for which there is a good deal of uncertainty in the mathematical model used to describe the appropriate physical, chemical, and biological processes. The method is based on a binary classification of Monte Carlo simulation results as being either satisfactory or unsatisfactory in terms of controller performance. Contrasts in parameters associated with the two classes are elucidated by statistical analysis. This allows the selection of a set of control parameters that maximizes the probability of acceptable behavior in the presence of uncertainty in process parameters. The method is applied to the problem of regulating the discharge from a lagoon with the object of preventing DO from falling below a predetermined standard. It was found that for this system the desired behavior of the controlled process can be achieved with a probability of 0.84 with a particularly simple controller design. Nevertheless, the results suggest that even modest levels of uncertainty in the process parameters can have a considerable effect on the controller performance and that additional attention should be devoted to the design of robust controllers for environmental systems.

“…The calculations of the optimal, continuous BOD waste distribution control policies are important because they set upper limits to performance and assimilative capacity of the receiving stream against which noncontinuous or discrete stage dumping policies can be compared. A somewhat less obvious justification 567 for using optimally distributed BOD waste loads in polluted waters was shown by Tarassov et al [1969] and later by Davidson a•d Bradshaw [1970]. In their studies, use was made of Pontryagin's extremum principle for distributed parameter and lumped parameter systems, respectively.…”

mentioning

confidence: 99%

Best combination of waste treatment and spatially distributed discharge of effluent

Mendiratta¹,

Davidson²

1972

Self Cite

The optimal control theory based on Pontryagin's minimum principle has been applied to the problem of specifying the best combination of minimum percent BOD waste treatment and effluent distribution policies for a single plant on a specified polluted river segment. The analysis features a new dual water quality stream standard consisting of a minimum allowable DO concentration at every point in the river segment combined with a maximum allowable BOD concentration at a specified downstream point. The optimal BOD effluent distribution policies are compared with choice suboptimal effluent discharge patterns associated with best single point, best uniform, and best bang‐bang injection policies. The inequality constraints, the nonlinearities in the system model, and the synthesis of the optimal controls were handled in a direct manner by using the Pontryagin control theory principle combined with gradient search and penalty function techniques. The results of the study very definitely established certain guidelines for increasing the assimilative capacity of a given river segment through judicious combinations of minimum percent BOD waste treatment and continuous BOD effluent distribution or dumping patterns associated with single plant effluents. The systems analysis was made tractable by incorporating a no short‐circuit constraint into the dumping policies.