Optimal periodic control

Optimal periodic control involves a periodic process, which is characterized by a repetition of its state over a fixed time period. Examples from nature include the circadian rhythm of the core body temperature of mammals and the cycle of seasons. Man-made processes are run periodically by enforcing periodic control inputs such as periodic feed rate to a chemical reactor or cyclical injection of steam to heavy oil reservoirs inside the earth s crust. The motivation is to obtain performance that would be better than that imder optimal steady state conditions. 

In this chapter, we first describe how to solve an optimal periodic control problem. Next, we derive the pi criterion to determine whether better periodic operation is possible in the vicinity of an optimal steady state operation. 

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This reactor poses a optimal periodic control problem, which involves periodic control functions. Their application can result in better performance relative to steady state operation and help achieve difficult performance criteria such as those involving molecular weight distribution (MWD). 

What new conditions would be required if the optimal periodic control problem of Section 1.3.5 (p. 11) is changed to a non-periodic one ... 

The solution of a optimal periodic control problem requires the integration of state and costate equations, both subject to periodicity conditions. Other than this integration aspect, the solution methods for optimal periodic control problems are similar to those for non-periodic problems. Therefore, we will focus on the methods to integrate state and costate equations under periodicity conditions. 

A periodicity condition implies that the initial and final values of a state (or costate) variable are equal to a single value. Thus, in a optimal periodic control problem, the set of state as well as costate equations poses a two point boundary value problem. Either successive substitution or the shooting Newton-Raphson method may be used to integrate the periodic state and costate equations. 

Optimal Periodic Control 4. Repeat the above steps for costate equations. 

Optimal Periodic Control Derivative Costate Equations... 

Following is the computational algorithm of the shooting Newton-Raphson method to solve the optimal periodic control problem. 

Using the above algorithm, the optimal periodic control problem was solved for the parameters listed in Table 8.1. The objective is to find two control functions and the time period that maximize the average reaction rate. 

The pi criterion is a sufficient condition for the existence of a periodic solution that is better than the neighboring optimal steady state solution of an optimal periodic control problem. Using the criterion, we would hke to know, for example, whether the time-averaged product concentration in a periodic process can be more than what the optimal steady state operation can provide. In other words, we would hke to check if oscillating the optimal steady state control with some frequency and time period improves on the steady state solution. 

Optimal Periodic Control subject to the following constraints ... 

F. Colonius. Optimal Periodic Control, Chapter VII, pages 115-119. Springer-Verlag, Berlin, Germany, 1988. 

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