Optimal periodic control algorithm

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. 

Here we examine the control of migration in a periodically driven nonlinear oscillator. Our aim is to demonstrate that application of the approximate solution found from the statistical analysis of fluctuational trajectories optimizes (minimizes) the energy of the control function. We compare the performance of some known adaptive control algorithms to that of the control function found through our analysis. 

The second important difference between human and automated controllers is that, as noted by Thomas [199], while automated systems have basically static control algorithms (although they may be updated periodically), humans employ dynamic control algorithms that they change as a result of feedback and changes in goals. Human error is best modeled and understood using feedback loops, not as a chain of directly related events or errors as found in traditional accident causality models. Less successful actions are a natural part of the search by operators for optimal performance [164]. 

A general approach to the analysis of low amplitude periodic operation based on the so-called Il-criterion is described in Refs. 11. The shape of the optimal control function can be found numerically using an algorithm by Horn and Lin [12]. In Refs. 9 and 13, this technique was extended to the simultaneous optimization of a forcing function shape and cycle period. The technique is based on periodic solution of the original system for state variables coupled with the solution of equations for adjoin variables [Aj, A2,..., A ], These adjoin equations are... 

Shape of the optimal control function can be found numerically using an algorithm by Horn and Lin [23]. In Refs. 24 and 25 this technique was extended to the simultaneous optimization of a forcing function shape and cycle period. 

Usually in the newsvendor setting, it is assumed that if any inventory remains at the end of the period, one discount is used to sell it or it is disposed of. Khouja [79] extends the newsvendor problem to the case in which multiple discounts with prices under the control of the newsvendor are used to sell excess inventory. They develop two algorithms, under two different assumptions about the behavior of the decision maker, for determining the optimal number of discounts under fixed discounting cost for a given order quantity and realization of demand. Then, they identify the optimal order quantity before any demand is realized for Normal and Uniform demand distributions. They also show how to get the optimal order quantity and price for the Uniform demand case when the initial price is also a decision variable. 

Model Predictive Control (MPC) has been also appUed to SC problems as a reactive approach. MPC is a control sfiategy based on the explicit use of a process model to predict the process performance over a period of time (Camacho and Bordons 1995). The model attempts to predict the control variables for a set of time periods. Predicted confiol variables depend on disturbance forecasts (i.e., demand, prices, and interest rates) and also on a set of given parameters that are known as control inputs. The MPC algorithm attempts to optimize a performance criterion that is a function of the control variables. Only a portion of the control variables, the portion corresponding to the following time period, is applied to the system. Next, as new control input information and disturbance forecasts are collected, the whole procedure is repeated, which produces a feed-forward effect and enables the system to counteract the environment dynamics. The procedure is illustrated in Fig. 1.6. 

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