BILINEAR OPTIMAL CONTROL IN NUCLEAR REACTORS

The optimal control for the kinetic equation of the nuclear reactors is considered in this thesis. The significance of such a problem is due to two reasons the first is the importance of the energy obtained from the nuclear reactors that expected to replace the other conventional sources of energy. The second reason is that the kinetic equation represents a seventh order bilinear system and it can be reduced to a second order one, which makes the problem easy to be solved. The importance of the bilinear systems arises because they are more realistic than the linear systems for representing many natural processes in engineering, biology and ecology.In this thesis, the control signal was considered as a piece wise constant input.This assumption is appropriate for the digital computer control applications. Another Advantage of the piece-wise constant control is that the mathematical formulations of different algorithms are simple and straightforward.The aim is to apply the methods used for solving the optimal control problem of the linear systems to solve the optimal control problem of the bilinear systems. The optimal control of the linear quadratic regulator problem was firstly discussed, the algorithms used to solve such problem were the dynamic programming method and the direct inversion method, which is used only with linear systems and gives the control in only one step. The results were compared with those obtained using the Riccati matrix equation method and they coincided in the case of the dynamic programming and there was a difference in the case of the direct inversion method.After that, the dynamic programming algorithm is extended to be used for solving the optimal control problem of the first order and the second order bilinear systems. The algorithm of the dynamic programming had been dealt with two approaches; the first one assumes that the problem is investigated near the stable point and this is called the vicinity method. On the other hand, the second approach neglects the second and higher orders of sampling period h which also contains u and this is called the power series expansion method.The parameters of the Three Mile Island nuclear reactor were used to represent the point kinetic equation, the dynamic programming power series and vicinity methods were used to solve the optimal control problem of that system. Computer results show that the algorithms work well in simulation.