Optimal Control for Induction-Heating Processes under Constraints of General Form

Abstract

At present, optimal control theory is widely developing as applied to the optimization of processes whose mathematical models are described by systems of quasilinear partial differential equations. The subject of study is coupled fields of different character, and the problems are characterized by the presence of all possible types of constraints, both terminal and state ones, and also by constraints on the control. In solving applied problems, one encounters special control modes and phase irregular points (see [3,4,6]). Induction heating is an example of such processes. Its mathematical model should take proper account of interaction of fields of various structure, such as thermal, electromagnetic, mechanical (stress), acoustic etc. fields. The induction-heating process can be controlled; moreover, the control can be a function not only of time, but also of the spatial geometric coordinates. In the general case, the induction heating process is characterized by the following specific features: – the quality of heating is high, because the process is rapid; no contamination occurs; it is possible to reach a wide range of temperatures, to heat a body in the presence of inert gases, in vacuum, etc; – the control is a flexible and high-precision one, because the process is not sluggish; it is possible to exactly measure out in doses the energy, to use several control channels; – material, labor, and, in many cases, energy resources are saved owing to lowered loss of material in the process of heating, improved quality of end products, increased productivity; – detrimental effects on the environment are moderated and the working conditions of operators are improved. The advantages of the induction-heating process listed above call for optimal control of this process. Induction heating is used to obtain materials with certain properties (for instance, surface hardening of ferromagnetic blanks, aluminum casting in a strong magnetic field, etc.); to ensure a given temperature condition in complex technological processes such as application of glass layers on the inner surfaces of titanium pipes, preparatory heating of blanks before further processing them by plastic deformation, and so on. In analyzing and implementing the induction-heating processes (as was already noted above), it is very important to optimize the control of these processes. By what was said above, the solution of the problem of optimal control for the induction-heating processes includes the solution of the following two main problems: the construction of a numerical model of the heating process and the synthesis of the control process. Specific features of the model describing a heating process are the following ones: due account should be taken of coupled fields (thermal, electromagnetic and thermal stress ones) with allowance made for the multivariable character of each of the fields listed above; the consideration of nonlinear dependences of properties of the material on temperature and electromagnetic field strength. A peculiar feature of the control process is the necessity of taking into account terminal and state constraints of the Chebyshev type. Several problems pertaining to the theme considered in this paper have already been studied by a number of authors. Thus, general control problems for distributed parameter systems were studied by J.-L. Lions, A. G. Butkovskii, T. K. Sirazerdinov, F. P. Vasil’ev, E. P. Chubarov, and others. It should be