Abstract
When optimal control law of system which is described by linear partial differential equations is found, its Riccati equation becomes usually non-linear partial differential equation. Since the optimal feedback gain is the solution of the infinite dimensional Riccati equation or Riccati type partial differential equation, exact solution can not be solved and it is impossible to construct closed-loop system analytically. In this paper, the computational problems associated with the numerical solutions of a linear diffusion process are treated and finite approximation taking N divisions can be shown to be solved and then by increasing the number of division, the solution to the exact feedback gain is approached, and might be shown to be considered as optimal feedback gain in terms of distributed parameter systems. Moreover, infinity of state variables and control might be estimated using those results from practical standpoint. While it is natural to consider that accuracy increases according to the number of division in such an above method, the resulting controller will become complex.
The accuracy of a finite difference solution to a partial differential equation in terms of the convergence of the difference scheme is discussed. Those problems are concluded numerically with the aid of performance criterion's convergence that this procedure produces results within acceptable engineering accuracy.
