First order hyperbolic equations appear often as control systems' models of such plants as heat exchangers, tubular reactors, metallurgical and ceramic kilns, aerated rivers, inventory and demographical processes. Application of orthogonal collocation to simulation and control of systems governed by these equations is presented in the paper. The collocation method is one of several Methods of Weighted Residuals. It makes possible the easy transition of a partial differential equation to an ordinary one, the solution of which approximates the exact solution in some discrete spatial collocation points.
The results of numerical experiments are given. They serve to evaluate usefulness of the method. In the examples the approximation accuracy in dependence of input signal frequency and accuracy of linear-quadratic optimal control solution for different collocation approximations are considered. The best results have been obtained by choosing the collocation points as zeroes of the Legendre orthogonal polynomials.
In practice it is sufficient to approximate one scalar partial differential equation by 4 to 5 ordinary ones. So the orthogonal collocation can be recommended to simulation and control of systems governed by first order hyperbolic equations. The formulae and collocation constants given in the appendix make the application easier.