Estimation of solutions to a second order control problem by finite element method with different boundary conditions

Abstract

Summary form only given. A brief introduction to the finite element method is given and it is applied to solve a real control problem with different boundary conditions and inputs. The problem considered is a 1D 2-point boundary-value problem characterized by a second-order linear ordinary differential equation with a pair of boundary conditions. Although the problem is not complex, the mathematical structure and our approach in formulating the finite element approximation are essentially the same as those in more complex problems. A mechanical system of an automobile containing a spring and shock-absorber is used. Two types of system linear equations, symmetric tridiagonal and nonsymmetric tridiagonal, are used to study this problem. The mid-point integral method is used in all simulations. Three software programs from the ESSL FORTRAN library are used: DPTSL is used in the symmetric tridiagonal system; DGTF and DGTS are used in the nonsymmetric system. Given linear and nonlinear inputs with different boundary conditions, the accuracy of the estimate solutions and the convergence speed are different. From the simulations, we know that using nonlinear input will increase output response error. To compensate for this problem, a small element (interval) is required. From the simulation results, we also know that the accuracy of the response is highly dependent on the essential boundary conditions.