Reduced ordered modelling of time delay systems using galerkin approximations and eigenvalue decomposition

In this paper, an r -dimensional reduced-order model (ROM) for infinite-dimensional delay differential equations (DDEs) is developed. The eigenvalues of the ROM match the r rightmost characteristic roots of the DDE with a user-specified tolerance of ε . Initially, the DDE is approximated by an N -dimensional set of ordinary differential equations using Galerkin approximations. However, only N c ( < N ) eigenvalues of this N -dimensional model match (with a tolerance of ε ) the rightmost characteristic roots of the DDEs. By performing numerical simulations, an empirical relationship for N c is obtained as a function of N and ε for a scalar DDE with multiple delays. Using eigenvalue decomposition, an r ( = N c ) dimensional model is constructed. First, an appropriate r is chosen, and then the minimum value of N at which at least r roots converge is selected. For each of the test cases considered, the time and frequency responses of the original DDE obtained using direct numerical simulations are compared with the corresponding r - and N -dimensional systems. By judiciously selecting r , solutions of the ROM and DDE match closely. Next, an r -dimensional model is developed for an experimental 3D hovercraft in the presence of delay. The time responses of the r -dimensional model compared favorably with the experimental results.