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

Abstract

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 $$\varepsilon $$. 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 $$\varepsilon $$) 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 $$\varepsilon $$ 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.