Convergence analysis of least-squares identification algorithm for unstable systems

The convergence behaviour of a least-squares equation error identification algorithm applied to an unstable time-invariant system is analysed. It is shown that, to obtain a uniform rate of convergence for all the parameter estimates of an ARMA model, an unbounded probing signal input is necessary. However, with the bounded probing signal input, the convergence of AR parameter estimates alone is ensured. It is also shown explicitly that, for unstable systems with a bounded probing signal input, the speed of convergence of AR parameter estimates is much faster than that for stable systems, but the convergence rate of MA parameter estimates is so slow that it is difficult to obtain them in a reasonable time span. The practical issues of unstable system identification, as well as the relationship between the robustness of the algorithm with respect to measurement noise and the relative stability of the system are also discussed. Finally, simulation results are included to illustrate the convergence behaviours.