Identification of Unstable Systems using Least-Squares Algorithm

The convergence behaviour of least-squares equation error identification algorithm applied to an unstable linear time-invariant system is analysed. It is shown that in order 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 relations between the robustness of the algorithm with respect to measurement noise and the relative stability of the system are also discussed.