Nuclear norm-enhanced recursive subspace identification: Closed-loop estimation of rapid variations in system dynamics

For a time-varying plant operating in closed-loop with a stabilising controller, rapid changes in system dynamics can be detected online using recursive subspace identification methods to estimate the open-loop system behaviour. However, these methods usually involve a speed-accuracy trade-off: accurate identification can often only be achieved by slow updates, which increases the lag in the detection of changes in system dynamics. In this paper, a closed-loop, recursive subspace identification algorithm is extended with a convex cost function based on the nuclear norm. The nuclear norm heuristic exploits structure in the algorithm by enforcing a low-rank condition on the state predictor matrix. This condition reduces the variance of the estimates at the price of introducing a bias. The new algorithm is demonstrated for a system where the damping changes from positive to negative, and it is shown to successfully and consistently estimate the onset of open-loop instability, outperforming conventional recursive identification. Further, by tuning the forgetting factor in the estimation algorithm, a favourable speed-accuracy trade-off can be achieved.