New method identifies unstable periodically time-varying systems from closed-loop data

A new paper by Hiroshi Okajima (arXiv:2605.29385) tackles the long-standing challenge of identifying linear periodically time-varying (LPTV) systems when the plant itself is unstable in open loop. Traditional methods require open-loop experiments, but for unstable plants those are impractical. Okajima's solution exploits the fact that in closed-loop operation, the overall system can be made stable even if the plant is not. The central contribution is an exact algebraic plant-extraction theorem based on cycled closed-loop realizations: for square strictly proper plants and a controller path meeting an invertibility condition, the cycled plant transfer matrix can be recovered directly from a shared state-space realization of stable closed-loop maps from reference to output and from reference to control input. No state augmentation is required, and the recovered plant realization need not be stable.

Building on this theorem, the author constructs a complete closed-loop identification algorithm. The algorithm takes the reference, output, and input signals as data, applies standard subspace identification to cycled versions of those signals, performs the algebraic extraction step, and then recovers the LPTV plant's state-space parameters via a coordinate transformation. The reliability of the extraction depends heavily on the conditioning of the inverse controller path. Numerical examples successfully demonstrate recovery of both stable and open-loop unstable single-input single-output (SISO) LPTV plants. A multiple-input multiple-output (MIMO) case is validated through coordinate-invariant Markov-parameter comparisons.

This work matters because LPTV systems appear in many real-world engineering applications—such as rotating machinery, power electronics, and chemical processes with periodic disturbances—where the plant itself may be unstable. Being able to identify such plants using only safe, stable closed-loop experiments removes a major practical barrier. The approach is exact (not approximate) and avoids the computational overhead of state augmentation, making it suitable for implementation in control design and system monitoring. The paper has been submitted to the journal Automatica.