Equation-error closed-loop system identification using cyclic spectral analysis

The problem of closed-loop system identification given noisy input-output measurements is considered. The closed-loop system operates under an external cyclostationary input which is not measured. Noisy measurements of the (direct) input and output of the plant are assumed to be available. The various disturbances affecting the system are either stationary or cyclostationary with cycle frequencies different from the input cycle frequencies. The closed-loop system must be stable but it is allowed to be unstable in open-loop. A frequency-domain parametric solution is proposed and analyzed using an equation error formulation, and cyclic spectrum and cross-spectrum of the input-output measurements. The parameter estimator is shown to be consistent. A simulation example using an unstable open-loop system is presented to illustrate the proposed approach.