Closed-loop identification of unstable systems using noncausal FIR models

Abstract

Motivated by the potential advantages of FIR model structures, the present paper considers the applicability of FIR models to closed-loop identification of open-loopunstable plants. We show that FIR models can be used effectively for closed-loop identification of open-loop-unstable plants.The key insight in this regard is to realize that a noncausal FIR model can serve as a truncated Laurent expansion inside the annulus between the asymptotically stable pole of largest modulus and the unstable pole of smallest modulus. The key to identifying the noncausal plant model is to delay the measured output relative to the measured input. With this techniques, the identified FIR model is precisely a noncausal approximation of the unstable plant, that is, an approximation of the Laurent expansion of the plant inside the annulus of analyticity lying between the disk of stable poles and the punctured plane of unstable poles.

Non-causal regularized least-squares for continuous-time system identification with band-limited input excitations

“…Proposition 3.3: Consider the system with sampled output as in (6), and assume {w(kh)} is white noise of variance σ 2 . The regularization term that minimizes the MSE matrix in a positive definite sense is given by γ opt = σ 2 and P opt r = ρ * ρ * , and the corresponding optimal regularized estimate is (11) Proof: The proof follows by the same reasoning as in the proof of Theorem 1 of [18]. Since the optimal regularization matrix is not known a priori, the matrix P r is typically parameterized by a lowdimensional hyperparameter vector β ∈ B according to what can be assumed about the impulse response.…”

“…The causal TC and SS-regularized least-squares estimators are obtained via the impulseest command in MATLAB, while the non-causal TC and SS kernels are tuned by solving (14). The unrealizable oracle is computed by (11). The performance of these estimators is compared via Monte Carlo simulations with 300 different noise realizations.…”

“…Non-causal system identification has been recently studied in [9], in which a Maximum Likelihood estimator is proposed for symmetric non-causal systems with applications to cross direction modeling of paper machines. In particular, noncausal FIR models have been used for identifying systems in closed-loop in [10], [11].…”

Aircraft Sensor Health Monitoring Based on Transmissibility Operators

“…This class of models consists of noncausal finite impulse response (FIR) models based on a truncated Laurent expansion. The causal (backward-shift) part of the Laurent expansion is asymptotically stable because all of its poles are zero, whereas the noncausal (forwardshift) part of the Laurent expansion captures the unstable and noncausal components of the transmissibility operator [22].…”