Optimal Experiment Design for the Identification of One Module in the Interconnection of Locally Controlled Systems

Abstract

In this paper, we consider the problem of designing the least costly experiment that leads to a sufficiently accurate estimate of one module in a network of locally controlled systems. A module in such a network can be identified by exciting the corresponding local closed loop system. Such an excitation signal will not only perturb the input/output of the to-be-identified module, but also other modules due to the interconnection. Consequently, the cost of the identification can be expressed as the sum of the influence of the excitation signal on the inputs and outputs of all locally controlled systems. We develop a methodology to design the spectrum of the excitation signal in such a way that this cost is minimized while guaranteeing a certain accuracy for the identified model. We also propose an alternative identification configuration which can further reduce the propagation of the excitation signal to other modules and we make steps to robustify this optimal experiment design problem with respect to the cost of the identification.

Robust optimal identification experiment design for multisine excitation

“…Remark. In (15), the actual value of the variance σ 2 e of the white noise e (see (1)) is generally unknown. However, we can also robustify the optimal experiment design problem against this uncertainty.…”

“…The constraints ( 13)-( 14) are indeed equivalent to (10) when the parametrization ( 12) is used for Φ u . In the paper [15], we have proposed the following tractable alternative for ( 13)-( 14):…”

“…Proposition 2 Consider an initial uncertainty set U init (see (5)) satisfying Assumption 2 and the robustified accuracy constraint (11) when the spectrum Φ u is parametrized as in (12) and where, for this reason, P −1 ρ (θ, Φ u ) has the expression given in (15). Consider the LFT (26) in I f L ⊗ θ which is the LFT representation fors = (F 1 (θ) T , F 2 (θ) T , ..., F k G (θ) T ) T and consider the set of multipliers A k G f L associated with U init (see Definition 1 with n = k G f L).…”

“…Let us then pre-and post-multiply the LMI constraint ( 27) with (I k G ⊗ (p T F , 1) T ) * and I k G ⊗ (p T F , 1) T , respectively. Using (26), (15) and the lemma in Appendix B , this yields:…”

“…This is due to the dependency on θ of the frequency response elements G(e jωm , θ) in G(θ). The approach in[15] (see Section 4.1) would in fact correspond to a relation betweenp andq of the form diag(θ(ω 1 ), θ(ω 2 ), ..., θ(ω L )) where θ(ω m ) ∈ U init for all ω m (m = 1, ..., L), but can be different for each frequency ω m (m = 1, ..., L). Indeed α G (ω m ) can be rewritten as sup θ(ωm)∈Uinit |G(e jωm , θ(ω m ))| 2 (m = 1, ..., L).…”