A new frequency-domain subspace algorithm with restricted poles location through LMI regions and its application to a wind tunnel test

ABSTRACT

In this paper, an innovative method is presented to identify models with a modified frequency-domain subspace method. This new approach allows to introduce in the subspace resolution constraints on the identified model poles location. Here, a very general formulation is proposed to take into account regions in continuous/discrete map. This formulation is based on an LMI (linear matrix inequalities) description where the stability domain represents a particular case. These LMI constraints are combined with the frequency-domain subspace resolution to obtain identified models whose poles are situated in the specified LMI regions. This approach is benchmarked with the Loewner one, which belongs to the class of frequency-domain input–output model identification and approximation methods. Besides the fact that they both belong to the data-driven model approximation class, they result to have slightly different objectives and show complementary performances. This discussion is illustrated in practice with experimentations that have been performed for the identification and control of the gust disturbance over a 2D wing span, from sub to transonic, in a wind tunnel facility.

KEYWORDS:

• Frequency data identification
• model approximation
• subspace
• Loewner
• aerospace

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Let also assume that the matrix pencil is regular, i.e. the matrix $\widehat{\mathbf{A}}-\lambda I_r$ is nonsingular for some finite $\lambda \in \mathbb{C}$.

2. Tangential interpolation vectors are standard tools in model reduction to handle the MIMO cases (see e.g. Van-Dooren, Gallivan, & Absil, 2008).

3. As made clearer latter on, this is also known as the Loewner pencil.

4. Similarly to their standard versions, these are called the generalised tangential observability and generalised tangential controllability matrices.

5. Note that once ${\mathcal{S}}_{\delta }$ is available, any model approximation/reduction technique can be applied. However, as stressed in the introduction, this paper focusses on the direct data-based identification and approximation process, only.

6. Generalities on the subspace approaches is let to the reader discretion.

7. For clarity purpose, only the noise-free case is considered. Reader is invited to refer McKelvey et al. (1996) for the noisy case.

8. Note that to avoid complex arithmetic, one should consider the real representation of G = [Re(G) Im(G)], W = [Re(W) Im(W)] and X = [Re(X^c) Im(X^c)].

9. The eigenvalues of a dynamical system $\dot{x}\left(t\right)=Ax\left(t\right)$ lie in LMI regions defined by $\mathcal{D}$ if these poles satisfied the characteristic function $f_{\mathcal{D}}$.

10. As the sampling frequency (F_s = 4096Hz) is widely superior to the main system dynamics, the continuous and discrete frequency-domain responses are strictly equal for frequencies inferior to 100 Hz.

11. Without lack of generalities, by designing adequate weighting filters to generate the generalised plant for transfer minimisation, the ${\mathcal{H}}_2$ objective can be – in some sense – recast as an ${\mathcal{H}}_{\infty }$ one. As it will be made clear in the following, this formulation also allows to handle additional constraints on the controller properties.

12. Five configurations of open-loop systems for Mach number going from 0.30 to 0.73, and AoA from 0 to 2 degrees. The controller has been obtained using the hinfstruct.m function of Matlab.

13. Note that here the exact Loewner model is used since, even is nominally unstable, it perfectly captures the frequency data over all the considered frequency range.

14. Note that this dynamic is linked to the pitch behaviour (see Figure 8) which is, after recurrent wind tunnel experimental tests, affected by viscous friction effect.

15. It was experimentally impossible to generate the same kind of test in WT due to the limited amplitude range of the gust generator. Indeed, this remark is also a justification for the frequency-domain-oriented method.

Additional information

Funding

The research leading to these results has received funding from the European Union's Seventh Framework Program (FP7/2007-2013) for the Clean Sky Joint Technology Initiative [grant number CSJU-GAM-SFWA-2008-001].