On the robustness of linear and non-linear fractional-order systems with non-linear uncertain parameters
Type
Article
Date
2016Permanent link to this record
http://hdl.handle.net/10754/670036Metadata
Show full item recordAbstract
This paper presents the robust stabilization problem of linear and non-linear fractional-order systems with non-linear uncertain parameters. The uncertainty in the model appears in the form of the combination of 'additive perturbation' and 'multiplicative perturbation'. Sufficient conditions for the robust asymptotical stabilization of linear fractional-order systems are presented in terms of linear matrix inequalities (LMIs) with the fractional-order 0 < α < 1. Sufficient conditions for the robust asymptotical stabilization of non-linear fractional-order systems are then derived using a generalization of the Gronwall-Bellman approach. Finally, a numerical example is given to illustrate the effectiveness of the proposed results.Citation
N’Doye, I., Darouach, M., Voos, H., & Zasadzinski, M. (2015). On the robustness of linear and non-linear fractional-order systems with non-linear uncertain parameters. IMA Journal of Mathematical Control and Information, 33(4), 997–1014. doi:10.1093/imamci/dnv022Publisher
Oxford University Press (OUP)Scopus Count
Related items
Showing items related by title, author, creator and subject.
-
Hybrid linear and non-linear full-waveform inversion of Gulf of Mexico data(SEG Technical Program Expanded Abstracts 2013, Society of Exploration Geophysicists, 2013-08-19) [Conference Paper]Standard Full-waveform inversion (FWI) often suffers from poor sensitivity to deep features of the subsurface model. To alleviate this problem, we propose a hybrid linear and non-linear optimization method to enhance the FWI results. In this method, iterative least-squares reverse-time migration (LSRTM) is used to estimate the model update at each nonlinear iteration, and the number of LSRTM iterations is progressively increased after each non-linear iteration. With this method, model updating along deep reflection wavepaths are automatically enhanced, which in turn improves imaging below the reach of diving-waves. This hybrid linear and non-linear FWI algorithm is implemented in the space-time domain to simultaneously invert the data over a range of frequencies. A multiscale approach is used where higher frequencies are iteratively incorporated into the inversion. Synthetic data are used to test the effectiveness of reconstructing both the high- and low-wavenumber features in the model without relying on diving waves in the inversion. We apply the method to Gulf of Mexico field data and illustrate the improvements after several iterations. Results show a significantly improved migration image in both the shallow and deep sections.
-
Linear deflectometry - Regularization and experimental design [Lineare deflektometrie - Regularisierung und experimentelles design](tm - Technisches Messen, Walter de Gruyter GmbH, 2011-01) [Article]Specular surfaces can be measured with deflectometric methods. The solutions form a one-parameter family whose properties are discussed in this paper. We show in theory and experiment that the shape sensitivity of solutions decreases with growing distance from the optical center of the imaging component of the sensor system and propose a novel regularization strategy. Recommendations for the construction of a measurement setup aim for benefiting this strategy as well as the contrarian standard approach of regularization by specular stereo. © Oldenbourg Wissenschaftsverlag.
-
LASSO with non-linear measurements is equivalent to one with linear measurements(Neural information processing systems foundation, 2015-01-01) [Conference Paper]Consider estimating an unknown, but structured (e.g. sparse, low-rank, etc.), signal x0 ∈ ℝn from a vector y ∈ ℝm of measurements of the form yi = gi(aiT x0), where the ai's are the rows of a known measurement matrix A, and, g(·) is a (potentially unknown) nonlinear and random link-function. Such measurement functions could arise in applications where the measurement device has nonlinearities and uncertainties. It could also arise by design, e.g., gi(x) = sign(x+zi), corresponds to noisy 1-bit quantized measurements. Motivated by the classical work of Brillinger, and more recent work of Plan and Vershynin, we estimate x0 via solving the Generalized-LASSO, i.e., ◯:= arg minx ||y-Ax0||2 + λf(x) for some regularization parameter λ > 0 and some (typically non-smooth) convex regularizer f(·) that promotes the structure of x0, e.g. ℓ1-norm, nuclear-norm, etc. While this approach seems to naively ignore the nonlinear function g(·), both Brillinger (in the non-constrained case) and Plan and Vershynin have shown that, when the entries of A are iid standard normal, this is a good estimator of x0 up to a constant of proportionality μ, which only depends on g(·). In this work, we considerably strengthen these results by obtaining explicit expressions for ||◯-μx0||2, for the regularized Generalized-LASSO, that are asymptotically precise when m and n grow large. A main result is that the estimation performance of the Generalized LASSO with non-linear measurements is asymptotically the same as one whose measurements are linear yi = μaiT x0 + σzi, with μ = Eγg(γ) and σ2 = E(g(γ)-μγ)2, and, γ standard normal. To the best of our knowledge, the derived expressions on the estimation performance are the first-known precise results in this context. One interesting consequence of our result is that the optimal quantizer of the measurements that minimizes the estimation error of the Generalized LASSO is the celebrated Lloyd-Max quantizer.
