Disturbance Modelling for Minimum Variance Control in Adaptive Optics Systems Using Wavefront Sensor Sampled-Data
<p>Block diagram for an AO closed-loop system.</p> "> Figure 2
<p>Wavefront sensor.</p> "> Figure 3
<p>Operation of a deformable mirror.</p> "> Figure 4
<p>Equivalent block diagram model for AO system. The auxiliary variable <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>k</mi> </msub> <mo>=</mo> <mo>−</mo> <msubsup> <mi>φ</mi> <mi>k</mi> <mi>cor</mi> </msubsup> </mrow> </semantics></math>.</p> "> Figure 5
<p>Frequency response of the multiplicative model error <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mi mathvariant="script">E</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>z</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mi>ϑ</mi> </mrow> </semantics></math> for different values of <math display="inline"><semantics> <mi>ϑ</mi> </semantics></math>.</p> "> Figure 6
<p>Frequency Response of the <span class="html-italic">true</span> discrete-time disturbance transfer function and the estimated discrete-time disturbance transfer function. The solid blue line represents the frequency response of the <span class="html-italic">true</span> disturbance model, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>(</mo> <msup> <mi>z</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </semantics></math>, and the dashed red line represents the frequency response of the estimated model with <math display="inline"><semantics> <mrow> <mi>ϑ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 7
<p>Discrete-time PSD <math display="inline"><semantics> <mfenced separators="" open="(" close=")"> <mrow> <mo>|</mo> </mrow> <mrow> <mi>H</mi> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi mathvariant="normal">i</mi> <msub> <mi>ω</mi> <mi>j</mi> </msub> <mo>Δ</mo> </mrow> </msup> <mo>)</mo> </mrow> <msup> <mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msup> <mi>λ</mi> <mn>2</mn> </msup> </mfenced> </semantics></math> of the disturbance signal.</p> "> Figure 8
<p>Discrete-time PSD of the turbulence-plus-vibrations perturbation signal in the Clay Telescope at the Las Campanas Observatory. 2014b observing run by the University of Arizona MagAO Team on the night of 31 October 2014.</p> "> Figure 9
<p>Monte-Carlo simulation results to discrete-time PSD of six disturbance for (<b>a</b>) Whittle’s likelihood method and (<b>b</b>) NLS fitting method. We use for simulation the values show in <a href="#sensors-21-03054-t001" class="html-table">Table 1</a>, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> ms, data length <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>. The solid blue line represents the average of all the periodograms, the dashed black line represents the <span class="html-italic">true</span> discrete-time PSD, the dotted red line represents the average of all the estimated PSDs. The gray shaded region represents the area in which all the estimated spectra lie.</p> "> Figure 10
<p>Discrete-time PSD of the disturbance (input signal) and controlled outputs. We use the estimated values that are shown in <a href="#sensors-21-03054-t003" class="html-table">Table 3</a> and <a href="#sensors-21-03054-t004" class="html-table">Table 4</a>, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> ms, and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>. The densely dotted red line represents the discrete-time PSD of disturbance or input signal, the solid orange line with square mark represents discrete-time PSD of the controlled output using the <span class="html-italic">true</span> model, the dashed blue line represents discrete-time PSD of the controlled output using the model obtained by Whittle’s likelihood method, the dotted black line represents discrete-time PSD of the controlled output using the model obtained by NLS fitting method.</p> ">
Abstract
:1. Introduction
2. AO Systems
2.1. Wavefront Sensor
2.2. Deformable Mirror
2.3. AO Controller
3. Disturbance Model in AO Systems
3.1. Equivalent AO System Model
3.2. Classical Sampled-Data Model for Disturbances in AO Systems
4. Proposed Modelling for Disturbances
Algorithm 1 Discrete-time PSD |
5. Identification of Disturbances
5.1. Nonlinear Least Square Fitting Method
5.2. Whittle’s Likelihood
Algorithm 2 Identification algorithm |
|
6. MVC Performance in AO Systems
6.1. Minimum Variance Control Design
6.2. Performance of MVC Subject to Model Error
6.3. Control Performance under Model Mismatch
7. Numerical Example
7.1. Disturbance Identification
7.2. Performance of MVC in AO System
Algorithm 3 MVC algorithm |
|
8. Discussion and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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2 | 14 | 20 | 29 | 43 | 60 | |
0.9 | 0.05 | 0.05 | 0.05 | 0.05 | 0.05 | |
1.439 | 3.493 | 3.734 | 4.253 | 5.540 | 9.711 |
0 | 0.5 | |
0 | 3.85 |
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6 |
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1 | |||
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6 |
Proposed Method | NLS [35] | |
0 | 0 | 0 |
1 | ||
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Coronel, M.; Carvajal, R.; Escárate, P.; Agüero, J.C. Disturbance Modelling for Minimum Variance Control in Adaptive Optics Systems Using Wavefront Sensor Sampled-Data. Sensors 2021, 21, 3054. https://doi.org/10.3390/s21093054
Coronel M, Carvajal R, Escárate P, Agüero JC. Disturbance Modelling for Minimum Variance Control in Adaptive Optics Systems Using Wavefront Sensor Sampled-Data. Sensors. 2021; 21(9):3054. https://doi.org/10.3390/s21093054
Chicago/Turabian StyleCoronel, María, Rodrigo Carvajal, Pedro Escárate, and Juan C. Agüero. 2021. "Disturbance Modelling for Minimum Variance Control in Adaptive Optics Systems Using Wavefront Sensor Sampled-Data" Sensors 21, no. 9: 3054. https://doi.org/10.3390/s21093054
APA StyleCoronel, M., Carvajal, R., Escárate, P., & Agüero, J. C. (2021). Disturbance Modelling for Minimum Variance Control in Adaptive Optics Systems Using Wavefront Sensor Sampled-Data. Sensors, 21(9), 3054. https://doi.org/10.3390/s21093054