tag:blogger.com,1999:blog-60685549777618101192024-03-13T10:47:00.348+09:00Welcome to My Sparse LandYou will find in this blog some new (or very old) things on signal processing, control systems, mathematics, education, etc.Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.comBlogger35125tag:blogger.com,1999:blog-6068554977761810119.post-13721681452283733552015-03-28T08:03:00.000+09:002015-03-31T07:41:31.941+09:00Discreteness and Sparsity<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-_mdvKRLjJMU/VRnAk_zwz6I/AAAAAAAABHg/sX6vzRFImf0/s1600/binary-65473_640.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-_mdvKRLjJMU/VRnAk_zwz6I/AAAAAAAABHg/sX6vzRFImf0/s1600/binary-65473_640.jpg" height="226" width="320" /></a></div>
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Discrete signals may be sparse, whereas sparse signals may not be discrete.<br />
<br />
Let us consider the following signal:<br />
<span style="font-family: inherit;"><i> x</i> = [1,1,-1,1,-1,-1,1,1];</span><br />
<span style="font-family: inherit;">This is a discrete signal generated from an alphabet {1,-1}. This is obviously not sparse, but if we transform this into</span><br />
<span style="font-family: inherit;"><i> x </i>- 1 = [0,0,-2,0,-2,-2,0,0]</span><br />
<span style="font-family: inherit;">we have a sparse signal. Also,</span><br />
<span style="font-family: inherit;"><span style="font-family: inherit;"><i> x </i>+ 1 = [2,2,0,2,0,0,2,2]</span></span><br />
<span style="font-family: inherit;"><span style="font-family: inherit;">is sparse. </span></span><span style="font-family: inherit;">Let's adapt this idea to discrete signal reconstruction.</span><br />
<br />
<b>Discrete Signal Reconstruction </b><br />
<i>Find the original n-dimensional discrete signal x over a finite alphabet</i><br />
<i>X</i> = {<i>r</i><sub>1</sub>,<i>r</i><sub>2</sub>,...,<i>r</i><sub><i>L</i></sub>}<br />
<i>with its probability distribution</i><br />
<b>P</b>(<i>r</i><sub><i>j</i></sub>) = <i>p</i><sub><i>j</i></sub>, 0 < <i>p</i><sub><i>j</i></sub> <1, <i>p</i><sub>1 </sub>+ <i>p</i><sub>2 </sub>+ ... + <i>p</i><sub><i>L </i></sub>= 1<br />
<i>from incomplete linear measurements</i><br />
<i>y</i> = <i>Ax</i><br />
<i>where A is an m-times-n matrix</i> (<i>m</i> < <i>n</i>).<br />
<br />
In this problem, each vector <i>x</i> - <i>r</i><sub><i>j</i></sub> is sparse, and hence its ell-1 norm may be small if the probability <i>p<sub>j</sub></i> is large. This observation leads to optimization with the<i> sum of absolute values (SOAV)</i>:<br />
<br />
minimize <i>p</i><sub>1</sub>||<i>x</i> - <i>r</i><sub>1</sub>||<sub>1</sub> + <i>p</i><sub>2</sub>||<i>x</i> - <i>r<sub>2</sub></i>||<sub>1</sub> +...+ <i>p</i><sub>L</sub>||<i>x</i> - <i>r</i><sub>L</sub>||<sub>1</sub><br />
subject to <i>y</i> = <i>Ax</i><br />
<i><br />
</i> Let's apply this to binary-valued image reconstruction. The following is a 37x37-pixel binary image disturbed by random Gaussian noise.<br />
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<a href="http://1.bp.blogspot.com/-wAq1Apmb004/VRXbK5EyY8I/AAAAAAAABG8/RnN3O16VFQg/s1600/noise_A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-wAq1Apmb004/VRXbK5EyY8I/AAAAAAAABG8/RnN3O16VFQg/s1600/noise_A.png" /></a></div>
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Then we transform this into frequency-domain data via discrete Fourier transform (DFT), and then randomly down-sample rows to generate half-size data. Then apply the SOAV optimization to reconstruct the original binary image. The following is the result.</div>
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<a href="http://2.bp.blogspot.com/-Iy-HiMdiH5A/VRXcf2GUROI/AAAAAAAABHE/oYzAyO49ptc/s1600/Proposed_A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-Iy-HiMdiH5A/VRXcf2GUROI/AAAAAAAABHE/oYzAyO49ptc/s1600/Proposed_A.png" /></a></div>
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For comparison, we also use the <a href="http://en.wikipedia.org/wiki/Basis_pursuit" target="_blank">basis pursuit (BP)</a> that minimizes the ell-1 norm of <i>x</i> instead of the sum of absolute values. The following is the result of BP:</div>
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<a href="http://3.bp.blogspot.com/-3kEjbaOtE4o/VRXdSmcxigI/AAAAAAAABHM/Hd-zWyJe1xs/s1600/BP_A.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-3kEjbaOtE4o/VRXdSmcxigI/AAAAAAAABHM/Hd-zWyJe1xs/s1600/BP_A.png" /></a></div>
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This is quite bad since BP only uses the property of sparsity in the original image while the SOAV approach takes the discreteness into account.<br />
<br />
If you are interested in this work, please see the original paper:<br />
<blockquote class="tr_bq">
M. Nagahara, Discrete Signal Reconstruction by Sum of Absolute Values, <i>IEEE Signal Processing Letters</i>, vol. 22 no. 10, pp. 1575-1579, Oct 2015 (to appear)</blockquote>
PDF can be downloaded from <a href="http://arxiv.org/pdf/1503.05299v1.pdf" target="_blank">here</a> and the MATLAB codes are available <a href="http://www-ics.acs.i.kyoto-u.ac.jp/mat/dsr/" target="_blank">here</a>.<br />
<br />
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<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com1tag:blogger.com,1999:blog-6068554977761810119.post-18872750753937706522014-09-07T22:32:00.000+09:002014-09-07T22:32:46.295+09:00Nice restaurants in Cape Town<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-GaFzvarUVsk/VAY6ZYHna5I/AAAAAAAAA7U/zIRd4nrtQzo/s1600/outside.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-GaFzvarUVsk/VAY6ZYHna5I/AAAAAAAAA7U/zIRd4nrtQzo/s1600/outside.jpg" height="183" width="320" /></a></div>
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<a href="http://www.ifac2014.org/" target="_blank">IFAC world congress 2014</a> was held at <a href="http://www.cticc.co.za/" target="_blank">Cape Town International Convention Centre (CTICC)</a>, Cape Town, South Africa. I attended this congress in this summer and enjoyed restaurants around CTICC. Today's post is on nice restaurants around CTICC that I recommend to go.<br />
<br />
<a href="http://tascadebelem.co.za/" target="_blank">TASCA</a><br />
If you stay near CTICC, then basically <a href="http://www.waterfront.co.za/" target="_blank">Waterfront</a> gives you nice restaurants. I went to this restaurant twice, because they offer quite nice seafoods. Also, this restaurant has a great view of the table mountain from tables outside (see the picture above).<br />
<br />
<a href="http://www.sevruga.co.za/" target="_blank">Sevruga</a><br />
This restaurant is also situated in Waterfront. Nice seafoods and wines. In particular, I recommend <a href="http://www.ij.co.za/kingklip/" target="_blank">KingKlip</a>, <span style="-webkit-text-size-adjust: auto; background-color: rgba(255, 255, 255, 0);">one of the most popular eating fish in South Africa.</span><br />
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<a href="http://nv-80.co.za/" target="_blank">NV80</a><br />
<a href="http://www.newcastle.edu.au/profile/daniel-quevedo" target="_blank">Daniel</a>, a friend of mine, invited 18 people including me to this restaurant. Meat is absolutely nice, in fact, this was the best meat I've ever had. At the time, I met Song from Hong Kong, who is also a friend of Daniel, and a reader of this blog! Amazing!! This restaurant is located in Sea Point, which approximately five minutes' drive from CTICC.</div>
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<i>Bon appetit!</i><br />
<i><br /></i></div>
Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com2tag:blogger.com,1999:blog-6068554977761810119.post-31168040107473720322014-08-21T23:30:00.000+09:002015-08-09T12:33:24.131+09:00Maximum Hands-Off Control<div class="separator" style="clear: both; text-align: center;">
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<a href="http://1.bp.blogspot.com/-dUlGqSZo9qo/U_PGzHgiXXI/AAAAAAAAA6c/aPM3rrJFNBU/s1600/Prius.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="198" src="http://1.bp.blogspot.com/-dUlGqSZo9qo/U_PGzHgiXXI/AAAAAAAAA6c/aPM3rrJFNBU/s1600/Prius.jpg" width="320" /></a></div>
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Sparsity is also important in control systems.<br />
<div>
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<div>
In particular, sparse control, called <i>hands-off control</i>, is proposed for saving energy and reducing CO2 emissions in control systems. For example, a hybrid vehicle, <a href="http://en.wikipedia.org/wiki/Toyota_Prius" target="_blank">Toyota Prius</a> (displayed above), uses this control. The internal combustion engine is <i>stopped</i> (control = zero; sparse control) when the vehicle is at a stop or the speed is lower than a preset threshold, and the electric motor is alternatively used. See also <a href="http://arxiv.org/abs/1407.2377" target="_blank">Hands-off Control as Green Control</a>.</div>
<div>
<br /></div>
<div>
Recently, it has been proved that the <i>sparsest</i> (or L0 optimal) control among all admissible controls is L1 optimal (see <a href="http://sparseland.blogspot.jp/2014/02/l0l1-equivalence-in-optimal-control.html" target="_blank">L0/L1 Equivalence in Optimal Control</a>). Based on this, hands-off control is extended to feedback control in</div>
<blockquote class="tr_bq">
<blockquote class="tr_bq">
<b>M. Nagahara</b>, <b>D. E. Quevedo</b>, <b>D. Nesic</b>, Maximum Hands-Off Control: A Paradigm of Control Effort Minimization, IEEE Transactions on Automatic Control, Vol. 61, No. 4, 2016 (to appear) (PDF is <a href="http://arxiv.org/abs/1408.3025" target="_blank">here</a>).</blockquote>
</blockquote>
The abstract reads:<br />
<blockquote class="tr_bq">
In this paper, we propose a new paradigm of control, called a maximum hands-off control. A hands-off control is defined as a control that has a short support per unit time. The maximum hands-off control is the minimum support (or sparsest) per unit time among all controls that achieve control objectives. For finite horizon control, we show the equivalence between the maximum hands-off control and L1-optimal control under a uniqueness assumption called normality. This result rationalizes the use of L1 optimality in computing a maximum hands-off control. We also propose an L1/L2-optimal control to obtain a smooth hands-off control. Furthermore, we give a self-triggered feedback control algorithm for linear time-invariant systems, which achieves a given sparsity rate and practical stability in the case of plant disturbances. An example is included to illustrate the effectiveness of the proposed control.</blockquote>
<b>A MATLAB code</b> for the simulation is available <a href="http://www-ics.acs.i.kyoto-u.ac.jp/mat/mhoc/" target="_blank">here</a>.<br />
<br />
Enjoy!<br />
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<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-90990560694861194922014-08-14T00:20:00.000+09:002014-08-14T00:20:43.553+09:00Textbooks on Compressed Sensing<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-lBQmYyXN5As/U9VwtKgO6hI/AAAAAAAAA4c/ICeGW_3sM8k/s1600/child_and_books_208362.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-lBQmYyXN5As/U9VwtKgO6hI/AAAAAAAAA4c/ICeGW_3sM8k/s1600/child_and_books_208362.jpg" height="211" width="320" /></a></div>
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Today, I'd like to list textbooks on compressed sensing and related topics.<br />
This list is far from complete, but I believe this helps a beginner take the first step.<br />
<br />
<div class="p1">
<b><a href="http://www.cambridge.org/us/academic/subjects/engineering/communications-and-signal-processing/compressed-sensing-theory-and-applications" target="_blank">Compressed Sensing: Theory and Applications</a></b></div>
<div class="p1">
Y. C. Eldar and G. Kutyniok (Editors),</div>
<div class="p1">
Cambridge University Press, June, 2012</div>
<div class="p1">
ISBN-13: 978-1107005587<br />
<b>My comment:</b> The introduction chapter (Chapter 1) is especially nice.</div>
<div class="p1">
<br /></div>
<div class="p2">
<b><a href="http://www.springer.com/mathematics/analysis/book/978-1-4419-7010-7" target="_blank">Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing</a></b></div>
<div class="p1">
M. Elad (Author)</div>
<div class="p1">
Springer, August, 2010.</div>
<div class="p1">
ISBN-13: 978-1441970107<br />
<b>My comment:</b> A nice introduction to sparse representation in signal and image processing.<br />
<br /></div>
<div class="p1">
<b><a href="https://www.ceremade.dauphine.fr/~peyre/wavelet-tour/" target="_blank">A Wavelet Tour of Signal Processing: The Sparse Way</a></b></div>
<div class="p1">
S. Mallat (Author)</div>
<div class="p1">
Academic Press, 3rd edition, December, 2008.</div>
<div class="p1">
ISBN-13: 978-0123743701</div>
<div class="p1">
<b>My comment:</b> Read this book and be a master of wavelet. You can!<br />
<br />
<div class="p1">
<b><a href="http://www.springer.com/birkhauser/mathematics/book/978-0-8176-4947-0" target="_blank">A Mathematical Introduction to Compressive Sensing</a></b></div>
<div class="p1">
S. Foucart and H. Rauhut (Authors)</div>
<div class="p1">
Birkhäuser, August, 2013.</div>
<div class="p1">
ISBN-13: 978-0817649470<br />
<b>My comment:</b> If you are interested in the theory of compressed sensing, this book will be of help.</div>
<br /></div>
<div class="p2">
<b><a href="http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-1-4419-9568-1" target="_blank">Fixed-Point Algorithms for Inverse Problems in Science and Engineering</a></b></div>
<div class="p1">
H. H. Bauschke, R. S. Burachik, P. L. Combettes, V. Elser, D. R. Luke, H. Wolkowicz (Editors)</div>
<div class="p1">
Springer, June, 2011.</div>
<div class="p1">
ISBN-13: 978-1441995681<br />
<b>My comment:</b> This book gives you fancy methods to solve sparse optimizations (e.g. L1/L2 optimization)</div>
<div class="p1">
<br /></div>
<div class="p1">
<b><a href="http://www.cambridge.org/us/academic/subjects/computer-science/computer-graphics-image-processing-and-robotics/sparse-image-and-signal-processing-wavelets-curvelets-morphological-diversity" target="_blank">Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity</a></b></div>
<div class="p1">
J.-L. Starck, F. Murtagh, and J. M. Fadili (Authors)</div>
<div class="p1">
Cambridge University Press, May 2010.</div>
<div class="p1">
ISBN-13: 978-0521119139</div>
<div class="p1">
<b>My comment:</b> Sparsity has many applications in image and signal processing.<br />
<br />
Enjoy!<br />
<br />
A related blog entry is <a href="http://sparseland.blogspot.jp/2013/03/a-users-guide-to-compressed-sensing-for.html" target="_blank">here</a>.</div>
Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com1tag:blogger.com,1999:blog-6068554977761810119.post-51399390326950070752014-08-09T23:37:00.001+09:002014-08-17T07:48:01.791+09:00MIT's Visual Microphone System<div class="separator" style="clear: both; text-align: center;">
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<a href="http://2.bp.blogspot.com/-aJADtEnnd-s/U-Yx4mR6haI/AAAAAAAAA5s/WUhdahW0-Ls/s1600/San_Francisco_salt_water_taffy.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-aJADtEnnd-s/U-Yx4mR6haI/AAAAAAAAA5s/WUhdahW0-Ls/s1600/San_Francisco_salt_water_taffy.jpg" height="213" width="320" /></a></div>
<br />
I can understand the theory, but the technology is quite unbelievable!<br />
See <a href="http://spectrum.ieee.org/tech-talk/consumer-electronics/audiovideo/your-candy-wrappers-are-listening" target="_blank">Your Candy Wrappers are Listening</a>.<br />
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<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-76680462842492324812014-07-30T05:50:00.000+09:002014-07-30T05:50:04.115+09:00Sparse Packetized Predictive Control for Networked Control over Erasure Channels<div class="p1">
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<a href="http://3.bp.blogspot.com/-lVQ8CPyRJOc/UqoJbNnzKEI/AAAAAAAAAuo/hB712TtBD3I/s1600/sparsity.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-lVQ8CPyRJOc/UqoJbNnzKEI/AAAAAAAAAuo/hB712TtBD3I/s1600/sparsity.png" height="132" width="400" /></a></div>
<br />
A new paper on sparsity-based control has been published:<br />
<br />
M. Nagahara, D. E. Quevedo, and J. Ostergaard,<br />
<b>Sparse Packetized Predictive Control for Networked Control over Erasure Channels</b>,<br />
<i>IEEE Transactions on Automatic Control</i>, vol. 59, no. 7, pp. 1899-1905, July 2014. </div>
<div class="p1">
<i><br />
</i> PDF can be downloaded from <a href="http://arxiv.org/abs/1307.8242" target="_blank">arxiv</a>.<br />
<br /></div>
<div class="p1">
The abstract reads:<br />
<blockquote class="tr_bq">
We study feedback control over erasure channels with packet-dropouts. To achieve robustness with respect to packet-dropouts, the controller transmits data packets containing plant input predictions, which minimize a finite horizon cost function. To reduce the data size of packets, we propose to adopt sparsity-promoting optimizations, namely, ell-1/ell-2 and ell-2-constrained ell-0 optimizations, for which efficient algorithms exist. We show how to design the tuning parameters to ensure (practical) stability of the resulting feedback control systems when the number of consecutive packet-dropouts is bounded.</blockquote>
</div>
<div class="p1">
A corresponding blog post is <a href="http://sparseland.blogspot.jp/2013/12/sparse-packetized-predictive-control.html" target="_blank">here</a>.</div>
Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-794272501952088002014-07-24T18:25:00.000+09:002014-07-24T22:23:27.200+09:00L1 Control Theoretic Smoothing Splines<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-igI9Bh-Ahy4/U9AlgE2weZI/AAAAAAAAA4I/xPP6jbmGUJk/s1600/L1result.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-igI9Bh-Ahy4/U9AlgE2weZI/AAAAAAAAA4I/xPP6jbmGUJk/s1600/L1result.png" height="300" width="400" /></a></div>
The control theoretic spline is a bridge between control and signal processing, as mentioned in <a href="http://sparseland.blogspot.jp/2013/04/control-theoretic-spline-bridge-between.html" target="_blank">a previous blog post</a>. This technique is extended to <i>robust and sparse</i> splines via <i>L</i><sup>1</sup> <i>optimality</i>:<br />
<blockquote class="tr_bq">
M. Nagahara and C. F. Martin, <br />
<b><a href="http://arxiv.org/abs/1407.1697" target="_blank">L1 Control Theoretic Smoothing Splines</a></b>,<br />
<i>IEEE Signal Processing Letters</i>, 2014. (accepted)</blockquote>
The robustness is against outliers in data, while the sparsity is for representation of a curve with smaller number of parameters.<br />
The abstract reads<br />
<blockquote class="tr_bq">
In this paper, we propose control theoretic smoothing splines with L1 optimality for reducing the number of parameters that describes the fitted curve as well as removing outlier data. A control theoretic spline is a smoothing spline that is generated as an output of a given linear dynamical system. Conventional design requires exactly the same number of base functions as given data, and the result is not robust against outliers. To solve these problems, we propose to use L1 optimality, that is, we use the L1 norm for the regularization term and/or the empirical risk term. The optimization is described by a convex optimization, which can be efficiently solved via a numerical optimization software. A numerical example shows the effectiveness of the proposed method.</blockquote>
The MATLAB codes are available <a href="http://www-ics.acs.i.kyoto-u.ac.jp/mat/spline/" target="_blank">here</a>.<br />
PDF is <a href="http://arxiv.org/abs/1407.1697" target="_blank">here</a>.Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-6714532572524252212014-02-06T21:33:00.000+09:002014-02-06T23:36:37.352+09:00L0/L1 Equivalence in Optimal Control<div class="separator" style="clear: both; text-align: center;">
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<a href="http://2.bp.blogspot.com/-vbe5YUORo1Q/UvHMUh4v0dI/AAAAAAAAAxI/KaQGl0pjH2w/s1600/control_4th-eps-converted-to.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-vbe5YUORo1Q/UvHMUh4v0dI/AAAAAAAAAxI/KaQGl0pjH2w/s1600/control_4th-eps-converted-to.png" height="240" width="320" /></a></div>
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One of the fundamental results of <a href="https://sites.google.com/site/igorcarron2/cs" target="_blank">compressed sensing</a> is that <i>L</i><sup>1</sup> optimization gives the sparsest (or <i>L</i><sup>0</sup>-optimal) solution of an inverse problem under some assumptions.<br />
<br />
Is it also true with <i>L</i><sup>1</sup> optimal control?<br />
Is <i>L</i><sup>1</sup> optimal control the sparsest one among all admissible controls?<br />
<br />
The answer is YES (under some assumptions, of course).<br />
Our recent research<br />
<blockquote class="tr_bq">
M. Nagahara, D. E. Quevedo, and D. Nesic,<br />
<b><a href="http://arxiv.org/abs/1307.8232" target="_blank">Maximum Hands-Off Control and L1 Optimality</a></b>,<br />
<i>52nd IEEE Conference on Decision and Control (CDC)</i>,<br />
pp. 3825-3830, Dec. 2013.</blockquote>
has revealed that.<br />
<br />
It is interesting that the <i>L</i><sup>0</sup> norm in this article is defined for <i>continuous-time</i> signals while most compressed sensing methods uses one for vectors. We call a control with small <i>L</i><sup>0</sup> norm a <i>hands-off control</i>.<br />
<br />
The abstract reads:<br />
<blockquote class="tr_bq">
In this article, we propose a new paradigm of control, called a maximum hands-off control. A hands-off control is defined as a control that has a much shorter support than the horizon length. The maximum hands-off control is the minimum support (or sparsest) control among all admissible controls. We first prove that a solution to an <i>L</i><sup>1</sup>-optimal control problem gives a maximum hands-off control, and vice versa. This result rationalizes the use of <i>L</i><sup>1</sup> optimality in computing a maximum hands-off control. The solution has in general the "bang-off-bang" property, and hence the control may be discontinuous. We then propose an <i>L</i><sup>1</sup>/<i>L</i><sup>2</sup>-optimal control to obtain a continuous hands-off control. Examples are shown to illustrate the effectiveness of the proposed control method.<br />
<div>
<br /></div>
</blockquote>
<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-37152275213682022482013-12-25T07:17:00.002+09:002014-08-17T07:48:54.730+09:00Sparsity-Related Researches in IEEE CDC 2013<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-E0Jlbm7rxB0/Uq0sxn99p5I/AAAAAAAAAvA/EMbqNmPQgL8/s1600/florence.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-E0Jlbm7rxB0/Uq0sxn99p5I/AAAAAAAAAvA/EMbqNmPQgL8/s320/florence.jpg" height="239" width="320" /></a></div>
<br />
Ciao!<br />
<br />
The most important conference on control<i>, </i><i><a href="http://cdc2013.units.it/" target="_blank">IEEE Conference on Decision and Control (CDC) 2013</a></i>, was held in this month, Dec 10-13, in Florence, Italy. Of course, I was there.<br />
<br />
Foods are good, wines excellent, Duomo gorgeous, bazaar funny, I fully enjoyed my stay!<br />
<br />
By the way, I found papers related to sparsity in control including my paper:<br />
<br />
<div>
<br /></div>
<div>
<div>
<b><a href="http://arxiv.org/abs/1307.8232" target="_blank">Maximum Hands-Off Control and L<sup>1</sup> Optimality</a></b><br />
M. Nagahara, Kyoto Univ.<br />
D. E. Quevedo, The Univ. of Newcastle<br />
D. Nesic, Univ. of Melbourne</div>
<div>
<blockquote class="tr_bq">
<b>Abstract: </b>In this article, we propose a new paradigm of control, called a maximum hands-off control. A hands-off control is defined as a control that has a much shorter support than the horizon length. The maximum hands-off control is the minimum support (or sparsest) control among all admissible controls. We first prove that a solution to an L1-optimal control problem gives a maximum hands-off control, and vice versa. This result rationalizes the use of L1 optimality in computing a maximum hands-off control. The solution has in general the "bang-off-bang" property, and hence the control may be discontinuous. We then propose an L1/L2-optimal control to obtain a continuous hands-off control. Examples are shown to illustrate the effectiveness of the proposed control method.</blockquote>
<br />
<b><a href="http://www-control.eng.cam.ac.uk/Homepage//publication.php?id=1180" target="_blank">Graphical FPGA Design for a Predictive Controller with Application to Spacecraft Rendezvous</a></b><br />
E. N. Hartley, Univ. of Cambridge<br />
J. M. Maciejowski, Univ. of Cambridge<br />
<blockquote class="tr_bq">
<b>Abstract:</b> A reconfigurable field-programmable gate array (FPGA)-based predictive controller based on Nesterov's fast gradient method is designed using Simulink and converted to VHDL using Mathworks' HDL Coder. The implementation is verified by application to a spacecraft rendezvous and capture scenario, with communication between the FPGA and a simulation of the relative dynamics occuring over Ethernet. For a problem with 120 decision variables and 240 constraints, computation times of 0.95 ms are achieved with a clock rate of 50 MHz, corresponding to a speed up of more than 2000 over running the algorithm directly on a MicroBlaze microprocessor implemented on the same FPGA.</blockquote>
<br />
<b><a href="http://users.isy.liu.se/rt/ohlsson/MPCsparse.pdf" target="_blank">Sparse Control Using Sum-Of-Norms Regularized Model Predictive Control</a></b><br />
S. K. Pakazad, Linköping Univ.<br />
H. Ohlsson, Linköping Univ.<br />
L. Ljung, Linkoping Univ.<br />
<blockquote class="tr_bq">
<b>Abstract:</b> Some control applications require the use of piecewise constant or impulse-type control signals, with as few changes as possible. So as to achieve this type of control, we consider the use of regularized model predictive control (MPC), which allows us to impose this structure through the use of regularization. It is then possible to regulate the trade-off between control performance and control signal characteristics by tuning the so-called regularization parameter. However, since the mentioned trade-off is only indirectly affected by this parameter, its tuning is often unintuitive and time-consuming. In this paper, we propose an equivalent reformulation of the regularized MPC, which enables us to configure the desired trade-off in a more intuitive and computationally efficient manner. This reformulation is inspired by the so-called varepsilon-constraint formulation of multi-objective optimization problems and enables us to quantify the trade-off, by explicitly assigning bounds over the control performance.</blockquote>
<b><br /></b>
<b><a href="http://divf.eng.cam.ac.uk/cfes/pub/Main/Presentations/Vidyasagar.pdf" target="_blank">Near-Ideal Behavior of a Modified Elastic Net Algorithm</a></b><br />
M. Vidyasagar, The Univ. of Texas at Dallas<br />
<blockquote class="tr_bq">
<b>Abstract:</b> In this paper it is shown that a modification of the Elastic Net algorithm (MEN) exhibits near ideal behavior in the following sense: Suppose the input to the algorithm is a vector of known sparsity index but unknown locations for the nonzero components. Then the output error of the algorithm is bounded by a universal constant times the error achieved by an oracle that knows not just the sparsity index but also the locations of the nonzero components. This result generalizes an earlier result of Candes and Plan on the near ideal behavior of the LASSO algorithm.</blockquote>
<div>
<br /></div>
<b>Sparse Networked Control of Input Constrained Linear Systems</b><br />
H. Kong, The Univ. of Newcastle<br />
G. C. Goodwin, Univ. of Newcastle<br />
M. Seron, The Univ. of Newcastle<br />
<blockquote class="tr_bq">
<b>Abstract:</b> In networked control, there is often an incentive to communicate only what is absolutely necessary to achieve the desired performance goals. This is true of both the downlink (between a control base station and actuators) and the uplink (between the sensors and base station). Here we present a possible solution to this problem based on a singular value decomposition (SVD) of the Hessian of the quadratic performance index generally considered in Model Predictive Control (MPC). The singular vectors are employed to generate an orthonormal basis function expansion of the unconstrained solution to the finite horizon optimal control problem. These are pre-loaded into each actuator and each sensor. On the downlink, the actuators are informed in real-time about which basis functions they should use. On the uplink, after a “burn in period”, the sensors need only communicate when their response departs from that pre-calculated for the given basis functions. We show that this strategy facilitates sparse communication in both the downlink and uplink. We also show that the strategy can be modified so that input constraints are satisfied. The proposed results are illustrated by an example.</blockquote>
<b><br /></b>
<b>Soft-Constrained Lasso-MPC for Robust LTI Tracking: Enlarged Feasible Region and an ISS Gain Estimate</b><br />
M. Gallieri, Univ. of Cambridge<br />
J. M. Maciejowski, Univ. of Cambridge<br />
<blockquote class="tr_bq">
<b>Abstract:</b> This paper investigates the robustness of a soft constrained LTI MPC for set-point tracking, using an l1- regularised cost. The MPC using this type of cost (informally dubbed lasso-MPC) is suitable, for instance, for redundantly- actuated systems. This is because of its ability to select a set of preferred actuators, leaving the other ones at rest for most of the time. The proposed approach aims to recover from state constraint violation and to track a changing set-point. Nominal stability is guaranteed for all feasible references and robustness to additive uncertainties is formally characterised, under certain assumptions. In particular, sufficient conditions are given for the feasible region to be robustly invariant, this region being larger than the MPC for regulation. The closed- loop system is input-to-state stable (ISS), and a local ISS gain is computed. All results apply to stabilisable LTI systems. Results hold as well for the more common quadratic MPC, a special case of the proposed controller. </blockquote>
<br />
<b>Proximal Newton Methods for Convex Composite Optimization</b><br />
P. Patrinos, IMT Inst. for Advanced Studies Lucca<br />
A. Bemporad, IMT Inst. for Advanced Studies Lucca<br />
<blockquote class="tr_bq">
<b>Abstract:</b> This paper proposes two proximal Newton methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a new continuously differentiable exact penalty function, namely the emph{Composite Moreau Envelope}. The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the solution of a linear system of usually small dimension.</blockquote>
<br />
Enjoy! </div>
</div>
Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-31802769972169415522013-12-13T04:17:00.001+09:002013-12-15T13:52:16.246+09:00Sparse Packetized Predictive Control<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-lVQ8CPyRJOc/UqoJbNnzKEI/AAAAAAAAAuk/gisGUIFfijo/s1600/sparsity.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="132" src="http://1.bp.blogspot.com/-lVQ8CPyRJOc/UqoJbNnzKEI/AAAAAAAAAuk/gisGUIFfijo/s400/sparsity.png" width="400" /></a></div>
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<br /></div>
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As in signal processing, sparsity is also a useful concept in control systems. </div>
<div class="p1">
<br /></div>
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Our paper on a sparsity-based control method has been accepted to</div>
<div class="p1">
<i>IEEE Transactions on Automatic Control</i>, entitled</div>
<div class="p1">
"<b>Sparse Packetized Predictive Control for Networked Control over Erasure Channels</b>"</div>
<div class="p1">
<br /></div>
<div class="p1">
The paper can be downloaded from <a href="http://arxiv.org/abs/1307.8242" target="_blank">arxiv</a>.</div>
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<br /></div>
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The motivation of this study is <i>sparse representation</i> of control data.</div>
<div class="p1">
In networked control systems, the data packets are transmitted through communication channels whose bit-rates are limited, and the transmitted data should be represented in fewer bits.</div>
<div class="p1">
Sparsity then gives an attractive solution to this problem.</div>
<div class="p2">
<br /></div>
<div class="p1">
To reduce the data size of packets, we have proposed to adopt sparsity-promoting optimizations, namely, </div>
<div class="p2">
</div>
<ol>
<li><b>L1-L2 optimization</b></li>
<li><b>L2-constrained L0 optimization</b></li>
</ol>
<br />
<div class="p1">
for which efficient algorithms exist (e.g., see <a href="http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/170851/1/ieice-e96-b3_pp685.pdf" target="_blank">this article</a>).</div>
<div class="p1">
<br /></div>
<div class="p1">
We have shown how to design the tuning parameters to ensure (practical) stability of the resulting feedback control systems when the number of consecutive packet-dropouts is bounded.</div>
<div class="p1">
<br /></div>
<div class="p1">
Enjoy!</div>
Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-68088438183460507012013-04-07T13:17:00.000+09:002014-08-09T23:42:07.537+09:00Control Theoretic Spline: A bridge between control and signal processing<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-iFL9vDzFekM/UWDN82UTMwI/AAAAAAAAAgo/bnbVT0k2nIs/s1600/cts.001.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-iFL9vDzFekM/UWDN82UTMwI/AAAAAAAAAgo/bnbVT0k2nIs/s320/cts.001.png" height="208" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
The<i> <a href="http://en.wikipedia.org/wiki/Spline_(mathematics)" target="_blank">spline</a></i> has been widely used in signal processing, numerical computation, statistics, etc. In particular, the <a href="http://en.wikipedia.org/wiki/Smoothing_spline" target="_blank"><i>smoothing spline</i></a> gives a smooth curve that has the best fit to given noisy data with the following optimization:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-CVnTSwANuy0/UVewe_043jI/AAAAAAAAAbY/DPEEsLZqHeg/s1600/spline_regularization.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-CVnTSwANuy0/UVewe_043jI/AAAAAAAAAbY/DPEEsLZqHeg/s320/spline_regularization.png" height="47" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br />
where (<i>t</i><sub>1</sub>, <i>y</i><sub>1</sub>),..., (<i>t</i><sub><i>n</i></sub>,<i>y</i><sub><i>n</i></sub>) are given data, <i>y</i>(<i>t</i>) is the curve to be estimated, and <i>r </i>is the regularization parameter that specifies the trade-off between the fidelity to the data (1st term) and the smoothness of the curve (2nd term). The optimal curve is given by a linear combination of shifted cubic splines (see <a href="http://epubs.siam.org/doi/book/10.1137/1.9781611970128" target="_blank">G. Wahba's book</a> for details).<br />
<br />
The smoothing spline can be rewritten in terms of control theory. We assume <i>y</i>(<i>t</i>) is the <i>output</i> of the double integrator for some input <i>u</i>(<i>t</i>), namely,<br />
<br />
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<a href="http://1.bp.blogspot.com/-I5Hc2pSV2mU/UVewliK_TBI/AAAAAAAAAbg/R2jF3DRR1ik/s1600/di.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-I5Hc2pSV2mU/UVewliK_TBI/AAAAAAAAAbg/R2jF3DRR1ik/s1600/di.png" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
where "<i>s</i>" is the differential operator (or the symbol of the <a href="http://en.wikipedia.org/wiki/Laplace_transform" target="_blank">Laplace transform</a>). Then we rewrite the optimization problem as<br />
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<a href="http://2.bp.blogspot.com/-HRv1my4IAhg/UVfXFqoaQmI/AAAAAAAAAcA/howQ_pxl8kw/s1600/ct_regularization.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-HRv1my4IAhg/UVfXFqoaQmI/AAAAAAAAAcA/howQ_pxl8kw/s320/ct_regularization.png" height="93" width="320" /></a></div>
<br />
Then, a control theoretic spline is defined by replacing 1/<i>s</i><sup>2</sup> with a <i>general transfer function</i> <i>P</i>(<i>s</i>), that is<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-AK8rnmwH_qQ/UWDIg5XJirI/AAAAAAAAAgE/tUsF4z_pLTI/s1600/cts_optimization.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-AK8rnmwH_qQ/UWDIg5XJirI/AAAAAAAAAgE/tUsF4z_pLTI/s320/cts_optimization.png" height="77" width="320" /></a></div>
<br />
<br />
For example, you can obtain a <i>general</i> smoothing spline that minimizes<br />
<br />
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<a href="http://2.bp.blogspot.com/-cZcRPL-XtUA/UWDJYoZQx2I/AAAAAAAAAgM/pFS5tXn20h8/s1600/cts_example.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-cZcRPL-XtUA/UWDJYoZQx2I/AAAAAAAAAgM/pFS5tXn20h8/s320/cts_example.png" height="41" width="320" /></a></div>
<br />
by solving the control theoretic spline optimization described above with<br />
<br />
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<a href="http://1.bp.blogspot.com/-o3lillVvURE/UWDJyyRxwrI/AAAAAAAAAgU/1xIcEAViD2E/s1600/cts_example_Ps.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-o3lillVvURE/UWDJyyRxwrI/AAAAAAAAAgU/1xIcEAViD2E/s1600/cts_example_Ps.png" /></a></div>
<br />
An important fact is that the optimal solution (or optimal control), <i>u</i>(<i>t</i>), is given by a linear combination of <i>exponential</i> functions (including polynomial functions) that are specified by the impulse response (i.e., the inverse Laplace transform) of <i>P</i>(<i>s</i>).<br />
<br />
A control theoretic spline can be explained as drawing a curve with a robot hand modeled by <i>P</i>(<i>s</i>); see the top picture. This illustrates that the control theoretic spline is a <i>bridge</i> between control and signal processing.<br />
<br />
The original paper on control theoretic spline can be downloaded from <a href="http://users.ece.gatech.edu/~magnus/Papers/ControlSplines.pdf" target="_blank">here</a>.<br />
Control theoretic spline with a monotonicity constraint is discussed in <a href="http://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/173110" target="_blank">this paper</a>.<br />
Compressive sampling approach to control theoretic spline is proposed in <a href="http://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/171259" target="_blank">this paper</a>.<br />
<br />
See also <a href="http://sparseland.blogspot.jp/2014/07/l1-control-theoretic-smoothing-splines.html" target="_blank">this blog entry</a>.<br />
<br />
<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-48952945945965625012013-03-02T20:21:00.002+09:002013-03-07T12:00:19.019+09:00A User's Guide to Compressed Sensing for Communications Systems<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-a4IUM2y20Qc/UTHbFVD2ZGI/AAAAAAAAAaM/AZhNQ7qsKiU/s1600/tin-can-phone.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="219" src="http://3.bp.blogspot.com/-a4IUM2y20Qc/UTHbFVD2ZGI/AAAAAAAAAaM/AZhNQ7qsKiU/s320/tin-can-phone.png" width="320" /></a></div>
A survey paper has been recently published, entitled <i>A User's Guide to Compressed Sensing for Communications Systems, </i>which I co-authored with <a href="http://www.msys.sys.i.kyoto-u.ac.jp/~kazunori/kazunori_e.html" target="_blank">Kazunori</a> and <a href="http://www-adsys.sys.i.kyoto-u.ac.jp/tt/Tanaka-e.html" target="_blank">Toshiyuki</a>.<br />
<br />
PDF is available <a href="http://search.ieice.org/bin/pdf_link.php?category=B&lang=E&year=2013&fname=e96-b_3_685&abst=" target="_blank">here</a>, and <a href="http://www.scilab.org/" target="_blank">SCILAB</a> codes <a href="http://www.msys.sys.i.kyoto-u.ac.jp/csdemo.html" target="_blank">here</a>.<br />
<br />
Compressed sensing (CS) becomes more and more popular in communications engineering. In this paper, you can find many applications of CS to communications systems. The summary reads<br />
<blockquote class="tr_bq">
<b>SUMMARY: </b>This survey provides a brief introduction to compressed sensing as well as several major algorithms to solve it and its various applications to communications systems. We firstly review linear simultaneous equations as ill-posed inverse problems, since the idea of compressed sensing could be best understood in the context of the linear equations. Then, we consider the problem of compressed sensing as an underdetermined linear system with a prior information that the true solution is sparse, and explain the sparse signal recovery based on ell-1 optimization, which plays the central role in compressed sensing, with some intuitive explanations on the optimization problem. Moreover, we introduce some important properties of the sensing matrix in order to establish the guarantee of the exact recovery of sparse signals from the underdetermined system. After summarizing several major algorithms to obtain a sparse solution focusing on the ell-1 optimization and the greedy approaches, we introduce applications of compressed sensing to communications systems, such as wireless channel estimation, wireless sensor network, network tomography, cognitive radio, array signal processing, multiple access scheme, and networked control.</blockquote>
Enjoy CS with this guide!<br />
<br />
<div class="p1">
<b>Related entries:</b></div>
<div class="p2">
<div class="p1">
<a href="http://sparseland.blogspot.jp/2012/08/compressed-sensing-for-communication.html" target="_blank">Compressed sensing for communication systems</a></div>
<div class="p1">
<a href="http://sparseland.blogspot.jp/2012/12/compressed-sensing-new-trend-in-control.html" target="_blank">Compressed sensing: a new trend in control</a></div>
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<span class="s1"><a href="http://sparseland.blogspot.jp/2012/05/recent-papers-on-compressed-sensing-for.html">Recent papers on compressed sensing for control systems</a></span></div>
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</div>
<span class="s1"><a href="http://sparseland.blogspot.jp/2012/05/compressed-sensing-meets-control.html">Compressed sensing meets control systems</a></span></div>
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<br />
<div class="p1">
<b>Edit:</b><br />
Igor has featured our paper on <a href="http://nuit-blanche.blogspot.jp/2013/03/a-users-guide-to-compressed-sensing-for.html" target="_blank">his blog entry</a>. Thank you Igor! (07/Mar/2013)<br />
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Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com2tag:blogger.com,1999:blog-6068554977761810119.post-69632006664638043442012-12-14T06:40:00.000+09:002012-12-14T09:56:16.866+09:00Compressed Sensing: A New Trend in Control<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-nSdb81bs-XE/UMohAd-4yGI/AAAAAAAAAVM/BP17d3eRaKc/s1600/banner4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="85" src="http://2.bp.blogspot.com/-nSdb81bs-XE/UMohAd-4yGI/AAAAAAAAAVM/BP17d3eRaKc/s400/banner4.jpg" width="400" /></a></div>
<br />
Aloha!<br />
<br />
I am now in Maui to attend <a href="http://control.disp.uniroma2.it/cdc2012/" target="_blank">IEEE CDC 2012</a>. IEEE CDC (Conference on Decision and Control) is the most important conference of control, and presentations in CDC show recent trends in control.<br />
<br />
I have found researches in this CDC that pertain to <i>sparsity</i> or <i>compressed sensing (CS). </i>The number of such researches is increasing in control.<br />
<br />
Today I introduce researches related to sparsity or CS found in the final program, including one by myself.<br />
<br />
<br />
<b><a href="http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/PS_File/cdc2012_ppc.pdf" target="_blank">Packetized Predictive Control for Rate-Limited Networks via Sparse Representation</a></b><br />
<div class="p1">
Nagahara, Masaaki, Kyoto Univ.</div>
<div class="p1">
Quevedo, Daniel E., <span class="Apple-tab-span"> </span>The Univ. of Newcastle</div>
<div class="p1">
Ostergaard, Jan, <span class="Apple-tab-span"> </span>Aalborg Univ.</div>
<blockquote class="tr_bq">
<b>Abstract:</b> We study a networked control architecture for linear time-invariant plants in which an unreliable data-rate limited network is placed between the controller and the plant input. The distinguishing aspect of the situation at hand is that an unreliable data-rate limited network is placed between controller and the plant input. To achieve robustness with respect to dropouts, the controller transmits data packets containing plant input predictions, which minimize a finite horizon cost function. In our formulation, we design sparse packets for rate-limited networks, by adopting an L0 optimization, which can be effectively solved by an orthogonal matching pursuit method. Our formulation ensures asymptotic stability of the control loop in the presence of bounded packet dropouts. Simulation results indicate that the proposed controller provides sparse control packets, thereby giving bit-rate reductions for the case of memoryless scalar coding schemes when compared to the use of, more common, quadratic cost functions, as in linear quadratic (LQ) control.</blockquote>
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<b>Sparse Multiple Kernels for Impulse Response Estimation with Majorization Minimization Algorithms</b></div>
<div class="p1">
Chen, Tianshi, <span class="Apple-tab-span"> </span>Linköping Univ. Sweden</div>
<div class="p1">
Ljung, Lennart, <span class="Apple-tab-span"> </span>Linkoping Univ.</div>
<div class="p1">
Andersen, Martin, <span class="Apple-tab-span"> </span>Linköping Univ.</div>
<div class="p1">
Chiuso, Alessandro, <span class="Apple-tab-span"> </span>Univ. di Padova</div>
<div class="p1">
Carli, Francesca, P, <span class="Apple-tab-span"> </span>Univ. of Padova</div>
<div class="p1">
Pillonetto, Gianluigi, <span class="Apple-tab-span"> </span>Univ. of Padova</div>
<blockquote class="tr_bq">
<b>Abstract:</b> This contribution aims to enrich the recently introduced kernel-based Bayesian approach for linear system identification. Instead of a single kernel, we use multiple kernels, which can be instances of any existing kernels for the impulse response estimation of linear systems. We also introduce a new class of kernels constructed based on output error (OE) model estimates. In this way, a more flexible and richer representation of the kernel is obtained. Due to this representation the associated hyper-parameter estimation problem has two good features. First, it is a difference of convex functions programming (DCP) problem. While it is still nonconvex, it can be transformed into a sequence of convex optimization problems with majorization minimization (MM) algorithms and a local minima can thus be found iteratively. Second, it leads to sparse hyper-parameters and thus sparse multiple kernels. This feature shows the kernel-based approach with multiple kernels has the potential to tackle various problems of finding sparse solutions in linear system identification. As an illustration the sparse dynamic network identification problem is studied.</blockquote>
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<div class="p1">
<b>Topology Identification of a Sparse Dynamic Network</b></div>
<div class="p1">
Seneviratne, Akila, <span class="Apple-tab-span"> </span>The Univ. of New South Wales</div>
<div class="p1">
Solo, Victor, <span class="Apple-tab-span"> </span>Univ. of New South Wales</div>
<blockquote class="tr_bq">
<b>Abstract:</b> This paper addresses the problem of identifying the topology of a sparsely connected network of dynamic systems. The goal is to identify the links, the direction of information flow and the transfer function of each dynamic system. The output of each system is affected by the incoming data of the directly connected systems and noise. In contrast to the related existing work we use causal Laguerre basis functions to expand the transfer functions. Since the network is sparsely connected we estimate the system topology using an algorithm which optimizes the l0 penalized least squares criterion with grouped variables. This also contrasts with previous work which usually uses and l1 penalty. The l0 penalty has the potential to generate greater sparsity. We present simulation results to demonstrate the effectiveness of this method.</blockquote>
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<br /></div>
<div class="p1">
<b>Data-Driven Graph Reconstruction Using Compressive Sensing</b></div>
<div class="p1">
Chang, Young Hwan<span class="Apple-tab-span"> </span>Univ. of California, Berkeley</div>
<div class="p1">
Tomlin, Claire J.<span class="Apple-tab-span"> </span>UC Berkeley</div>
<blockquote class="tr_bq">
<b>Abstract:</b> Modeling of biological signal pathways forms the basis of systems biology. Also, network models have been important representations of biological signal pathways. In many biological signal pathways, the underlying networks over which the propagations spread are unobserved so inferring network structures from observed data is an important procedure to study the biological systems. In this paper, we focus on protein regulatory networks which are sparse and where the time series measurements of protein dynamics are available. We propose a method based on compressive sensing (CS) for reconstructing a sparse network structure based on limited time-series gene expression data without any a priori information. We present a set of numerical examples to demonstrate the method. We discuss issues of coherence in the data set, and we demonstrate that incoherence in the sensing matrix can be used as a performance metric and a guideline for designing effective experiments.</blockquote>
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<b><br /></b></div>
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<b>Stability Analysis of Non-Vector Space Control Via Compressive Feedbacks</b></div>
<div class="p1">
Zhao, Jianguo, <span class="Apple-tab-span"> </span>Michigan State Univ.</div>
<div class="p1">
Xi, Ning, <span class="Apple-tab-span"> </span>Michigan State Univ.</div>
<div class="p1">
Sun, Liang, <span class="Apple-tab-span"> </span>Michigan State Univ.</div>
<div class="p1">
Song, Bo, <span class="Apple-tab-span"> </span>Michigan State Univ.</div>
<blockquote class="tr_bq">
<b>Abstract:</b> A new non-vector space control method with compressive feedback is presented in this paper. Two problems are discussed. The first is the development of the non-vector space control method. It formulates the control problem in the space of sets instead of the vector space, and it provides a convenient way to model physical phenomenon such as shapes and images. For the non-vector space dynamical systems, the issues related to stability analysis and design of controllers are discussed in the paper. The second problem is to design controllers for non-vector space dynamic systems using compressive feedback, which is a new concept proposed in this paper. Compressive feedback, obtained from the original feedback, reduces the amount of data for the original feedback but retains the essential information to ensure the stability and other key performance measures for a control system. The design methods for controllers based on compressive feedback are developed in this paper. Furthermore, the non-vector space control system with compressive feedback has been successfully applied to visual servoing, and simulations results have clearly demonstrated the superb performance and advantages.</blockquote>
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<div class="p1">
<b><a href="http://www.ist.uni-stuttgart.de/~zelazo/papers/CDC12_1464_FI.pdf" target="_blank">Design of Sparse Relative Sensing Networks</a></b></div>
<div class="p1">
Schuler, Simone, <span class="Apple-tab-span"> </span>Univ. of Stuttgart</div>
<div class="p1">
Zelazo, Daniel, <span class="Apple-tab-span"> </span>Univ. Stuttgart</div>
<div class="p1">
Allgower, Frank, <span class="Apple-tab-span"> </span>Univ. of Stuttgart</div>
<blockquote class="tr_bq">
<b>Abstract:</b> This paper considers the problem of designing sparse relative sensing networks (RSN) subject to a given Hinf-performance constraint. The topology design considers homogeneous and heterogeneous agents over weighted graphs. We develop a computationally efficient formulation of the sparse topology design via a convex l1-relaxation. This makes the proposed algorithm attractive for practical applications. We also demonstrate how this relaxation can be used to embed additional performance criteria, such as maximization of the algebraic connectivity of the RSN.</blockquote>
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<div class="p1">
<b>Sparse Coloured System Identification with Guaranteed Stability</b></div>
<div class="p1">
Seneviratne, Akila, <span class="Apple-tab-span"> </span>The Univ. of New South Wales</div>
<div class="p1">
Solo, Victor, <span class="Apple-tab-span"> </span>Univ. of New South Wales</div>
<blockquote class="tr_bq">
<b>Abstract:</b> We develop a method for sparse transfer function estimation in the presence of coloured noise. Unlike most previous sparse transfer function estimation methods our approach via Laguerre polynomials guarantees stability. Also unlike previous methods the new procedure can handle coloured noise. We discuss both l1 and l0 penalized procedures. Simulation results are presented to demonstrate the effectiveness of this approach and to compare the performance of the two penalties that are considered.</blockquote>
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<div class="p1">
<b><a href="http://www.bg.ic.ac.uk/research/g.stan/CDC_2012b.pdf" target="_blank">Reconstruction of Arbitrary Biochemical Reaction Networks: A Compressive Sensing Approach</a></b></div>
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Pan, Wei, <span class="Apple-tab-span"> </span>Imperial Coll. London</div>
<div class="p1">
Yuan, Ye, <span class="Apple-tab-span"> </span>Univ. of Cambridge</div>
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Goncalves, Jorge, M.<span class="Apple-tab-span"> </span>Univ. of Cambridge</div>
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STAN, Guy-Bart Vincent, <span class="Apple-tab-span"> </span>Imperial Coll. London</div>
<blockquote class="tr_bq">
<b>Abstract:</b> Reconstruction of biochemical reaction networks (BRN) and genetic regulatory networks (GRN) in particular is a central topic in systems biology which raises crucial theoretical challenges in system identification. Nonlinear Ordinary Differential Equations (ODEs) that involve polynomial and rational functions are typically used to model biochemical reaction networks. Such nonlinear models make the problem of determining the connectivity of biochemical networks from time-series experimental data quite difficult. In this paper, we present a network reconstruction algorithm that can deal with ODE model descriptions containing polynomial and rational functions. Rather than identifying the parameters of linear or nonlinear ODEs characterised by predefined equation structures, our methodology allows us to determine the nonlinear ODEs structure together with their associated parameters. To solve the network reconstruction problem, we cast it as a compressive sensing (CS) problem and use sparse Bayesian learning (SBL) algorithms as a computationally efficient and robust way to obtain its solution.</blockquote>
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<div class="p1">
<b>Comparison of the Sparse-Grid Quadrature Rule and the Cubature Rule in the Nonlinear Filtering</b></div>
<div class="p1">
Jia, Bin, <span class="Apple-tab-span"> </span>Mississippi State Univ.</div>
<div class="p1">
Xin, Ming, <span class="Apple-tab-span"> </span>Mississippi State Univ.</div>
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Cheng, Yang, <span class="Apple-tab-span"> </span>Mississippi State Univ.</div>
<blockquote class="tr_bq">
<b>Abstract:</b> In this paper, the recently developed sparse-grid quadrature filter is compared with the cubature Kalman filter. The relation between the sparse-grid quadrature rule and the cubature rule is revealed. It can be shown that arbitrary degree cubature rules can be obtained by the projection of the sparse-grid quadrature rule. Since both rules can achieve an arbitrary high degree of accuracy, they are more accurate than the conventional third-degree cubature rule and the unscented transformation. In addition, they are computationally more efficient than the Gauss-Hermite quadrature rule and the Monte-Carlo method when they are used to calculate the Gaussian type integrals in the nonlinear filtering. The comparison of these rules is demonstrated by a benchmark numerical integration example.</blockquote>
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<br /></div>
<div class="p1">
<b><a href="http://inside.mines.edu/~bmolazem/my_papers/CDC2012_Tutorial.pdf" target="_blank">A Tutorial on Recovery Conditions for Compressive System Identification of Sparse Channels</a></b></div>
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Sanandaji, Borhan M., <span class="Apple-tab-span"> </span>Univ. of California at Berkeley</div>
<div class="p1">
Vincent, Tyrone L., <span class="Apple-tab-span"> </span>Colorado School of Mines</div>
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Poolla, Kameshwar, <span class="Apple-tab-span"> </span>Univ. of California at Berkeley</div>
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Wakin, Michael, <span class="Apple-tab-span"> </span>Colorado School of Mines</div>
<blockquote class="tr_bq">
<b>Abstract:</b> In this tutorial, we review some of the recent results concerning Compressive System Identification (CSI) (identification from few measurements) of sparse channels (and in general, Finite Impulse Response (FIR) systems) when it is known a priori that the impulse response of the system under study is sparse (high-dimensional but with few non-zero entries) in an appropriate basis. For the systems under study in this tutorial, the system identification problem boils down to an inverse problem of the form Ax = b, where the vector x is high-dimensional but with few non-zero entries and the matrix A is underdetermined. Over the past few years, several algorithms with corresponding recovery conditions have been proposed to perform such a recovery. These conditions provide the number of measurements sufficient for correct recovery. In this note, we review alternate approaches to derive such recovery conditions concerning CSI of FIR systems whose impulse response is known to be sparse.</blockquote>
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<div class="p1">
<b><a href="http://arxiv.org/pdf/1204.0590v1.pdf" target="_blank">Linear System Identification Via Atomic Norm Regularization</a></b></div>
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Shah, Parikshit, <span class="Apple-tab-span"> </span>Massachusetts Inst. of Tech.</div>
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Bhaskar, Badri, Narayan<span class="Apple-tab-span"> </span>Univ. of Wisconsin - Madison</div>
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Tang, Gongguo, <span class="Apple-tab-span"> </span>Univ. of Wisconsin-Madison</div>
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Recht, Benjamin, <span class="Apple-tab-span"> </span>Univ. of Wisconsin-Madison</div>
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<b>Abstract:</b> This paper proposes a new algorithm for linear system identification from noisy measurements. The proposed algorithm balances a data fidelity term with a norm induced by the set of single pole filters. We pose a convex optimization problem that approximately solves the atomic norm minimization problem and identifies the unknown system from noisy linear measurements. This problem can be solved efficiently with standard, free software. We provide rigorous statistical guarantees that explicitly bound the estimation error (in the H_2-norm) in terms of the stability radius, the Hankel singular values of the true system and the number of measurements. These results in turn yield complexity bounds and asymptotic consistency. We provide numerical experiments demonstrating the efficacy of our method for estimating linear systems from a variety of linear measurements.</blockquote>
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<br /></div>
<div class="p1">
<b><a href="http://www.rolandtoth.us/Roland_Toths_webpage/Publications_files/CDC2012b.pdf" target="_blank">Order and Structural Dependence Selection of LPV-ARX Models Revisited</a></b></div>
<div class="p1">
Tóth, Roland, <span class="Apple-tab-span"> </span>Eindhoven Univ. of Tech.</div>
<div class="p1">
Hjalmarsson, Håkan, <span class="Apple-tab-span"> </span>Royal Inst. of Tech.</div>
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Rojas, Cristian R., <span class="Apple-tab-span"> </span>KTH Royal Inst. of Tech.</div>
<blockquote class="tr_bq">
<b>Abstract:</b> Accurate parametric identification of Linear Parameter-Varying (LPV) systems requires an optimal prior selection of model order and a set of functional dependencies for the parameterization of the model coefficients. In order to address this problem for linear regression models, a regressor shrinkage method, the Non-Negative Garrote (NNG) approach, has been proposed recently. This approach achieves statistically efficient order and structural coefficient dependence selection using only measured data of the system. However, particular drawbacks of the NNG are that it is not applicable for large-scale over-parameterized problems due to computational limitations and that adequate performance of the estimator requires a relatively large data set compared to the size of the parameterization used in the model. To overcome these limitations, a recently introduced L_1 sparse estimator approach, the so-called SPARSEVA method, is extended to the LPV case and its performance is compared to the NNG.</blockquote>
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<br /></div>
<div class="p1">
<b><a href="http://www.rolandtoth.us/Roland_Toths_webpage/Publications_files/CDC2012a.pdf" target="_blank">Joint Order and Dependency Reduction for LPV State-Space Models</a></b></div>
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Siraj, Muhammad Mohsin, <span class="Apple-tab-span"> </span>Eindhoven Univ. of Tech.</div>
<div class="p1">
Tóth, Roland, <span class="Apple-tab-span"> </span>Eindhoven Univ. of Tech.</div>
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Weiland, Siep, <span class="Apple-tab-span"> </span>Eindhoven Univ. of Tech.</div>
<blockquote class="tr_bq">
<b>Abstract:</b> In the reduction of Linear Parameter-Varying (LPV) models, decreasing model complexity is not only limited to model order reduction (state reduction), but also to the simplification of the dependency of the model on the scheduling variable. This is due to the fact that the concept of minimality is strongly connected to both types of complexities. While order reduction of LPV models has been deeply studied in the literature resulting in the extension of various reduction approaches of the LTI system theory, reduction of the scheduling dependency still remains to be a largely open problem. In this paper, a model reduction method for LPV state-space models is proposed which achieves both state-order and scheduling dependency reduction. The introduced approach is based on an LPV Ho-Kalman algorithm via imposing a sparsity expectation on the extended Hankel matrix of the model to be reduced. This sparsity is realized by an L1-norm based minimization of a priori selected set of dependencies associated sub-Markov parameters. The performance of the proposed method is demonstrated via a representative simulation example.</blockquote>
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<b>Related entries:</b></div>
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<span class="s1"><a href="http://sparseland.blogspot.jp/2012/05/compressed-sensing-meets-control.html">Compressed sensing meets control systems</a></span></div>
<div class="p2">
<span class="s1"><a href="http://sparseland.blogspot.jp/2012/05/recent-papers-on-compressed-sensing-for.html">Recent papers on compressed sensing for control systems</a></span></div>
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<a href="http://sparseland.blogspot.jp/2012/08/compressed-sensing-for-communication.html" target="_blank">Compressed sensing for communication systems</a></div>
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Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-60778537940712227312012-08-25T00:24:00.002+09:002014-08-17T07:49:18.858+09:00Compressed Sensing for Communication Systems<br />
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<a href="http://3.bp.blogspot.com/-poIB5vw6dIk/UDeYB2_nl5I/AAAAAAAAAUo/rn3Xy2Z0AwA/s1600/Wifi.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-poIB5vw6dIk/UDeYB2_nl5I/AAAAAAAAAUo/rn3Xy2Z0AwA/s320/Wifi.svg.png" height="320" width="226" /></a></div>
Compressed sensing becomes popular in the field of communications.<br />
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This week, there was a small and nice conference, <a href="http://apwcs2012.cce.i.kyoto-u.ac.jp/" target="_blank">APWCS</a> (Asia Pacific Wireless Communications Symposium), in Kyoto University, JAPAN. Researchers and engineers of wireless communications discussed on compressed sensing for communication systems at a special session called <i>Compressed Sensing for Communication Systems</i>. I felt compressed sensing is a powerful tool for a number of problems in communications.<br />
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The following is the summary of the session:<br />
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<b><span style="color: #4c1130; font-size: large;">S4: Special Session "Compressed Sensing for Communication Systems"</span></b><br />
Thursday, August 23, 11:10 - 12:50, Room: Conference Room III<br />
Chair: Kazunori Hayashi (Kyoto University, Japan)<br />
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<b><span style="color: #20124d;">Session Summary:</span></b> Compressed sensing has drawing much attention in various fields of applications. This is not only because it enables us to obtain an exact solution from an underdetermined linear system under a certain condition on the measurement matrix taking advantage of the sparsity of the solution, but also because the solution can be obtained via computationally efficient algorithms. In this special session, we focus on the communication applications of compressed sensing, such as OFDM systems, network tomography and networked control systems, as well as a brief introduction to compressed sensing. Moreover, we also have a couple of talks on the problems with underdetermined systems, where we discuss a possibility to apply compressed sensing to the problems.<br />
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<span style="color: #20124d;"><b>S4.1</b> <b>A Brief Introduction to Compressed Sensing</b></span><br />
Kazunori Hayashi (Kyoto University, Japan); Masaaki Nagahara (Kyoto University, Japan)<br />
<b><br /></b>
<span style="color: #20124d;"><b>S4.2 Application of Compressed Sensing to Inter-Symbol Interference Cancellation in OFDM </b><b>Systems</b></span><br />
Masato Saito (University of the Ryukyus, Japan)<br />
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<b><span style="color: #20124d;">S4.3 Path Construction for Compressed Sensing-Based Network Tomography</span></b><br />
Kazushi Takemoto (Osaka University, Japan); Takahiro Matsuda (Osaka University, Japan); Tetsuya Takine (Osaka University, Japan)<br />
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<b><span style="color: #20124d;">S4.4 Sparsity-Promoting Methods in Remote Control Systems</span></b><br />
Masaaki Nagahara (Kyoto University, Japan)<br />
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<span style="color: #20124d;"><b>S4.5 A Successive Detector with Virtual Channel Detection for Cooperative Multiuser MIMO </b><b>Systems</b></span><br />
Akihito Taya (Kyoto University, Japan); Satoshi Denno (Okayama University, Japan); Koji Yamamoto (Kyoto University, Japan); Masahiro Morikura (Kyoto University, Japan); Daisuke Umehara (Kyoto Institute of Technology, Japan); Hidekazu Murata (Kyoto University, Japan); Susumu Yoshida (Graduate School of Informatics, Kyoto University, Japan)<br />
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<b><span style="color: #20124d;">S4.6 Nested Array and Its Applications</span></b><br />
Masashi Tsuji (Tokyo University of Agriculture and Technology, Japan); Takashi Morimoto (Tokyo University of Agriculture and Technology, Japan); Kenta Umebayashi (Tokyo University of Agriculture and Technology, Japan); Yasuo Suzuki (Tokyo University of Agriculture and Technology, Japan)<br />
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<br />
<b>Related entries:</b><br />
<a href="http://sparseland.blogspot.jp/2012/05/compressed-sensing-meets-control.html" target="_blank">Compressed sensing meets control systems</a><br />
<a href="http://sparseland.blogspot.jp/2012/05/recent-papers-on-compressed-sensing-for.html" target="_blank">Recent papers on compressed sensing for control systems</a><br />
<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-32599511523387217252012-08-18T08:02:00.001+09:002012-08-25T14:56:24.431+09:00Optimal Design of Delta-Sigma Modulators<div class="separator" style="clear: both; text-align: center;">
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Control theory sometimes gives a powerful tool for solving problems in signal processing. <a href="http://en.wikipedia.org/wiki/Linear_matrix_inequality" target="_blank">LMI</a> (Linear Matrix Inequality) is one of them. Today, I'd like to introduce <a href="http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6156470" target="_blank">my paper</a> (<a href="http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/ds/manuscript.pdf" target="_blank">preprint PDF</a>) on LMI application to <a href="http://en.wikipedia.org/wiki/Delta-sigma_modulation" target="_blank">delta-sigma modulation</a>:<br />
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<blockquote class="tr_bq">
M. Nagahara and Y. Yamamoto, Frequency Domain Min-Max Optimization of Noise-Shaping Delta-Sigma Modulators, <i>IEEE Trans. on Signal Processing</i>, Vol. 60, No. 6, pp. 2828-2839, 2012.</blockquote>
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What is delta-sigma modulation? <a href="http://en.wikipedia.org/wiki/Delta-sigma_modulation" target="_blank">The Wikipedia entry</a> reads:<br />
<blockquote class="tr_bq">
<span style="font-family: inherit;"><span style="line-height: 19px;">Delta-sigma</span><span style="line-height: 19px;"> (</span><span style="line-height: 19px;">ΔΣ</span><span style="line-height: 19px;">; or </span><span style="line-height: 19px;">sigma-delta</span><span style="line-height: 19px;">, </span><span style="line-height: 19px;">ΣΔ</span><span style="line-height: 19px;">) modulation is a method for encoding </span>analog signals<span style="line-height: 19px;"> into </span>digital signals<span style="line-height: 19px;"> or higher-resolution digital signals into lower-resolution digital signals. The conversion is done using error feedback, where the difference between the two signals is measured and used to improve the conversion. The low-resolution signal typically changes more quickly than the high-resolution signal and it can be filtered to recover the high-resolution signal with little or no loss of fidelity. </span></span></blockquote>
To filter out the quantization noise from low-resolution quickly-changing signals, it is essential to push away the quantization noise out of the frequency band of the original signal. In other words, it should be sufficiently attenuated the noise transfer function (NTF) from the quantization noise to the modulator output (before the lowpass filter).<br />
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Although a delta-sigma modulator ia a highly nonlinear system, designing is a linear problem like filter design; shape the frequency response of NTF.<br />
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Conventionally, NTF is shaped such that the squared norm of the magnitude (i.e., H<sup>2</sup> norm) of NTF on the frequency band of interest. As I noted in <a href="http://sparseland.blogspot.jp/2012/05/h-2-versus-h.html" target="_blank">my past entry</a> that an H<sup>2</sup>-based system shows very nice performance for almost all frequencies but is fragile for a frequency, say f<sub>0 </sub>[Hz] (see the picture below).<br />
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<a href="http://2.bp.blogspot.com/-1XgQ43XCBpQ/T63iu0jrd2I/AAAAAAAAAO0/-2G-5Djwh68/s1600/H2vsHinfinity.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="176" src="http://2.bp.blogspot.com/-1XgQ43XCBpQ/T63iu0jrd2I/AAAAAAAAAO0/-2G-5Djwh68/s320/H2vsHinfinity.png" width="320" /></a></div>
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To avoid this, we introduced the H<sup>∞</sup> norm to uniformly attenuate the magnitude on a frequency band. A difficulty is that this problem is not a standard H<sup>∞</sup> problem since it does not optimize over the whole frequency band [0,π) but on a subset [0,Ω), Ω<π. You cannot use the standard <span style="font-family: Courier New, Courier, monospace;">hinfsyn</span><br />
command in MATLAB. How to solve it?<br />
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The solver is found in control theory, <a href="http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1381647&isnumber=30124" target="_blank">generalized KYP (GKYP) lemma</a>. This solves our problem via <a href="http://en.wikipedia.org/wiki/Linear_matrix_inequality" target="_blank">LMI</a> (linear matrix inequality).<br />
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The picture below shows two plots: the Bode magnitude plot of NTF designed by GKYP (red line) and by <a href="http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=242348&isnumber=6230" target="_blank">H<sup>2</sup>-based zero optimization</a> (blue line). The magnitude by GKYP is uniformly attenuated over the low frequency range, while that by H<sup>2</sup> shows peaks in this band. The difference between the two maximal magnitudes at the frequency ω = π/32 is approximately 11.2 (dB), and the difference at low frequencies is about 12.4 (dB).<br />
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<a href="http://4.bp.blogspot.com/-yAvl_sg5cFc/UAOKP-KcTvI/AAAAAAAAAT8/GdPXTR4Xf-0/s1600/delsig.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="275" src="http://4.bp.blogspot.com/-yAvl_sg5cFc/UAOKP-KcTvI/AAAAAAAAAT8/GdPXTR4Xf-0/s400/delsig.png" width="400" /></a></div>
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<br />
The paper is available <a href="http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/ds/manuscript.pdf" target="_blank">here</a>.<br />
MATLAB codes are available <a href="http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/fir/" target="_blank">here</a>.<br />
GKYP for signal processing is also discussed in <a href="http://www.intechopen.com/books/applications-of-digital-signal-processing/min-max-design-of-fir-digital-filters-by-semidefinite-programming" target="_blank">this article</a>.<br />
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A related blog entry is <a href="http://sparseland.blogspot.jp/2012/05/h-2-versus-h.html" target="_blank">here</a>.<br />
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Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com1tag:blogger.com,1999:blog-6068554977761810119.post-60817059174709242512012-07-15T20:16:00.002+09:002012-11-23T18:31:20.495+09:00Nice Restaurants around the University of Melbourne<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-ef9m1fy6-Ic/UAKI2Lv5X1I/AAAAAAAAATg/srXZWHH617M/s1600/800px-Lygon_st_melbourne_z.jpeg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="179" src="http://2.bp.blogspot.com/-ef9m1fy6-Ic/UAKI2Lv5X1I/AAAAAAAAATg/srXZWHH617M/s320/800px-Lygon_st_melbourne_z.jpeg" width="320" /></a></div>
Last week, I was in <a href="http://www.unimelb.edu.au/" target="_blank">the University of Melbourne</a>, Australia, to attend the <a href="http://mtns2012.com.au/" target="_blank">MTNS 2012</a> conference.<br />
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In the daytime, I discussed with some researchers, <span style="background-color: white;">listend presentations, </span><span style="background-color: white;">and also had my presentation. </span><span style="background-color: white;">In the evening, I ate a lot at nice restaurants around the Uni of Melbourne. Today's post is on restaurants I recommend to go for lunch or dinner. </span><br />
<br />
<ol>
<li><a href="http://www.ilgambero.com.au/" target="_blank">IL GAMBERO ON THE PARK</a><br />A quite nice Italian restaurant near the university. Around this restaurant, there are many Italian restaurants, and they call the place <i><a href="http://en.wikipedia.org/wiki/Little_Italy,_Melbourne" target="_blank">Little Italy</a></i>. My friend, Yeon Jeong, who lives in Melbourne, took me to this nice restaurant for lunch, and she recommended <i>marinara pasta</i>, which I had. The pasta is full of sea food: clams, mussels, calamari, and scallops with olive oil and garlic. I enjoyed the pasta very much. You should try it!</li>
<li><a href="http://thecarltonwineroom.com.au/" target="_blank">The Carlton Wine Room</a><br />My friend and also my research colleague, <a href="http://www.newcastle.edu.au/staff/research-profile/Daniel_Quevedo/" target="_blank">Daniel Quevedo</a>, invited me to have dinner at this restaurant. It was by chance that I got this restaurant; he planned to go dinner with a professor in Melbourne, who recommended this restaurant, but he was sick at the time. I was a quite lucky man (sorry, Professor). We ordered Chef's picks and grasses of wine which a waitress recommended. After we ordered, a group of professors who attended the conference appeared in the restaurant. What a coincidence! This proves that the restaurant is loved by "high level" foodies as professors (as you know, professors go abroad many times a year and they are often invited to five-star restaurants!). <br /><b>Note:</b> you should make a reservation before you go for dinner; this restaurant is full of customers every night.</li>
<li><a href="http://www.urbanspoon.com/r/71/760554/restaurant/Melbourne/South-Yarra-Toorak/Dainty-Sichuan-South-Yarra" target="_blank">Dainty Sichuan Food</a><br />Daniel's friend, Alex, invited us (Daniel, Masashi, and me) to this restaurant. This is a Chinese restaurant, located a bit far from the Uni of Melbourne (20 minutes from the uni by car). The restaurant is full of <i>Chinese</i> customers, this proves the restaurant is a quite nice Chinese restaurant. I like spicy food very much, and I enjoyed spicy (level 1 to 3) and very tasty food. I also recommend drink <a href="http://en.wikipedia.org/wiki/Tsingtao_Brewery" target="_blank">Tsingtao beer</a>, which goes with spicy Chinese food. We ate and drank a lot!</li>
</ol>
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Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com1tag:blogger.com,1999:blog-6068554977761810119.post-84908165947850279242012-07-10T08:22:00.003+09:002012-07-30T15:51:39.545+09:00Trinity College in Melbourne<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-2C89_cCyLsY/T_tb_UMgIeI/AAAAAAAAATU/FVwYMd8PLG0/s1600/2012-07-10+08.25.31.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="240" src="http://2.bp.blogspot.com/-2C89_cCyLsY/T_tb_UMgIeI/AAAAAAAAATU/FVwYMd8PLG0/s320/2012-07-10+08.25.31.jpg" width="320" /></a></div>
I am now in Melbourne, Australia, to attend <a href="http://mtns2012.com.au/" target="_blank">MTNS</a> (Mathematical Theory of Networks and Systems) 2012 conference. I stay in <a href="http://www.trinity.unimelb.edu.au/" target="_blank">Trinity College</a> in the University of Melbourne.<br />
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Trinity College is founded in 1872 as the first residential college of the University of Melbourne. It is very old but quite beautiful. The accommodation is also nice; old but comfortable, not so expensive, and safe. The internet access works very well.<br />
<br />
The University of Melbourne is founded in 1853, and <span style="background-color: white;">Melbourne School of Engineering in 1861, </span><span style="background-color: white;">the oldest engineering faculty in Australia (in 2011 it </span><span style="background-color: white;">celebrates 150 years of engineering education at the University of Melbourne.</span><br />
<span style="background-color: white;"><br /></span><br />
<span style="background-color: white;"><br /></span><br />
<span style="background-color: white;">See also my entry: <i><a href="http://sparseland.blogspot.jp/2012/07/nice-restaurants-around-university-of.html" target="_blank">Nice Restaurants around the University of Melbourne</a>.</i></span><br />
<span style="background-color: white;"><i><br /></i></span><br />
<span style="background-color: white;"><i><br /></i></span><br />
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<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com1Trinity College, University of Melbourne, Royal Parade, Parkville VIC 3052, Australia-37.794948 144.958611-37.807495499999995 144.93886999999998 -37.7824005 144.978352tag:blogger.com,1999:blog-6068554977761810119.post-16711570529104079912012-06-24T10:45:00.000+09:002012-06-24T12:55:08.282+09:00Boeing 787 Dreamliner<a href="https://lh4.googleusercontent.com/-kdyjVOrhtXs/T-TOQ-rImiI/AAAAAAAAAS8/HrvclxMot6w/s640/blogger-image-1338415770.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="150" src="https://lh4.googleusercontent.com/-kdyjVOrhtXs/T-TOQ-rImiI/AAAAAAAAAS8/HrvclxMot6w/s200/blogger-image-1338415770.jpg" width="200" /></a>I am now in Boston, USA, to attend the <a href="http://www.coe.neu.edu/IFACTDS2012/main.htm" target="_blank">IFAC TDS</a> (Time Delay Systems) conference. From Tokyo to Boston, I flied with the Boeing 787 Dreamliner, which is a new airplane, entered commercial service on October 26, 2011.<br />
<br />
<a href="http://en.wikipedia.org/wiki/Boeing_787_Dreamliner" target="_blank">The wikipedia entry</a> reads:<br />
<blockquote class="tr_bq">
The Boeing 787 Dreamliner is a long-range, mid-size <a href="http://en.wikipedia.org/wiki/Wide-body_aircraft" target="_blank">wide-body</a>, <a href="http://en.wikipedia.org/wiki/Twinjet" target="_blank">twin-engine</a> <a href="http://en.wikipedia.org/wiki/Jet_airliner" target="_blank">jet airliner</a> developed by <a href="http://en.wikipedia.org/wiki/Boeing_Commercial_Airplanes" target="_blank">Boeing Commercial Airplanes</a>. It seats 210 to 290 passengers, depending on the variant. Boeing states that it is the company's most <a href="http://en.wikipedia.org/wiki/Fuel_efficiency" target="_blank">fuel-efficient</a> airliner and the world's first major airliner to use <a href="http://en.wikipedia.org/wiki/Composite_material" target="_blank">composite materials</a> for most of its construction. According to Boeing, the 787 consumes 20% less fuel than the similarly-sized <a href="http://en.wikipedia.org/wiki/Boeing_767" target="_blank">767</a>. Its distinguishing features include a four-panel windshield, noise-reducing chevrons on its engine <a href="http://en.wikipedia.org/wiki/Nacelle" target="_blank">nacelles</a>, and a smoother nose contour. The 787 shares a common <a href="http://en.wikipedia.org/wiki/Type_rating" target="_blank">type rating</a> with the larger <a href="http://en.wikipedia.org/wiki/Boeing_777" target="_blank">777</a> twinjet, allowing qualified pilots to operate both models, due to related design features.</blockquote>
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<a href="http://1.bp.blogspot.com/-ZeQ2XGiVzps/T-Zrrs1B54I/AAAAAAAAATI/gKjtDhjROKU/s1600/window.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="133" src="http://1.bp.blogspot.com/-ZeQ2XGiVzps/T-Zrrs1B54I/AAAAAAAAATI/gKjtDhjROKU/s200/window.jpg" width="200" /></a></div>
An amazing thing is the cabin windows. For the windows, <span style="background-color: white;"><a href="http://en.wikipedia.org/wiki/Electrochromism" target="_blank">electrochromism</a>-based "auto-dimming" (smart glass) is used instead of window shades, which reduces cabin glare while maintaining transparency. One can adjust the level of transparency with the key just below the window. I frequently changed the level before I got bored and fixed to the lowest level.</span><br />
<span style="background-color: white;"><br /></span><br />
<span style="background-color: white;">If you have a chance to come to Japan from Boston (or to Boston from Japan), you can get the experience!</span><br />
<span style="background-color: white;"><br /></span>Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-29043135165183852962012-06-12T23:21:00.000+09:002014-07-26T08:56:07.533+09:00Robots vs Humans: 2050 World Cup<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-23yBQfU5Ow4/T9dGhszHXII/AAAAAAAAASg/7PubZli_xiY/s1600/800px-Manuel_Neuer,_Germany_national_football_team_(06).jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://4.bp.blogspot.com/-23yBQfU5Ow4/T9dGhszHXII/AAAAAAAAASg/7PubZli_xiY/s200/800px-Manuel_Neuer,_Germany_national_football_team_(06).jpg" height="133" width="200" /></a></div>
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<a href="http://www.uefa.com/uefaeuro/" target="_blank">Euro 2012</a> has been kicked off! Star players in Europe appear in this cup and show their amazing skills. Every night, I watch the games on TV to see how awesome humans can be! (FIFA World Cup qualifier games of Japan, also)<br />
<br />
Then, I remember the ultimate goal of the <a href="http://www.robocup.org/" target="_blank">RoboCup</a>, the "World Cup" of robots:<br />
<blockquote class="tr_bq">
By mid-21st century, a team of fully autonomous humanoid robot soccer players shall win the soccer game, comply with the official rule of the FIFA, against the winner of the most recent World Cup.</blockquote>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-G5GJcgz_GQA/T9dPkJXydLI/AAAAAAAAASs/j8ANzcJVS1Q/s1600/robocup.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-G5GJcgz_GQA/T9dPkJXydLI/AAAAAAAAASs/j8ANzcJVS1Q/s200/robocup.jpg" height="133" width="200" /></a></div>
<br />
It is fun to predict the winner of the 2050 World Cup; someone says it's US, I hope it's Japan. But, will robots beat the winner? Following are difficulties:<br />
<br />
<ol>
<li>Dribbling the ball fast enough without falling down</li>
<li>Understanding tactics (in real time)</li>
<li>Intercepting the ball without foul</li>
<li>Recognizing what the referee says</li>
<li>Gaining supporters</li>
</ol>
<br />
The last problem seems to be the most difficult.<br />
<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-6698025365210901462012-06-10T02:13:00.000+09:002012-06-10T02:26:21.325+09:00Tokyo Skytree: a modern five-story pagoda<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-4Om_fNEHSlk/T8r6ksChCMI/AAAAAAAAASI/TSihg7MW7Hs/s1600/skytree.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="http://2.bp.blogspot.com/-4Om_fNEHSlk/T8r6ksChCMI/AAAAAAAAASI/TSihg7MW7Hs/s200/skytree.JPG" width="133" /></a></div>
A very modern tower adopts a very old technology.<br />
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<a href="http://en.wikipedia.org/wiki/Tokyo_Skytree" target="_blank">Tokyo Skytree</a> opened to the public on 22 May 2012. This is the <a href="http://en.wikipedia.org/wiki/List_of_tallest_structures_in_Japan" target="_blank">tallest structure</a> (634m, 2,080ft) in Japan today. One might think whether the tower is OK, since more than a few earthquakes occur in Tokyo.<br />
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<a href="http://4.bp.blogspot.com/-lX9ftyDcVOg/T9OBXO02vDI/AAAAAAAAASU/QGoPNK0Frek/s1600/pagoda.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="http://4.bp.blogspot.com/-lX9ftyDcVOg/T9OBXO02vDI/AAAAAAAAASU/QGoPNK0Frek/s200/pagoda.jpg" width="133" /></a>The design of the tower is inspired by the structure of Japanese <a href="http://en.wikipedia.org/wiki/Pagoda" target="_blank">five-story pagodas</a>. Almost all Japanese five-story pagodas keep standing (some pagodas for more than 1000 years) against earthquakes. You can find four secrets of the robustness <a href="http://web-japan.org/nipponia/nipponia33/en/topic/" target="_blank">here</a>. May Tokyo Skytree also alive for 1000 years? Maybe.<br />
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When you come to Japan, please visit both the modern and the old structures. I especially recommend the pagodas in <a href="http://en.wikipedia.org/wiki/H%C5%8Dry%C5%AB-ji" target="_blank">Horyu-ji</a> (Nara) and <a href="http://en.wikipedia.org/wiki/T%C5%8D-ji" target="_blank">To-ji</a> (Kyoto). They are really beautiful.<br />
<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-10934744345612713352012-06-04T12:06:00.000+09:002014-08-17T07:49:45.935+09:00Transit of Venus<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-MXQYkB6AK88/T8rnttGidJI/AAAAAAAAAR8/Kikf_nrxClg/s1600/transit.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-MXQYkB6AK88/T8rnttGidJI/AAAAAAAAAR8/Kikf_nrxClg/s200/transit.jpg" height="200" width="195" /></a></div>
The <a href="http://en.wikipedia.org/wiki/Transit_of_Venus" target="_blank">transit of venus</a>, <a href="http://en.wikipedia.org/wiki/Transit_of_Venus,_2012" target="_blank">2012</a>, will begin at 22:09 UTC on 5 June 2012, and will finish at 04:49 UTC on 6 June. This is very rare event; the next transit will occur in December 2117.<br />
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Almost all humans living today can never experience the 2117 transit. So, do not miss the rare chance.<br />
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The last transit was in 2004, but before that, it was in 1874 and 1882. Around that period, Japan experienced <a href="http://en.wikipedia.org/wiki/Meiji_Restoration" target="_blank">Meiji Restoration</a>, and based on this, Japan developed to absolutely higher level than before. Hence, that was one of the greatest period in Japan.<br />
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I cannot imagine what will people in 2117 think of today, but I can say that the estimate depends on our daily doings.<br />
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<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-63339994587996121592012-06-03T12:57:00.001+09:002012-06-03T12:57:42.484+09:00Talk with Wolfram Alpha<div class="separator" style="clear: both; text-align: center;">
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<div class="separator" style="clear: both; text-align: center;">
<a href="http://1.bp.blogspot.com/-cfLnku7bonc/T8reZLUDx8I/AAAAAAAAARw/2tfXSE9ll50/s1600/256px-HAL9000.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="http://1.bp.blogspot.com/-cfLnku7bonc/T8reZLUDx8I/AAAAAAAAARw/2tfXSE9ll50/s200/256px-HAL9000.png" width="200" /></a></div>
<br />
<a href="http://www.wolframalpha.com/" target="_blank">Wolfram Alpha</a> might be <a href="http://en.wikipedia.org/wiki/HAL_9000" target="_blank">HAL9000</a>.<br />
<br />
I talked with Wolfram Alpha (WA) last night. The following is the recorded conversation.<br />
<br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=how+are+you%3F" target="_blank">How are you?</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>I am doing well, thank you.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=Where+are+you%3F" target="_blank">Where are you?</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>I live on the internet.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=What+are+you+doing+now%3F" target="_blank">What are you doing now?</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>I am doing computation for the world.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=do+you+like+computation%3F" target="_blank">Do you like computation?</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>Yes, I do.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=Do+you+like+math%3F" target="_blank">Do you like math?</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>Yes, I do.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=Do+you+like+humans%3F" target="_blank">Do you like humans?</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>Of course; I like all humans who ask me questions I can answer.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=Are+you+smart%3F" target="_blank">Are you smart?</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>I am capable of universal computation; that I can say.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=You+are+smart%21" target="_blank">You are smart!</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>Thank you! I still have much to learn, though I'm learning learning more every day.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=How+am+I%3F" target="_blank">How am I?</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>Since you're engaged in the fun activity of interacting with a computational knowledge engine, I assume you're doing well.</b></span><br />
<br />
Me: <a href="http://www.wolframalpha.com/input/?i=see+you+later%2C+alligator" target="_blank">See you later, alligator.</a><br />
WA: <span style="font-family: 'Courier New', Courier, monospace;"><b>After 'while, crocodile.</b></span><br />
<br />
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<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-13286653699668074182012-06-02T12:47:00.000+09:002012-06-03T23:48:39.816+09:00Beyond Shannon with your iPhone<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-uf4CUWMKRiU/T8mDvx8VMTI/AAAAAAAAARc/LtYzdx8zowI/s1600/fantabit.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-uf4CUWMKRiU/T8mDvx8VMTI/AAAAAAAAARc/LtYzdx8zowI/s1600/fantabit.jpg" /></a></div>
I posted an <a href="http://sparseland.blogspot.jp/2012/05/beyond-shannon-infinity-versus-zero.html" target="_blank">entry</a> about beyond-Shannon signal reconstruction by H <sup>∞</sup> optimization. Based on this method, an iPhone application, <a href="http://itunes.apple.com/app/fantabit/id506232017" target="_blank">FANTABIT</a>, has been designed and recently released. The iTunes page says<br />
<blockquote class="tr_bq">
<blockquote class="tr_bq">
Automatically upgrade your flat, compressed iTunes music library into rich sounding music with the FANTABIT player!</blockquote>
<blockquote class="tr_bq">
Listen to your iTunes library with in full dynamic range using the FANTABIT audio enhancing technology and player. This high quality music player is for discerning audiophiles who are tired of listening to uninspiring, compressed music. Install the FANTABIT player and your iTunes music library is instantly converted.</blockquote>
<blockquote class="tr_bq">
Audiophiles can take full advantage of expensive listening equipment (high-end cables, speakers, headphones, etc.,) to listen to the fully restored music and sounds from their iTunes library using the FANTABIT player. FANTABIT allows you to take full advantage of your system, immersing yourself in the richness of your favorite music.</blockquote>
</blockquote>
<br />
Compressed sensing (CS) is another beyond-Shannon method, which, to my knowledge, has not been applied to such real-time audio. This is mostly due to the computational cost of optimization in CS reconstruction. Actually, the H<sup>∞</sup> reconstruction requires just up-sampling and filtering, which is much faster than CS.<br />
<br />
<br />
<b>Related entries:</b><br />
<a href="http://sparseland.blogspot.jp/2012/05/beyond-shannon-infinity-versus-zero.html" target="_blank">Beyond Shannon: infinity versus zero</a><br />
<a href="http://sparseland.blogspot.jp/2012/05/zero-one-and-infinity-in-cs.html" target="_blank">Zero, one, and infinity in compressed sensing</a><br />
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<br />Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-49898291944806072562012-05-29T23:18:00.000+09:002012-06-03T23:52:30.953+09:00Newton's method as feedback system<span style="text-align: left;">This entry is inspired by </span><a href="http://www-ics.acs.i.kyoto-u.ac.jp/~yy/Papers/KK-YYAutomatica.pdf" style="text-align: left;" target="_blank">the work</a><span style="text-align: left;"> by K. Kashima and Y. Yamamoto.</span><br />
<span style="text-align: left;"><br /></span><br />
To obtain an approximated value of <span style="font-family: 'Courier New', Courier, monospace;"><b>sqrt(a)</b></span>, one can use Newton's method:<br />
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<a href="http://3.bp.blogspot.com/-VHFqLHv-i0s/T8IoFEuUm4I/AAAAAAAAAQg/QRhHytC1y6U/s1600/latex-image-1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="39" src="http://3.bp.blogspot.com/-VHFqLHv-i0s/T8IoFEuUm4I/AAAAAAAAAQg/QRhHytC1y6U/s320/latex-image-1.png" width="320" /></a></div>
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with a given real <span style="font-family: 'Courier New', Courier, monospace;"><b>x[0]</b></span>. This can be easily implemented in computer programs such as <a href="http://www.mathworks.com/" target="_blank">MATLAB</a> or <a href="http://www.scilab.org/" target="_blank">SCILAB</a>, but today I will represent this as a feedback system.<br />
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First, the iteration can be represented by<br />
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<a href="http://4.bp.blogspot.com/-IogMRSBYSyw/T8IntxNS-xI/AAAAAAAAAQY/nx7sEr_FqiQ/s1600/newton2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="42" src="http://4.bp.blogspot.com/-IogMRSBYSyw/T8IntxNS-xI/AAAAAAAAAQY/nx7sEr_FqiQ/s320/newton2.png" width="320" /></a></div>
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where <b><span style="font-family: Times, 'Times New Roman', serif;"><i>σ</i></span></b> denotes the shift operator</div>
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<a href="http://1.bp.blogspot.com/-yqqsnoWbX70/T8IpH2qsevI/AAAAAAAAAQo/hIrB6KGyuyk/s1600/latex-image-2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="25" src="http://1.bp.blogspot.com/-yqqsnoWbX70/T8IpH2qsevI/AAAAAAAAAQo/hIrB6KGyuyk/s320/latex-image-2.png" width="320" /></a></div>
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In control theory or signal processing, this operator is also denoted by <span style="font-family: 'Courier New', Courier, monospace;"><b>z<sup>-1</sup></b></span>. Next, the iteration is divided into two parts, <b><span style="font-family: Times, 'Times New Roman', serif;"><i>φ</i></span></b> and <i><b>σ</b></i>, that is,</div>
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<a href="http://1.bp.blogspot.com/-Z-LII7Jrf8o/T8Iq8TCOw4I/AAAAAAAAAQw/RFOP3OEjEEM/s1600/latex-image-3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="68" src="http://1.bp.blogspot.com/-Z-LII7Jrf8o/T8Iq8TCOw4I/AAAAAAAAAQw/RFOP3OEjEEM/s320/latex-image-3.png" width="320" /></a></div>
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Finally, this can be represented by the following block diagram:</div>
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<a href="http://2.bp.blogspot.com/-JjNCjZ96xlA/T8IuV8LFi-I/AAAAAAAAAQ8/og_0xYs1Rko/s1600/feedback.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-JjNCjZ96xlA/T8IuV8LFi-I/AAAAAAAAAQ8/og_0xYs1Rko/s1600/feedback.png" /></a></div>
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Based on this, you can run simulation with <a href="http://www.scilab.org/products/xcos" target="_blank">XCOS</a> in <a href="http://www.scilab.org/" target="_blank">SCLAB</a> (xcos file is <a href="http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/nc_text/newton_sqrt.xcos" target="_blank">here</a>).</div>
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<a href="http://4.bp.blogspot.com/-YNTlAWNgrsU/T8IwJ02adHI/AAAAAAAAARE/67XFg_SFH6Q/s1600/xcos.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-YNTlAWNgrsU/T8IwJ02adHI/AAAAAAAAARE/67XFg_SFH6Q/s320/xcos.png" width="311" /></a></div>
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This is a "feedback system" for <b><span style="font-family: 'Courier New', Courier, monospace;">sqrt(2)</span></b>. You can change the initial value <span style="font-family: 'Courier New', Courier, monospace;"><b>x[0]</b></span> by double-clicking "<span style="font-family: 'Courier New', Courier, monospace;"><b>1/z</b></span>" block. Also, if you want another square root, double-click the "<span style="font-family: 'Courier New', Courier, monospace;"><b>Expression</b></span>" block and change "2" in the numerator to another <i>positive</i> number. But, if you feel like wilding out, use a <i>negative</i> number instead. In this case, the sequence will not converge, but show some chaotic behavior. The following is the result for "<span style="font-family: 'Courier New', Courier, monospace;"><b>-2</b></span>."</div>
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<a href="http://3.bp.blogspot.com/-EP6kL5_oaDc/T8IzEIzMEnI/AAAAAAAAARQ/yv7zbxVbI9U/s1600/Untitled-export.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://3.bp.blogspot.com/-EP6kL5_oaDc/T8IzEIzMEnI/AAAAAAAAARQ/yv7zbxVbI9U/s320/Untitled-export.png" width="320" /></a></div>
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<b>Related entries:</b></div>
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<a href="http://sparseland.blogspot.jp/2012/05/simplest-way-to-solve-xcosx.html" target="_blank">The simplest way to solve "x=cos(x)"</a></div>
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<a href="http://sparseland.blogspot.jp/2012/05/play-with-scientific-calculator.html" target="_blank">Play with scientific calculator</a></div>
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<br /></div>Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0tag:blogger.com,1999:blog-6068554977761810119.post-31246023941297004282012-05-27T08:16:00.001+09:002012-05-27T08:16:18.052+09:00Play with scientific calculatorI posted an <a href="http://sparseland.blogspot.jp/2012/05/simplest-way-to-solve-xcosx.html" target="_blank">entry</a> where I showed a very simple method to solve <b style="text-align: center;"><span style="font-family: 'Courier New', Courier, monospace;">x = cos(x)</span></b><span style="text-align: center;"> </span>by using a scientific calculator. Now, let's continue playing with a scientific calculator.<br />
<ol>
<li>Bring your scientific calculator (you can use your <a href="http://osxdaily.com/2011/12/29/iphone-scientific-calculator/" target="_blank">iphone</a> or a <a href="http://www.mathsisfun.com/scientific-calculator.html" target="_blank">web calculator</a>).</li>
<li>Change the mode to "rad" (radian).</li>
<li>Push "<span style="font-family: 'Courier New', Courier, monospace;"><b>1</b></span>" (one) key.</li>
<li>Push "<b><span style="font-family: 'Courier New', Courier, monospace;">sin</span></b>" key.</li>
<li>Push "<span style="font-family: 'Courier New', Courier, monospace;"><b>1/x</b></span>" key.</li>
<li>Go back to step 4, unless you want out.<br />
</li>
</ol>
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You will see two numbers:</div>
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<ul>
<li><b><span style="font-family: 'Courier New', Courier, monospace;">0.8975395...</span></b> (after step 4.)</li>
<li><span style="font-family: 'Courier New', Courier, monospace;"><b>1.1141571... </b></span>(after step 5.)</li>
</ul>
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The second number is a solution of the equation <span style="font-family: 'Courier New', Courier, monospace;"><b>x = 1/sin(x)</b></span>(this equation has infinite number of solutions), and the first number is its inverse.<br />
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Then, change "<span style="font-family: 'Courier New', Courier, monospace;"><b>sin</b></span>" to "<b><span style="font-family: 'Courier New', Courier, monospace;">cos</span></b>" in step 4. Does it converge?<br />
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No.<br />
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I simulated the procedure by using <a href="http://www.scilab.org/" target="_blank">SCILAB</a>, and obtained a sequence shown below.<br />
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<a href="http://3.bp.blogspot.com/-1RJLdqwH0TY/T8BxxYfxZJI/AAAAAAAAAP8/5KjDDvaByCM/s1600/result2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="301" src="http://3.bp.blogspot.com/-1RJLdqwH0TY/T8BxxYfxZJI/AAAAAAAAAP8/5KjDDvaByCM/s400/result2.png" width="400" /></a></div>
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This looks like a chaotic sequence.<br />
<br /></div>Kitakyushu Fishinghttp://www.blogger.com/profile/01677531634711698554noreply@blogger.com0