tdehaeze/lecture-h-infinity
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Re-organize subsections

Browse Source
This commit is contained in:
Thomas Dehaeze 2020-12-01 18:49:40 +01:00
parent 7a1817c20d
commit 21c874f66a
15 changed files with 1369 additions and 869 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
figs/bode_plot_example_Gd.pdf
View File

Binary file not shown.

BIN
figs/bode_plot_example_Gd.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 63 KiB

After

Width:  |  Height:  |  Size: 62 KiB

BIN
figs/h-infinity-4-blocs-constrains.pdf
View File

Binary file not shown.

BIN
figs/h-infinity-4-blocs-constrains.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 32 KiB

After

Width:  |  Height:  |  Size: 45 KiB

57
figs/h-infinity-4-blocs-constrains.svg
View File

@ -56,8 +56,37 @@
</clipPath>
</defs>
<g id="surface1">
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,79.998779%);fill-opacity:1;" d="M 0.996094 117.195312 L 170.832031 117.195312 L 170.832031 1.140625 L 0.996094 1.140625 Z M 0.996094 117.195312 "/>
<g clip-path="url(#clip1)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -14.173475 -76.535669 L 155.906532 -76.535669 L 155.906532 39.6858 L -14.173475 39.6858 Z M -14.173475 -76.535669 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="82.577329" y="15.863267"/>
<use xlink:href="#glyph0-1" x="82.577329" y="11.255888"/>
</g>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,79.998779%);fill-opacity:1;" d="M 182.15625 117.195312 L 351.992188 117.195312 L 351.992188 1.140625 L 182.15625 1.140625 Z M 182.15625 117.195312 "/>
<g clip-path="url(#clip2)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M 167.247025 -76.535669 L 337.327032 -76.535669 L 337.327032 39.6858 L 167.247025 39.6858 Z M 167.247025 -76.535669 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-2" x="259.822611" y="11.255888"/>
<use xlink:href="#glyph0-1" x="267.644962" y="11.255888"/>
</g>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,79.998779%);fill-opacity:1;" d="M 0.996094 241.742188 L 170.832031 241.742188 L 170.832031 125.6875 L 0.996094 125.6875 Z M 0.996094 241.742188 "/>
<g clip-path="url(#clip3)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -14.173475 -201.261529 L 155.906532 -201.261529 L 155.906532 -85.04006 L -14.173475 -85.04006 Z M -14.173475 -201.261529 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="77.996911" y="135.800906"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="87.162283" y="135.800906"/>
</g>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,79.998779%);fill-opacity:1;" d="M 182.15625 241.742188 L 351.992188 241.742188 L 351.992188 125.6875 L 182.15625 125.6875 Z M 182.15625 241.742188 "/>
<g clip-path="url(#clip4)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M 167.247025 -201.261529 L 337.327032 -201.261529 L 337.327032 -85.04006 L 167.247025 -85.04006 Z M 167.247025 -201.261529 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-4" x="263.473365" y="135.800906"/>
</g>
<path style="fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -5.669083 -70.867378 L -5.669083 14.172625 L 39.685064 14.172625 L 39.685064 -70.867378 Z M -5.669083 -70.867378 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
@ -83,13 +112,7 @@
<path style="fill-rule:nonzero;fill:rgb(100%,79.998779%,79.998779%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 75.118725 4.252139 C 75.118725 6.599257 73.213647 8.504335 70.866529 8.504335 C 68.519411 8.504335 66.614333 6.599257 66.614333 4.252139 C 66.614333 1.905021 68.519411 -0.0000566533 70.866529 -0.0000566533 C 73.213647 -0.0000566533 75.118725 1.905021 75.118725 4.252139 Z M 75.118725 4.252139 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -11.337374 -0.0000566533 L 153.074343 -0.0000566533 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.000792824 -56.694696 C 22.199033 -34.49487 34.494021 -22.195971 56.693847 -0.0000566533 C 61.258992 4.565088 64.411954 8.504335 70.866529 8.504335 C 82.41435 8.504335 87.664071 -0.0000566533 99.211893 -0.0000566533 C 115.794283 -0.0000566533 125.15146 -0.0000566533 141.73385 -0.0000566533 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<g clip-path="url(#clip1)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -14.173475 -76.535669 L -14.173475 39.6858 L 155.906532 39.6858 L 155.906532 -76.535669 Z M -14.173475 -76.535669 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-2" x="259.822611" y="15.863267"/>
<use xlink:href="#glyph0-1" x="267.644962" y="15.863267"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -0.000792824 -0.0000566533 Z M -0.000792824 -0.0000566533 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<path style="fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 175.751417 -70.867378 L 175.751417 14.172625 L 221.105564 14.172625 L 221.105564 -70.867378 Z M 175.751417 -70.867378 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="198.118286" y="58.971317"/>
@ -112,15 +135,6 @@
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M 170.079214 -0.0000566533 L 334.490931 -0.0000566533 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 181.419707 -56.694696 C 192.517664 -45.596739 198.667114 -39.44729 209.765071 -28.345421 C 223.100614 -15.013789 233.431846 -5.668347 252.287029 -5.668347 C 271.142211 -5.668347 281.473443 -15.013789 294.805074 -28.345421 C 305.906943 -39.44729 312.056393 -45.596739 323.15435 -56.694696 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<g clip-path="url(#clip2)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M 167.247025 -76.535669 L 167.247025 39.6858 L 337.327032 39.6858 L 337.327032 -76.535669 Z M 167.247025 -76.535669 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="77.996911" y="140.408285"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="87.162283" y="140.408285"/>
</g>
<path style="fill-rule:nonzero;fill:rgb(100%,79.998779%,79.998779%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -5.669083 -195.593238 L -5.669083 -110.553235 L 39.685064 -110.553235 L 39.685064 -195.593238 Z M -5.669083 -195.593238 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="17.631662" y="220.314452"/>
@ -143,12 +157,6 @@
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -11.337374 -124.725917 L 153.074343 -124.725917 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.000792824 -167.247874 C 11.097164 -156.146006 17.246614 -149.996556 28.348483 -138.898599 C 41.680114 -125.566967 52.011346 -116.221525 70.866529 -116.221525 C 89.721712 -116.221525 100.052943 -125.566967 113.388486 -138.898599 C 124.486444 -149.996556 130.635893 -156.146006 141.73385 -167.247874 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<g clip-path="url(#clip3)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -14.173475 -201.261529 L -14.173475 -85.04006 L 155.906532 -85.04006 L 155.906532 -201.261529 Z M -14.173475 -201.261529 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-4" x="263.473365" y="140.408285"/>
</g>
<path style="fill-rule:nonzero;fill:rgb(100%,79.998779%,79.998779%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 175.751417 -195.593238 L 175.751417 -110.553235 L 221.105564 -110.553235 L 221.105564 -195.593238 Z M 175.751417 -195.593238 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="205.552602" y="182.60065"/>
@ -167,8 +175,5 @@
<path style="fill-rule:nonzero;fill:rgb(100%,79.998779%,79.998779%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 256.539224 -120.473721 C 256.539224 -118.126603 254.634147 -116.221525 252.287029 -116.221525 C 249.93991 -116.221525 248.034833 -118.126603 248.034833 -120.473721 C 248.034833 -122.820839 249.93991 -124.725917 252.287029 -124.725917 C 254.634147 -124.725917 256.539224 -122.820839 256.539224 -120.473721 Z M 256.539224 -120.473721 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M 170.079214 -124.725917 L 334.490931 -124.725917 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 181.419707 -124.725917 C 198.002097 -124.725917 207.359275 -124.725917 223.937753 -124.725917 C 235.489486 -124.725917 240.739207 -116.221525 252.287029 -116.221525 C 258.741604 -116.221525 271.024855 -129.291062 266.459711 -124.725917 C 288.655625 -146.921831 300.954524 -159.22073 323.15435 -181.420556 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
<g clip-path="url(#clip4)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M 167.247025 -201.261529 L 167.247025 -85.04006 L 337.327032 -85.04006 L 337.327032 -201.261529 Z M 167.247025 -201.261529 " transform="matrix(0.998565,0,0,-0.998565,15.149229,40.769475)"/>
</g>
</g>
</svg>

Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 22 KiB

After

Width:  |  Height:  |  Size: 24 KiB

BIN
figs/loop_shaping_S_with_W.pdf
View File

Binary file not shown.

BIN
figs/loop_shaping_S_with_W.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 18 KiB

After

Width:  |  Height:  |  Size: 18 KiB

183
figs/loop_shaping_S_with_W.svg
View File

@ -1,5 +1,5 @@
<?xml version="1.0" encoding="UTF-8"?>
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="206.943pt" height="156.445pt" viewBox="0 0 206.943 156.445" version="1.2">
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="214.977pt" height="156.445pt" viewBox="0 0 214.977 156.445" version="1.2">
<defs>
<g>
<symbol overflow="visible" id="glyph0-0">
@ -81,15 +81,18 @@
<path style="stroke:none;" d="M 3.546875 -0.390625 C 3.546875 -0.421875 3.546875 -0.53125 3.453125 -0.53125 C 3.453125 -0.53125 3.40625 -0.53125 3.328125 -0.46875 C 3.015625 -0.28125 2.65625 -0.109375 2.28125 -0.109375 C 1.703125 -0.109375 1.203125 -0.53125 1.203125 -1.40625 C 1.203125 -1.75 1.296875 -2.125 1.3125 -2.25 L 2.953125 -2.25 C 3.109375 -2.25 3.296875 -2.25 3.296875 -2.40625 C 3.296875 -2.546875 3.171875 -2.546875 3 -2.546875 L 1.40625 -2.546875 C 1.640625 -3.390625 2.1875 -3.96875 3.09375 -3.96875 L 3.40625 -3.96875 C 3.578125 -3.96875 3.734375 -3.96875 3.734375 -4.140625 C 3.734375 -4.28125 3.609375 -4.28125 3.4375 -4.28125 L 3.078125 -4.28125 C 1.796875 -4.28125 0.46875 -3.28125 0.46875 -1.765625 C 0.46875 -0.671875 1.203125 0.109375 2.265625 0.109375 C 2.90625 0.109375 3.546875 -0.28125 3.546875 -0.390625 Z M 3.546875 -0.390625 "/>
</symbol>
<symbol overflow="visible" id="glyph1-7">
<path style="stroke:none;" d="M 4.640625 -3.6875 C 4.640625 -4.234375 4.390625 -4.390625 4.203125 -4.390625 C 3.953125 -4.390625 3.71875 -4.125 3.71875 -3.90625 C 3.71875 -3.78125 3.765625 -3.71875 3.875 -3.609375 C 4.09375 -3.40625 4.21875 -3.15625 4.21875 -2.796875 C 4.21875 -2.375 3.609375 -0.109375 2.453125 -0.109375 C 1.9375 -0.109375 1.71875 -0.453125 1.71875 -0.96875 C 1.71875 -1.53125 1.984375 -2.25 2.296875 -3.078125 C 2.359375 -3.25 2.40625 -3.390625 2.40625 -3.578125 C 2.40625 -4.015625 2.09375 -4.390625 1.59375 -4.390625 C 0.671875 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.859375 -3.9375 1.28125 -4.171875 1.5625 -4.171875 C 1.640625 -4.171875 1.8125 -4.171875 1.8125 -3.84375 C 1.8125 -3.609375 1.71875 -3.328125 1.640625 -3.15625 C 1.203125 -2 1.078125 -1.546875 1.078125 -1.125 C 1.078125 -0.046875 1.953125 0.109375 2.40625 0.109375 C 4.078125 0.109375 4.640625 -3.171875 4.640625 -3.6875 Z M 4.640625 -3.6875 "/>
<path style="stroke:none;" d="M 4.3125 -1.421875 C 4.3125 -1.46875 4.28125 -1.515625 4.203125 -1.515625 C 4.109375 -1.515625 4.09375 -1.453125 4.0625 -1.390625 C 3.828125 -0.75 3.1875 -0.5625 2.875 -0.5625 C 2.671875 -0.5625 2.484375 -0.609375 2.28125 -0.6875 C 1.9375 -0.8125 1.796875 -0.859375 1.59375 -0.859375 C 1.59375 -0.859375 1.40625 -0.859375 1.3125 -0.828125 C 1.859375 -1.40625 2.140625 -1.640625 2.5 -1.953125 C 2.5 -1.953125 3.109375 -2.484375 3.46875 -2.84375 C 4.421875 -3.765625 4.640625 -4.25 4.640625 -4.28125 C 4.640625 -4.390625 4.53125 -4.390625 4.53125 -4.390625 C 4.453125 -4.390625 4.421875 -4.359375 4.375 -4.28125 C 4.078125 -3.796875 3.875 -3.640625 3.625 -3.640625 C 3.390625 -3.640625 3.28125 -3.796875 3.125 -3.953125 C 2.9375 -4.1875 2.765625 -4.390625 2.4375 -4.390625 C 1.703125 -4.390625 1.234375 -3.46875 1.234375 -3.25 C 1.234375 -3.203125 1.265625 -3.140625 1.359375 -3.140625 C 1.453125 -3.140625 1.46875 -3.1875 1.484375 -3.25 C 1.671875 -3.703125 2.25 -3.71875 2.328125 -3.71875 C 2.546875 -3.71875 2.734375 -3.65625 2.953125 -3.578125 C 3.359375 -3.421875 3.46875 -3.421875 3.71875 -3.421875 C 3.359375 -3 2.53125 -2.28125 2.34375 -2.125 L 1.453125 -1.296875 C 0.78125 -0.625 0.421875 -0.0625 0.421875 0.015625 C 0.421875 0.109375 0.546875 0.109375 0.546875 0.109375 C 0.625 0.109375 0.640625 0.09375 0.703125 -0.015625 C 0.9375 -0.359375 1.234375 -0.640625 1.546875 -0.640625 C 1.78125 -0.640625 1.875 -0.546875 2.125 -0.265625 C 2.296875 -0.046875 2.46875 0.109375 2.765625 0.109375 C 3.734375 0.109375 4.3125 -1.15625 4.3125 -1.421875 Z M 4.3125 -1.421875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-8">
<path style="stroke:none;" d="M 5.390625 -1.421875 C 5.390625 -1.515625 5.296875 -1.515625 5.265625 -1.515625 C 5.171875 -1.515625 5.15625 -1.484375 5.125 -1.34375 C 4.984375 -0.78125 4.796875 -0.109375 4.390625 -0.109375 C 4.171875 -0.109375 4.078125 -0.234375 4.078125 -0.5625 C 4.078125 -0.78125 4.203125 -1.25 4.28125 -1.59375 L 4.546875 -2.671875 C 4.578125 -2.8125 4.6875 -3.1875 4.71875 -3.34375 C 4.765625 -3.578125 4.875 -3.953125 4.875 -4.015625 C 4.875 -4.1875 4.734375 -4.28125 4.578125 -4.28125 C 4.53125 -4.28125 4.28125 -4.265625 4.203125 -3.921875 L 3.453125 -0.9375 C 3.4375 -0.90625 3.046875 -0.109375 2.328125 -0.109375 C 1.8125 -0.109375 1.703125 -0.5625 1.703125 -0.921875 C 1.703125 -1.484375 1.984375 -2.265625 2.25 -2.953125 C 2.359375 -3.25 2.40625 -3.390625 2.40625 -3.578125 C 2.40625 -4.015625 2.09375 -4.390625 1.59375 -4.390625 C 0.65625 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.8125 -3.796875 1.1875 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.15625 1.8125 -3.84375 C 1.8125 -3.609375 1.703125 -3.3125 1.640625 -3.171875 C 1.28125 -2.1875 1.078125 -1.5625 1.078125 -1.078125 C 1.078125 -0.140625 1.75 0.109375 2.296875 0.109375 C 2.953125 0.109375 3.296875 -0.34375 3.46875 -0.5625 C 3.578125 -0.15625 3.921875 0.109375 4.359375 0.109375 C 4.703125 0.109375 4.9375 -0.125 5.09375 -0.4375 C 5.265625 -0.796875 5.390625 -1.421875 5.390625 -1.421875 Z M 5.390625 -1.421875 "/>
<path style="stroke:none;" d="M 4.640625 -3.6875 C 4.640625 -4.234375 4.390625 -4.390625 4.203125 -4.390625 C 3.953125 -4.390625 3.71875 -4.125 3.71875 -3.90625 C 3.71875 -3.78125 3.765625 -3.71875 3.875 -3.609375 C 4.09375 -3.40625 4.21875 -3.15625 4.21875 -2.796875 C 4.21875 -2.375 3.609375 -0.109375 2.453125 -0.109375 C 1.9375 -0.109375 1.71875 -0.453125 1.71875 -0.96875 C 1.71875 -1.53125 1.984375 -2.25 2.296875 -3.078125 C 2.359375 -3.25 2.40625 -3.390625 2.40625 -3.578125 C 2.40625 -4.015625 2.09375 -4.390625 1.59375 -4.390625 C 0.671875 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.859375 -3.9375 1.28125 -4.171875 1.5625 -4.171875 C 1.640625 -4.171875 1.8125 -4.171875 1.8125 -3.84375 C 1.8125 -3.609375 1.71875 -3.328125 1.640625 -3.15625 C 1.203125 -2 1.078125 -1.546875 1.078125 -1.125 C 1.078125 -0.046875 1.953125 0.109375 2.40625 0.109375 C 4.078125 0.109375 4.640625 -3.171875 4.640625 -3.6875 Z M 4.640625 -3.6875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-9">
<path style="stroke:none;" d="M 6.859375 -3.6875 C 6.859375 -4.234375 6.59375 -4.390625 6.421875 -4.390625 C 6.171875 -4.390625 5.921875 -4.125 5.921875 -3.90625 C 5.921875 -3.78125 5.984375 -3.71875 6.078125 -3.640625 C 6.1875 -3.53125 6.421875 -3.28125 6.421875 -2.796875 C 6.421875 -2.453125 6.140625 -1.484375 5.890625 -0.984375 C 5.625 -0.453125 5.28125 -0.109375 4.796875 -0.109375 C 4.328125 -0.109375 4.0625 -0.40625 4.0625 -0.96875 C 4.0625 -1.25 4.125 -1.5625 4.171875 -1.703125 L 4.578125 -3.359375 C 4.640625 -3.578125 4.734375 -3.953125 4.734375 -4.015625 C 4.734375 -4.1875 4.59375 -4.28125 4.4375 -4.28125 C 4.328125 -4.28125 4.140625 -4.203125 4.078125 -4 C 4.046875 -3.921875 3.578125 -2.03125 3.515625 -1.78125 C 3.4375 -1.484375 3.421875 -1.296875 3.421875 -1.125 C 3.421875 -1.015625 3.421875 -0.984375 3.4375 -0.9375 C 3.203125 -0.421875 2.90625 -0.109375 2.515625 -0.109375 C 1.71875 -0.109375 1.71875 -0.84375 1.71875 -1.015625 C 1.71875 -1.328125 1.78125 -1.71875 2.25 -2.9375 C 2.34375 -3.234375 2.40625 -3.375 2.40625 -3.578125 C 2.40625 -4.015625 2.078125 -4.390625 1.59375 -4.390625 C 0.65625 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.828125 -3.859375 1.21875 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.15625 1.8125 -3.84375 C 1.8125 -3.59375 1.703125 -3.3125 1.640625 -3.140625 C 1.203125 -1.96875 1.078125 -1.515625 1.078125 -1.140625 C 1.078125 -0.234375 1.75 0.109375 2.484375 0.109375 C 2.65625 0.109375 3.125 0.109375 3.515625 -0.578125 C 3.78125 0.046875 4.46875 0.109375 4.765625 0.109375 C 5.5 0.109375 5.9375 -0.515625 6.203125 -1.109375 C 6.53125 -1.890625 6.859375 -3.21875 6.859375 -3.6875 Z M 6.859375 -3.6875 "/>
<path style="stroke:none;" d="M 5.390625 -1.421875 C 5.390625 -1.515625 5.296875 -1.515625 5.265625 -1.515625 C 5.171875 -1.515625 5.15625 -1.484375 5.125 -1.34375 C 4.984375 -0.78125 4.796875 -0.109375 4.390625 -0.109375 C 4.171875 -0.109375 4.078125 -0.234375 4.078125 -0.5625 C 4.078125 -0.78125 4.203125 -1.25 4.28125 -1.59375 L 4.546875 -2.671875 C 4.578125 -2.8125 4.6875 -3.1875 4.71875 -3.34375 C 4.765625 -3.578125 4.875 -3.953125 4.875 -4.015625 C 4.875 -4.1875 4.734375 -4.28125 4.578125 -4.28125 C 4.53125 -4.28125 4.28125 -4.265625 4.203125 -3.921875 L 3.453125 -0.9375 C 3.4375 -0.90625 3.046875 -0.109375 2.328125 -0.109375 C 1.8125 -0.109375 1.703125 -0.5625 1.703125 -0.921875 C 1.703125 -1.484375 1.984375 -2.265625 2.25 -2.953125 C 2.359375 -3.25 2.40625 -3.390625 2.40625 -3.578125 C 2.40625 -4.015625 2.09375 -4.390625 1.59375 -4.390625 C 0.65625 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.8125 -3.796875 1.1875 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.15625 1.8125 -3.84375 C 1.8125 -3.609375 1.703125 -3.3125 1.640625 -3.171875 C 1.28125 -2.1875 1.078125 -1.5625 1.078125 -1.078125 C 1.078125 -0.140625 1.75 0.109375 2.296875 0.109375 C 2.953125 0.109375 3.296875 -0.34375 3.46875 -0.5625 C 3.578125 -0.15625 3.921875 0.109375 4.359375 0.109375 C 4.703125 0.109375 4.9375 -0.125 5.09375 -0.4375 C 5.265625 -0.796875 5.390625 -1.421875 5.390625 -1.421875 Z M 5.390625 -1.421875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-10">
<path style="stroke:none;" d="M 6.859375 -3.6875 C 6.859375 -4.234375 6.59375 -4.390625 6.421875 -4.390625 C 6.171875 -4.390625 5.921875 -4.125 5.921875 -3.90625 C 5.921875 -3.78125 5.984375 -3.71875 6.078125 -3.640625 C 6.1875 -3.53125 6.421875 -3.28125 6.421875 -2.796875 C 6.421875 -2.453125 6.140625 -1.484375 5.890625 -0.984375 C 5.625 -0.453125 5.28125 -0.109375 4.796875 -0.109375 C 4.328125 -0.109375 4.0625 -0.40625 4.0625 -0.96875 C 4.0625 -1.25 4.125 -1.5625 4.171875 -1.703125 L 4.578125 -3.359375 C 4.640625 -3.578125 4.734375 -3.953125 4.734375 -4.015625 C 4.734375 -4.1875 4.59375 -4.28125 4.4375 -4.28125 C 4.328125 -4.28125 4.140625 -4.203125 4.078125 -4 C 4.046875 -3.921875 3.578125 -2.03125 3.515625 -1.78125 C 3.4375 -1.484375 3.421875 -1.296875 3.421875 -1.125 C 3.421875 -1.015625 3.421875 -0.984375 3.4375 -0.9375 C 3.203125 -0.421875 2.90625 -0.109375 2.515625 -0.109375 C 1.71875 -0.109375 1.71875 -0.84375 1.71875 -1.015625 C 1.71875 -1.328125 1.78125 -1.71875 2.25 -2.9375 C 2.34375 -3.234375 2.40625 -3.375 2.40625 -3.578125 C 2.40625 -4.015625 2.078125 -4.390625 1.59375 -4.390625 C 0.65625 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.828125 -3.859375 1.21875 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.15625 1.8125 -3.84375 C 1.8125 -3.59375 1.703125 -3.3125 1.640625 -3.140625 C 1.203125 -1.96875 1.078125 -1.515625 1.078125 -1.140625 C 1.078125 -0.234375 1.75 0.109375 2.484375 0.109375 C 2.65625 0.109375 3.125 0.109375 3.515625 -0.578125 C 3.78125 0.046875 4.46875 0.109375 4.765625 0.109375 C 5.5 0.109375 5.9375 -0.515625 6.203125 -1.109375 C 6.53125 -1.890625 6.859375 -3.21875 6.859375 -3.6875 Z M 6.859375 -3.6875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-11">
<path style="stroke:none;" d="M 4.328125 -3.734375 C 4.328125 -4.09375 4.015625 -4.390625 3.5 -4.390625 C 2.859375 -4.390625 2.421875 -3.90625 2.234375 -3.625 C 2.15625 -4.0625 1.796875 -4.390625 1.328125 -4.390625 C 0.875 -4.390625 0.6875 -4 0.59375 -3.8125 C 0.421875 -3.484375 0.28125 -2.890625 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.765625 0.578125 -2.984375 C 0.75 -3.6875 0.9375 -4.171875 1.296875 -4.171875 C 1.46875 -4.171875 1.609375 -4.09375 1.609375 -3.703125 C 1.609375 -3.5 1.578125 -3.390625 1.453125 -2.875 L 0.875 -0.578125 C 0.84375 -0.4375 0.78125 -0.203125 0.78125 -0.15625 C 0.78125 0.015625 0.921875 0.109375 1.078125 0.109375 C 1.1875 0.109375 1.375 0.03125 1.4375 -0.171875 C 1.453125 -0.203125 1.796875 -1.5625 1.828125 -1.734375 L 2.15625 -3.03125 C 2.1875 -3.15625 2.46875 -3.625 2.703125 -3.84375 C 2.78125 -3.90625 3.078125 -4.171875 3.5 -4.171875 C 3.765625 -4.171875 3.921875 -4.046875 3.921875 -4.046875 C 3.625 -4 3.40625 -3.765625 3.40625 -3.5 C 3.40625 -3.34375 3.515625 -3.15625 3.78125 -3.15625 C 4.046875 -3.15625 4.328125 -3.390625 4.328125 -3.734375 Z M 4.328125 -3.734375 "/>
</symbol>
<symbol overflow="visible" id="glyph2-0">
@ -115,159 +118,159 @@
<path d="M 93 126 L 129 126 L 129 155.890625 L 93 155.890625 Z M 93 126 "/>
</clipPath>
<clipPath id="clip2">
<path d="M 179 26 L 206.546875 26 L 206.546875 58 L 179 58 Z M 179 26 "/>
</clipPath>
<clipPath id="clip3">
<path d="M 115 125 L 149 125 L 149 155.890625 L 115 155.890625 Z M 115 125 "/>
</clipPath>
</defs>
<g id="surface1">
<path style="fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,79.998779%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -25.475029 -22.641073 L 122.850086 -22.641073 L 122.850086 72.703786 L -25.475029 72.703786 Z M -25.475029 -22.641073 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,79.998779%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -25.474919 -22.641073 L 122.850197 -22.641073 L 122.850197 72.703786 L -25.474919 72.703786 Z M -25.474919 -22.641073 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="38.528889" y="12.568413"/>
<use xlink:href="#glyph0-1" x="38.560029" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-2" x="47.908279" y="12.568413"/>
<use xlink:href="#glyph0-3" x="52.321" y="12.568413"/>
<use xlink:href="#glyph0-4" x="55.078827" y="12.568413"/>
<use xlink:href="#glyph0-5" x="60.042518" y="12.568413"/>
<use xlink:href="#glyph0-2" x="47.939419" y="12.568413"/>
<use xlink:href="#glyph0-3" x="52.35214" y="12.568413"/>
<use xlink:href="#glyph0-4" x="55.109967" y="12.568413"/>
<use xlink:href="#glyph0-5" x="60.073658" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-6" x="65.280205" y="12.568413"/>
<use xlink:href="#glyph0-2" x="69.140963" y="12.568413"/>
<use xlink:href="#glyph0-7" x="73.553685" y="12.568413"/>
<use xlink:href="#glyph0-6" x="65.311345" y="12.568413"/>
<use xlink:href="#glyph0-2" x="69.172104" y="12.568413"/>
<use xlink:href="#glyph0-7" x="73.584825" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-8" x="82.375156" y="12.568413"/>
<use xlink:href="#glyph0-2" x="90.165173" y="12.568413"/>
<use xlink:href="#glyph0-9" x="94.577894" y="12.568413"/>
<use xlink:href="#glyph0-2" x="100.093547" y="12.568413"/>
<use xlink:href="#glyph0-10" x="104.506269" y="12.568413"/>
<use xlink:href="#glyph0-11" x="108.394824" y="12.568413"/>
<use xlink:href="#glyph0-12" x="113.358515" y="12.568413"/>
<use xlink:href="#glyph0-3" x="116.116342" y="12.568413"/>
<use xlink:href="#glyph0-13" x="118.874168" y="12.568413"/>
<use xlink:href="#glyph0-2" x="123.28689" y="12.568413"/>
<use xlink:href="#glyph0-7" x="127.699611" y="12.568413"/>
<use xlink:href="#glyph0-8" x="82.406296" y="12.568413"/>
<use xlink:href="#glyph0-2" x="90.196313" y="12.568413"/>
<use xlink:href="#glyph0-9" x="94.609034" y="12.568413"/>
<use xlink:href="#glyph0-2" x="100.124688" y="12.568413"/>
<use xlink:href="#glyph0-10" x="104.537409" y="12.568413"/>
<use xlink:href="#glyph0-11" x="108.425964" y="12.568413"/>
<use xlink:href="#glyph0-12" x="113.389655" y="12.568413"/>
<use xlink:href="#glyph0-3" x="116.147482" y="12.568413"/>
<use xlink:href="#glyph0-13" x="118.905309" y="12.568413"/>
<use xlink:href="#glyph0-2" x="123.31803" y="12.568413"/>
<use xlink:href="#glyph0-7" x="127.730751" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-14" x="136.521083" y="12.568413"/>
<use xlink:href="#glyph0-12" x="143.277659" y="12.568413"/>
<use xlink:href="#glyph0-11" x="146.035485" y="12.568413"/>
<use xlink:href="#glyph0-9" x="150.999176" y="12.568413"/>
<use xlink:href="#glyph0-14" x="136.552223" y="12.568413"/>
<use xlink:href="#glyph0-12" x="143.308799" y="12.568413"/>
<use xlink:href="#glyph0-11" x="146.066625" y="12.568413"/>
<use xlink:href="#glyph0-9" x="151.030316" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-6" x="156.236863" y="12.568413"/>
<use xlink:href="#glyph0-6" x="156.268003" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-15" x="165.629988" y="10.059314"/>
<use xlink:href="#glyph0-15" x="165.661128" y="10.059314"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="163.407871" y="12.568413"/>
<use xlink:href="#glyph1-1" x="163.439011" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-16" x="171.160369" y="12.568413"/>
<use xlink:href="#glyph0-16" x="171.191509" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="175.020674" y="12.568413"/>
<use xlink:href="#glyph1-2" x="175.051814" y="12.568413"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-17" x="179.674166" y="12.568413"/>
<use xlink:href="#glyph0-17" x="179.705306" y="12.568413"/>
</g>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -17.007596 -14.173641 L 17.007178 -14.173641 L 17.007178 14.172658 L -17.007596 14.172658 Z M -17.007596 -14.173641 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -17.007486 -14.173641 L 17.007289 -14.173641 L 17.007289 14.172658 L -17.007486 14.172658 Z M -17.007486 -14.173641 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-3" x="52.427582" y="94.280533"/>
<use xlink:href="#glyph1-3" x="52.458722" y="94.280533"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-16" x="60.232892" y="94.280533"/>
<use xlink:href="#glyph0-16" x="60.264032" y="94.280533"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="64.093197" y="94.280533"/>
<use xlink:href="#glyph1-2" x="64.124338" y="94.280533"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-17" x="68.746689" y="94.280533"/>
<use xlink:href="#glyph0-17" x="68.777829" y="94.280533"/>
</g>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.440287 0.00146892 C 54.440287 5.50138 49.979204 9.962463 44.475373 9.962463 C 38.975462 9.962463 34.514379 5.50138 34.514379 0.00146892 C 34.514379 -5.502362 38.975462 -9.963445 44.475373 -9.963445 C 49.979204 -9.963445 54.440287 -5.502362 54.440287 0.00146892 Z M 54.440287 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.440397 0.00146892 C 54.440397 5.50138 49.979315 9.962463 44.475483 9.962463 C 38.975572 9.962463 34.51449 5.50138 34.51449 0.00146892 C 34.51449 -5.502362 38.975572 -9.963445 44.475483 -9.963445 C 49.979315 -9.963445 54.440397 -5.502362 54.440397 0.00146892 Z M 54.440397 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="100.662465" y="95.949612"/>
<use xlink:href="#glyph2-1" x="100.693605" y="95.949612"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph3-1" x="87.202216" y="88.986315"/>
<use xlink:href="#glyph3-1" x="87.233357" y="88.986315"/>
</g>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 80.367879 35.890055 L 114.382654 35.890055 L 114.382654 64.236354 L 80.367879 64.236354 Z M 80.367879 35.890055 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 80.36799 35.890055 L 114.382764 35.890055 L 114.382764 64.236354 L 80.36799 64.236354 Z M 80.36799 35.890055 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-4" x="146.549677" y="44.395504"/>
<use xlink:href="#glyph1-4" x="146.580817" y="44.395504"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph4-1" x="155.925416" y="45.884222"/>
<use xlink:href="#glyph4-1" x="155.956556" y="45.884222"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-16" x="160.16936" y="44.395504"/>
<use xlink:href="#glyph0-16" x="160.2005" y="44.395504"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="164.029666" y="44.395504"/>
<use xlink:href="#glyph1-2" x="164.060806" y="44.395504"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-17" x="168.683157" y="44.395504"/>
<use xlink:href="#glyph0-17" x="168.714297" y="44.395504"/>
</g>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 94.085938 154.898438 L 127.980469 154.898438 L 127.980469 126.652344 L 94.085938 126.652344 Z M 94.085938 154.898438 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 94.117188 154.898438 L 128.011719 154.898438 L 128.011719 126.652344 L 94.117188 126.652344 Z M 94.117188 154.898438 "/>
<g clip-path="url(#clip1)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 31.680142 -63.323951 L 65.694916 -63.323951 L 65.694916 -34.977652 L 31.680142 -34.977652 Z M 31.680142 -63.323951 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 31.680252 -63.323951 L 65.695026 -63.323951 L 65.695026 -34.977652 L 31.680252 -34.977652 Z M 31.680252 -63.323951 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-5" x="100.273844" y="143.25579"/>
<use xlink:href="#glyph1-5" x="100.304984" y="143.25579"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-16" x="109.415414" y="143.25579"/>
<use xlink:href="#glyph0-16" x="109.446554" y="143.25579"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="113.276715" y="143.25579"/>
<use xlink:href="#glyph1-2" x="113.307855" y="143.25579"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-17" x="117.930207" y="143.25579"/>
<use xlink:href="#glyph0-17" x="117.961347" y="143.25579"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 17.505032 0.00146892 L 29.880812 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.05294 0.00146892 L 1.607538 1.683195 L 3.089339 0.00146892 L 1.607538 -1.684177 Z M 6.05294 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,89.464551,91.798339)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 65.777238 0.00146892 L 65.777238 50.061244 L 75.236458 50.061244 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.054267 -0.00183568 L 1.608864 1.683811 L 3.086745 -0.00183568 L 1.608864 -1.683562 Z M 6.054267 -0.00183568 " transform="matrix(0.996465,0,0,-0.996465,134.658542,41.912233)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 130.050781 91.796875 C 130.050781 90.703125 129.160156 89.8125 128.0625 89.8125 C 126.96875 89.8125 126.078125 90.703125 126.078125 91.796875 C 126.078125 92.894531 126.96875 93.785156 128.0625 93.785156 C 129.160156 93.785156 130.050781 92.894531 130.050781 91.796875 Z M 130.050781 91.796875 "/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 17.505142 0.00146892 L 29.880922 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053051 0.00146892 L 1.607649 1.683195 L 3.089449 0.00146892 L 1.607649 -1.684177 Z M 6.053051 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,89.495691,91.798339)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 65.777349 0.00146892 L 65.777349 50.061244 L 75.236569 50.061244 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.054377 -0.00183568 L 1.608975 1.683811 L 3.086855 -0.00183568 L 1.608975 -1.683562 Z M 6.054377 -0.00183568 " transform="matrix(0.996465,0,0,-0.996465,134.689682,41.912233)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 130.082031 91.796875 C 130.082031 90.703125 129.191406 89.8125 128.09375 89.8125 C 127 89.8125 126.109375 90.703125 126.109375 91.796875 C 126.109375 92.894531 127 93.785156 128.09375 93.785156 C 129.191406 93.785156 130.082031 92.894531 130.082031 91.796875 Z M 130.082031 91.796875 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-6" x="134.268246" y="38.10781"/>
<use xlink:href="#glyph1-6" x="134.299386" y="38.10781"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 114.880507 50.061244 L 132.924763 50.061244 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 198.175781 41.914062 L 193.746094 40.234375 L 195.222656 41.914062 L 193.746094 43.589844 Z M 198.175781 41.914062 "/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 114.880618 50.061244 L 132.924874 50.061244 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.052782 -0.00183568 L 1.60738 1.683811 L 3.08918 -0.00183568 L 1.60738 -1.683562 Z M 6.052782 -0.00183568 " transform="matrix(0.996465,0,0,-0.996465,192.175646,41.912233)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-7" x="188.45925" y="38.10781"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-18" x="196.270539" y="38.10781"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-15" x="206.831288" y="38.10781"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-6" x="206.749365" y="38.10781"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.440397 0.00146892 L 137.558441 0.00146892 L 137.558441 -49.148841 L 70.826447 -49.148841 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 129.921875 140.773438 L 134.351562 142.453125 L 132.878906 140.773438 L 134.351562 139.097656 Z M 129.921875 140.773438 "/>
<g clip-path="url(#clip2)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.052672 -0.00183568 L 1.607269 1.683811 L 3.08907 -0.00183568 L 1.607269 -1.683562 Z M 6.052672 -0.00183568 " transform="matrix(0.996465,0,0,-0.996465,192.144506,41.912233)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053771 -0.00111851 L 1.608369 1.684528 L 3.086249 -0.00111851 L 1.608369 -1.682845 Z M 6.053771 -0.00111851 " transform="matrix(-0.996465,0,0,0.996465,135.954246,140.774552)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-15" x="191.838013" y="38.10781"/>
<use xlink:href="#glyph1-8" x="203.424161" y="118.423883"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 31.182398 -49.148841 L -40.183163 -49.148841 L -40.183163 0.00146892 L -22.138907 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053841 0.00146892 L 1.608439 1.683195 L 3.08632 0.00146892 L 1.608439 -1.684177 Z M 6.053841 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,37.658965,91.798339)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-9" x="13.019635" y="118.423883"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -45.851638 50.061244 L 44.475483 50.061244 L 44.475483 14.59603 " transform="matrix(0.996465,0,0,-0.996465,62.548926,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053298 -0.00171665 L 1.607896 1.68393 L 3.085777 -0.00171665 L 1.607896 -1.683443 Z M 6.053298 -0.00171665 " transform="matrix(0,0.996465,0.996465,0,106.868898,74.425131)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-10" x="4.176009" y="38.10781"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-6" x="191.753314" y="38.10781"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.440287 0.00146892 L 131.71737 0.00146892 L 131.71737 -49.148841 L 70.826337 -49.148841 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 129.890625 140.773438 L 134.320312 142.453125 L 132.847656 140.773438 L 134.320312 139.097656 Z M 129.890625 140.773438 "/>
<g clip-path="url(#clip3)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053881 -0.00111851 L 1.608479 1.684528 L 3.08636 -0.00111851 L 1.608479 -1.682845 Z M 6.053881 -0.00111851 " transform="matrix(-0.996465,0,0,0.996465,135.923106,140.774552)"/>
<use xlink:href="#glyph0-18" x="14.307068" y="38.10781"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-7" x="197.573666" y="118.423883"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 31.182288 -49.148841 L -34.514797 -49.148841 L -34.514797 0.00146892 L -22.139017 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053731 0.00146892 L 1.608329 1.683195 L 3.08621 0.00146892 L 1.608329 -1.684177 Z M 6.053731 0.00146892 " transform="matrix(0.996465,0,0,-0.996465,37.627825,91.798339)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-8" x="18.637455" y="118.423883"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -45.851749 50.061244 L 44.475373 50.061244 L 44.475373 14.59603 " transform="matrix(0.996465,0,0,-0.996465,62.517786,91.798339)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053298 -0.00182691 L 1.607896 1.683819 L 3.085777 -0.00182691 L 1.607896 -1.683553 Z M 6.053298 -0.00182691 " transform="matrix(0,0.996465,0.996465,0,106.837758,74.425131)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-9" x="4.144868" y="38.10781"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-18" x="14.275928" y="38.10781"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-10" x="24.75575" y="38.10781"/>
<use xlink:href="#glyph1-11" x="24.78689" y="38.10781"/>
</g>
</g>
</svg>

Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 45 KiB

After

Width:  |  Height:  |  Size: 46 KiB

BIN
figs/loop_shaping_S_without_W.pdf
View File

Binary file not shown.

BIN
figs/loop_shaping_S_without_W.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 15 KiB

After

Width:  |  Height:  |  Size: 15 KiB

143
figs/loop_shaping_S_without_W.svg
View File

@ -1,5 +1,5 @@
<?xml version="1.0" encoding="UTF-8"?>
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="160.592pt" height="119.836pt" viewBox="0 0 160.592 119.836" version="1.2">
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="168.627pt" height="119.836pt" viewBox="0 0 168.627 119.836" version="1.2">
<defs>
<g>
<symbol overflow="visible" id="glyph0-0">
@ -63,18 +63,21 @@
<path style="stroke:none;" d="M 7.359375 -0.203125 C 7.359375 -0.3125 7.25 -0.3125 7.15625 -0.3125 C 6.75 -0.3125 6.625 -0.40625 6.46875 -0.75 L 5.0625 -4.015625 C 5.046875 -4.046875 5.015625 -4.125 5.015625 -4.15625 C 5.015625 -4.15625 5.1875 -4.296875 5.296875 -4.375 L 7.03125 -5.71875 C 7.96875 -6.40625 8.359375 -6.453125 8.65625 -6.484375 C 8.734375 -6.484375 8.828125 -6.5 8.828125 -6.671875 C 8.828125 -6.71875 8.796875 -6.78125 8.71875 -6.78125 C 8.5 -6.78125 8.265625 -6.75 8.015625 -6.75 C 7.65625 -6.75 7.28125 -6.78125 6.921875 -6.78125 C 6.84375 -6.78125 6.734375 -6.78125 6.734375 -6.59375 C 6.734375 -6.515625 6.78125 -6.484375 6.84375 -6.484375 C 7.0625 -6.453125 7.15625 -6.40625 7.15625 -6.265625 C 7.15625 -6.09375 6.859375 -5.859375 6.796875 -5.8125 L 2.921875 -2.828125 L 3.71875 -6.015625 C 3.8125 -6.375 3.828125 -6.484375 4.546875 -6.484375 C 4.796875 -6.484375 4.890625 -6.484375 4.890625 -6.671875 C 4.890625 -6.765625 4.8125 -6.78125 4.75 -6.78125 L 3.484375 -6.75 L 2.203125 -6.78125 C 2.125 -6.78125 2 -6.78125 2 -6.59375 C 2 -6.484375 2.09375 -6.484375 2.28125 -6.484375 C 2.421875 -6.484375 2.59375 -6.46875 2.71875 -6.453125 C 2.875 -6.4375 2.9375 -6.40625 2.9375 -6.296875 C 2.9375 -6.265625 2.921875 -6.234375 2.890625 -6.109375 L 1.5625 -0.78125 C 1.453125 -0.390625 1.4375 -0.3125 0.65625 -0.3125 C 0.484375 -0.3125 0.375 -0.3125 0.375 -0.125 C 0.375 0 0.5 0 0.53125 0 L 1.78125 -0.03125 L 2.421875 -0.015625 C 2.640625 -0.015625 2.859375 0 3.078125 0 C 3.140625 0 3.265625 0 3.265625 -0.203125 C 3.265625 -0.3125 3.1875 -0.3125 2.984375 -0.3125 C 2.625 -0.3125 2.34375 -0.3125 2.34375 -0.484375 C 2.34375 -0.5625 2.40625 -0.78125 2.4375 -0.921875 L 2.828125 -2.484375 L 4.3125 -3.640625 L 5.46875 -0.96875 C 5.578125 -0.703125 5.578125 -0.671875 5.578125 -0.609375 C 5.578125 -0.3125 5.15625 -0.3125 5.0625 -0.3125 C 4.953125 -0.3125 4.84375 -0.3125 4.84375 -0.109375 C 4.84375 0 4.984375 0 4.984375 0 C 5.390625 0 5.796875 -0.03125 6.203125 -0.03125 C 6.421875 -0.03125 6.953125 0 7.171875 0 C 7.21875 0 7.359375 0 7.359375 -0.203125 Z M 7.359375 -0.203125 "/>
</symbol>
<symbol overflow="visible" id="glyph1-5">
<path style="stroke:none;" d="M 3.5625 -0.390625 C 3.5625 -0.421875 3.546875 -0.53125 3.453125 -0.53125 C 3.453125 -0.53125 3.421875 -0.53125 3.328125 -0.484375 C 3.015625 -0.28125 2.65625 -0.109375 2.28125 -0.109375 C 1.703125 -0.109375 1.21875 -0.53125 1.21875 -1.40625 C 1.21875 -1.75 1.296875 -2.125 1.328125 -2.25 L 2.96875 -2.25 C 3.125 -2.25 3.296875 -2.25 3.296875 -2.421875 C 3.296875 -2.546875 3.1875 -2.546875 3.015625 -2.546875 L 1.40625 -2.546875 C 1.640625 -3.40625 2.203125 -3.96875 3.09375 -3.96875 L 3.40625 -3.96875 C 3.578125 -3.96875 3.734375 -3.96875 3.734375 -4.140625 C 3.734375 -4.28125 3.609375 -4.28125 3.4375 -4.28125 L 3.09375 -4.28125 C 1.796875 -4.28125 0.46875 -3.296875 0.46875 -1.765625 C 0.46875 -0.671875 1.21875 0.109375 2.265625 0.109375 C 2.90625 0.109375 3.5625 -0.28125 3.5625 -0.390625 Z M 3.5625 -0.390625 "/>
<path style="stroke:none;" d="M 4.328125 -1.421875 C 4.328125 -1.46875 4.28125 -1.515625 4.203125 -1.515625 C 4.109375 -1.515625 4.09375 -1.453125 4.0625 -1.390625 C 3.828125 -0.75 3.203125 -0.5625 2.875 -0.5625 C 2.671875 -0.5625 2.5 -0.609375 2.28125 -0.6875 C 1.953125 -0.8125 1.796875 -0.859375 1.59375 -0.859375 C 1.59375 -0.859375 1.40625 -0.859375 1.328125 -0.828125 C 1.859375 -1.40625 2.140625 -1.65625 2.5 -1.953125 C 2.5 -1.953125 3.125 -2.5 3.484375 -2.859375 C 4.421875 -3.78125 4.640625 -4.25 4.640625 -4.296875 C 4.640625 -4.390625 4.53125 -4.390625 4.53125 -4.390625 C 4.46875 -4.390625 4.4375 -4.375 4.375 -4.28125 C 4.078125 -3.8125 3.875 -3.640625 3.640625 -3.640625 C 3.40625 -3.640625 3.28125 -3.796875 3.125 -3.96875 C 2.9375 -4.1875 2.765625 -4.390625 2.4375 -4.390625 C 1.703125 -4.390625 1.25 -3.46875 1.25 -3.265625 C 1.25 -3.203125 1.265625 -3.15625 1.359375 -3.15625 C 1.453125 -3.15625 1.46875 -3.203125 1.484375 -3.265625 C 1.671875 -3.71875 2.25 -3.734375 2.328125 -3.734375 C 2.546875 -3.734375 2.734375 -3.65625 2.96875 -3.578125 C 3.359375 -3.421875 3.46875 -3.421875 3.734375 -3.421875 C 3.375 -3 2.53125 -2.28125 2.34375 -2.125 L 1.453125 -1.296875 C 0.78125 -0.625 0.421875 -0.0625 0.421875 0.015625 C 0.421875 0.109375 0.546875 0.109375 0.546875 0.109375 C 0.625 0.109375 0.640625 0.09375 0.703125 -0.015625 C 0.9375 -0.375 1.234375 -0.640625 1.546875 -0.640625 C 1.78125 -0.640625 1.875 -0.546875 2.125 -0.265625 C 2.296875 -0.046875 2.46875 0.109375 2.765625 0.109375 C 3.75 0.109375 4.328125 -1.15625 4.328125 -1.421875 Z M 4.328125 -1.421875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-6">
<path style="stroke:none;" d="M 4.65625 -3.703125 C 4.65625 -4.234375 4.390625 -4.390625 4.21875 -4.390625 C 3.96875 -4.390625 3.734375 -4.140625 3.734375 -3.921875 C 3.734375 -3.78125 3.78125 -3.734375 3.890625 -3.625 C 4.09375 -3.421875 4.21875 -3.15625 4.21875 -2.796875 C 4.21875 -2.390625 3.625 -0.109375 2.453125 -0.109375 C 1.953125 -0.109375 1.71875 -0.453125 1.71875 -0.96875 C 1.71875 -1.53125 1.984375 -2.25 2.296875 -3.078125 C 2.359375 -3.25 2.421875 -3.390625 2.421875 -3.578125 C 2.421875 -4.03125 2.09375 -4.390625 1.59375 -4.390625 C 0.671875 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.859375 -3.9375 1.28125 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.171875 1.8125 -3.859375 C 1.8125 -3.609375 1.71875 -3.34375 1.65625 -3.15625 C 1.21875 -2 1.078125 -1.546875 1.078125 -1.125 C 1.078125 -0.046875 1.953125 0.109375 2.421875 0.109375 C 4.078125 0.109375 4.65625 -3.1875 4.65625 -3.703125 Z M 4.65625 -3.703125 "/>
<path style="stroke:none;" d="M 3.5625 -0.390625 C 3.5625 -0.421875 3.546875 -0.53125 3.453125 -0.53125 C 3.453125 -0.53125 3.421875 -0.53125 3.328125 -0.484375 C 3.015625 -0.28125 2.65625 -0.109375 2.28125 -0.109375 C 1.703125 -0.109375 1.21875 -0.53125 1.21875 -1.40625 C 1.21875 -1.75 1.296875 -2.125 1.328125 -2.25 L 2.96875 -2.25 C 3.125 -2.25 3.296875 -2.25 3.296875 -2.421875 C 3.296875 -2.546875 3.1875 -2.546875 3.015625 -2.546875 L 1.40625 -2.546875 C 1.640625 -3.40625 2.203125 -3.96875 3.09375 -3.96875 L 3.40625 -3.96875 C 3.578125 -3.96875 3.734375 -3.96875 3.734375 -4.140625 C 3.734375 -4.28125 3.609375 -4.28125 3.4375 -4.28125 L 3.09375 -4.28125 C 1.796875 -4.28125 0.46875 -3.296875 0.46875 -1.765625 C 0.46875 -0.671875 1.21875 0.109375 2.265625 0.109375 C 2.90625 0.109375 3.5625 -0.28125 3.5625 -0.390625 Z M 3.5625 -0.390625 "/>
</symbol>
<symbol overflow="visible" id="glyph1-7">
<path style="stroke:none;" d="M 5.390625 -1.421875 C 5.390625 -1.515625 5.3125 -1.515625 5.28125 -1.515625 C 5.171875 -1.515625 5.171875 -1.484375 5.140625 -1.34375 C 5 -0.78125 4.8125 -0.109375 4.390625 -0.109375 C 4.1875 -0.109375 4.078125 -0.234375 4.078125 -0.5625 C 4.078125 -0.78125 4.203125 -1.25 4.28125 -1.59375 L 4.5625 -2.671875 C 4.59375 -2.828125 4.6875 -3.203125 4.734375 -3.34375 C 4.78125 -3.578125 4.875 -3.953125 4.875 -4.015625 C 4.875 -4.1875 4.734375 -4.28125 4.59375 -4.28125 C 4.546875 -4.28125 4.28125 -4.265625 4.203125 -3.9375 L 3.453125 -0.9375 C 3.453125 -0.90625 3.046875 -0.109375 2.328125 -0.109375 C 1.8125 -0.109375 1.703125 -0.5625 1.703125 -0.921875 C 1.703125 -1.484375 1.984375 -2.265625 2.25 -2.953125 C 2.359375 -3.25 2.421875 -3.390625 2.421875 -3.578125 C 2.421875 -4.03125 2.09375 -4.390625 1.59375 -4.390625 C 0.65625 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.8125 -3.8125 1.1875 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.15625 1.8125 -3.84375 C 1.8125 -3.609375 1.703125 -3.3125 1.65625 -3.171875 C 1.28125 -2.1875 1.078125 -1.5625 1.078125 -1.078125 C 1.078125 -0.140625 1.765625 0.109375 2.296875 0.109375 C 2.953125 0.109375 3.3125 -0.34375 3.484375 -0.5625 C 3.59375 -0.15625 3.9375 0.109375 4.359375 0.109375 C 4.703125 0.109375 4.9375 -0.125 5.09375 -0.4375 C 5.265625 -0.796875 5.390625 -1.421875 5.390625 -1.421875 Z M 5.390625 -1.421875 "/>
<path style="stroke:none;" d="M 4.65625 -3.703125 C 4.65625 -4.234375 4.390625 -4.390625 4.21875 -4.390625 C 3.96875 -4.390625 3.734375 -4.140625 3.734375 -3.921875 C 3.734375 -3.78125 3.78125 -3.734375 3.890625 -3.625 C 4.09375 -3.421875 4.21875 -3.15625 4.21875 -2.796875 C 4.21875 -2.390625 3.625 -0.109375 2.453125 -0.109375 C 1.953125 -0.109375 1.71875 -0.453125 1.71875 -0.96875 C 1.71875 -1.53125 1.984375 -2.25 2.296875 -3.078125 C 2.359375 -3.25 2.421875 -3.390625 2.421875 -3.578125 C 2.421875 -4.03125 2.09375 -4.390625 1.59375 -4.390625 C 0.671875 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.859375 -3.9375 1.28125 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.171875 1.8125 -3.859375 C 1.8125 -3.609375 1.71875 -3.34375 1.65625 -3.15625 C 1.21875 -2 1.078125 -1.546875 1.078125 -1.125 C 1.078125 -0.046875 1.953125 0.109375 2.421875 0.109375 C 4.078125 0.109375 4.65625 -3.1875 4.65625 -3.703125 Z M 4.65625 -3.703125 "/>
</symbol>
<symbol overflow="visible" id="glyph1-8">
<path style="stroke:none;" d="M 6.859375 -3.703125 C 6.859375 -4.25 6.59375 -4.390625 6.421875 -4.390625 C 6.1875 -4.390625 5.9375 -4.140625 5.9375 -3.921875 C 5.9375 -3.78125 6 -3.734375 6.078125 -3.640625 C 6.1875 -3.53125 6.4375 -3.28125 6.4375 -2.796875 C 6.4375 -2.46875 6.15625 -1.484375 5.890625 -0.984375 C 5.640625 -0.453125 5.28125 -0.109375 4.796875 -0.109375 C 4.328125 -0.109375 4.0625 -0.40625 4.0625 -0.96875 C 4.0625 -1.25 4.140625 -1.5625 4.171875 -1.703125 L 4.59375 -3.375 C 4.640625 -3.59375 4.734375 -3.953125 4.734375 -4.015625 C 4.734375 -4.1875 4.59375 -4.28125 4.453125 -4.28125 C 4.328125 -4.28125 4.15625 -4.203125 4.078125 -4 C 4.046875 -3.9375 3.59375 -2.03125 3.515625 -1.78125 C 3.453125 -1.484375 3.421875 -1.296875 3.421875 -1.125 C 3.421875 -1.015625 3.421875 -1 3.4375 -0.9375 C 3.203125 -0.421875 2.90625 -0.109375 2.53125 -0.109375 C 1.734375 -0.109375 1.734375 -0.84375 1.734375 -1.015625 C 1.734375 -1.328125 1.78125 -1.71875 2.25 -2.9375 C 2.359375 -3.234375 2.421875 -3.375 2.421875 -3.578125 C 2.421875 -4.03125 2.09375 -4.390625 1.59375 -4.390625 C 0.65625 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.828125 -3.859375 1.21875 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.15625 1.8125 -3.84375 C 1.8125 -3.59375 1.703125 -3.3125 1.640625 -3.15625 C 1.203125 -1.984375 1.078125 -1.515625 1.078125 -1.140625 C 1.078125 -0.234375 1.75 0.109375 2.5 0.109375 C 2.65625 0.109375 3.125 0.109375 3.53125 -0.59375 C 3.78125 0.046875 4.46875 0.109375 4.765625 0.109375 C 5.515625 0.109375 5.953125 -0.515625 6.21875 -1.109375 C 6.546875 -1.890625 6.859375 -3.21875 6.859375 -3.703125 Z M 6.859375 -3.703125 "/>
<path style="stroke:none;" d="M 5.390625 -1.421875 C 5.390625 -1.515625 5.3125 -1.515625 5.28125 -1.515625 C 5.171875 -1.515625 5.171875 -1.484375 5.140625 -1.34375 C 5 -0.78125 4.8125 -0.109375 4.390625 -0.109375 C 4.1875 -0.109375 4.078125 -0.234375 4.078125 -0.5625 C 4.078125 -0.78125 4.203125 -1.25 4.28125 -1.59375 L 4.5625 -2.671875 C 4.59375 -2.828125 4.6875 -3.203125 4.734375 -3.34375 C 4.78125 -3.578125 4.875 -3.953125 4.875 -4.015625 C 4.875 -4.1875 4.734375 -4.28125 4.59375 -4.28125 C 4.546875 -4.28125 4.28125 -4.265625 4.203125 -3.9375 L 3.453125 -0.9375 C 3.453125 -0.90625 3.046875 -0.109375 2.328125 -0.109375 C 1.8125 -0.109375 1.703125 -0.5625 1.703125 -0.921875 C 1.703125 -1.484375 1.984375 -2.265625 2.25 -2.953125 C 2.359375 -3.25 2.421875 -3.390625 2.421875 -3.578125 C 2.421875 -4.03125 2.09375 -4.390625 1.59375 -4.390625 C 0.65625 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.8125 -3.8125 1.1875 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.15625 1.8125 -3.84375 C 1.8125 -3.609375 1.703125 -3.3125 1.65625 -3.171875 C 1.28125 -2.1875 1.078125 -1.5625 1.078125 -1.078125 C 1.078125 -0.140625 1.765625 0.109375 2.296875 0.109375 C 2.953125 0.109375 3.3125 -0.34375 3.484375 -0.5625 C 3.59375 -0.15625 3.9375 0.109375 4.359375 0.109375 C 4.703125 0.109375 4.9375 -0.125 5.09375 -0.4375 C 5.265625 -0.796875 5.390625 -1.421875 5.390625 -1.421875 Z M 5.390625 -1.421875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-9">
<path style="stroke:none;" d="M 6.859375 -3.703125 C 6.859375 -4.25 6.59375 -4.390625 6.421875 -4.390625 C 6.1875 -4.390625 5.9375 -4.140625 5.9375 -3.921875 C 5.9375 -3.78125 6 -3.734375 6.078125 -3.640625 C 6.1875 -3.53125 6.4375 -3.28125 6.4375 -2.796875 C 6.4375 -2.46875 6.15625 -1.484375 5.890625 -0.984375 C 5.640625 -0.453125 5.28125 -0.109375 4.796875 -0.109375 C 4.328125 -0.109375 4.0625 -0.40625 4.0625 -0.96875 C 4.0625 -1.25 4.140625 -1.5625 4.171875 -1.703125 L 4.59375 -3.375 C 4.640625 -3.59375 4.734375 -3.953125 4.734375 -4.015625 C 4.734375 -4.1875 4.59375 -4.28125 4.453125 -4.28125 C 4.328125 -4.28125 4.15625 -4.203125 4.078125 -4 C 4.046875 -3.9375 3.59375 -2.03125 3.515625 -1.78125 C 3.453125 -1.484375 3.421875 -1.296875 3.421875 -1.125 C 3.421875 -1.015625 3.421875 -1 3.4375 -0.9375 C 3.203125 -0.421875 2.90625 -0.109375 2.53125 -0.109375 C 1.734375 -0.109375 1.734375 -0.84375 1.734375 -1.015625 C 1.734375 -1.328125 1.78125 -1.71875 2.25 -2.9375 C 2.359375 -3.234375 2.421875 -3.375 2.421875 -3.578125 C 2.421875 -4.03125 2.09375 -4.390625 1.59375 -4.390625 C 0.65625 -4.390625 0.28125 -2.953125 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.78125 0.5625 -2.9375 C 0.828125 -3.859375 1.21875 -4.171875 1.5625 -4.171875 C 1.65625 -4.171875 1.8125 -4.15625 1.8125 -3.84375 C 1.8125 -3.59375 1.703125 -3.3125 1.640625 -3.15625 C 1.203125 -1.984375 1.078125 -1.515625 1.078125 -1.140625 C 1.078125 -0.234375 1.75 0.109375 2.5 0.109375 C 2.65625 0.109375 3.125 0.109375 3.53125 -0.59375 C 3.78125 0.046875 4.46875 0.109375 4.765625 0.109375 C 5.515625 0.109375 5.953125 -0.515625 6.21875 -1.109375 C 6.546875 -1.890625 6.859375 -3.21875 6.859375 -3.703125 Z M 6.859375 -3.703125 "/>
</symbol>
<symbol overflow="visible" id="glyph1-10">
<path style="stroke:none;" d="M 4.328125 -3.75 C 4.328125 -4.09375 4.015625 -4.390625 3.515625 -4.390625 C 2.859375 -4.390625 2.421875 -3.90625 2.234375 -3.625 C 2.15625 -4.078125 1.796875 -4.390625 1.328125 -4.390625 C 0.875 -4.390625 0.6875 -4 0.59375 -3.828125 C 0.421875 -3.484375 0.28125 -2.890625 0.28125 -2.859375 C 0.28125 -2.765625 0.40625 -2.765625 0.40625 -2.765625 C 0.5 -2.765625 0.515625 -2.765625 0.578125 -2.984375 C 0.75 -3.703125 0.9375 -4.171875 1.296875 -4.171875 C 1.46875 -4.171875 1.609375 -4.09375 1.609375 -3.71875 C 1.609375 -3.515625 1.578125 -3.40625 1.453125 -2.875 L 0.875 -0.59375 C 0.84375 -0.4375 0.78125 -0.203125 0.78125 -0.15625 C 0.78125 0.015625 0.921875 0.109375 1.078125 0.109375 C 1.1875 0.109375 1.375 0.03125 1.4375 -0.171875 C 1.453125 -0.203125 1.796875 -1.5625 1.84375 -1.734375 L 2.15625 -3.03125 C 2.203125 -3.15625 2.46875 -3.625 2.71875 -3.84375 C 2.796875 -3.921875 3.078125 -4.171875 3.515625 -4.171875 C 3.765625 -4.171875 3.921875 -4.046875 3.921875 -4.046875 C 3.625 -4 3.40625 -3.765625 3.40625 -3.515625 C 3.40625 -3.34375 3.515625 -3.15625 3.78125 -3.15625 C 4.046875 -3.15625 4.328125 -3.390625 4.328125 -3.75 Z M 4.328125 -3.75 "/>
</symbol>
<symbol overflow="visible" id="glyph2-0">
@ -91,127 +94,127 @@
</symbol>
</g>
<clipPath id="clip1">
<path d="M 32 0.0703125 L 143 0.0703125 L 143 88 L 32 88 Z M 32 0.0703125 "/>
<path d="M 32 0.0507812 L 140 0.0507812 L 140 88 L 32 88 Z M 32 0.0507812 "/>
</clipPath>
<clipPath id="clip2">
<path d="M 64 89 L 100 89 L 100 119.601562 L 64 119.601562 Z M 64 89 "/>
<path d="M 64 89 L 100 89 L 100 119.621094 L 64 119.621094 Z M 64 89 "/>
</clipPath>
<clipPath id="clip3">
<path d="M 133 9 L 160.183594 9 L 160.183594 41 L 133 41 Z M 133 9 "/>
</clipPath>
<clipPath id="clip4">
<path d="M 86 88 L 120 88 L 120 119.601562 L 86 119.601562 Z M 86 88 "/>
<path d="M 86 88 L 120 88 L 120 119.621094 L 86 119.621094 Z M 86 88 "/>
</clipPath>
</defs>
<g id="surface1">
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,79.998779%);fill-opacity:1;" d="M 32.855469 86.746094 L 142.183594 86.746094 L 142.183594 1.066406 L 32.855469 1.066406 Z M 32.855469 86.746094 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(79.998779%,79.998779%,79.998779%);fill-opacity:1;" d="M 32.867188 86.757812 L 138.898438 86.757812 L 138.898438 1.046875 L 32.867188 1.046875 Z M 32.867188 86.757812 "/>
<g clip-path="url(#clip1)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -29.462059 -26.625253 L 80.143825 -26.625253 L 80.143825 59.272112 L -29.462059 59.272112 Z M -29.462059 -26.625253 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:2.98883,2.98883;stroke-miterlimit:10;" d="M -29.461111 -26.628182 L 76.804678 -26.628182 L 76.804678 59.272347 L -29.461111 59.272347 Z M -29.461111 -26.628182 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="36.888282" y="11.831169"/>
<use xlink:href="#glyph0-2" x="44.686123" y="11.831169"/>
<use xlink:href="#glyph0-3" x="49.103276" y="11.831169"/>
<use xlink:href="#glyph0-2" x="54.62447" y="11.831169"/>
<use xlink:href="#glyph0-4" x="59.041623" y="11.831169"/>
<use xlink:href="#glyph0-5" x="62.934084" y="11.831169"/>
<use xlink:href="#glyph0-6" x="67.902761" y="11.831169"/>
<use xlink:href="#glyph0-7" x="70.663358" y="11.831169"/>
<use xlink:href="#glyph0-8" x="73.423954" y="11.831169"/>
<use xlink:href="#glyph0-2" x="77.841108" y="11.831169"/>
<use xlink:href="#glyph0-9" x="82.258261" y="11.831169"/>
<use xlink:href="#glyph0-1" x="35.235061" y="11.815402"/>
<use xlink:href="#glyph0-2" x="43.035459" y="11.815402"/>
<use xlink:href="#glyph0-3" x="47.45406" y="11.815402"/>
<use xlink:href="#glyph0-2" x="52.977064" y="11.815402"/>
<use xlink:href="#glyph0-4" x="57.395666" y="11.815402"/>
<use xlink:href="#glyph0-5" x="61.289404" y="11.815402"/>
<use xlink:href="#glyph0-6" x="66.259709" y="11.815402"/>
<use xlink:href="#glyph0-7" x="69.021211" y="11.815402"/>
<use xlink:href="#glyph0-8" x="71.782713" y="11.815402"/>
<use xlink:href="#glyph0-2" x="76.201315" y="11.815402"/>
<use xlink:href="#glyph0-9" x="80.619917" y="11.815402"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-10" x="91.088593" y="11.831169"/>
<use xlink:href="#glyph0-6" x="97.851956" y="11.831169"/>
<use xlink:href="#glyph0-5" x="100.612553" y="11.831169"/>
<use xlink:href="#glyph0-3" x="105.581229" y="11.831169"/>
<use xlink:href="#glyph0-10" x="89.453144" y="11.815402"/>
<use xlink:href="#glyph0-6" x="96.218724" y="11.815402"/>
<use xlink:href="#glyph0-5" x="98.980226" y="11.815402"/>
<use xlink:href="#glyph0-3" x="103.950532" y="11.815402"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-11" x="110.824177" y="11.831169"/>
<use xlink:href="#glyph0-11" x="109.195199" y="11.815402"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="118.002204" y="11.831169"/>
<use xlink:href="#glyph1-1" x="116.375579" y="11.815402"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-12" x="125.76149" y="11.831169"/>
<use xlink:href="#glyph0-12" x="124.138408" y="11.815402"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="129.626671" y="11.831169"/>
<use xlink:href="#glyph1-2" x="128.003858" y="11.815402"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-13" x="134.284836" y="11.831169"/>
<use xlink:href="#glyph0-13" x="132.663551" y="11.815402"/>
</g>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -17.008625 -14.171819 L 17.007265 -14.171819 L 17.007265 14.17345 L -17.008625 14.17345 Z M -17.008625 -14.171819 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -17.007844 -14.174915 L 17.00864 -14.174915 L 17.00864 14.172808 L -17.007844 14.172808 Z M -17.007844 -14.174915 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-3" x="52.142527" y="62.673001"/>
<use xlink:href="#glyph1-3" x="52.159624" y="62.673904"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-12" x="59.955677" y="62.673001"/>
<use xlink:href="#glyph0-12" x="59.975336" y="62.673904"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="63.819859" y="62.673001"/>
<use xlink:href="#glyph1-2" x="63.840785" y="62.673904"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-13" x="68.478025" y="62.673001"/>
<use xlink:href="#glyph0-13" x="68.500478" y="62.673904"/>
</g>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.438058 0.000815308 C 54.438058 5.50304 49.981452 9.963563 44.475311 9.963563 C 38.973086 9.963563 34.512563 5.50304 34.512563 0.000815308 C 34.512563 -5.501409 38.973086 -9.961932 44.475311 -9.961932 C 49.981452 -9.961932 54.438058 -5.501409 54.438058 0.000815308 Z M 54.438058 0.000815308 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.438909 0.000903871 C 54.438909 5.501325 49.979848 9.9643 44.475512 9.9643 C 38.975091 9.9643 34.516031 5.501325 34.516031 0.000903871 C 34.516031 -5.503432 38.975091 -9.962493 44.475512 -9.962493 C 49.979848 -9.962493 54.438909 -5.503432 54.438909 0.000903871 Z M 54.438909 0.000903871 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="100.425858" y="64.344753"/>
<use xlink:href="#glyph2-1" x="100.458787" y="64.346205"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph3-1" x="86.95209" y="57.373465"/>
<use xlink:href="#glyph3-1" x="86.9806" y="57.37263"/>
</g>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 65.152344 118.605469 L 99.082031 118.605469 L 99.082031 90.332031 L 65.152344 90.332031 Z M 65.152344 118.605469 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 65.175781 118.625 L 99.117188 118.625 L 99.117188 90.34375 L 65.175781 90.34375 Z M 65.175781 118.625 "/>
<g clip-path="url(#clip2)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 2.91687 -58.56557 L 36.932759 -58.56557 L 36.932759 -30.220301 L 2.91687 -30.220301 Z M 2.91687 -58.56557 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 2.918949 -58.565859 L 36.935433 -58.565859 L 36.935433 -30.222051 L 2.918949 -30.222051 Z M 2.918949 -58.565859 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-4" x="71.348731" y="106.953502"/>
<use xlink:href="#glyph1-4" x="71.372126" y="106.968924"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-12" x="80.499483" y="106.953502"/>
<use xlink:href="#glyph0-12" x="80.525878" y="106.968924"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-2" x="84.364663" y="106.953502"/>
<use xlink:href="#glyph1-2" x="84.392326" y="106.968924"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-13" x="89.022829" y="106.953502"/>
<use xlink:href="#glyph0-13" x="89.052018" y="106.968924"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 17.504619 0.000815308 L 29.879729 0.000815308 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.052548 0.000815308 L 1.607691 1.68477 L 3.088004 0.000815308 L 1.607691 -1.68314 Z M 6.052548 0.000815308 " transform="matrix(0.997466,0,0,-0.997466,89.216696,60.188313)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 65.779298 0.000815308 L 65.779298 35.391282 L 86.574183 35.391282 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 151.804688 24.886719 L 147.371094 23.207031 L 148.84375 24.886719 L 147.371094 26.566406 Z M 151.804688 24.886719 "/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 17.505831 0.000903871 L 29.8808 0.000903871 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.052565 0.000903871 L 1.609165 1.684307 L 3.088993 0.000903871 L 1.609165 -1.682499 Z M 6.052565 0.000903871 " transform="matrix(0.997793,0,0,-0.997793,89.24595,60.188402)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 65.780347 0.000903871 L 65.780347 35.391515 L 86.57233 35.391515 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.05514 -0.0000454916 L 1.607825 1.683357 L 3.087653 -0.0000454916 L 1.607825 -1.683448 Z M 6.05514 -0.0000454916 " transform="matrix(0.997793,0,0,-0.997793,145.813693,24.874955)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 129.886719 60.1875 C 129.886719 59.089844 128.996094 58.199219 127.898438 58.199219 C 126.796875 58.199219 125.910156 59.089844 125.910156 60.1875 C 125.910156 61.285156 126.796875 62.175781 127.898438 62.175781 C 128.996094 62.175781 129.886719 61.285156 129.886719 60.1875 Z M 129.886719 60.1875 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-5" x="142.092693" y="21.06594"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-14" x="149.914392" y="21.06594"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-6" x="160.407182" y="21.06594"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.438909 0.000903871 L 91.207561 0.000903871 L 91.207561 -44.393955 L 42.067855 -44.393955 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 101.027344 104.484375 L 105.464844 106.164062 L 103.988281 104.484375 L 105.464844 102.804688 Z M 101.027344 104.484375 "/>
<g clip-path="url(#clip3)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.054132 -0.000278285 L 1.609275 1.683677 L 3.085672 -0.000278285 L 1.609275 -1.684233 Z M 6.054132 -0.000278285 " transform="matrix(0.997466,0,0,-0.997466,145.765897,24.886441)"/>
</g>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 129.84375 60.1875 C 129.84375 59.089844 128.953125 58.199219 127.855469 58.199219 C 126.757812 58.199219 125.867188 59.089844 125.867188 60.1875 C 125.867188 61.285156 126.757812 62.175781 127.855469 62.175781 C 128.953125 62.175781 129.84375 61.285156 129.84375 60.1875 Z M 129.84375 60.1875 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-5" x="145.374661" y="21.078675"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 54.438058 0.000815308 L 85.368002 0.000815308 L 85.368002 -44.392936 L 42.066863 -44.392936 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 100.996094 104.46875 L 105.429688 106.148438 L 103.953125 104.46875 L 105.429688 102.789062 Z M 100.996094 104.46875 "/>
<g clip-path="url(#clip4)" clip-rule="nonzero">
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.052846 -0.000864374 L 1.607988 1.683091 L 3.088302 -0.000864374 L 1.607988 -1.684819 Z M 6.052846 -0.000864374 " transform="matrix(-0.997466,0,0,0.997466,107.0336,104.469612)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.054716 0.000155049 L 1.6074 1.683558 L 3.087228 0.000155049 L 1.6074 -1.683248 Z M 6.054716 0.000155049 " transform="matrix(-0.997793,0,0,0.997793,107.068696,104.48422)"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-6" x="151.200859" y="84.468627"/>
<use xlink:href="#glyph1-7" x="157.078545" y="84.476677"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 2.419516 -44.392936 L -34.513923 -44.392936 L -34.513923 0.000815308 L -22.138813 0.000815308 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.054626 0.000815308 L 1.609768 1.68477 L 3.086166 0.000815308 L 1.609768 -1.68314 Z M 6.054626 0.000815308 " transform="matrix(0.997466,0,0,-0.997466,37.327905,60.188313)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 2.421758 -44.393955 L -43.018377 -44.393955 L -43.018377 0.000903871 L -22.140266 0.000903871 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.05212 0.000903871 L 1.608719 1.684307 L 3.088547 0.000903871 L 1.608719 -1.682499 Z M 6.05212 0.000903871 " transform="matrix(0.997793,0,0,-0.997793,37.340144,60.188402)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-7" x="18.31846" y="84.468627"/>
<use xlink:href="#glyph1-8" x="9.839236" y="84.476677"/>
</g>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -45.851248 35.391282 L 44.475311 35.391282 L 44.475311 14.596397 " transform="matrix(0.997466,0,0,-0.997466,62.242866,60.188313)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053622 -0.00188942 L 1.608765 1.682065 L 3.089079 -0.00188942 L 1.608765 -1.681928 Z M 6.053622 -0.00188942 " transform="matrix(0,0.997466,0.997466,0,106.607353,42.797656)"/>
<path style="fill:none;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -45.852758 35.391515 L 44.475512 35.391515 L 44.475512 14.595616 " transform="matrix(0.997793,0,0,-0.997793,62.263275,60.188402)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.99628;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 6.053349 -0.00168779 L 1.609948 1.681715 L 3.085862 -0.00168779 L 1.609948 -1.681176 Z M 6.053349 -0.00168779 " transform="matrix(0,0.997793,0.997793,0,106.642309,42.792042)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-8" x="3.811317" y="21.078675"/>
<use xlink:href="#glyph1-9" x="3.812567" y="21.06594"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-14" x="13.952552" y="21.078675"/>
<use xlink:href="#glyph0-14" x="13.957127" y="21.06594"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-9" x="24.4429" y="21.078675"/>
<use xlink:href="#glyph1-10" x="24.450915" y="21.06594"/>
</g>
</g>
</svg>

Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 35 KiB

After

Width:  |  Height:  |  Size: 36 KiB

BIN
figs/results_sensitivity_hinf.pdf
View File

Binary file not shown.

BIN
figs/results_sensitivity_hinf.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 76 KiB

After

Width:  |  Height:  |  Size: 73 KiB

1353
index.html
View File

File diff suppressed because it is too large Load Diff

502
index.org
View File

@ -67,6 +67,12 @@ This document is structured as follows:
* Introduction to Model Based Control
<<sec:model_based_control>>
** Introduction :ignore:
- Section [[sec:model_based_control_methodology]]
- Section [[sec:comp_classical_modern_robust_control]]
- Section [[sec:example_system]]
** Model Based Control - Methodology
<<sec:model_based_control_methodology>>
@ -396,6 +402,8 @@ And now the system dynamics $G(s)$ and $G_d(s)$ (their bode plots are shown in F
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz'))));
hold off;
@ -417,6 +425,13 @@ And now the system dynamics $G(s)$ and $G_d(s)$ (their bode plots are shown in F
* Classical Open Loop Shaping
<<sec:open_loop_shaping>>
** Introduction :ignore:
- Section [[sec:open_loop_shaping_introduction]]
- Section [[sec:loop_shaping_example]]
- Section [[sec:h_infinity_open_loop_shaping]]
- Section [[sec:h_infinity_open_loop_shaping_example]]
** Introduction to Loop Shaping
<<sec:open_loop_shaping_introduction>>
@ -631,15 +646,13 @@ The $\mathcal{H}_\infty$ optimal open loop shaping synthesis is performed using
[K, ~, GAM] = loopsyn(G, Lw);
#+end_src
The Bode plot of the obtained controller is shown in Figure [[fig:open_loop_shaping_hinf_K]].
#+begin_important
It is always important to analyze the controller after the synthesis is performed.
In the end, a synthesize controller is just a combination of low pass filters, high pass filters, notches, leads, etc.
#+end_important
Let's briefly analyze this controller:
Let's briefly analyze the obtained controller which bode plot is shown in Figure [[fig:open_loop_shaping_hinf_K]]:
- two integrators are used at low frequency to have the wanted low frequency high gain
- a lead is added centered with the crossover frequency to increase the phase margin
- a notch is added at the resonance of the plant to increase the gain margin (this is very typical of $\mathcal{H}_\infty$ controllers, and can be an issue, more info on that latter)
@ -675,7 +688,7 @@ Let's briefly analyze this controller:
#+RESULTS:
[[file:figs/open_loop_shaping_hinf_K.png]]
The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]].
The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]] and matches the specified one by a factor $\gamma$.
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
@ -736,9 +749,17 @@ Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ con
| Phase Margin $> 30$ [deg] | 35.4 | 54.7 |
| Crossover $\approx 10$ [Hz] | 10.1 | 9.9 |
* First Steps in the $\mathcal{H}_\infty$ world
* A first Step into the $\mathcal{H}_\infty$ world
<<sec:h_infinity_introduction>>
** Introduction :ignore:
- Section [[sec:h_infinity_norm]]
- Section [[sec:h_infinity_synthesis]]
- Section [[sec:generalized_plant]]
- Section [[sec:h_infinity_general_synthesis]]
- Section [[sec:generalized_plant_derivation]]
** The $\mathcal{H}_\infty$ Norm
<<sec:h_infinity_norm>>
@ -810,7 +831,7 @@ Note that there are many ways to use the $\mathcal{H}_\infty$ Synthesis:
- Open Loop Shaping $\mathcal{H}_\infty$ Synthesis (=loopsyn= [[https://www.mathworks.com/help/robust/ref/loopsyn.html][doc]])
- Mixed Sensitivity Loop Shaping (=mixsyn= [[https://www.mathworks.com/help/robust/ref/lti.mixsyn.html][doc]])
- Fixed-Structure $\mathcal{H}_\infty$ Synthesis (=hinfstruct= [[https://www.mathworks.com/help/robust/ref/lti.hinfstruct.html][doc]])
- Signal Based $\mathcal{H}_\infty$ Synthesis
- Signal Based $\mathcal{H}_\infty$ Synthesis, and many more...
** The Generalized Plant
<<sec:generalized_plant>>
@ -1007,18 +1028,21 @@ Using Matlab, the generalized plant can be defined as follows:
P.OutputName = {'e', 'u', 'v'};
#+end_src
* Modern Interpretation of the Control Specifications
* Modern Interpretation of Control Specifications
<<sec:modern_interpretation_specification>>
** Introduction
** Introduction :ignore:
As shown in Section [[sec:open_loop_shaping]], the loop gain $L(s) = G(s) K(s)$ is a useful and easy tool for the manual design of controllers.
- Section [[sec:closed_loop_tf]]
- Section [[sec:sensitivity_transfer_functions]]
- Section [[sec:module_margin]]
- Section [[sec:other_requirements]]
$L(s)$ is very easy to shape as it depends linearly on $K(s)$.
As shown in Section [[sec:open_loop_shaping]], the loop gain $L(s) = G(s) K(s)$ is a useful and easy tool when manually designing controllers.
This is mainly due to the fact that $L(s)$ is very easy to shape as it depends /linearly/ on $K(s)$.
Moreover, important quantities such as the stability margins and the control bandwidth can be estimated from the shape/phase of $L(s)$.
However, the loop gain $L(s)$ does *not* directly give the performances of the closed-loop system.
The closed loop system behavior is indeed determined by the *closed-loop* transfer functions.
However, the loop gain $L(s)$ does *not* directly give the performances of the closed-loop system, which are determined by the *closed-loop* transfer functions.
If we consider the feedback system shown in Figure [[fig:gang_of_four_feedback]], we can link to the following specifications to closed-loop transfer functions.
This is summarized in Table [[tab:spec_closed_loop_tf]].
@ -1082,7 +1106,7 @@ Isolate $y$ at the right hand side, and finally obtain:
Do the same procedure for $u$ and $\epsilon$
#+HTML: </details>
#+HTML: <details><summary>Anwser</summary>
#+HTML: <details><summary>Answer</summary>
The following equations should be obtained:
\begin{align}
y &= \frac{GK}{1 + GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \\
@ -1115,12 +1139,16 @@ And we have:
u &= KS r - S d - KS n
\end{align}
Thus, for reference tracking, we want to shape the /closed-loop/ transfer function from $r$ to $\epsilon$, that is the sensitivity function $S(s)$.
Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we want to act on the complementary sensitivity function $T(s)$.
Thus, for reference tracking, we have to shape the /closed-loop/ transfer function from $r$ to $\epsilon$, that is the sensitivity function $S(s)$.
Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we have to act on the complementary sensitivity function $T(s)$.
** Sensitivity Function
<<sec:sensitivity_transfer_functions>>
The sensitivity function is indisputably the most important closed-loop transfer function of a feedback system.
In this section, we will see how the shape of the sensitivity function will impact the performances of the closed-loop system.
Suppose we have developed a "/reference/" controller $K_r(s)$ and made three small changes to obtained three controllers $K_1(s)$, $K_2(s)$ and $K_3(s)$.
The obtained sensitivity functions are shown in Figure [[fig:sensitivity_shape_effect]] and the corresponding step responses are shown in Figure [[fig:sensitivity_shape_effect_step]].
@ -1342,6 +1370,9 @@ This might indicate very good robustness properties of the closed-loop system.
Now let's suppose the "real" plant $G_r(s)$ as a slightly lower damping factor:
#+begin_src matlab
xi = 0.03;
#+end_src
#+begin_src matlab :exports none
Gr = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1);
#+end_src
@ -1350,7 +1381,7 @@ At a frequency little bit above 100Hz, the phase of the loop gain reaches -180 d
It is confirmed by checking the stability of the closed loop system:
#+begin_src matlab :results value replace
isstable(feedback(Gp,K))
isstable(feedback(Gr,K))
#+end_src
#+RESULTS:
@ -1413,23 +1444,18 @@ Let's now determine a new robustness indicator based on the Nyquist Stability Cr
For more information about the /general/ Nyquist Stability Criteria, you may want to look at [[https://www.youtube.com/watch?v=sof3meN96MA][this]] video.
#+end_seealso
From the Nyquist stability criteria, it is clear that we want $L(j\omega)$ to be as far away from the $-1$ point (called the /unstable point/) in the complex plane.
From this, we define the *module margin*.
From the Nyquist stability criteria, it is clear that we want $L(j\omega)$ to be as far as possible from the $-1$ point (called the /unstable point/) in the complex plane.
This minimum distance is called the *module margin*.
#+begin_definition
- Module Margin ::
The Module Margin $\Delta M$ is defined as the *minimum distance* between the point $-1$ and the loop gain $L(j\omega)$ in the complex plane.
#+end_definition
#+begin_exampl
A typical Nyquist plot is shown in Figure [[fig:module_margin_example]].
The gain, phase and module margins are graphically shown to have an idea of what they represent.
As expected from Figure [[fig:module_margin_example]], there is a close relationship between the module margin and the gain and phase margins.
We can indeed show that for a given value of the module margin $\Delta M$, we have:
\begin{equation}
\Delta G \ge \frac{\Delta M}{\Delta M - 1}; \quad \Delta \phi \ge \frac{1}{\Delta M}
\end{equation}
#+begin_src matlab :exports none
% Example Plant
k = 1e6; % Stiffness [N/m]
@ -1485,6 +1511,14 @@ We can indeed show that for a given value of the module margin $\Delta M$, we ha
#+caption: Nyquist plot with visual indication of the Gain margin $\Delta G$, Phase margin $\Delta \phi$ and Module margin $\Delta M$
#+RESULTS:
[[file:figs/module_margin_example.png]]
#+end_exampl
As expected from Figure [[fig:module_margin_example]], there is a close relationship between the module margin and the gain and phase margins.
We can indeed show that for a given value of the module margin $\Delta M$, we have:
\begin{equation}
\Delta G \ge \frac{\Delta M}{\Delta M - 1}; \quad \Delta \phi \ge \frac{1}{\Delta M}
\end{equation}
Let's now try to express the Module margin $\Delta M$ as an $\mathcal{H}_\infty$ norm of a closed-loop transfer function:
\begin{align*}
@ -1506,6 +1540,7 @@ The wanted robustness of the closed-loop system can be specified by setting a ma
#+end_important
Note that this is why large peak value of $|S(j\omega)|$ usually indicate robustness problems.
And we know understand why setting an upper bound on the magnitude of $S$ is generally a good idea.
#+begin_exampl
Typical, we require $\|S\|_\infty < 2 (6dB)$ which implies $\Delta G \ge 2$ and $\Delta \phi \ge 29^o$
@ -1515,12 +1550,59 @@ Note that this is why large peak value of $|S(j\omega)|$ usually indicate robust
To learn more about module/disk margin, you can check out [[https://www.youtube.com/watch?v=XazdN6eZF80][this]] video.
#+end_seealso
** How to *Shape* transfer function? Using of Weighting Functions!
** TODO Other Requirements
<<sec:other_requirements>>
Interpretation of the $\mathcal{H}_\infty$ norm of systems:
- frequency by frequency attenuation / amplification
Let's note $G_t(s)$ the closed-loop transfer function from $w$ to $z$.
Consider an input sinusoidal signal $w(t) = \sin\left( \omega_0 t \right)$, then the output signal $z(t)$ will be equal to:
\[ z(t) = A \sin\left( \omega_0 t + \phi \right) \]
with:
- $A = |G_t(j\omega_0)|$ is the magnitude of $G_t(s)$ at $\omega_0$
- $\phi = \angle G_t(j\omega_0)$ is the phase of $G_t(s)$ at $\omega_0$
Noise Attenuation: typical wanted shape for $T$
#+name: tab:specification_modern
#+caption: Typical Specifications and corresponding wanted norms of open and closed loop tansfer functions
| | Open-Loop Shaping | Closed-Loop Shaping |
|-----------------------------+--------------------+--------------------------------------------|
| Reference Tracking | $L$ large | $S$ small |
| Disturbance Rejection | $L$ large | $GS$ small |
| Measurement Noise Filtering | $L$ small | $T$ small |
| Small Command Amplitude | $K$ and $L$ small | $KS$ small |
| Robustness | Phase/Gain margins | Module margin: $\Vert S\Vert_\infty$ small |
* $\mathcal{H}_\infty$ Shaping of closed-loop transfer functions
<<sec:closed-loop-shaping>>
** Introduction :ignore:
In the previous sections, we have seen that the performances of the system depends on the *shape* of the closed-loop transfer function.
Therefore, the synthesis problem is to design $K(s)$ such that closed-loop system is stable and such that various closed-loop transfer functions such as $S$, $KS$ and $T$ are shaped as wanted.
This is clearly not simple as these closed-loop transfer functions does not depend linearly on $K$.
But don't worry, the $\mathcal{H}_\infty$ synthesis will do this job for us!
This
Section [[sec:weighting_functions]]
Section [[sec:weighting_functions_design]]
Section [[sec:sensitivity_shaping_example]]
Section [[sec:shaping_multiple_tf]]
** How to Shape closed-loop transfer function? Using Weighting Functions!
<<sec:weighting_functions>>
- [ ] Maybe put this section in Previous chapter
If the $\mathcal{H}_\infty$ synthesis is applied on the generalized plant $P(s)$ shown in Figure [[fig:loop_shaping_S_without_W]], it will generate a controller $K(s)$ such that the $\mathcal{H}_\infty$ norm of closed-loop transfer function from $r$ to $\epsilon$ is minimized.
This closed-loop transfer function actually correspond to the sensitivity function.
Therefore, it will minimize the the $\mathcal{H}_\infty$ norm of the sensitivity function: $\|S\|_\infty$.
Let's say we want to shape the sensitivity transfer function corresponding to the transfer function from $r$ to $\epsilon$ of the control architecture shown in Figure [[fig:loop_shaping_S_without_W]].
However, as the $\mathcal{H}_\infty$ norm is the maximum peak value of the transfer function's magnitude, this synthesis is quite useless and clearly does not allow to *shape* the norm of $S(j\omega)$ over all frequencies.
#+begin_src latex :file loop_shaping_S_without_W.pdf
\begin{tikzpicture}
@ -1534,10 +1616,10 @@ Let's say we want to shape the sensitivity transfer function corresponding to th
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (epsilon) node[above left](z1){$\epsilon$};
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (epsilon) node[above](z1){$z = \epsilon$};
\draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.4, 0)$) -- (G.west);
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.1, 0)$) -- (G.west);
\draw[->] (w) node[above]{$w = r$} -| (addw.north);
@ -1553,25 +1635,24 @@ Let's say we want to shape the sensitivity transfer function corresponding to th
#+RESULTS:
[[file:figs/loop_shaping_S_without_W.png]]
If the $\mathcal{H}_\infty$ synthesis is directly applied on the generalized plant $P(s)$ shown in Figure [[fig:loop_shaping_S_without_W]], if will minimize the $\mathcal{H}_\infty$ norm of transfer function from $r$ to $\epsilon$ (the sensitivity transfer function).
However, as the $\mathcal{H}_\infty$ norm is the maximum peak value of the transfer function's magnitude, it does not allow to *shape* the norm over all frequencies.
#+begin_important
The /trick/ is to include a *weighting function* $W_S(s)$ in the generalized plant as shown in Figure [[fig:loop_shaping_S_with_W]].
Now, the closed-loop transfer function from $w$ to $z$ is equal to $W_s(s)S(s)$ and applying the $\mathcal{H}_\infty$ synthesis to the /weighted/ generalized plant $\tilde{P}(s)$ will generate a controller $K(s)$ such that $\|W_s(s)S(s)\|_\infty$ is minimized.
#+end_important
A /trick/ is to include a *weighting function* in the generalized plant as shown in Figure [[fig:loop_shaping_S_with_W]].
Applying the $\mathcal{H}_\infty$ synthesis to the /weighted/ generalized plant $\tilde{P}(s)$ (Figure [[fig:loop_shaping_S_with_W]]) will generate a controller $K(s)$ that minimizes the $\mathcal{H}_\infty$ norm between $r$ and $\tilde{\epsilon}$:
Let's now show how this is equivalent as *shaping* the sensitivity function:
\begin{align}
& \left\| \frac{\tilde{\epsilon}}{r} \right\|_\infty < \gamma (=1)\nonumber \\
\Leftrightarrow & \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\
& \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\
\Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega\nonumber \\
\Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega \label{eq:sensitivity_shaping}
\end{align}
#+begin_important
As shown in Equation eqref:eq:sensitivity_shaping, the $\mathcal{H}_\infty$ synthesis allows to *shape* the magnitude of the sensitivity transfer function.
Therefore, the choice of the weighting function $W_s(s)$ is very important.
Its inverse magnitude will define the frequency dependent upper bound of the sensitivity transfer function magnitude.
As shown in Equation eqref:eq:sensitivity_shaping, the $\mathcal{H}_\infty$ synthesis applying on the /weighted/ generalized plant allows to *shape* the magnitude of the sensitivity transfer function.
Therefore, the choice of the weighting function $W_s(s)$ is very important: its inverse magnitude will define the wanted *upper bound* of the sensitivity function magnitude.
#+end_important
#+begin_src latex :file loop_shaping_S_with_W.pdf
@ -1593,10 +1674,10 @@ Applying the $\mathcal{H}_\infty$ synthesis to the /weighted/ generalized plant
% Connections
\draw[->] (G.east) -- (addw.west);
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (Ws.west)node[above left]{$\epsilon$};
\draw[->] (Ws.east) -- (epsilon) node[above left](z1){$\tilde{\epsilon}$};
\draw[->] (Ws.east) -- (epsilon) node[above](z1){$z = \tilde{\epsilon}$};
\draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.4, 0)$) -- (G.west);
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.2, 0)$) -- (G.west);
\draw[->] (w) node[above]{$w = r$} -| (addw.north);
\end{tikzpicture}
@ -1607,41 +1688,59 @@ Applying the $\mathcal{H}_\infty$ synthesis to the /weighted/ generalized plant
#+RESULTS:
[[file:figs/loop_shaping_S_with_W.png]]
Once the weighting function is designed, it should be added to the generalized plant as shown in Figure [[fig:loop_shaping_S_with_W]].
#+begin_exercice
Using matlab, compute the weighted generalized plant shown in Figure [[fig:first_order_weight]] as a function of $G(s)$ and $W_S(s)$.
The weighted generalized plant can be defined in Matlab by either re-defining all the inputs or by pre-multiplying the (non-weighted) generalized plant by a block-diagonal MIMO transfer function containing the weights for the outputs $z$ and =1= for the outputs $v$.
#+HTML: <details><summary>Hint</summary>
The weighted generalized plant can be defined in Matlab using two techniques:
- by writing manually the 4 transfer functions from $[w, u]$ to $[\tilde{\epsilon}, v]$
- by pre-multiplying the (non-weighted) generalized plant by a block-diagonal transfer function matrix containing the weights for the outputs $z$ and =1= for the outputs $v$
#+HTML: </details>
#+HTML: <details><summary>Answer</summary>
The two solutions below can be used.
#+begin_src matlab :tangle no :eval no
Pw = [Ws -Ws*G;
1 -G]
1 -G];
#+end_src
% Alternative
#+begin_src matlab :tangle no :eval no
Pw = blkdiag(Ws, 1)*P;
#+end_src
The second solution is however more general, and can also be used when weights are added at the inputs by post-multiplying instead of pre-multiplying.
#+HTML: </details>
#+end_exercice
** Design of Weighting Functions
<<sec:weighting_functions_design>>
Weighting function used must be *proper*, *stable* and *minimum phase* transfer functions.
Weighting function included in the generalized plant must be *proper*, *stable* and *minimum phase* transfer functions.
#+begin_definition
- proper ::
more poles than zeros, this implies $\lim_{\omega \to \infty} |W(j\omega)| < \infty$
- stable ::
no poles in the right half plane
- minimum phase ::
no zeros in the right half plane
#+end_definition
Matlab is providing the =makeweight= function that creates a first-order weights by specifying the low frequency gain, high frequency gain, and a gain at a specific frequency:
Matlab is providing the =makeweight= function that allows to design first-order weights by specifying the low frequency gain, high frequency gain, and the gain at a specific frequency:
#+begin_src matlab :tangle no :eval no
W = makeweight(dcgain,[freq,mag],hfgain)
#+end_src
with:
- =dcgain=
- =freq=
- =mag=
- =hfgain=
- =dcgain=: low frequency gain
- =[freq,mag]=: frequency =freq= at which the gain is =mag=
- =hfgain=: high frequency gain
#+begin_exampl
The Matlab code below produces a weighting function with a magnitude shape shown in Figure [[fig:first_order_weight]].
The Matlab code below produces a weighting function with the following characteristics (Figure [[fig:first_order_weight]]):
- Low frequency gain of 100
- Gain of 1 at 10Hz
- High frequency gain of 0.5
#+begin_src matlab
Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
@ -1672,7 +1771,7 @@ The Matlab code below produces a weighting function with a magnitude shape shown
#+begin_seealso
Quite often, higher orders weights are required.
In such case, the following formula can be used the design of these weights:
In such case, the following formula can be used:
\begin{equation}
W(s) = \left( \frac{
@ -1752,52 +1851,103 @@ The obtained shapes are shown in Figure [[fig:high_order_weight]].
[[file:figs/high_order_weight.png]]
#+end_seealso
** Sensitivity Function Shaping - Example
** Shaping the Sensitivity Function
<<sec:sensitivity_shaping_example>>
Let's design a controller using the $\mathcal{H}_\infty$ synthesis that fulfils the following requirements:
1. Bandwidth of at least 10Hz
2. Small static errors for step responses
3. Robustness: Large module margin $\Delta M > 0.5$ ($\Rightarrow \Delta G > 2$ and $\Delta \phi > 29^o$)
- Robustness: Module margin > 2 ($\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o$)
- Bandwidth:
- Slope of -2
As usual, the plant used is the one presented in Section [[sec:example_system]].
First, the weighting functions is generated.
#+begin_src matlab
Ws = generateWeight('G0', 1e3, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
#+begin_exercice
Translate the requirements as upper bounds on the Sensitivity function and design the corresponding Weight using Matlab.
#+HTML: <details><summary>Hint</summary>
The typical wanted upper bound of the sensitivity function is shown in Figure [[fig:h-infinity-spec-S-bis]].
More precisely:
1. Recall that the closed-loop bandwidth is defined as the frequency $|S(j\omega)|$ first crosses $1/\sqrt{2} = -3dB$ from below
2. For the small static error, -60dB is usually enough as other factors (measurement noise, disturbances) will anyhow limit the performances
3. Recall that the module margin is equal to the inverse of the $\mathcal{H}_\infty$ norm of the sensitivity function:
\[ \Delta M = \frac{1}{\|S\|_\infty} \]
Remember that the wanted upper bound of the sensitivity function is defined by the *inverse* magnitude of the weight.
#+name: fig:h-infinity-spec-S-bis
#+caption: Typical wanted shape of the Sensitivity transfer function
[[file:figs/h-infinity-spec-S.png]]
#+HTML: </details>
#+HTML: <details><summary>Answer</summary>
1. $|W_s(j \cdot 2 \pi 10)| = \sqrt{2}$
2. $|W_s(j \cdot 0)| = 10^3$
3. $\|W_s\|_\infty = 0.5$
Using Matlab, such weighting function can be generated using the =makeweight= function as shown below:
#+begin_src matlab :eval no :tangle no
Ws = makeweight(1e3, [2*pi*10, sqrt(2)], 1/2);
#+end_src
It is then added to the generalized plant.
Or using the =generateWeight= function:
#+begin_src matlab :eval no :tangle no
Ws = generateWeight('G0', 1e3, ...
'G1', 1/2, ...
'Gc', sqrt(2), 'wc', 2*pi*10, ...
'n', 2);
#+end_src
#+HTML: </details>
#+end_exercice
Let's say we came up with the following weighting function:
#+begin_src matlab
Ws = generateWeight('G0', 1e3, ...
'G1', 1/2, ...
'Gc', sqrt(2), 'wc', 2*pi*10, ...
'n', 2);
#+end_src
The weighting function is then added to the generalized plant.
#+begin_src matlab
P = [1 -G;
1 -G];
Pw = blkdiag(Ws, 1)*P;
#+end_src
And the $\mathcal{H}_\infty$ synthesis is performed.
And the $\mathcal{H}_\infty$ synthesis is performed on the /weighted/ generalized plant.
#+begin_src matlab :results output replace
K = hinfsyn(Pw, 1, 1, 'Display', 'on');
#+end_src
#+RESULTS:
#+begin_example
K = hinfsyn(Pw, 1, 1, 'Display', 'on');
Test bounds: 0.5 <= gamma <= 0.51
gamma X>=0 Y>=0 rho(XY)<1 p/f
5.05e-01 0.0e+00 0.0e+00 4.497e-28 p
5.05e-01 0.0e+00 0.0e+00 3.000e-16 p
Limiting gains...
5.05e-01 0.0e+00 0.0e+00 0.000e+00 p
5.05e-01 -1.8e+01 # -2.9e-15 1.514e-15 f
5.05e-01 0.0e+00 0.0e+00 3.461e-16 p
5.05e-01 -3.5e+01 # -4.9e-14 1.732e-26 f
Best performance (actual): 0.504
Best performance (actual): 0.503
#+end_example
The obtained $\gamma \approx 0.5$ means that it found a controller $K(s)$ that stabilize the closed-loop system, and such that:
$\gamma \approx 0.5$ means that the $\mathcal{H}_\infty$ synthesis generated a controller $K(s)$ that stabilizes the closed-loop system, and such that:
\begin{aligned}
& \| W_s(s) S(s) \|_\infty < 0.5 \\
& \| W_s(s) S(s) \|_\infty \approx 0.5 \\
& \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
\end{aligned}
This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:results_sensitivity_hinf]].
#+begin_important
Having $\gamma < 1$ means that the $\mathcal{H}_\infty$ synthesis found a controller such that the specified closed-loop transfer functions are bellow the specified upper bounds.
Having $\gamma$ slightly above one does not necessary means the obtained controller is not "good".
It just means that at some frequency, one of the closed-loop transfer functions is above the specified upper bound by a factor $\gamma$.
#+end_important
#+begin_src matlab :exports none
figure;
hold on;
@ -1817,89 +1967,18 @@ This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:re
#+RESULTS:
[[file:figs/results_sensitivity_hinf.png]]
** Complementary Sensitivity Function
** Shaping multiple closed-loop transfer functions
<<sec:shaping_multiple_tf>>
As was shown in Section [[sec:modern_interpretation_specification]], depending on the specifications, up to four closed-loop transfer function may be shaped (the Gang of four).
This was summarized in Table [[tab:specification_modern]].
** Summary
For instance to limit the control input $u$, $KS$ should be shaped while to filter measurement noise, $T$ should be shaped.
#+name: tab:specification_modern
#+caption: Table caption
| | Open-Loop Shaping | Closed-Loop Shaping |
|-----------------------------+--------------------+--------------------------------------------|
| Reference Tracking | $L$ large | $S$ small |
| Disturbance Rejection | $L$ large | $GS$ small |
| Measurement Noise Filtering | $L$ small | $T$ small |
| Small Command Amplitude | $K$ and $L$ small | $KS$ small |
| Robustness | Phase/Gain margins | Module margin: $\Vert S\Vert_\infty$ small |
When multiple closed-loop transfer function are shaped at the same time, it is refereed to as "Mixed-Sensitivity $\mathcal{H}_\infty$ Control" and is the subject of Section [[sec:h_infinity_mixed_sensitivity]].
#+begin_src latex :file h-infinity-4-blocs-constrains.pdf
\begin{tikzpicture}
\begin{scope}[shift={(0, 0)}]
\draw[] (2.5, 1.0) node[]{$S$};
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim GK^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim 1$};
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3) to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
Depending on the closed-loop transfer function being shaped, different general control configuration are used and are described below.
\begin{scope}[shift={(6.4, 0)}]
\draw[] (2.5, 1.0) node[]{$GS$};
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim K^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim G$};
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (1, -1) to[out=45, in=180] (2.5,-0.2) to[out=0,in=180-45] (4,-1) to[out=-45,in=180-45] (5, -2);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\begin{scope}[shift={(0, -4.4)}]
\draw[] (2.5, 1.0) node[]{$KS$};
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -1.8) node[]{$\sim G^{-1}$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.3) node[]{$\sim K$};
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-1.5) to[out=45,in=180+45] (1, -0.5) to[out=45, in=180] (2.5,0.3) to[out=0,in=180-45] (4,-0.5) to[out=-45,in=180-45] (5, -1.5);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\begin{scope}[shift={(6.4, -4.4)}]
\draw[] (2.5, 1.0) node[]{$T$};
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim 1$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim GK$};
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:h-infinity-4-blocs-constrains
#+caption: Shaping the Gang of Four: Limitations
#+RESULTS:
[[file:figs/h-infinity-4-blocs-constrains.png]]
* $\mathcal{H}_\infty$ Mixed-Sensitivity Synthesis
<<sec:h_infinity_mixed_sensitivity>>
** Problem
** Typical Procedure
** Step 1 - Shaping of the Sensitivity Function
** Step 2 - Shaping of
** General Configuration for various shaping
*** S KS :ignore:
#+HTML: <details><summary>Shaping of S and KS</summary>
#+begin_src latex :file general_conf_shaping_S_KS.pdf
@ -2132,15 +2211,63 @@ This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:re
- $W_2W_3$ is used to shape $T$
#+HTML: </details>
* Conclusion
<<sec:conclusion>>
* Things to add :noexport:
** 2 blocs criterion - constrains
#+begin_src latex :file h-infinity-2-blocs-constrains.pdf
*** Limitation :ignore:
When shaping multiple closed-loop transfer functions, one should be verify careful about the three following points that are further discussed:
- The shaped closed-loop transfer functions are linked by mathematical relations and cannot be shaped
- Closed-loop transfer function can only be shaped in certain frequency range.
- The size of the obtained controller may be very large and not implementable in practice
#+begin_warning
Mathematical relations are linking the closed-loop transfer functions.
For instance, the sensitivity function $S(s)$ and the complementary sensitivity function $T(s)$ as link by the following well known relation:
\begin{equation}
S(s) + T(s) = 1
\end{equation}
This means that $|S(j\omega)|$ and $|T(j\omega)|$ cannot be made small at the same time!
It is therefore *not* possible to shape the four closed-loop transfer functions independently.
The weighting function should be carefully design such as these fundamental relations are not violated.
#+end_warning
The control bandwidth is clearly limited by physical constrains such as sampling frequency, electronics bandwidth,
\begin{align*}
&|G(j\omega) K(j\omega)| \ll 1 \Longrightarrow |S(j\omega)| = \frac{1}{1 + |G(j\omega)K(j\omega)|} \approx 1 \\
&|G(j\omega) K(j\omega)| \gg 1 \Longrightarrow |S(j\omega)| = \frac{1}{1 + |G(j\omega)K(j\omega)|} \approx \frac{1}{|G(j\omega)K(j\omega)|}
\end{align*}
Similar relationship can be found for $T$, $KS$ and $GS$.
#+begin_exercice
Determine the approximate norms of $T$, $KS$ and $GS$ for large loop gains ($|G(j\omega) K(j\omega)| \gg 1$) and small loop gains ($|G(j\omega) K(j\omega)| \ll 1$).
#+HTML: <details><summary>Hint</summary>
You can follows this procedure for $T$, $KS$ and $GS$:
1. Write the closed-loop transfer function $T(s)$ as a function of $K(s)$ and $G(s)$
2. Take $|K(j\omega)G(j\omega)| \gg 1$ and conclude on $|T(j\omega)|$
3. Take $|K(j\omega)G(j\omega)| \ll 1$ and conclude on $|T(j\omega)|$
#+HTML: </details>
#+HTML: <details><summary>Answer</summary>
The obtained constrains are shown in Figure [[fig:h-infinity-4-blocs-constrains]].
#+HTML: </details>
#+end_exercice
Depending on the frequency band, the norms of the closed-loop transfer functions depend on the controller $K$ and therefore can be shaped.
However, in some frequency bands, the norms do not depend on the controller and therefore *cannot* be shaped.
Therefore the weighting functions should only focus on certainty frequency range depending on the transfer function being shaped.
These regions are summarized in Figure [[fig:h-infinity-4-blocs-constrains]].
#+begin_src latex :file h-infinity-4-blocs-constrains.pdf
\begin{tikzpicture}
\begin{scope}[shift={(0, 0)}]
\draw[] (2.5, 1.0) node[]{$S$};
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim GK^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
@ -2148,11 +2275,40 @@ This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:re
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3) to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\draw[dashed] rectangle ;
\begin{scope}[on background layer]
\node[fit={(-0.5, -2.7) (5.5, 1.4)}, inner sep=0pt, draw, dashed, fill=black!20!white] (S) {};
\node[below] at (S.north) {$S$};
\end{scope}
\end{scope}
\begin{scope}[shift={(6.4, 0)}]
\draw[] (2.5, 1.0) node[]{$T$};
\draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim K^{-1}$};
\draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.5) node[]{$\sim G$};
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-2) to[out=45,in=180+45] (1, -1) to[out=45, in=180] (2.5,-0.2) to[out=0,in=180-45] (4,-1) to[out=-45,in=180-45] (5, -2);
\begin{scope}[on background layer]
\node[fit={(-0.5, -2.7) (5.5, 1.4)}, inner sep=0pt, draw, dashed, fill=black!20!white] (GS) {};
\node[below] at (GS.north) {$GS$};
\end{scope}
\end{scope}
\begin{scope}[shift={(0, -4.4)}]
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -1.8) node[]{$\sim G^{-1}$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
\draw[] (4.5, -0.3) node[]{$\sim K$};
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,-1.5) to[out=45,in=180+45] (1, -0.5) to[out=45, in=180] (2.5,0.3) to[out=0,in=180-45] (4,-0.5) to[out=-45,in=180-45] (5, -1.5);
\begin{scope}[on background layer]
\node[fit={(-0.5, -2.7) (5.5, 1.4)}, inner sep=0pt, draw, dashed, fill=black!20!white] (KS) {};
\node[below] at (KS.north) {$KS$};
\end{scope}
\end{scope}
\begin{scope}[shift={(6.4, -4.4)}]
\draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5);
\draw[] (0.6, -0.5) node[]{$\sim 1$};
\draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5);
@ -2160,16 +2316,46 @@ This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:re
\draw[fill=red!20] (2.5, 0.15) circle (0.15);
\draw[dashed] (-0.4, 0) -- (5.4, 0);
\draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2);
\draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4);
\begin{scope}[on background layer]
\node[fit={(-0.5, -2.7) (5.5, 1.4)}, inner sep=0pt, draw, dashed, fill=black!20!white] (T) {};
\node[below] at (T.north) {$T$};
\end{scope}
\end{scope}
\end{tikzpicture}
#+end_src
#+name: fig:h-infinity-4-blocs-constrains
#+caption: Shaping the Gang of Four: Limitations
#+RESULTS:
[[file:figs/h-infinity-2-blocs-constrains.png]]
[[file:figs/h-infinity-4-blocs-constrains.png]]
#+begin_warning
The order (resp. number of state) of the controller given by the $\mathcal{H}_\infty$ synthesis is equal to the order (resp. number of state) of the weighted generalized plant.
It is thus equal to the *sum* of the number of state of the non-weighted generalized plant and the number of state of all the weighting functions.
Two approaches can be used to obtain controllers with reasonable order:
1. use simple weights (usually first order)
2. perform a model reduction on the obtained high order controller
#+end_warning
* Mixed-Sensitivity $\mathcal{H}_\infty$ Control - Example
<<sec:h_infinity_mixed_sensitivity>>
** Problem
** Typical Procedure
** Step 1 - Shaping of the Sensitivity Function
** Step 2 - Shaping of
* Conclusion
<<sec:conclusion>>
* Resources
:PROPERTIES:
:UNNUMBERED: notoc
:END:
yt:?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin