A Constructive Positive Realization with Sparse Matrices for
a Continuous-Time Positive Linear System

<table>The geometric interpretation of <svg height="16.625" id="M367" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 103.4125 16.625" width="103.4125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path d="M471 401q0 -32 -14 -91l-83 -376q-30 -146 -21 -180l-7 -15q-22 0 -47.5 23.5t-25.5 40.5q0 49 53 271l53 224q4 18 6.5 31t3.5 22.5t1 15t-2 9t-4.5 4.5t-6.5 1q-31 0 -100.5 -63.5t-126.5 -145.5q-16 -53 -40 -163q-48 -11 -74 -21l-7 8l82 341q9 36 -4 36
q-8 0 -28.5 -14t-38.5 -32l-17 22q41 43 85 71t66 28q23 0 23 -39q0 -19 -10 -64l-24 -103h2q77 101 201 182q37 24 74 24q31 0 31 -47z" id="x1D702"></path></g><g transform="matrix(.017,-0,0,-.017,14.341,12.138)"><path d="M414 650l-6 -28q-63 -4 -78.5 -17.5t-27.5 -71.5l-77 -416q-11 -58 0.5 -71.5t77.5 -17.5l-5 -28h-275l8 28q63 4 79.5 18.5t27.5 70.5l80 416q11 57 -2 71t-81 18l6 28h273z" id="x1D43C"></path></g><g transform="matrix(.017,-0,0,-.017,25.544,12.138)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.017,-0,0,-.017,39.296,12.138)"><path d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" id="x1D434"></path></g><g transform="matrix(.017,-0,0,-.017,51.756,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g><g transform="matrix(.017,-0,0,-.017,57.637,12.138)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" id="x1D443"></path></g><g transform="matrix(.017,-0,0,-.017,73.174,12.138)"><path d="M512 1h-130q-127 0 -217 74.5t-90 179.5t90 179.5t217 74.5h130v-50h-131q-103 0 -175.5 -59.5t-72.5 -144.5t72.5 -144.5t175.5 -59.5h131v-50z" id="x2282"></path></g><g transform="matrix(.017,-0,0,-.017,87.861,12.138)"><path d="M856 669l7 -19q-33 -13 -66 -38q89 -68 89 -165q0 -67 -44 -122q-44 -53 -105 -53q-34 0 -53 18.5t-19 47.5q0 22 16 43.5t42 35.5l10 -18q-36 -21 -36 -54q0 -13 9 -21t25 -8q41 0 67.5 42.5t26.5 106.5q0 75 -55 123q-40 -38 -81 -119l-39 -81q-32 -70 -40 -86
q-83 -166 -179.5 -241.5t-219.5 -75.5q-77 0 -125 36t-48 87q0 31 17 51t42 20q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q95 0 156 52q62 53 146 203l64 113q23 40 54 86q55 84 129 139q-76 48 -192 48
q-150 0 -250 -75q-101 -75 -101 -174q0 -41 30 -68.5t73 -27.5q158 0 205 249h20q-7 -112 -67 -196q-59 -83 -160 -83q-63 0 -102.5 35.5t-39.5 94.5q0 107 111 191t282 84q124 0 221 -59q38 24 84 41z" id="x1D4AB"></path></g>
</svg> using <svg height="16.625" id="M368" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 45.049999 16.625" width="45.049999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path d="M769 521v-16q-67 -12 -67 -95q-1 -26 17 -68l36.5 -89.5t17.5 -87.5q0 -69 -75 -129t-148 -60q-66 0 -114 22v-202h-117v235q-24 21 -56 21q-72 0 -114 -75l-20 9q18 64 52 110q43 58 117 58q16 0 21 -1v250q0 139 -30 197q-33 64 -83 64q-29 -1 -48 -20t-19 -46
t12 -47.5t28.5 -34l33.5 -27.5t29 -36.5t12 -53.5q0 -51 -50 -99q-49 -47 -116 -64l-9 22q23 7 50 34q49 53 7 109q-16 23 -39 41q-57 43 -57 99q0 71 53 112q54 42 133 42q62 0 111.5 -24t74.5 -66h2q40 88 143 88q87 0 117 -56q6 -10 17 -34t18.5 -37.5t23 -27t36.5 -17.5
zM695 503v2q-41 4 -67 20.5t-36 37t-17 40.5t-20.5 33.5t-36.5 13.5q-32 0 -52 -24q-30 -37 -30 -112v-379q18 -13 43 -45q27 -34 48 -50t47 -16q38 0 65.5 24.5t27.5 71.5q0 34 -41 114t-41 130q0 41 29 78.5t81 60.5z" id="x1D513"></path></g>
<g transform="matrix(.012,-0,0,-.012,14.05,16.213)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" id="x1D45A"></path></g>
<g transform="matrix(.017,-0,0,-.017,24.062,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,29.944,12.138)"><path d="M516 277q0 -126 -140 -231q-36 -27 -75.5 -42.5t-63.5 -15.5q-38 0 -73 43q-10 -44 -18 -106q-18 -134 -32 -151q-12 -16 -40 -25t-46 -10l-5 14q22 38 84 406q16 92 44.5 146.5t88.5 95.5q68 47 136 47q63 0 101.5 -47.5t38.5 -123.5zM432 259q0 61 -27 101t-79 40
t-82 -49t-49 -148l-24 -127q42 -36 88 -36q79 0 126 61.5t47 157.5z" id="x1D70C"></path></g><g transform="matrix(.017,-0,0,-.017,39.106,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg>.</table>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: A Constructive Positive Realization with Sparse Matrices for
a Continuous-Time Positive Linear System 

Mathematical Problems in Engineering

A Constructive Positive Realization with Sparse Matrices for a Continuous-Time Positive Linear System

Figure 1