Online Health Management for Complex Nonlinear Systems Based on Hidden Semi-Markov Model Using Sequential Monte Carlo Methods

<table>Prognostic structure based on HSMM using SMC method. Life time of a component <svg height="16.4125" id="M156" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 271.08749 16.4125" width="271.08749" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q166 0 262.5 95t96.5 244z" id="x1D437"></path></g><g transform="matrix(.017,-0,0,-.017,34.077,12.138)"><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,-.017,39.959,12.138)"><path d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81
q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z" id="x1D460"></path></g>
<g transform="matrix(.012,-0,0,-.012,46.338,16.212)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
<g transform="matrix(.017,-0,0,-.017,52.675,12.138)"><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,-.017,62.313,12.138)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.017,-0,0,-.017,76.082,12.138)"><use xlink:href="#x1D437"></use></g><g transform="matrix(.017,-0,0,-.017,89.732,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,95.614,12.138)"><use xlink:href="#x1D460"></use></g>
<g transform="matrix(.012,-0,0,-.012,101.988,16.212)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
<g transform="matrix(.017,-0,0,-.017,108.325,12.138)"><use xlink:href="#x29"></use></g><g transform="matrix(.017,-0,0,-.017,117.98,12.138)"><use xlink:href="#x2B"></use></g><g transform="matrix(.017,-0,0,-.017,135.693,12.138)"><use xlink:href="#x1D437"></use></g><g transform="matrix(.017,-0,0,-.017,149.343,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,155.225,12.138)"><use xlink:href="#x1D460"></use></g>
<g transform="matrix(.012,-0,0,-.012,161.612,16.212)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g>
<g transform="matrix(.017,-0,0,-.017,167.95,12.138)"><use xlink:href="#x29"></use></g><g transform="matrix(.017,-0,0,-.017,177.588,12.138)"><use xlink:href="#x2B"></use></g><g transform="matrix(.017,-0,0,-.017,191.357,12.138)"><path d="M215 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5zM541 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5zM867 255q0 -26 -16 -42t-40 -16q-25 0 -40.5 16t-15.5 42
q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5z" id="x22EF"></path></g><g transform="matrix(.017,-0,0,-.017,214.459,12.138)"><use xlink:href="#x2B"></use></g><g transform="matrix(.017,-0,0,-.017,228.211,12.138)"><use xlink:href="#x1D437"></use></g><g transform="matrix(.017,-0,0,-.017,241.861,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,247.742,12.138)"><use xlink:href="#x1D460"></use></g>
<g transform="matrix(.012,-0,0,-.012,254.125,16.212)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g>
<g transform="matrix(.017,-0,0,-.017,265.15,12.138)"><use xlink:href="#x29"></use></g>
</svg>. <svg height="16.4125" id="M157" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 35.787498 16.4125" width="35.787498" 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)"><use xlink:href="#x1D437"></use></g><g transform="matrix(.017,-0,0,-.017,13.712,12.138)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,19.594,12.138)"><use xlink:href="#x1D460"></use></g>
<g transform="matrix(.012,-0,0,-.012,25.963,16.212)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
<g transform="matrix(.017,-0,0,-.017,29.838,12.138)"><use xlink:href="#x29"></use></g>
</svg>: duration of a component staying at state <svg height="11.9625" id="M158" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 10.375 11.9625" width="10.375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D460"></use></g>
<g transform="matrix(.012,-0,0,-.012,6.438,11.762)"><use xlink:href="#x1D456"></use></g>
</svg>. <svg height="11.825" id="M159" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 12.8375 11.825" width="12.8375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D460"></use></g>
<g transform="matrix(.012,-0,0,-.012,6.438,11.762)"><use xlink:href="#x31"></use></g>
</svg>: health state 1 (baseline), <svg height="11.825" id="M160" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 12.8375 11.825" width="12.8375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D460"></use></g>
<g transform="matrix(.012,-0,0,-.012,6.438,11.762)"><use xlink:href="#x32"></use></g>
</svg>: health state 2 (contamination 1), …, <svg height="11.9125" id="M161" style="vertical-align:-3.35245pt" version="1.1" viewbox="0 0 17.525 11.9125" width="17.525" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D460"></use></g>
<g transform="matrix(.012,-0,0,-.012,6.438,11.762)"><use xlink:href="#x1D441"></use></g>
</svg>: health state <svg height="11.3125" id="M162" style="vertical-align:-0.1092pt" version="1.1" viewbox="0 0 14.9875 11.3125" width="14.9875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D441"></use></g>
</svg> (failure), <svg height="14.925" id="M163" style="vertical-align:-5.76984pt" version="1.1" viewbox="0 0 17.625 14.925" width="17.625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M483 97q-42 -50 -88.5 -79.5t-68.5 -29.5q-37 0 -17 93l22 102h-2q-54 -79 -144 -149q-59 -46 -100 -46q-24 0 -43 29t-19 86q0 78 34.5 153t94.5 120q41 31 94 51.5t98 20.5q29 0 72 -9q26 -6 39 -6l2 -4q-30 -117 -67 -323q-8 -41 2 -41q16 0 79 58zM374 387
q-32 15 -73 15q-52 0 -83 -23q-48 -36 -78 -108.5t-30 -152.5q0 -33 8.5 -50.5t20.5 -17.5q31 0 107 79t99 132q15 40 29 126z" id="x1D44E"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.663,11.762)"><use xlink:href="#x1D456"></use></g><g transform="matrix(.012,-0,0,-.012,11.911,11.762)"><path d="M394 607q0 -25 -15.5 -43t-39.5 -18q-19 0 -32 13t-13 35t16 41.5t40 19.5q20 0 32 -13.5t12 -34.5zM360 349l-57 -283q-38 -188 -122 -263q-72 -64 -114 -64q-16 0 -30 10.5t-14 22.5q0 24 21 45q15 13 26 6q30 -20 54 -20q41 0 65 95q37 147 86 415q6 34 6.5 42.5
t-5.5 11.5q-18 0 -79 -54l-14 25q48 48 96.5 79t71.5 31q29 0 9 -99z" id="x1D457"></path></g>
</svg>: transition probability.</table>

Mathematical Problems in Engineering

fig4

Figure 4

Figure 4: Online Health Management for Complex Nonlinear Systems Based on Hidden Semi-Markov Model Using Sequential Monte Carlo Methods 