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Documentation(info = "<html><head></head><body>This model demonstrates the less obvious charactristics of <u>periodic</u> table interpolation.<div>This is relevant to both 1D and 2D tables.</div><div><br></div><div><strong>Periodicity</strong></div><div>The periodicity of a one-dimensional table is defined as table[end,1]-table[1,1].</div><div>This implies that the first and last points in the table 'overlap' when extrapolating. </div><div>The top model in this example, 'discontinuousExtrapol', illustrates how this works out during simulation. It defines a saw-tooth function by its minimum and maximum value and linear interpolation.</div><div>The values at both ends of the definition interval (at <em>t</em>=0.5s and <em>t</em>=0.9s) are equal to the values defined in table[1,2] and table[end,2] respectively. Thus the table is evaluated including the interval limits: [start, end]</div><div><br></div><div>Outside of the definition interval, the <em>limit</em> towards the definition interval is used. On the left side, the table is evaluated excluding the end value: [start, end>. On the right side it is evaluated excluding the start value: <start, end].</div><div>This effect is deliberately exaggerated in this model by choosing a large simulation interval.</div><div>It is clear that for <em>t</em> > 0.9s, the table output approaches -1 when <em>t</em> decreases, but <em>at</em> t=0.9s (and t=1.3s, etc.), the output will be exactly equal to 1.</div><div>Likewise, for <em>t</em><0.5s the table output will be <1 but never equal to 1. Instead at <iem>t</em>=0.5 (and <em>t</em>=0.1 etc.), the output will be exactly -1.</div><div><br></div><div><strong>Differentiability</strong></div><div>The second model 'continuousC0Extrapolation' demonstrates that the derivative is not defined in the edges of the definition interval. The table definition [0, -1; 1, 0; 2, 1; 3, 0; 4, -1] defines a function that would be a triangle with linear interpolation.</div><div>With continuous derivative, it is smooth in the interval, but not in the edges (<em>t</em>=0.5s, <em>t</em>= 0.9s, etc.).</div><div><br></div><div>The bottom model 'continuousC1Extrapol' with table [0, -1; 0.25, -1; 0.5, -1; 2, 1; 3.5, -1; 3.75, -1; 4, -1] defines a function which is continuous in the interval edge, as well as its first 2 derivatives: around the interval edge there are 5 consequtive points at -1.</div><div>This results in a smooth function.</div><div><br></div><div><br></div><div>For more information, see <em>Proceedings of </em><em>the 10th International Modelica Conference</em>. Ed. by Hubertus Tummescheit and Karl-Erik Årzén. Lund, Sweden, March 2014.</div>
Documentation(info="<html><head></head><body>This model demonstrates the less obvious charactristics of <u>periodic</u> table interpolation.<div>This is relevant to both 1D and 2D tables.</div><div><br></div><div><strong>Periodicity</strong></div><div>The periodicity of a one-dimensional table is defined as table[end,1]-table[1,1].</div><div>This implies that the first and last points in the table 'overlap' when extrapolating. </div><div>The top model in this example, 'discontinuousExtrapol', illustrates how this works out during simulation. It defines a saw-tooth function by its minimum and maximum value and linear interpolation.</div><div>The values at both ends of the definition interval (at <em>t</em>=0.5s and <em>t</em>=0.9s) are equal to the values defined in table[1,2] and table[end,2] respectively. Thus the table is evaluated including the interval limits: [start, end]</div><div><br></div><div>Outside of the definition interval, the <em>limit</em> towards the definition interval is used. On the left side, the table is evaluated excluding the end value: [start, end>. On the right side it is evaluated excluding the start value: <start, end].</div><div>This effect is deliberately exaggerated in this model by choosing a large simulation interval.</div><div>It is clear that for <em>t</em> > 0.9s, the table output approaches -1 when <em>t</em> decreases, but <em>at</em> t=0.9s (and t=1.3s, etc.), the output will be exactly equal to 1.</div><div>Likewise, for <em>t</em><0.5s the table output will be <1 but never equal to 1. Instead at <iem>t</em>=0.5 (and <em>t</em>=0.1 etc.), the output will be exactly -1.</div><div><br></div><div><strong>Differentiability</strong></div><div>The second model 'continuousC0Extrapolation' demonstrates that the derivative is not defined in the edges of the definition interval. The table definition [0, -1; 1, 0; 2, 1; 3, 0; 4, -1] defines a function that would be a triangle with linear interpolation.</div><div>With continuous derivative, it is smooth in the interval, but not in the edges (<em>t</em>=0.5s, <em>t</em>= 0.9s, etc.).</div><div><br></div><div>The bottom model 'continuousC1Extrapol' with table [0, -1; 0.25, -1; 0.5, -1; 2, 1; 3.5, -1; 3.75, -1; 4, -1] defines a function which is continuous in the interval edge, as well as its first 2 derivatives: around the interval edge there are 5 consequtive points at -1.</div><div>This results in a smooth function.</div><div><br></div><div><br></div><div>For more information, see <em>Proceedings of </em><em>the 10th International Modelica Conference</em>. Ed. by Hubertus Tummescheit and Karl-Erik Årzén. Lund, Sweden, March 2014.</div>
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