{"id":5976,"date":"2023-10-14T23:25:03","date_gmt":"2023-10-14T22:25:03","guid":{"rendered":"http:\/\/mosaic-modeling.de\/?page_id=5976"},"modified":"2024-07-12T11:58:04","modified_gmt":"2024-07-12T10:58:04","slug":"use-of-connectors-vi-dynamic-system","status":"publish","type":"page","link":"https:\/\/mosaic-modeling.de\/?page_id=5976","title":{"rendered":"Use of Connectors VI &#8211; Dynamic System"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">On this page<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Model-description\" >Model description<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Modeling-workflow\" >Modeling workflow<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Notation-of-model\" >Notation of model<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Base-names\" >Base names<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Superscripts\" >Superscripts<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Subscripts\" >Subscripts<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Indices\" >Indices<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Notation-of-controller\" >Notation of controller<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Base-names-2\" >Base names<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Superscripts-2\" >Superscripts<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Subscripts-2\" >Subscripts<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equations-of-the-model\" >Equations of the model<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equations-of-the-controller\" >Equations of the controller<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Connectors\" >Connectors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equation-system-for-open-loop\" >Equation system for open loop<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equation-system-for-PI-controller\" >Equation system for PI controller<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equation-system-for-closed-loop\" >Equation system for closed loop<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Simulation-workflow\" >Simulation workflow<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equation-system-and-Indexing\" >Equation system and Indexing<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equation-system-for-open-loop-2\" >Equation system for open loop<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equation-system-for-closed-loop-2\" >Equation system for closed loop<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Initialization-and-results\" >Initialization and results<\/a><ul class='ez-toc-list-level-5' ><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equation-system-for-open-loop-3\" >Equation system for open loop<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Equation-system-for-closed-loop-3\" >Equation system for closed loop<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mosaic-modeling.de\/?page_id=5976\/#Use-of-functions\" >Use of functions<\/a><\/li><\/ul><\/nav><\/div>\n<p>This tutorial demonstrates how to use connectors for a dynamic system. This could, for example, be necessary because you want to use a dynamic sub-model of a colleague. Here, we want to use the sub-system of a PI controller. If we did not use a connector for the time variable, MOSAICmodeling would issue a warning because we had two different time variables. MOSAICmodeling would interpret this as a partial differential equation, which can currently not be solved.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Model-description\"><\/span>Model description<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>This example is based on the article by Asprey and Macchietto (2002), DOI: <span class=\"anchor-text\">10.1016\/S0959-1524(01)00020-8<\/span>. It features two differential equations representing the concentrations of the components as well as an expression for the reaction rate:<\/p>\n<p><p class=\"ql-left-displayed-equation\" style=\"line-height: 167px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-5aaea1c733604287482e28e2d0c268da_l3.png\" height=\"167\" width=\"214\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#116;&#125;&#32;&#61;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#32;&#114;&#32;&#45;&#32;&#117;&#32;&#45;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#52;&#125;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;&#32;&#92;&#92;&#91;&#50;&#101;&#120;&#93;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#116;&#125;&#32;&#61;&#32;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#32;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#51;&#125;&#125;&#32;&#43;&#32;&#117;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#32;&#120;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#70;&#125;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;&#32;&#45;&#32;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#125;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#92;&#91;&#50;&#101;&#120;&#93;&#114;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#49;&#125;&#32;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#50;&#125;&#32;&#43;&#32;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#125;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n<p>In addition, we want to add a PI controller to this system. This PI controller consists of the following generic equations, which have to be matched to the system above via a connector:<\/p>\n<p><p class=\"ql-left-displayed-equation\" style=\"line-height: 123px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-aee879336157ec9070f28ddc0ad7d9e3_l3.png\" height=\"123\" width=\"183\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#117;&#32;&#61;&#32;&#75;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#80;&#125;&#32;&#101;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#80;&#125;&#32;&#43;&#32;&#75;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#73;&#125;&#32;&#101;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#73;&#125;&#32;&#43;&#32;&#117;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#115;&#101;&#116;&#125;&#32;&#92;&#92;&#91;&#50;&#101;&#120;&#93;&#101;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#80;&#125;&#32;&#61;&#32;&#121;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#115;&#101;&#116;&#125;&#32;&#45;&#32;&#121;&#32;&#92;&#92;&#91;&#50;&#101;&#120;&#93;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#101;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#73;&#125;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#100;&#125;&#116;&#125;&#32;&#61;&#32;&#101;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#80;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n<p>Note that the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e817933126862db10ae510d35359568e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> in the PI equations is the same <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e817933126862db10ae510d35359568e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> in the model equations above.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Modeling-workflow\"><\/span>Modeling workflow<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>In the following, we show the workflow for setting up the model with both open loop and closed loop. We start by defining the notation for this example. Then, we define the notation of the controller equations.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Notation-of-model\"><\/span>Notation of model<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Set up a notation with the following base names and indices:<\/p>\n<h5><span class=\"ez-toc-section\" id=\"Base-names\"><\/span>Base names<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7b2034939b850e3311120fca462ab64e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/>, parameter in various units<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-01bcf7e9e043561da78fecf715c8a46e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/>, reaction rate in 1\/h<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>, time in h<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e817933126862db10ae510d35359568e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, dilution factor in 1\/h<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7e5fbfa0bbbd9f3051cd156a0f1b5e31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, molar concentration in g\/L<\/li>\n<\/ul>\n<h5><span class=\"ez-toc-section\" id=\"Superscripts\"><\/span>Superscripts<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li>B, biomass<\/li>\n<li>S, substate<\/li>\n<li>SP, setpoint<\/li>\n<\/ul>\n<h5><span class=\"ez-toc-section\" id=\"Subscripts\"><\/span>Subscripts<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li>F, feed<\/li>\n<\/ul>\n<h5><span class=\"ez-toc-section\" id=\"Indices\"><\/span>Indices<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-31318c5dcb226c69e0818e5f7d2422b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>, parameter index 1&#8230;NI<\/li>\n<\/ul>\n<p>The resulting notation has ID 2323.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Notation-of-controller\"><\/span>Notation of controller<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h5><span class=\"ez-toc-section\" id=\"Base-names-2\"><\/span>Base names<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-558a73dc6903ad0c33580b0ce9f110d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/>, control deviation<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7fb8d8d37cb2b48aee9e97aee7728d8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/>, gain factor<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>, time in h<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e817933126862db10ae510d35359568e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, manipulated variable<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-38461fc041e953482219abf5d4cce1cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/>, control variable<\/li>\n<\/ul>\n<h5><span class=\"ez-toc-section\" id=\"Superscripts-2\"><\/span>Superscripts<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li>set, setpoint<\/li>\n<\/ul>\n<h5><span class=\"ez-toc-section\" id=\"Subscripts-2\"><\/span>Subscripts<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<ul>\n<li>I, integral part<\/li>\n<li>P, proportional part<\/li>\n<\/ul>\n<p>The resulting notation has ID 186460.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Equations-of-the-model\"><\/span>Equations of the model<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Go to the Equation tab and perform the following steps:<\/p>\n<ol>\n<li>Load the model notation set up above<\/li>\n<li>Enter the model equations<\/li>\n<\/ol>\n<p>The model equations have IDs 2324, 2325, and 2326.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Equations-of-the-controller\"><\/span>Equations of the controller<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Go to the Equation tab and perform the following steps:<\/p>\n<ol>\n<li>Load the controller notation set up above<\/li>\n<li>Enter the model equations<\/li>\n<\/ol>\n<p>The model equations have IDs 186459, 186461, and 186462.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Connectors\"><\/span>Connectors<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>To connect these two equation systems, we will need a connector. One could suggest to simply use the naming policy integrate as this would create the same namespace for all variables and there would only be one variable <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>. However,\u00a0 the two models could also use completely different variable names for time, e.g., <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-ec5583fa081a1e03212c151e3c222412_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"\/> in a more generic case. In this case here, we also need to match the controlled variables: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-38461fc041e953482219abf5d4cce1cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/> in the generic model of the PI controller and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7f5a11f1d0cf1f201f033654b50122b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: 0px;\"\/> in the model equations. For this purpose, go to the Connector tab and do the following:<\/p>\n<ol>\n<li>Add a helpful description for the connector<\/li>\n<li>Activate the tab Edit Matching and select the controller equation as Sub Notation and the model notation of this example as Super Notation<\/li>\n<li>Generate the missing variables for the Sub Notation and the Super Notation, i.e.,\n<ul>\n<li>Sub Notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e817933126862db10ae510d35359568e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-38461fc041e953482219abf5d4cce1cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-a836978eb5686b0f0e7847a55d08a55e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#115;&#101;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"25\" style=\"vertical-align: -4px;\"\/><\/li>\n<li>Super Notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e817933126862db10ae510d35359568e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7f5a11f1d0cf1f201f033654b50122b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-b27946ec516dbfabac17dab456245e68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#44;&#83;&#80;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: 0px;\"\/><\/li>\n<\/ul>\n<\/li>\n<li>Select the analogous variables and click on Match to achieve the following matching:\n<ul>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-72c82bcb048a046f6f0de2b4e706e5af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#97;&#114;&#114;&#111;&#119;&#32;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\"\/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7f39753221b0706c1d0b67208c0158e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#97;&#114;&#114;&#111;&#119;&#32;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"48\" style=\"vertical-align: -1px;\"\/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7e04eca2509a7ee3babda7aa13f9ca3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#97;&#114;&#114;&#111;&#119;&#32;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -4px;\"\/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-edca584b8ed137448a7cf3d549d68dd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#115;&#101;&#116;&#125;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#97;&#114;&#114;&#111;&#119;&#32;&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#44;&#83;&#80;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -4px;\"\/><\/li>\n<\/ul>\n<\/li>\n<li>Save the connector<\/li>\n<\/ol>\n<p>The connector has ID 2351.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Equation-system-for-open-loop\"><\/span>Equation system for open loop<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>To construct this equation systems, go to the Equation System tab and take the following steps.<\/p>\n<ol>\n<li>Load the model notation and add a reasonable description<\/li>\n<li>Click on &#8220;Add EQU\/EQS&#8221; and add the three model equations<\/li>\n<li>Make sure that the Naming policy is integrate<\/li>\n<li>Save the equation system<\/li>\n<\/ol>\n<h4><span class=\"ez-toc-section\" id=\"Equation-system-for-PI-controller\"><\/span>Equation system for PI controller<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>To construct this equation systems, go to the Equation System tab and take the following steps.<\/p>\n<ol>\n<li>Load the controller notation and add a reasonable description<\/li>\n<li>Click on &#8220;Add EQU\/EQS&#8221; and add the three controller equations<\/li>\n<li>Make sure that the Naming policy is integrate<\/li>\n<li>Save the equation system<\/li>\n<\/ol>\n<h4><span class=\"ez-toc-section\" id=\"Equation-system-for-closed-loop\"><\/span>Equation system for closed loop<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>We now combine the two previous equation systems:<\/p>\n<ol>\n<li>Load the model notation and add a reasonable description<\/li>\n<li>Click on &#8220;Add EQU\/EQS&#8221; and add the equation system of the model<\/li>\n<li>Make sure that the Naming policy is integrate<\/li>\n<li>Click again on &#8220;Add EQU\/EQS&#8221; and add the equation system of the controller; select encapsulate as naming policy and select the connector from above; then confirm<\/li>\n<li>Save the equation system<\/li>\n<\/ol>\n<p>The naming policy encapsulate is chosen arbitrarily in this case and the policy integrate would also work. In general, we recommend encapsulate, though, as you avoid undesired matches between variables in your individual sub-models.<\/p>\n<p>The equation system for open loop is available with ID 2334, the equation system of the controller has ID 186463, and the equation system for closed loop has ID 2352.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Simulation-workflow\"><\/span>Simulation workflow<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h4><span class=\"ez-toc-section\" id=\"Equation-system-and-Indexing\"><\/span>Equation system and Indexing<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h5><span class=\"ez-toc-section\" id=\"Equation-system-for-open-loop-2\"><\/span>Equation system for open loop<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p>Go to the &#8220;Simulation&#8221; section and do the following:<\/p>\n<ol>\n<li>Load your equation system for the open loop in the tab Equation System<\/li>\n<li>Set the maximum value NI to 4 in the tab Indexing<\/li>\n<\/ol>\n<h5><span class=\"ez-toc-section\" id=\"Equation-system-for-closed-loop-2\"><\/span>Equation system for closed loop<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p>Go to the &#8220;Simulation&#8221; section and do the following:<\/p>\n<ol>\n<li>Load your equation system for the closed loop in the tab Equation System<\/li>\n<li>Set the maximum value NI to 4 in the tab Indexing<\/li>\n<\/ol>\n<h4><span class=\"ez-toc-section\" id=\"Initialization-and-results\"><\/span>Initialization and results<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h5><span class=\"ez-toc-section\" id=\"Equation-system-for-open-loop-3\"><\/span>Equation system for open loop<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p>Go to the tab Specifications and open the Variables tab. To initialize and specify the model, take the following steps:<\/p>\n<ol>\n<li>Initialize this example with the design values and initial guesses given in Table 1<\/li>\n<li>Save the variable specification<\/li>\n<li>Save the simulation<\/li>\n<li>Go to the Evaluation tab and generate the code for your preferred environment<\/li>\n<li>Solve the system using the generated code<\/li>\n<\/ol>\n<p>This simulation is available with ID 2335 with variable specification 2334.<\/p>\n<h5><span class=\"ez-toc-section\" id=\"Equation-system-for-closed-loop-3\"><\/span>Equation system for closed loop<span class=\"ez-toc-section-end\"><\/span><\/h5>\n<p>Go to the tab Specifications and open the Variables tab. To initialize and specify the model, take the following steps:<\/p>\n<ol>\n<li>Initialize this example with the design values and initial guesses given in Table 2<\/li>\n<li>Save the variable specification<\/li>\n<li>Save the simulation<\/li>\n<li>Go to the Evaluation tab and generate the code for your preferred environment<\/li>\n<li>Solve the system using the generated code<\/li>\n<\/ol>\n<p>This simulation is available with ID 2354 with variable specification 2353.<\/p>\n<p>When you simulate the system with open loop and closed loop, you can observe the impact of the controller on the model performance.<\/p>\n\n\n<figure class=\"wp-block-table is-style-stripes has-small-font-size\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Name<\/th><th class=\"has-text-align-left\" data-align=\"left\">Description<\/th><th class=\"has-text-align-center\" data-align=\"center\">Value \/ Initial condition \/ Initial guess<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-10b641bfb72a1ed548465165c36be16a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"30\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter 1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.325<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-0b1f911ea0f26f19f168c568bc798784_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter 2<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.267<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-88b5633dfb1862c635c2db8856835f1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter 3<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.559<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-eb40a8b10af9a9b64074f6de23d57fe8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter 4<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.055<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-01bcf7e9e043561da78fecf715c8a46e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Reaction rate<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.01<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Time, independent variable<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e817933126862db10ae510d35359568e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Control<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7f5a11f1d0cf1f201f033654b50122b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mass concentration of biomass<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-63691c9c1822296af9249588fe2ccd15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mass concentration of substrate<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.01<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-46d598792b155f149cc738256d7dfe6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#70;&#125;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"19\" style=\"vertical-align: -5px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mass concentration of substrate in feed<\/td><td class=\"has-text-align-center\" data-align=\"center\">30.0<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\">Table 1: Overview of design values, initial conditions, and initial guesses for the open loop.<\/figcaption><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes has-small-font-size\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Name<\/th><th class=\"has-text-align-left\" data-align=\"left\">Description<\/th><th class=\"has-text-align-center\" data-align=\"center\">Value \/ Initial condition \/ Initial guess<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-10b641bfb72a1ed548465165c36be16a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"30\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter 1<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.325<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-0b1f911ea0f26f19f168c568bc798784_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter 2<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.267<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-88b5633dfb1862c635c2db8856835f1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter 3<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.559<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-eb40a8b10af9a9b64074f6de23d57fe8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#123;&#105;&#61;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Parameter 4<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.055<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-0eb07a2703d5f4a3b262265416c00640_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#73;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Gain factor of integral controller<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.001<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-d33507340811d5d964dd5d585b413654_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#75;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#80;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"24\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Gain factor of proportional controller<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.001<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-c0a766345b26919b2583b51df4adf900_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#73;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"13\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Control deviation for integral controller<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-3ec051105f08276bfeee8a3d2a88d894_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#80;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Control deviation for proportional controller<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-01bcf7e9e043561da78fecf715c8a46e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Reaction rate<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.01<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Time, independent variable<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e817933126862db10ae510d35359568e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Control<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-e8359eb59810dd9daa5aa7296cf4d44e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#115;&#101;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Feedforward control value<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-7f5a11f1d0cf1f201f033654b50122b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mass concentration of biomass<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-b27946ec516dbfabac17dab456245e68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#66;&#44;&#83;&#80;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Controller setpoint<\/td><td class=\"has-text-align-center\" data-align=\"center\">12.0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-63691c9c1822296af9249588fe2ccd15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mass concentration of substrate<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.01<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/mosaic-modeling.de\/wp-content\/ql-cache\/quicklatex.com-46d598792b155f149cc738256d7dfe6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#70;&#125;&#94;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"19\" style=\"vertical-align: -5px;\"\/><\/td><td class=\"has-text-align-left\" data-align=\"left\">Mass concentration of substrate in feed<\/td><td class=\"has-text-align-center\" data-align=\"center\">30.0<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\">Table 2: Overview of design values, initial conditions, and initial guesses for the closed loop.<\/figcaption><\/figure>\n\n\n<h3><span class=\"ez-toc-section\" id=\"Use-of-functions\"><\/span>Use of functions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If you prefer to formulate an ODE system in which certain algebraic variables are calculated via functions, you can do so. For this example, the reaction rate was also formulated as function. In this case, the parameters 1 and 2 must be added to a parameter list (ID: 189115). This parameter list is then used in the function because non-generic indices cannot appear in the function body (ID: 189116). Finally, the function replaces the algebraic equation for the reaction rate in a new equation system (ID: 189114). This equation system is used in a new simulation with ID 189119, using the variable specification with ID 189117 and the parameter specification with ID 189118. The values are the same as given in Table 1. The results with be the same, but the system is now used with an ODE solver instead of a DAE solver.<\/p>","protected":false},"excerpt":{"rendered":"<p>This tutorial demonstrates how to use connectors for a dynamic system. This could, for example, be necessary because you want to use a dynamic sub-model of a colleague. Here, we want to use the sub-system of a PI controller. If we did not use a connector for the time variable, MOSAICmodeling would issue a warning [&hellip;]<\/p>\n","protected":false},"author":252,"featured_media":0,"parent":4741,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-5976","page","type-page","status-publish","czr-hentry"],"_links":{"self":[{"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages\/5976","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/users\/252"}],"replies":[{"embeddable":true,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5976"}],"version-history":[{"count":32,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages\/5976\/revisions"}],"predecessor-version":[{"id":6052,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages\/5976\/revisions\/6052"}],"up":[{"embeddable":true,"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=\/wp\/v2\/pages\/4741"}],"wp:attachment":[{"href":"https:\/\/mosaic-modeling.de\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}