Module 3: State-Space Systems; Translating Transfer Functions to State-Space Form and Back Again

Lectures

Folder description

Please click the folder link above to access the Module 3 instructional content. Module 3 contains four video lectures that present the following topics:

Linearization of a Multivariate State Space System (3A)
The State Transition Equation (3B)
An Example; Solving The State Transition Equation (3D)
Direct Decomposition (3D)

Technical

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Module at a Glance ObjectivesHow to Complete This Module:

Cell description

Objectives

By the end of this module, you will be able to:

Apply linearization techniques to nonlinear multivariable systems.
Compute the system State Transition Matrix for single variable and multivariable systems.
Determine the system State Transition Equation for single variable and multivariable systems.
Apply the method of Direct Decomposition to convert general LTIODE systems in transfer function form to State-Space form.

How to Complete This Module:

View Lecture 3A: Linearization of a Multivariate State Space System.
View Lecture 3B: The State Transition Equation.
View Lecture 3C: An Example; Solving The State Transition Equation.
View Lecture 3D: Direct Decomposition.
Complete the assigned reading: Chapter 8, Sections 8.1-8.8, 8.11 in the 10th Edition of the textbook (Chapter 10, Sections 10.1 through10.6, 10.10 in the 9th Ed.).
Complete the Module 3 Practice Assignment.
Take part in the Module 3 Discussion.
Complete Quiz 1.

Technical

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Readings

Cell description

To access EReserves readings, click the Readings title above or EReserves on the course menu.

This week's reading assignment includes the following:

Golnaraghi, Farid, and Benjamin C. Kuo. Automatic Control Systems (10th Edition).
- Chapter 8, Sections 8.1-8.8, 8.11.
- (Or, Chapter 10, sections 10.1 through 10.9 in the 9th Ed. of the textbook.)

Technical

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Module 3 Practice Assignment

Cell description

Instructions: Download and print the Module 3 Practice Assignment PDF, complete by hand, and then scan and upload your final document to the Blackboard course site. Please ensure that your scan is of readable quality and oriented in the proper direction for reading. All final submittals are to be in PDF format.

Submit your file using the following naming structure: Lastname_Firstname_M3Assignment (example: Palumbo_Neil_M3Assignment).

Technical

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Module 3 Discussion

Cell description

Students should respond to the following prompt and respond to at least two classmate’s post as well.

Given the Homogeneous State Space System,

, the multivariate State Transition Matrix is defined to be

Given a system matrix A, discuss in enough detail how one might write a small and efficient computer program to approximately calculate Φ(t)
If you are still having difficulty with a topic/concept, please elaborate so that I and/or your classmates can assist you.

Technical

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<span></span><span><span></span></span><span><span></span></span><span><span></span><span><span></span> ,<span></span> <span></span></span><span><span></span></span><span>the multivariate State <span></span></span><span></span></span><span><span></span></span><span><span>Transition Matrix is defined to be <span></span></span><span></span></span><p></p> \n <p><span><span></span><span></span></span><span class=\"axScoreIndicator axVtbeEmbedded image\"><span data-ally-scoreindicator=\"_86813289_2\"></span></span>\n<img src=\"https://blackboard.jhu.edu/bbcswebdav/pid-7500214-dt-content-rid-86813289_2/xid-86813289_2\" data-mathml=\"«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨»§#934;«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo mathvariant=¨bold¨»§#8801;«/mo»«mo»§#160;«/mo»«msup»«mi»e«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mi»t«/mi»«/mrow»«/msup»«mo»§#160;«/mo»«mo»§#8801;«/mo»«msup»«mo mathvariant=¨bold¨»§#8466;«/mo»«mrow»«mo»-«/mo»«mn»1«/mn»«mo»§#160;«/mo»«/mrow»«/msup»«mfenced open=¨{¨ close=¨}¨»«msup»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»s«/mi»«mi mathvariant=¨bold¨»I«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»A«/mi»«/mrow»«/mfenced»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/mfenced»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»I«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«mo mathvariant=¨bold¨»+«/mo»«mo mathvariant=¨bold¨»§#160;«/mo»«mi mathvariant=¨bold¨»A«/mi»«mi»t«/mi»«mo mathvariant=¨bold¨»+«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»2«/mn»«mo»!«/mo»«/mrow»«/mfrac»«msup»«mi mathvariant=¨bold¨»A«/mi»«mn»2«/mn»«/msup»«msup»«mi»t«/mi»«mrow»«mn»2«/mn»«mo»§#160;«/mo»«/mrow»«/msup»«mo»+«/mo»«mo»§#160;«/mo»«mfrac»«mn»1«/mn»«mrow»«mn»3«/mn»«mo»!«/mo»«/mrow»«/mfrac»«msup»«mi mathvariant=¨bold¨»A«/mi»«mn»3«/mn»«/msup»«msup»«mi»t«/mi»«mrow»«mn mathvariant=¨italic¨»3«/mn»«mo mathvariant=¨italic¨»§#160;«/mo»«/mrow»«/msup»«mo»+«/mo»«mo»§#160;«/mo»«mo»§#8943;«/mo»«/math»\" class=\"Wirisformula\" alt=\"bold capital phi open parentheses t close parentheses bold identical to space e to the power of bold A t end exponent space identical to bold script capital l to the power of minus 1 space end exponent open curly brackets open square brackets s bold I bold minus bold A close square brackets to the power of minus 1 end exponent close curly brackets bold equals bold I bold space bold plus bold space bold A t bold plus fraction numerator 1 over denominator 2 factorial end fraction bold A squared t to the power of 2 space end exponent plus space fraction numerator 1 over denominator 3 factorial end fraction bold A cubed t to the power of italic 3 italic space end exponent plus space midline horizontal ellipsis\" align=\"\" style=\"vertical-align: -22px;\"> \n<span class=\"contextMenuContainer image\" style=\" display: inline;\" 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bb:overwritenavitems=\"\" bb:beforeshowfunc=\"\"> <a id=\"cmlink_748a47fbe02547a7a732da1ecfcb14ca\" onfocus=\"page.ContextMenu.LI(event, 'cmdiv_748a47fbe02547a7a732da1ecfcb14ca',true);\" onmouseover=\"page.ContextMenu.LI(event, 'cmdiv_748a47fbe02547a7a732da1ecfcb14ca',true);\" class=\"cmimg editmode jsInit   cmimg-hide\" href=\"#contextMenu\" title=\"Click for more options\"> <img src=\"/images/ci/icons/cmlink_generic.gif\" alt=\"Click for more options\" onload=\"page.ContextMenu.LI(event, 'cmdiv_748a47fbe02547a7a732da1ecfcb14ca',true);\"></a> \n </span></p>  \n<input type=\"hidden\" name=\"86813289_2_fileReadWritePermAvl\" id=\"86813289_2_fileReadWritePermAvl\" value=\"false\"> \n<script type=\"text/javascript\" src=\"/javascript/cdn.js\"></script> \n<script type=\"text/javascript\" src=\"/groupjs/FF147D720E72E17F96127A7085584389.js?v=3800.0.9-rel.16+8f8c3e7\"></script> \n<script type=\"text/javascript\" src=\"/webapps/cmsmain/dirlistrightframe/dirlist.js?v=3800.0.9-rel.16+8f8c3e7\"></script> \n<script type=\"text/javascript\" src=\"/webapps/cmsmain/static/bbDriveUtil.js?v=3800.0.9-rel.16+8f8c3e7\"></script> \n<script type=\"text/javascript\">\n    (function(){\n     var f = function(){\n           if ( window.DWREngine )\n      {\n       if( DWREngine._batch )\n       {\n      \t try {DWREngine.endBatch();} catch(ignore) {}\n       }\n       try {DWREngine.beginBatch();} catch(ignore) {}\n      } \n             page.ContextMenu.hideMenuDiv('748a47fbe02547a7a732da1ecfcb14ca');\n            if ( window.DWREngine )\n      {\n        try {DWREngine.endBatch();} catch(ignore) {}\n      }\n               };\n     var f2 = function()\n     {\n       if ( page.isLoaded )\n       {\n         f();\n       }\n       else\n       {\n         setTimeout( f2, 500 );\n       }\n     }\n     f2();\n    })();\n   </script><p></p> \n <ul> \n  <li><span><span>Given a system matrix <span></span></span><strong>A</strong><span><span></span>,</span><span></span></span><span><span></span></span><span><span></span><span><span></span> discuss in enough detail how one might write a small and efficient computer program to approximately calculate <span></span></span><strong>Φ</strong><span><span></span>(</span><em>t</em><span><span></span>)</span><span></span></span><span><span></span></span></li> \n  <li><span><span>If you are still having difficulty with a topic/concept,<span></span></span><span></span></span><span><span></span><span><span></span> please elaborate so that I and/or your classmates can assist you.<span></span></span></span></li> \n </ul> \n <p><br></p> \n</div>\n  </div>\n\n <div id=\"_7500214_1previewModule\" class=\"moduleSample\">\n </div>\n",
        "url": "https://blackboard.jhu.edu/webapps/blackboard/content/launchLink.jsp?course_id=_197015_1&content_id=_7500214_1&mode=view",
        "title": "Module 3 Discussion"
    },
    "html": "\n\n<img alt=\"Discussion Board\" src=\"/images/ci/sets/set01/discussion_on.gif\" class=\"item_icon\">\n\n<div class=\"item clearfix\" id=\"_7500214_1\">\n\n <h3 id=\"anonymous_element_14\">\n  <span class=\"reorder editmode hideme\"><span><img src=\"/images/ci/icons/generic_updown.gif\" alt=\"\"></span></span>\n\n\n            <a href=\"/webapps/blackboard/content/launchLink.jsp?course_id=_197015_1&amp;content_id=_7500214_1&amp;mode=view\"><span style=\"color:#0067ac;\">Module 3 Discussion</span></a>\n      </h3><a href=\"#\" class=\"ally-alternative-formats ally-prominent-af-download-button\" data-ally-external-id=\"content:_7500214_1\" data-ally-upload-type=\"RichContent\" title=\"Alternative formats\" style=\"display: none\">\n    <span class=\"ally-sr-only\">\n  \n\n\n            Module 3 Discussion\n      </span>\n</a>\n\n   \n </div>\n\n\n\n<div class=\"details\">\n    <div class=\"vtbegenerated\"> \n <p><span><span>Students should respond to the following prompt and respond to at least two classmate’s post as well.<span></span></span><span></span></span></p> \n <p><span><span>Given the Homogeneous State Space System,<span></span></span><span></span></span><span><span></span><span><span></span>&nbsp;</span><span class=\"axScoreIndicator axVtbeEmbedded image\"><span data-ally-scoreindicator=\"_86813288_2\"></span></span>\n<img src=\"https://blackboard.jhu.edu/bbcswebdav/pid-7500214-dt-content-rid-86813288_2/xid-86813288_2\" data-mathml=\"«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mover»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#175;«/mo»«/mover»«mo»§#160;«/mo»«/mrow»«mo»§#183;«/mo»«/mover»«mfenced»«mi»t«/mi»«/mfenced»«mo»§#160;«/mo»«mo»=«/mo»«mi mathvariant=¨bold¨»A«/mi»«mover»«mi mathvariant=¨bold¨»x«/mi»«mo mathvariant=¨bold¨»§#175;«/mo»«/mover»«mo»§#160;«/mo»«mfenced»«mi»t«/mi»«/mfenced»«/math»\" class=\"Wirisformula\" alt=\"stack bold x with bold bar on top space with times on top open parentheses t close parentheses space equals bold A bold x with bold bar on top space open parentheses t close parentheses\" align=\"\" style=\"vertical-align: -23px;\"> \n<span class=\"contextMenuContainer image\" style=\" display: inline;\" bb:menugeneratorurl=\"/webapps/cmsmain/webui/courses/EN.525.609.81.SP20/Math%20Images/ae7dbc28b7a97d6ba075bd3b81c79139.png?action=details&amp;subaction=generateFileMenuItem&amp;uniq=vd00n0&amp;course_id=_197015_1&amp;ctxMenuXythosId=86813288_2&amp;course_id=_197015_1\" bb:navitem=\"\" bb:contextparameters=\"\" 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src=\"/webapps/cmsmain/dirlistrightframe/dirlist.js?v=3800.0.9-rel.16+8f8c3e7\"></script> \n<script type=\"text/javascript\" src=\"/webapps/cmsmain/static/bbDriveUtil.js?v=3800.0.9-rel.16+8f8c3e7\"></script> \n<script type=\"text/javascript\">\n    (function(){\n     var f = function(){\n           if ( window.DWREngine )\n      {\n       if( DWREngine._batch )\n       {\n      \t try {DWREngine.endBatch();} catch(ignore) {}\n       }\n       try {DWREngine.beginBatch();} catch(ignore) {}\n      } \n             page.ContextMenu.hideMenuDiv('4af5c16540514554820b14bcaa221f1c');\n            if ( window.DWREngine )\n      {\n        try {DWREngine.endBatch();} catch(ignore) {}\n      }\n               };\n     var f2 = function()\n     {\n       if ( page.isLoaded )\n       {\n         f();\n       }\n       else\n       {\n         setTimeout( f2, 500 );\n       }\n     }\n     f2();\n    })();\n   </script> <span><span></span></span> <span></span><span><span></span></span><span><span></span></span><span><span></span><span><span></span> ,<span></span> <span></span></span><span><span></span></span><span>the multivariate State <span></span></span><span></span></span><span><span></span></span><span><span>Transition Matrix is defined to be <span></span></span><span></span></span><p></p> \n <p><span><span></span><span></span></span><span class=\"axScoreIndicator axVtbeEmbedded image\"><span data-ally-scoreindicator=\"_86813289_2\"></span></span>\n<img src=\"https://blackboard.jhu.edu/bbcswebdav/pid-7500214-dt-content-rid-86813289_2/xid-86813289_2\" data-mathml=\"«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨bold¨»§#934;«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo mathvariant=¨bold¨»§#8801;«/mo»«mo»§#160;«/mo»«msup»«mi»e«/mi»«mrow»«mi mathvariant=¨bold¨»A«/mi»«mi»t«/mi»«/mrow»«/msup»«mo»§#160;«/mo»«mo»§#8801;«/mo»«msup»«mo 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bb:overwritenavitems=\"\" bb:beforeshowfunc=\"\"> <a id=\"cmlink_748a47fbe02547a7a732da1ecfcb14ca\" onfocus=\"page.ContextMenu.LI(event, 'cmdiv_748a47fbe02547a7a732da1ecfcb14ca',true);\" onmouseover=\"page.ContextMenu.LI(event, 'cmdiv_748a47fbe02547a7a732da1ecfcb14ca',true);\" class=\"cmimg editmode jsInit   cmimg-hide\" href=\"#contextMenu\" title=\"Click for more options\"> <img src=\"/images/ci/icons/cmlink_generic.gif\" alt=\"Click for more options\" onload=\"page.ContextMenu.LI(event, 'cmdiv_748a47fbe02547a7a732da1ecfcb14ca',true);\"></a> \n </span></p>  \n<input type=\"hidden\" name=\"86813289_2_fileReadWritePermAvl\" id=\"86813289_2_fileReadWritePermAvl\" value=\"false\"> \n<script type=\"text/javascript\" src=\"/javascript/cdn.js\"></script> \n<script type=\"text/javascript\" src=\"/groupjs/FF147D720E72E17F96127A7085584389.js?v=3800.0.9-rel.16+8f8c3e7\"></script> \n<script type=\"text/javascript\" src=\"/webapps/cmsmain/dirlistrightframe/dirlist.js?v=3800.0.9-rel.16+8f8c3e7\"></script> \n<script type=\"text/javascript\" src=\"/webapps/cmsmain/static/bbDriveUtil.js?v=3800.0.9-rel.16+8f8c3e7\"></script> \n<script type=\"text/javascript\">\n    (function(){\n     var f = function(){\n           if ( window.DWREngine )\n      {\n       if( DWREngine._batch )\n       {\n      \t try {DWREngine.endBatch();} catch(ignore) {}\n       }\n       try {DWREngine.beginBatch();} catch(ignore) {}\n      } \n             page.ContextMenu.hideMenuDiv('748a47fbe02547a7a732da1ecfcb14ca');\n            if ( window.DWREngine )\n      {\n        try {DWREngine.endBatch();} catch(ignore) {}\n      }\n               };\n     var f2 = function()\n     {\n       if ( page.isLoaded )\n       {\n         f();\n       }\n       else\n       {\n         setTimeout( f2, 500 );\n       }\n     }\n     f2();\n    })();\n   </script><p></p> \n <ul> \n  <li><span><span>Given a system matrix <span></span></span><strong>A</strong><span><span></span>,</span><span></span></span><span><span></span></span><span><span></span><span><span></span> discuss in enough detail how one might write a small and efficient computer program to approximately calculate <span></span></span><strong>Φ</strong><span><span></span>(</span><em>t</em><span><span></span>)</span><span></span></span><span><span></span></span></li> \n  <li><span><span>If you are still having difficulty with a topic/concept,<span></span></span><span></span></span><span><span></span><span><span></span> please elaborate so that I and/or your classmates can assist you.<span></span></span></span></li> \n </ul> \n <p><br></p> \n</div>\n  </div>\n\n <div id=\"_7500214_1previewModule\" class=\"moduleSample\">\n </div>\n",
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        "title": "Module 3 Discussion",
        "body": "\n    <div> \n <p><span><span>Students should respond to the following prompt and respond to at least two classmate’s post as well.</span></span></p> \n <p><span><span>Given the Homogeneous State Space System,</span></span><span><span>&nbsp;</span>\n \n<span bb:menugeneratorurl=\"/webapps/cmsmain/webui/courses/EN.525.609.81.SP20/Math%20Images/ae7dbc28b7a97d6ba075bd3b81c79139.png?action=details&amp;subaction=generateFileMenuItem&amp;uniq=vd00n0&amp;course_id=_197015_1&amp;ctxMenuXythosId=86813288_2&amp;course_id=_197015_1\" bb:navitem=\"\" bb:contextparameters=\"\" bb:menuorder=\"openId_4af5c16540514554820b14bcaa221f1c,open_with_browser_4af5c16540514554820b14bcaa221f1c,open_with_bbdrive_4af5c16540514554820b14bcaa221f1c,openWebId_4af5c16540514554820b14bcaa221f1c,view360Id_4af5c16540514554820b14bcaa221f1c,*separator*,*separator*,ax-ally-placeholder_4af5c16540514554820b14bcaa221f1c,ax-ally-markForAxAnalysis_4af5c16540514554820b14bcaa221f1c,*separator*,*separator*,content_settings_4af5c16540514554820b14bcaa221f1c,content_editSettings_4af5c16540514554820b14bcaa221f1c,content_edit_4af5c16540514554820b14bcaa221f1c,properties_folder_4af5c16540514554820b14bcaa221f1c,itemcarot_managerecord_4af5c16540514554820b14bcaa221f1c,comments_tool_4af5c16540514554820b14bcaa221f1c,subscriptions_list_4af5c16540514554820b14bcaa221f1c,content_readfile_4af5c16540514554820b14bcaa221f1c,content_overwrite_4af5c16540514554820b14bcaa221f1c,passes_tool_4af5c16540514554820b14bcaa221f1c,permissions_tool_4af5c16540514554820b14bcaa221f1c,content_add_alignment_4af5c16540514554820b14bcaa221f1c,content_tracking_4af5c16540514554820b14bcaa221f1c,content_versions_4af5c16540514554820b14bcaa221f1c,*separator*,*separator*,bookmarkId_4af5c16540514554820b14bcaa221f1c,itemsId_4af5c16540514554820b14bcaa221f1c,packagesId_4af5c16540514554820b14bcaa221f1c,emailId_4af5c16540514554820b14bcaa221f1c,removeContentFolderId_4af5c16540514554820b14bcaa221f1c,*separator*,copyContentId_4af5c16540514554820b14bcaa221f1c,moveContentId_4af5c16540514554820b14bcaa221f1c,*separator*,removeContentId_4af5c16540514554820b14bcaa221f1c,*separator*,Metadata_General_4af5c16540514554820b14bcaa221f1c,Metadata_Dublin_Core_4af5c16540514554820b14bcaa221f1c,Metadata_IMS_4af5c16540514554820b14bcaa221f1c,Metadata_Full_IMS_4af5c16540514554820b14bcaa221f1c,\" bb:overwritenavitems=\"\" bb:beforeshowfunc=\"\"> <a id=\"cmlink_4af5c16540514554820b14bcaa221f1c\" onfocus=\"page.ContextMenu.LI(event, 'cmdiv_4af5c16540514554820b14bcaa221f1c',true);\" onmouseover=\"page.ContextMenu.LI(event, 'cmdiv_4af5c16540514554820b14bcaa221f1c',true);\" href=\"#contextMenu\" title=\"Click for more options\"> </a> \n </span></span></p>  \n \n \n \n \n \n  <span><span> , </span><span>the multivariate State </span></span><span><span>Transition Matrix is defined to be </span></span> \n <p>\n \n<span bb:menugeneratorurl=\"/webapps/cmsmain/webui/courses/EN.525.609.81.SP20/Math%20Images/9e578739f1a6ab534d9a6376826f044b.png?action=details&amp;subaction=generateFileMenuItem&amp;uniq=vd00n2&amp;course_id=_197015_1&amp;ctxMenuXythosId=86813289_2&amp;course_id=_197015_1\" bb:navitem=\"\" bb:contextparameters=\"\" bb:menuorder=\"openId_748a47fbe02547a7a732da1ecfcb14ca,open_with_browser_748a47fbe02547a7a732da1ecfcb14ca,open_with_bbdrive_748a47fbe02547a7a732da1ecfcb14ca,openWebId_748a47fbe02547a7a732da1ecfcb14ca,view360Id_748a47fbe02547a7a732da1ecfcb14ca,*separator*,*separator*,ax-ally-placeholder_748a47fbe02547a7a732da1ecfcb14ca,ax-ally-markForAxAnalysis_748a47fbe02547a7a732da1ecfcb14ca,*separator*,*separator*,content_settings_748a47fbe02547a7a732da1ecfcb14ca,content_editSettings_748a47fbe02547a7a732da1ecfcb14ca,content_edit_748a47fbe02547a7a732da1ecfcb14ca,properties_folder_748a47fbe02547a7a732da1ecfcb14ca,itemcarot_managerecord_748a47fbe02547a7a732da1ecfcb14ca,comments_tool_748a47fbe02547a7a732da1ecfcb14ca,subscriptions_list_748a47fbe02547a7a732da1ecfcb14ca,content_readfile_748a47fbe02547a7a732da1ecfcb14ca,content_overwrite_748a47fbe02547a7a732da1ecfcb14ca,passes_tool_748a47fbe02547a7a732da1ecfcb14ca,permissions_tool_748a47fbe02547a7a732da1ecfcb14ca,content_add_alignment_748a47fbe02547a7a732da1ecfcb14ca,content_tracking_748a47fbe02547a7a732da1ecfcb14ca,content_versions_748a47fbe02547a7a732da1ecfcb14ca,*separator*,*separator*,bookmarkId_748a47fbe02547a7a732da1ecfcb14ca,itemsId_748a47fbe02547a7a732da1ecfcb14ca,packagesId_748a47fbe02547a7a732da1ecfcb14ca,emailId_748a47fbe02547a7a732da1ecfcb14ca,removeContentFolderId_748a47fbe02547a7a732da1ecfcb14ca,*separator*,copyContentId_748a47fbe02547a7a732da1ecfcb14ca,moveContentId_748a47fbe02547a7a732da1ecfcb14ca,*separator*,removeContentId_748a47fbe02547a7a732da1ecfcb14ca,*separator*,Metadata_General_748a47fbe02547a7a732da1ecfcb14ca,Metadata_Dublin_Core_748a47fbe02547a7a732da1ecfcb14ca,Metadata_IMS_748a47fbe02547a7a732da1ecfcb14ca,Metadata_Full_IMS_748a47fbe02547a7a732da1ecfcb14ca,\" bb:overwritenavitems=\"\" bb:beforeshowfunc=\"\"> <a id=\"cmlink_748a47fbe02547a7a732da1ecfcb14ca\" onfocus=\"page.ContextMenu.LI(event, 'cmdiv_748a47fbe02547a7a732da1ecfcb14ca',true);\" onmouseover=\"page.ContextMenu.LI(event, 'cmdiv_748a47fbe02547a7a732da1ecfcb14ca',true);\" href=\"#contextMenu\" title=\"Click for more options\"> </a> \n </span></p>  \n \n \n \n \n \n \n <ul> \n  <li><span><span>Given a system matrix </span><strong>A</strong><span>,</span></span><span><span> discuss in enough detail how one might write a small and efficient computer program to approximately calculate </span><strong>Φ</strong><span>(</span><em>t</em><span>)</span></span></li> \n  <li><span><span>If you are still having difficulty with a topic/concept,</span></span><span><span> please elaborate so that I and/or your classmates can assist you.</span></span></li> \n </ul> \n  \n</div>\n  "
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Module 3 Discussion Assignment Solutions

Cell description

Click on the link below to review the Discussion Assignment Solutions.

(To be released on the first day of Module 4.)

Technical

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Module 3 Practice Assignment Solutions

Cell description

Click on the link below to review the Practice Assignment Solutions.

(To be released on the first day of Module 4.)

Technical

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