[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657433-105":59,"doc-detail-657433-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","introductory-control-systems-block-diagram-transformations","Introductory Control Systems - Block Diagram Transformations","","Introductory notes explain how complex control-system block diagrams can be reduced to simpler equivalent forms through algebraic manipulations. The material outlines common transformations, including combining series blocks, relocating summing points and pick-off points, and collapsing feedback loops. It emphasizes equivalence by preserving input and output signals and maintaining the same transfer function. The document also clarifies differences between negative and positive feedback and provides guidance for reducing multiple nested closed loops.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/introductory-control-systems-block-diagram-transformations/657433/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/introductory-control-systems-block-diagram-transformations/657433.png","ImageObject",300,407,{"name":92,"@type":93},"AfterMidnight","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How are equivalent block diagram transformations derived?","Question",{"text":112,"@type":113},"They are derived by manipulating the algebraic equations associated with the block diagram. The goal is to produce a reduced diagram with the same input-output behavior.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does “negative feedback” mean in the context of the feedback loop reduction?",{"text":117,"@type":113},"The signal is negated at the summing block, which leads to a specific form of the closed-loop behavior distinct from positive feedback.",{"name":119,"@type":110,"acceptedAnswer":120},"Why is it important not to change or eliminate signals from other parts of the diagram?",{"text":121,"@type":113},"Because those signals may be required elsewhere; altering them can make the transformed diagram not equivalent to the original, even if some local reduction was performed correctly.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657433,1792979036,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":52},962090896061,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Introductory Control Systems  \nBlock Diagram Transformations  \nReference: R.C. Dorf and R.H. Bishop, Modern Control Systems, 11th Ed., Pearson/Prentice-Hall, 2008.  \nComplex block diagrams can be transformed into simpler equivalent diagrams using basic block diagram transformations. These transformations are derived by simply manipulating the algebraic equations associated with the diagram. The list of transformations derived below is not meant to be all inclusive, but rather to encourage the reader to begin to understand block diagram transformations by reading the details of the block diagram.  \nMotivation: It is not intended here that the analyst become proficient at reducing large complex block diagrams, but rather should be able to read the details provided by the block diagram. If the details of the block diagram truly reflect the function of the system, then understanding the block diagram is the same as understanding the operation of the system itself. Automated procedures are available to assist the analyst in the reduction of complex block diagrams.  \nBlock Diagram Transformations  \nIn the transformations shown below, simple block diagrams are reduced to simpler forms by using the block diagram algebra associated with the diagram. In each case, the first figure is the starting diagram, and the second figure is the reduced “equivalent” diagram.  \n1. Combining blocks in a series:  \nAlgebra:  \nx3 􀀠 G2x2 􀀠 G2 G1x1  \n􀂟  \nx3  􀀠 G G 􀀠 G Gx1 2 1 1 2  \n􀂟  \n2. Moving a summing point behind a block:  \nAlgebra:  \nx3 􀀠 G 􀀋x1 􀀐 x2 􀀌 􀀠 G x1 􀀐 G x2  \n􀂟  \nKamman – Introductory Control Systems – Block Diagram Transformations – page: 1/4  \n3. Moving a summing point in front of a block:  \nAlgebra:  \n􀂧 􀂧 1 􀂷 􀂷 x3 􀀠 G x1 􀀐 x2 􀀠 G 􀂩􀂨 x1 􀀐 􀂩􀂨 G  􀂹􀂸 x2 􀂹􀂸  \n􀂟  \n4. Moving a pick-off point behind a block:  \nAlgebra:  \nx2 􀀠 Gx1  \n􀂟  \nx1 􀀠 􀂧􀂨􀂩 ~~ ~~1G~~ ~~ 􀂷􀂹􀂸x2  \n􀂟  \n5. Moving a pick-off point in front of a block:  \nAlgebra:  \nx2 􀀠 Gx1  \n􀂟  \nKamman – Introductory Control Systems – Block Diagram Transformations – page: 2/4  \n6. Collapsing a feedback loop:  \nAlgebra:  \nx2 􀀠 G e 􀀠 G 􀀋x1 􀀐 x3 􀀌 􀀠 Gx1 􀀐 Gx3 􀀠 Gx1 􀀐 GH x2  \n􀂟 􀀋1􀀎 GH􀀌x2 􀀠 Gx1  \n􀂟  \n\n| x2 | G\u003Cbr>􀀠 |\n| --- | --- |\n| x1 1 􀀎 GH |  |\n\nAs shown, the system is said to have “negative feedback”, because the signal x3 is negated at the summing block. If the system has “positive feedback”, then it is easy to show that the transfer function becomes  \nx2  \nx1  \nG  \n􀀠  \n1 􀀐 GH  \n(for positive feedback)  \nNotes:  \no In each transformation, signals may be modified or even eliminated from the system. So, the transformed system is not “identical” to the original. The transformed system is“equivalent” to the original system in that it has the same input and output signals. Of course, the reduced system must have the same transfer function as the original system.  \no When transforming a block diagram, it is important not to change or eliminate signals required by other portions ofthe diagram. These types of changes will generally render the transformed diagram to be not equivalent to the original.  \no When reducing systems that have multiple closed loops, the original system should be transformed into one whose closed loops are nested. This is accomplished using transformations like items (1) through (5) above.  \no Once the system is in a nested form, it can be reduced by collapsing the inner-most loop first and then collapsing each successive inner-most loop until the outer-most loop is collapsed.  \nKamman – Introductory Control Systems – Block Diagram Transformations – page: 3/4  \no For the simple closed loop system of item (6), the following terminology is often used:  \nx   G   \n􀂾 Closed loop transfer function: 2 􀀠  \nx1 1 􀁲 GH  \n􀂾 Forward path transfer function: ~~ ~~x~~ 2~~e~~ ~~ 􀀠 G  \nx   \n􀂾 Feedback path transfer function: 3 􀀠 H  \n􀂾 Loop transfer function:  x 2x3e  􀀠 GH  \nKamman – Introductory Control Systems – Block Diagram Transformations – page: 4/4","cbCaisUIEVOVnF4B","https://ap.wps.com/l/cbCaisUIEVOVnF4B","pdf",623489,"English","# Block Diagram Transformations\n## Combining blocks in a series\n## Moving a summing point behind a block\n## Moving a summing point in front of a block\n## Moving a pick-off point behind a block\n## Moving a pick-off point in front of a block\n## Collapsing a feedback loop\n## Notes on equivalence and nested loop reduction","[{\"question\":\"How are equivalent block diagram transformations derived?\",\"answer\":\"They are derived by manipulating the algebraic equations associated with the block diagram. The goal is to produce a reduced diagram with the same input-output behavior.\"},{\"question\":\"What does “negative feedback” mean in the context of the feedback loop reduction?\",\"answer\":\"The signal is negated at the summing block, which leads to a specific form of the closed-loop behavior distinct from positive feedback.\"},{\"question\":\"Why is it important not to change or eliminate signals from other parts of the diagram?\",\"answer\":\"Because those signals may be required elsewhere; altering them can make the transformed diagram not equivalent to the original, even if some local reduction was performed correctly.\"}]","Introductory Control Systems - Block Diagram Transformations | PDF"]