Name: signal flow graph branch rules
Uploaded: 2024-11-21T14:54:07.000Z
Duration: 1 min 16 s
Channel: JoVE

In Signal Flow Graph algebra, the value of a variable a node represents equals the sum of all the signals entering the node. This value is then transmitted through all branches leaving that node.
Parallel branches in the same direction can be replaced with a single branch with a gain equal to the sum of the gains of the original branches.
Similarly, a cascaded connection of unidirectional branches can be replaced with a single branch with a gain equal to the product of the original branch gains.
Feedback systems in signal flow graphs or SFGs follow specific algebraic equations that can be used to derive the closed-loop transfer function.
A block diagram of a control system can be converted into an SFG format. This involves identifying variables, associating each variable to a node, interconnecting nodes with branches, and labeling each branch with the appropriate gain.
Lastly, SFGs can also be constructed from a set of algebraic equations. This process involves step-by-step construction, creating nodes for each variable, and interconnecting them based on the equations.

ssg-algebra

In de signaalstroomgrafiek (SSG) algebra wordt de waarde die een knooppunt vertegenwoordigt bepaald door de som van alle signalen die dat knooppunt binnenkomen. Deze opgetelde waarde wordt vervolgens verzonden via elke vertakking die het knooppunt verlaat, waardoor de SSG een krachtig hulpmiddel is voor het visualiseren en analyseren van regelsystemen.

Elk knooppunt in een SFG komt overeen met een variabele en de interacties tussen knooppunten worden weergegeven door vertakkingen met bijbehorende versterkingen. Wanneer meerdere vertakkingen naar een knooppunt leiden, is de waarde bij dat knooppunt de som van de signalen van alle inkomende vertakkingen. De uitgaande vertakkingen verzenden deze waarde, vermenigvuldigd met hun respectievelijke versterkingen, vervolgens naar de volgende knooppunten.

Parallelvertakkingen die signalen in dezelfde richting geleiden, kunnen worden samengevoegd tot één enkele vertakking. De versterking van deze nieuwe vertakking is de som van de versterkingen van de oorspronkelijke parallelvertakkingen. Als bijvoorbeeld twee vertakkingen met versterkingen G_1 en G_2 parallel zijn, kunnen ze worden vervangen door één vertakking met een versterking G=G_1+G_2.

Gecasceerde-vertakkingen, of vertakkingen die in serie zijn aangesloten, kunnen op dezelfde manier worden vereenvoudigd. De versterking van de resulterende vertakking is het product van de versterkingen van de oorspronkelijke vertakkingen. Als bijvoorbeeld twee vertakkingen met versterkingen G_1 en G_2 in cascade worden aangesloten, kunnen ze worden vervangen door een enkele vertakking met een versterking G=G_1×G_2.

In feedbacksystemen worden specifieke algebraïsche vergelijkingen gebruikt om de gesloten-lus overdrachtsfunctie af te leiden. Dit omvat het herkennen van de feedbacklus en het toepassen van de juiste formule om het gedrag van het systeem te bepalen. Om een blokdiagram van een besturingssysteem om te zetten in een SSG, volgt u deze stappen:

<ol style="padding-left: 35px; margin: 0;">
 <li>Variabelen identificeren: Elke variabele in het systeem is gekoppeld aan een knooppunt.</li>
 <li>Verbinding knooppunten: Teken vertakkingen tussen knooppunten om de relaties tussen variabelen weer te geven. Zo weet u zeker dat de richting van de signaalstroom nauwkeurig wordt weergegeven.</li>
 <li>Label vertakkingen: Wijs de juiste versterking toe aan elke vertakking, die de overdrachtsfunctie van het overeenkomstige deel van het systeem weerspiegelt.</li>
</ol>

SSG's kunnen ook worden afgeleid uit een set algebraïsche vergelijkingen. Dit proces omvat de volgende stappen:

<ol style="padding-left: 35px; margin: 0;">
 <li>Identificeer en creëer knooppunten voor elke variabele in de vergelijkingen.</li>
 <li>Verbind de knooppunten op basis van de relaties die door de vergelijkingen worden gedefinieerd. Gebruik vertakkingen om deze relaties weer te geven.</li>
 <li>Wijs versterkingen toe aan de vertakkingen volgens de coëfficiënten in de vergelijkingen.</li>
</ol>

Door gebruik te maken van deze principes bieden SSG's een veelzijdige en intuïtieve methode voor het analyseren van complexe besturingssystemen, waardoor het afleiden van overdrachtsfuncties wordt vergemakkelijkt en ons inzicht in de systeemdynamiek wordt vergroot.

SSG-algebra

In Signal Flow Graph algebra, the value of a variable a node represents equals the sum of all the signals entering the node. This value is then ...

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signal flow graph branch rules

Signal flow graph (SFG) branch rules show how signals move through a control system. Each node in an SFG stands for a variable. The value at a node is the sum of all signals entering it. That summed value then travels through every branch leaving the node.

Branches connect the nodes and show the relationship between variables. Each branch has a gain, which tells how strongly the signal is passed along. When several branches enter the same node, their signals are added together. When one node sends its value to the next nodes, each outgoing branch carries that value multiplied by its own gain.

Parallel branches point in the same direction and can be combined into one branch. The gain of the new branch is the sum of the original gains, such as G1 + G2. Cascaded branches are connected in series and can also be simplified. In that case, the gain of the new branch is the product of the original gains, such as G1 × G2.

SFG algebra is also used in feedback systems to find the closed-loop transfer function. This requires identifying the feedback loop and applying the correct formula for the system. A block diagram can be converted into an SFG by identifying the variables, drawing the branches between them, and labeling each branch with the right gain. An SFG can also be built directly from algebraic equations by creating nodes for each variable, linking them by the equation relationships, and assigning branch gains from the coefficients.

Signal Flow Graph Branch Rules

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Signal flow graph (SFG) branch rules show how signals move through a control system. Each node in an SFG stands for a variable. The value at a node is the sum of all signals entering it. That summed value then travels through every branch leaving the node. Branches connect the nodes and show the relationship between variables. Each branch has a gain, which tells how strongly the signal is passed along. When several branches enter the same node, their signals are added together. When one node...

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Signal Flow Graph Branch Rules

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