Dr.kouroshKianiLinearControlSystemsLecture09SignalFlowGraphs

*DREAMPLANIDEAIMPLEMENTATION

*Present to: Amirkabir University of Technology (Tehran Polytechnic) & Semnan University

Dr. Kourosh KianiEmail: [email protected]: [email protected]: www.kouroshkiani.com

*Lecture 9

*Signal Flow Graphs

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*Block Diagram

*Block Diagram

*Signal Flow Graphs

*Signal Flow Graphs

*SIGNAL FLOW DIAGRAMS: MASONS GAIN FORMULAR(s)C(s)G1(s)G2(s)+-Y(s)Instead of a Block diagram as shown below:We can represent the system as a signal flow graph:R(s)C(s)G1(s)G2(s)11-1Y(s)

*A signal flow graph and a block diagram contain exactly the same information ( there is no advantage to one over the other; there is only personal preference)Signal Flow Graphs

*Why SFG?Block Diagrams are adequate for representation, but cumbersome. SFG provides the relation between system variables without requiring any reduction procedure.

*Basic Elements of Signal Flow GraphA Signal flow graph is a diagram consisting of nodes that are connected by several directed branches. nodenodebranchbranch

*Basic Elements of Signal Flow GraphA node is used to represent a variable (inputs, outputs, other signals)A branch shows the dependence of one variable ( node) on another variable (node)Each branch has GAIN and DIRECTIONA signal can transmit through a branch only in the direction of the arrowIf gain is not specified gain =1

*NodesA node is used to represent a variableSource node (input node)All braches connected to the node are leaving the nodeInput signal is not affected by other signalsSink node (output node)All braches connected to the node are entering the nodeoutput signal is not affecting other signals

*Relationship Between VariablesU (input)X=AU+YY=BXZ=CY+DXV=Z (output)

*Another ExampleX=AU+YY=BX+KZZ=CY+DX+HWW=3UV=Z

*Basic Properties Signal flow graphs applies to linear systems onlyNodes are used to represent variablesA branch from node X to node Y means that Y depends on XValue of the variable (node) is the sum of gain of branch * value of nodeNon-input node cannot be converted to an input nodeWe can create an output node by connecting unit branch to any node

* Terminology: PathsA path: is a branch or a continuous sequence of branches that can be traversed from one node to another nodePaths from U to Z

* Terminology: PathsA path: is a branch or a continuous sequence of branches that can be traversed from one node to another nodeForward path: path from a source to a sinkPath gain: product of gains of the braches that make the path

* Terminology: loopA loop: is a closed path that originates and terminates on the same node, and along the path no node is met twice.Non-touching loops: two loops are said to be non-touching if they do not have a common node.

*An example

*An exampleThis have the solution

*An example

Self loops a11 , a22 , a12 a21 Product of non-touching loops a22 a11

*SGF : in generalThe linear dependence (Tij) between the independent variable xi (input) and the dependent variable (output) xj is given by Masons SF gain formula

*The determinant D

Or D= is the determinant of the graph = 1 - (sum of all individual loop gains) + ( sum of gain products of all possible combinations of two non-touching loops) - (sum of gain products of all possible combinations of three non-touching loops) + - . . .The cofactor is the determinant with loops touching the kth path removed

*Example The number of forward paths from U to V = ?Path Gains ?

Determine the transfer function between V and U

* A BV U C X Y Z 3 W H K Example

The number of forward paths from U to V = 2Path Gains ABC, 3HLoop Gains B, CK

Determine the transfer function between V and U

*Block Diagram and its corresponding Signal Flow Graph

*An example:Two paths :P1, P2Four loopsP1 = G1G2G3G4, P2= G5G6G7G8L1=G2H2 L2=G3H3 L3=G6H6 L4=G7H7D= 1 - (L1+L2+L3+L4)+(L1L3+L1L4+L2L3+L2L4)For path 1 is L1=L2 =0 and For Path 2 L3=L4 =0Cofactor for path 1: D1= 1- (L3+L4)Cofactor for path 2: D2= 1-(L1+L2)T(s) = (P1D1 + P2D2)/D

*Another example3 Paths8 loops

*Another example:

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*Block diagram and its equivalent signal-flow graph111

*Example :Find the transfer function for the following block diagramSolution.123612346123456

*Example :Find the transfer function for the following block diagramSolution.Applying Masons rule, we find the transfer function to be

*Example :Find the transfer function for the following SFGSolution.

*Solution.Applying Masons rule, we find the transfer function to beExample:Find the transfer function for the following SFG

*Block Diagram and its corresponding Signal Flow Graph

*Block Diagram and its corresponding Signal Flow Graph

*Block Diagram and its corresponding Signal Flow Graph

*Block Diagram and its corresponding Signal Flow Graph

*Questions? Discussion? Suggestions ?

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