NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 1- Lecture 3. Dr. Bishakh Bhattacharya

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1 Information Organization through Signal Flow Graph Dr. Bishakh Bhattacharya Professor, Department t of Mechanical Engineering i IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD

2 Discussion i of Last Assignment In this lecture, we will show another graphical method to represent a dynamic system. This is known as Signal Flow Diagram. Joint Initiative of IITs and IISc - Funded by MHRD 2

3 The Lecture Contains Introduction to Signal Flow Diagram Relationship between Signal Flow Graph and Block Diagram Elements of Signal Flow Graph Simple Rules of Developing Signal Flow Graphs Joint Initiative of IITs and IISc - Funded by MHRD

4 Basic Premises of Signal Flow Graph Signals travel along branches only in the direction of the arrows. A signal travelling along any branch is multiplied by the transmission of that branch. The value of any node variable is the sum of all signals entering the node. The value of any node variable is transmitted on all branches leaving that node. Joint Initiative of IITs and IISc - Funded by MHRD

5 Block Diagram Vs Signal Flow Graph The Figures below show Block Diagrams for different Dynamic Systems and their corresponding Signal Flow Graphs for quick comparison.

6 Elements of Signal-Flow Graph Node. A node is a point representing a variable of signal. Transmittance. The transmittance is a real gain or complex gain between two nodes. Such gains can be expressed in terms of the transfer function between two nodes. Branch. A branch is a directed line segment joining two nodes. The gain of branch is a transmittance. Input mode or source. An input node or source is a node that only have outgoing branches. This corresponds to an independent variable. Output node or sink. An output node or sink is II node that has only incoming branches. This corresponds to a dependent variable.

7 Mixed node. A mixed node is a node that has both incoming and outgoing branches. Path. A path is a traversal of connected branches-in the direction of the branch arrows. If no node is crossed more than once, the path is open. If the path ends at the same node from which it began and does not cross any other node more than once, it is closed. If a path crosses some node more than once but ends at a different node from which it began, it is neither open nor closed. Loop. A loop is a closed path. Loop gain. The loop gain is the product of the branch transmittances of a loop. Non touching loops. Loopsarenontouchingiftheydonotpossessanycommon nodes. Forward path. A forward path is apath from an input node (source) to an output node (sink) that does not cross any nodes more than once. Forward path gain. A forward path gain is the product of the branch transmittances of a forward path.

8 Signal-graph Sg g ap basic rules The value of a node with one incoming branch and gain a is x 2 =ax 1 The total transmittance of cascaded branches is equal to the product of the branch transmittances. Cascaded branches can thus be combined into a single branch by multiplying the transmittances Parallel branches may, be combined by adding the transmittances Mixed nodes and loops may be eliminated to calculate the complete transfer function, for example, a loop may be eliminated at junction 2 in the last figure by noting that x=bx 2, x 2 =ax 1 +cx 3

9 Signal Flow Graphs and Simplifications

10 For a complex signal flow graph, evaluation of the transfer function based on first principles is quite cumbersome. An algorithmic way of evaluating the same based on Graph Theory is known as Mason s rule. Mason s rule Key points Forward Path Gain Product of branch Gains found by traversing a path from the input to output node in the direction of signal flow Non-touching Loops Loops that do not have any nodes in common Non-touching Loop Gain The product of loop gains from non-touching loops

11 Mason s Rule k = number of forward path gain T i = the i th forward path gain Δ = 1 Σ loop gains + Σ non touching loop gains taken two at a time + Σ non touching loop gains taken three at a time etc. Δ i = Δ Σ loop gain terms in Δ that touch the i th forward path

12 Special Reference for this lecture Feedback Control of Dynamic Systems Franklin, Powell and Naeini, Pearson Education Asia Advanced Control Systems Dorf and Bishop, Pearson Education Asia Control Systems Engineering Norman S Nise, John Wiley & Sons Modern Control Engineering K. Ogata, Prentice Hall Joint Initiative of IITs and IISc Funded by MHRD 12

13 Assignment Find out the Transfer Function for the Signal Flow Graph Shown Below G 4 U 1 G 1 G 2 G H 1 H 2 H 3 R 13