Signal Flow Graph In Control System Pdf

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    Signal Flow Graph In Control System Pdf

      понедельник 04 мая
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    Signal-flow graphs are another method for visually representing a system. Signal Flow Diagrams are especially useful, because they allow for particular methods of analysis, such as Mason's Gain Formula. Signal flow diagrams typically use curved lines to represent wires and systems, instead of using lines at right-angles. UNIT I – CONTROL SYSTEM MODELLING Basic Elements of Control System – Open loop and Closed loop systems - Differential equation - Transfer function, Modeling of Electric systems, Translational and rotational mechanical systems - Block diagram reduction Techniques - Signal flow graph PART A Q.No Questions BT Level Domain 1. Compare the Open loop System with Closed loop System.


    Signal flow graph is a graphical representation of algebraic equations. In this chapter, let us discuss the basic concepts related signal flow graph and also learn how to draw signal flow graphs.

    Basic Elements of Signal Flow Graph

    Nodes and branches are the basic elements of signal flow graph.

    Node

    Node is a point which represents either a variable or a signal. There are three types of nodes — input node, output node and mixed node.

    • Input Node − It is a node, which has only outgoing branches.

    • Output Node − It is a node, which has only incoming branches.

    • Mixed Node − It is a node, which has both incoming and outgoing branches.

    Example

    Let us consider the following signal flow graph to identify these nodes.

    • The nodes present in this signal flow graph are y1, y2, y3 and y4.

    • y1 and y4 are the input node and output node respectively.

    • y2 and y3 are mixed nodes.

    Branch

    Branch is a line segment which joins two nodes. It has both gain and direction. For example, there are four branches in the above signal flow graph. These branches have gains of a, b, c and -d.

    Construction of Signal Flow Graph

    Let us construct a signal flow graph by considering the following algebraic equations −

    $$y_2=a_{12}y_1+a_{42}y_4$$

    $$y_3=a_{23}y_2+a_{53}y_5$$

    $$y_4=a_{34}y_3$$

    $$y_5=a_{45}y_4+a_{35}y_3$$

    $$y_6=a_{56}y_5$$

    There will be six nodes (y1, y2, y3, y4, y5 and y6) and eight branches in this signal flow graph. The gains of the branches are a12, a23, a34, a45, a56, a42, a53 and a35.

    To get the overall signal flow graph, draw the signal flow graph for each equation, then combine all these signal flow graphs and then follow the steps given below −

    Step 1 − Signal flow graph for $y_2 = a_{13}y_1 + a_{42}y_4$ is shown in the following figure.

    Step 2 − Signal flow graph for $y_3 = a_{23}y_2 + a_{53}y_5$ is shown in the following figure.

    Step 3Accounting an introduction atrilli pdf. − Signal flow graph for $y_4 = a_{34}y_3$ is shown in the following figure.

    Step 4 − Signal flow graph for $y_5 = a_{45}y_4 + a_{35}y_3$ is shown in the following figure.

    Step 5 − Signal flow graph for $y_6 = a_{56}y_5$ is shown in the following figure.

    Step 6 − Signal flow graph of overall system is shown in the following figure.

    Conversion of Block Diagrams into Signal Flow Graphs

    Follow these steps for converting a block diagram into its equivalent signal flow graph.

    • Represent all the signals, variables, summing points and take-off points of block diagram as nodes in signal flow graph.

    • Represent the blocks of block diagram as branches in signal flow graph.

    • Represent the transfer functions inside the blocks of block diagram as gains of the branches in signal flow graph.

    • Connect the nodes as per the block diagram. If there is connection between two nodes (but there is no block in between), then represent the gain of the branch as one. For example, between summing points, between summing point and takeoff point, between input and summing point, between take-off point and output.

    Example

    Let us convert the following block diagram into its equivalent signal flow graph.

    Represent the input signal $R(s)$ and output signal $C(s)$ of block diagram as input node $R(s)$ and output node $C(s)$ of signal flow graph.

    Just for reference, the remaining nodes (y1 to y9) are labelled in the block diagram. There are nine nodes other than input and output nodes. That is four nodes for four summing points, four nodes for four take-off points and one node for the variable between blocks $G_1$ and $G_2$.

    The following figure shows the equivalent signal flow graph.

    With the help of Mason’s gain formula (discussed in the next chapter), you can calculate the transfer function of this signal flow graph. This is the advantage of signal flow graphs. Here, we no need to simplify (reduce) the signal flow graphs for calculating the transfer function.

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    Let us now discuss the Mason’s Gain Formula. Suppose there are ‘N’ forward paths in a signal flow graph. The gain between the input and the output nodes of a signal flow graph is nothing but the transfer function of the system. It can be calculated by using Mason’s gain formula.

    Mason’s gain formula is

    $$T=frac{C(s)}{R(s)}=frac{Sigma ^N _{i=1}P_iDelta _i}{Delta}$$

    Where,

    • C(s) is the output node

    • R(s) is the input node

    • T is the transfer function or gain between $R(s)$ and $C(s)$

    • Pi is the ith forward path gain

    $Delta =1-(sum : of : all : individual : loop : gains)$

    $+(sum : of : gain : products : of : all : possible : two :nontouching : loops)$

    $$-(sum : of : gain : products : of : all : possible : three : nontouching : loops)+..$$

    Δi is obtained from Δ by removing the loops which are touching the ith forward path.

    Consider the following signal flow graph in order to understand the basic terminology involved here.

    Path

    It is a traversal of branches from one node to any other node in the direction of branch arrows. It should not traverse any node more than once.

    Examples − $y_2 rightarrow y_3 rightarrow y_4 rightarrow y_5$ and $y_5 rightarrow y_3 rightarrow y_2$

    Forward Path

    The path that exists from the input node to the output node is known as forward path.

    Graph

    Examples − $y_1 rightarrow y_2 rightarrow y_3 rightarrow y_4 rightarrow y_5 rightarrow y_6$ and $y_1 rightarrow y_2 rightarrow y_3 rightarrow y_5 rightarrow y_6$.

    Forward Path Gain

    It is obtained by calculating the product of all branch gains of the forward path.

    Examples − $abcde$ is the forward path gain of $y_1 rightarrow y_2 rightarrow y_3 rightarrow y_4 rightarrow y_5 rightarrow y_6$ and abge is the forward path gain of $y_1 rightarrow y_2 rightarrow y_3 rightarrow y_5 rightarrow y_6$.

    Loop

    The path that starts from one node and ends at the same node is known as loop. Hence, it is a closed path.

    Examples − $y_2 rightarrow y_3 rightarrow y_2$ and $y_3 rightarrow y_5 rightarrow y_3$.

    Loop Gain

    It is obtained by calculating the product of all branch gains of a loop.

    Examples − $b_j$ is the loop gain of $y_2 rightarrow y_3 rightarrow y_2$ and $g_h$ is the loop gain of $y_3 rightarrow y_5 rightarrow y_3$.

    Non-touching Loops

    These are the loops, which should not have any common node.

    Examples − The loops, $y_2 rightarrow y_3 rightarrow y_2$ and $y_4 rightarrow y_5 rightarrow y_4$ are non-touching.

    Calculation of Transfer Function using Mason’s Gain Formula

    Let us consider the same signal flow graph for finding transfer function.

    • Number of forward paths, N = 2.

    • First forward path is - $y_1 rightarrow y_2 rightarrow y_3 rightarrow y_4 rightarrow y_5 rightarrow y_6$.

    • First forward path gain, $p_1 = abcde$.

    • Second forward path is - $y_1 rightarrow y_2 rightarrow y_3 rightarrow y_5 rightarrow y_6$.

    • Second forward path gain, $p_2 = abge$.

    • Number of individual loops, L = 5.

    • Loops are - $y_2 rightarrow y_3 rightarrow y_2$, $y_3 rightarrow y_5 rightarrow y_3$, $y_3 rightarrow y_4 rightarrow y_5 rightarrow y_3$, $y_4 rightarrow y_5 rightarrow y_4$ and $y_5 rightarrow y_5$.

    • Loop gains are - $l_1 = bj$, $l_2 = gh$, $l_3 = cdh$, $l_4 = di$ and $l_5 = f$.

    • Number of two non-touching loops = 2.

    • First non-touching loops pair is - $y_2 rightarrow y_3 rightarrow y_2$, $y_4 rightarrow y_5 rightarrow y_4$.

    • Gain product of first non-touching loops pair, $l_1l_4 = bjdi$

    • Second non-touching loops pair is - $y_2 rightarrow y_3 rightarrow y_2$, $y_5 rightarrow y_5$.

    • Gain product of second non-touching loops pair is - $l_1l_5 = bjf$

    Higher number of (more than two) non-touching loops are not present in this signal flow graph.

    We know,

    $Delta =1-(sum : of : all : individual : loop : gains)$

    $+(sum : of : gain : products : of : all : possible : two :nontouching : loops)$

    $$-(sum : of : gain : products : of : all : possible : three : nontouching : loops)+..$$

    Substitute the values in the above equation,

    $Delta =1-(bj+gh+cdh+di+f)+(bjdi+bjf)-(0)$

    $Rightarrow Delta=1-(bj+gh+cdh+di+f)+bjdi+bjf$

    There is no loop which is non-touching to the first forward path.

    So, $Delta_1=1$.

    Similarly, $Delta_2=1$. Since, no loop which is non-touching to the second forward path.

    Substitute, N = 2 in Mason’s gain formula

    $$T=frac{C(s)}{R(s)}=frac{Sigma ^2 _{i=1}P_iDelta _i}{Delta}$$

    $$T=frac{C(s)}{R(s)}=frac{P_1Delta_1+P_2Delta_2}{Delta}$$

    Substitute all the necessary values in the above equation.

    $$T=frac{C(s)}{R(s)}=frac{(abcde)1+(abge)1}{1-(bj+gh+cdh+di+f)+bjdi+bjf}$$

    $$Rightarrow T=frac{C(s)}{R(s)}=frac{(abcde)+(abge)}{1-(bj+gh+cdh+di+f)+bjdi+bjf}$$

    Therefore, the transfer function is -

    $$T=frac{C(s)}{R(s)}=frac{(abcde)+(abge)}{1-(bj+gh+cdh+di+f)+bjdi+bjf}$$