Saturday, May 21, 2011

Control Systems- Block Diagrams and Signal Flow Graphs

Control Systems- Block Diagrams and Signal Flow Graphs
  • The block diagram modeling may provide control engineers with a better understanding of the composition and interconnection of the components of a system. 
  • It can be used, together with transfer functions, to describe the cause- effect relationships throughout the system. 
  •  The common elements in block diagrams of most control systems include:
    • Comparators
    • Blocks representing individual component transfer functions, including:
    • Reference sensor (or input sensor)
    • Output sensor
    • Actuator 
    • Controller
    • Plant (the component whose variables are to be controlled)
    • Input or reference signals
    • Output signals
    • Disturbance signal
    • Feedback loops
  •  Consider the block diagram of two transfer functions G1(s) and G2(s) that are connected in series.   
  • Basic block diagram of a feedback control system.
  • The feedback system have a negative feedback loop because the comparator subtracts. When the comparator adds the feedback, it is called positive feedback, and the transfer function.  
Block diagram reductions:  
  • An SFG may be defined as a graphical means of portraying the input-output relationships among the variables of a set of linear algebraic equations.
  • The important properties of the SFG that have been covered thus far are summarized as
    1. SFG applies only to linear systems.
    2. The equations for which an SFG is drawn must be algebraic equations in the form of cause-and-effect.
    3. Nodes are used to represent variables. Normally, the nodes are arranged from left to right, from the input to the output, following a succession of cause-and-effect relations through the system.
    4. Signals travel along branches only in the direction described by the arrows of the branches.
  • Input Node (Source): An input node is a node that has only outgoing branches.
  • Output Node (Sink): An output node is a node that has only incoming branches:
  • A path is any collection of a continuous succession of branches traversed in the same direction.
  • A forward path is a path that starts at an input node and ends at an output node and along which no node is traversed more than once.
  • The product of the branch gains encountered in traversing a path is called
    the path gain.
  • A loop is a path that originates and terminates on the same node and along which no other node is encountered more than once.
  • The forward-path gain is the path gain of a forward path.
  • The loop gain is the path gain of a loop.
  • Non-touching Loops: Two parts of an SFG are non-touching if they do not share a common node.
  Example: Solution of SFG.

Problems and Solutions: 
1. Reduce the Block Diagram shown below:

Solution: By eliminating the feed-back paths, we get

Combining the blocks in series, we get

Eliminating the feed back path, we get

Solution: Shifting the take-off  T1 beyond the block G3 we get

3. Obtain the closed loop TF, by using Mason’s gain formula.

4. Construct signal flow graph from the following equations & obtain the overall TF.

 1. The transfer function C/R of the system shown in the figure is:

Ans: b.

2.  When the signal flow graph is as shown in the figure, the overall transfer function of the system will be

3.  The block diagrams shown in figure-I and figure-II are equivalent if ‘X’ (in figure-II) is equal to

a. 1
b. 2
c. S + 1
d. S + 2

Ans: a.

4. The signal flow graph shown in the given
figure has

 a. three forward paths and two nontouching loops
b. three forward paths and three loops
c. two forward paths and two nontouching loops
d. two forward paths and three loops

5. A closed-loop system is shown in the given figure. The noise transfer function
C(s) / N(s) [Cn(s) = output corresponding to noise input N(s) is approximately.

6. A signal flow graph is shown in the given figure. The number of forward paths M and the number of individual loops P for this signal flow graph would be

a. M = 4 and P = 4
b. M = 6 and P = 4
c. M = 4 and P = 6
d. M = 6 and P = 6

7. Consider the following blocks diagrams:

 Which of these block diagrams can be reduced to transfer function

a. 1 and 3
b. 2 and 4
c. 1 and 4
d. 2 and 3

8. The number of forward paths and the number of non-touching loop pairs for the signal flow graph given in the figure are respectively,

a. 1, 3
b. 3, 2
c. 3, 1
d. 2, 4


10.  The gain C(s) / G(s) of the signal flow graph shown above is

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