Unit 4: Block Diagram Reduction Block Diagram Reduction by kxb86934

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									                   Block Diagram Reduction                                                       Block Diagram Reduction
                         Signal-Flow Graphs                                                            Signal-Flow Graphs




                                                                                1   Block Diagram Reduction
              Unit 4: Block Diagram Reduction                                         Cascade Form
                                                                                      Parallel Form
                               Engineering 5821:                                      Feedback Form
                               Control Systems I
                                                                                      Moving Blocks

                    Faculty of Engineering & Applied Science                          Example
                      Memorial University of Newfoundland


                                                                                1   Signal-Flow Graphs
                               February 27, 2009



                                ENGI 5821     Unit 4: Block Diagram Reduction                                 ENGI 5821     Unit 4: Block Diagram Reduction


                                              Cascade Form
                                              Parallel Form
                   Block Diagram Reduction
                                              Feedback Form
                         Signal-Flow Graphs
                                              Moving Blocks
                                              Example

                                                                                First we summarize the elements of block diagrams:
Block Diagram Reduction
  Subsystems are represented in block diagrams as blocks, each
  representing a transfer function. In this unit we will consider how
  to combine the blocks corresponding to individual subsystems so
  that we can represent a whole system as a single block, and
  therefore a single transfer function. Here is an example of this
  reduction:




                                                                                We now consider the forms in which blocks are typically connected
                                                                                and how these forms can be reduced to single blocks.
  Reduced Form:


                                ENGI 5821     Unit 4: Block Diagram Reduction
                                             Cascade Form
                                             Parallel Form
                  Block Diagram Reduction
                                             Feedback Form
                        Signal-Flow Graphs
                                             Moving Blocks
                                             Example


Cascade Form                                                                     When reducing subsystems in cascade form we make the
                                                                                 assumption that adjacent subsystems do not load each other.
                                                                                 That is, a subsystem’s output remains the same no matter what
  When multiple subsystems are connected such that the output of                 the output is connected to. If another subsystem connected to the
  one subsystem serves as the input to the next, these subsystems                output modifies that output, we say that it loads the first system.
  are said to be in cascade form.                                                Consider interconnecting the circuits (a) and (b) below:




                                                                                 The overall TF is not the product of the individual TF’s!
  The algebraic form of the final output clearly shows the equivalent
  system TF—the product of the cascaded subsystem TF’s.


                               ENGI 5821     Unit 4: Block Diagram Reduction


                                                                                                                            Cascade Form
                                                                                                                            Parallel Form
                                                                                                 Block Diagram Reduction
                                                                                                                            Feedback Form
                                                                                                       Signal-Flow Graphs
                                                                                                                            Moving Blocks
                                                                                                                            Example
  We can prevent loading by inserting an amplifier. This amplifier
  should have a high input impedance so it does not load its source,
                                                                               Parallel Form
  and low output impedance so it appears as a pure voltage source                Parallel subsystems have a common input and their outputs are
  to the subsystem it feeds into.                                                summed together.




  If no actual gain is desired then K = 1 and the “amplifier” is
  referred to as a buffer.
                                                                                 The equivalent TF is the sum of parallel TF’s (with matched signs
                                                                                 at summing junction).
                                                                                                              ENGI 5821     Unit 4: Block Diagram Reduction
Feedback Form

  Systems with feedback typically have the following form:




                                                                  We can easily establish the following two facts:

                                                                                     E (s) = R(s) ∓ C (s)H(s)
  Noticing the cascade form within the feedforward and feedback
                                                                                     C (s) = E (s)G (s)
  paths we can simplify:
                                                                  We can now eliminate E (s) to obtain,

                                                                                                       G (s)
                                                                                        Ge (s) =
                                                                                                   1 ± G (s)H(s)




Moving Blocks

  A system’s block diagram may require some modification before    Or we may need to move blocks to the left or right of a pickoff
  the reductions discussed above can be applied.                  point:
  We may need to move blocks either to the left or right of a
  summing junction:
                                             Cascade Form
                                             Parallel Form
                  Block Diagram Reduction
                                             Feedback Form
                        Signal-Flow Graphs
                                             Moving Blocks
                                             Example


Example


  Reduce the following system to a single TF:


                                                                               We can now recognize the parallel form in the feedback path:




  First we can combine the three summing junctions together...                 We now have G1 cascaded with a feedback subsystem:



                               ENGI 5821     Unit 4: Block Diagram Reduction



Example 2

  Reduce the following more complicated block diagram:




                                                                                   Reduce parallel form involving 1/G2 and unity
  Steps:
                                                                                   Push G1 to the right past the summing junction to create a
      Rightmost feedback loop can be reduced                                       parallel form in the feedback path
      Create parallel form by moving G2 left
      Reduce parallel form on left                                      Reduce feedback form on left
      Recognize cascade form on right




Signal-Flow Graphs

                                                                    We can convert the cascaded, parallel, and feedback forms into
                                                                    signal-flow graphs:




  Signal-flow graphs are an alternative to block diagrams. They
  consist of branches which represent systems (a) and nodes which
  represent signals (b). Multiple branches converging on a node
  implies summation.

          V (s) = R1 (s)G1 (s) − R2 (s)G2 (s) + R3 (s)G3 (s)
         C1 (s) = V (s)G4 (s)
         C2 (s) = V (s)G5 (s)
         C3 (s) = V (s)G6 (s)
e.g. Convert the following block diagram to a signal-flow graph:

								
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