Unit 4: Block Diagram Reduction. Block Diagram Reduction. Cascade Form Parallel Form Feedback Form Moving Blocks Example

Transcription

1 Engineering 5821: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland February 15, 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: First we summarize the elements of block diagrams: Reduced Form: We now consider the forms in which blocks are typically connected and how these forms can be reduced to single blocks.

2 When multiple subsystems are connected such that the output of one subsystem serves as the input to the next, these subsystems are said to be in 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 the output is connected to. If another subsystem connected to the output modifies that output, we say that it loads the first system. Consider interconnecting the circuits (a) and (b) below: The algebraic form of the final output clearly shows the equivalent system TF the product of the cascaded subsystem TF s. The overall TF is not the product of the individual TF s! We can prevent loading by inserting an amplifier. This amplifier should have a high input impedance so it does not load its source, and low output impedance so it appears as a pure voltage source to the subsystem it feeds into. Parallel subsystems have a common input and their outputs are 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).

3 Systems with feedback typically have the following form: We can easily establish the following two facts: Noticing the cascade form within the feedforward and feedback paths we can simplify: E(s) = R(s) C(s)H(s) C(s) = E(s)G(s) We can now eliminate E(s) to obtain, G(s) Ge(s) = 1 ± G(s)H(s) A system s block diagram may require some modification before the reductions discussed above can be applied. We may need to move blocks either to the left or right of a summing junction: Or we may need to move blocks to the left or right of a pickoff point:

4 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: 2 Reduce the following more complicated block diagram: Steps: Rightmost feedback loop can be reduced Create parallel form by moving G2 left Reduce parallel form involving 1/G2 and unity Push G1 to the right past the summing junction to create a parallel form in the feedback path

5 Reduce parallel form on left Recognize cascade form on right Reduce feedback form on left 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)

6 e.g. Convert the following block diagram to a signal-flow graph: