Block Diagram Reduction Signal-Flow Graphs
Unit 4: Block Diagram Reduction
Engineering 5821: Control Systems I
Faculty of Engineering & Applied Science Memorial University of Newfoundland
February 15, 2010
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Block Diagram Reduction Cascade Form Parallel Form Feedback Form Moving Blocks Example
Signal-Flow Graphs
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Block Diagram Reduction
Subsystems are represented in block diagrams as blocks, each representing a transfer function.
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
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.
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
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:
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
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:
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
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:
Reduced Form:
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
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:
Reduced Form:
ENGI 5821
Unit 4: Block Diagram Reduction
First we summarize the elements of block diagrams:
First we summarize the elements of block diagrams:
First we summarize the elements of block diagrams:
We now consider the forms in which blocks are typically connected and how these forms can be reduced to single blocks.
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Cascade Form
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.
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Cascade Form
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.
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Cascade Form
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.
The algebraic form of the nal output clearly shows the equivalent system TFthe product of the cascaded subsystem TFs.
ENGI 5821
Unit 4: Block Diagram Reduction
When reducing subsystems in cascade form we make the assumption that adjacent subsystems do not load each other.
When reducing subsystems in cascade form we make the assumption that adjacent subsystems do not load each other. That is, a subsystems output remains the same no matter what the output is connected to.
When reducing subsystems in cascade form we make the assumption that adjacent subsystems do not load each other. That is, a subsystems output remains the same no matter what the output is connected to. If another subsystem connected to the output modies that output, we say that it loads the rst system.
When reducing subsystems in cascade form we make the assumption that adjacent subsystems do not load each other. That is, a subsystems output remains the same no matter what the output is connected to. If another subsystem connected to the output modies that output, we say that it loads the rst system. Consider interconnecting the circuits (a) and (b) below:
When reducing subsystems in cascade form we make the assumption that adjacent subsystems do not load each other. That is, a subsystems output remains the same no matter what the output is connected to. If another subsystem connected to the output modies that output, we say that it loads the rst system. Consider interconnecting the circuits (a) and (b) below:
When reducing subsystems in cascade form we make the assumption that adjacent subsystems do not load each other. That is, a subsystems output remains the same no matter what the output is connected to. If another subsystem connected to the output modies that output, we say that it loads the rst system. Consider interconnecting the circuits (a) and (b) below:
The overall TF is not the product of the individual TFs!
We can prevent loading by inserting an amplier.
We can prevent loading by inserting an amplier. This amplier 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.
We can prevent loading by inserting an amplier. This amplier 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.
If no actual gain is desired then K = 1 and the amplier is referred to as a buer.
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Parallel Form
Parallel subsystems have a common input and their outputs are summed together.
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Parallel Form
Parallel subsystems have a common input and their outputs are summed together.
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Parallel Form
Parallel subsystems have a common input and their outputs are summed together.
The equivalent TF is the sum of parallel TFs (with matched signs at summing junction).
ENGI 5821 Unit 4: Block Diagram Reduction
Feedback Form
Systems with feedback typically have the following form:
Feedback Form
Systems with feedback typically have the following form:
Feedback Form
Systems with feedback typically have the following form:
Noticing the cascade form within the feedforward and feedback paths we can simplify:
Feedback Form
Systems with feedback typically have the following form:
Noticing the cascade form within the feedforward and feedback paths we can simplify:
We can easily establish the following two facts:
We can easily establish the following two facts: E (s ) = R (s ) C (s )H (s )
We can easily establish the following two facts: E (s ) = R (s ) C (s )H (s ) C (s ) = E (s )G (s )
We can easily establish the following two facts: E (s ) = R (s ) C (s )H (s ) C (s ) = E (s )G (s ) We can now eliminate E (s ) to obtain,
We can easily establish the following two facts: E (s ) = R (s ) C (s )H (s ) C (s ) = E (s )G (s ) We can now eliminate E (s ) to obtain, Ge (s ) = G (s ) 1 G (s )H (s )
We can easily establish the following two facts: E (s ) = R (s ) C (s )H (s ) C (s ) = E (s )G (s ) We can now eliminate E (s ) to obtain, Ge (s ) = G (s ) 1 G (s )H (s )
Moving Blocks
A systems block diagram may require some modication before the reductions discussed above can be applied.
Moving Blocks
A systems block diagram may require some modication before the reductions discussed above can be applied. We may need to move blocks either to the left or right of a summing junction:
Moving Blocks
A systems block diagram may require some modication 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 picko point:
Or we may need to move blocks to the left or right of a picko point:
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Example
Reduce the following system to a single TF:
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Example
Reduce the following system to a single TF:
ENGI 5821
Unit 4: Block Diagram Reduction
Block Diagram Reduction Signal-Flow Graphs
Cascade Form Parallel Form Feedback Form Moving Blocks Example
Example
Reduce the following system to a single TF:
First we can combine the three summing junctions together...
ENGI 5821
Unit 4: Block Diagram Reduction
We can now recognize the parallel form in the feedback path:
We can now recognize the parallel form in the feedback path:
We can now recognize the parallel form in the feedback path:
We now have G1 cascaded with a feedback subsystem:
We can now recognize the parallel form in the feedback path:
We now have G1 cascaded with a feedback subsystem:
Example 2
Reduce the following more complicated block diagram:
Example 2
Reduce the following more complicated block diagram:
Example 2
Reduce the following more complicated block diagram:
Steps:
Example 2
Reduce the following more complicated block diagram:
Steps: Rightmost feedback loop can be reduced
Example 2
Reduce the following more complicated block diagram:
Steps: Rightmost feedback loop can be reduced Create parallel form by moving G2 left
Example 2
Reduce the following more complicated block diagram:
Steps: Rightmost feedback loop can be reduced Create parallel form by moving G2 left
Example 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
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
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
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
Reduce parallel form on left
Reduce parallel form on left Recognize cascade form on right
Reduce parallel form on left Recognize cascade form on right
Reduce parallel form on left Recognize cascade form on right
Reduce feedback form on left
Reduce feedback form on left
Reduce feedback form on left
Reduce feedback form on left
Reduce feedback form on left
Signal-Flow Graphs
Signal-ow graphs are an alternative to block diagrams.
Signal-Flow Graphs
Signal-ow graphs are an alternative to block diagrams. They consist of branches which represent systems (a) and nodes which represent signals (b).
Signal-Flow Graphs
Signal-ow 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.
Signal-Flow Graphs
Signal-ow 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 )
Signal-Flow Graphs
Signal-ow 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 )
Signal-Flow Graphs
Signal-ow 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 )
Signal-Flow Graphs
Signal-ow 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 )
We can convert the cascaded, parallel, and feedback forms into signal-ow graphs:
We can convert the cascaded, parallel, and feedback forms into signal-ow graphs:
e.g. Convert the following block diagram to a signal-ow graph:
e.g. Convert the following block diagram to a signal-ow graph:
e.g. Convert the following block diagram to a signal-ow graph:
