Block Diagram ReductionSignal-Flow Graphs

Unit 4: Block Diagram Reduction

Engineering 5821:Control Systems I

Faculty of Engineering & Applied ScienceMemorial University of Newfoundland

February 15, 2010

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

1 Block Diagram Reduction

Cascade Form

Parallel Form

Feedback Form

Moving Blocks

Example

1 Signal-Flow Graphs

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Block Diagram Reduction

Subsystems are represented in block diagrams as blocks, eachrepresenting a transfer function.

In this unit we will consider howto combine the blocks corresponding to individual subsystems sothat we can represent a whole system as a single block, andtherefore a single transfer function. Here is an example of thisreduction:

Reduced Form:

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Block Diagram Reduction

Subsystems are represented in block diagrams as blocks, eachrepresenting a transfer function. In this unit we will consider howto combine the blocks corresponding to individual subsystems sothat we can represent a whole system as a single block, andtherefore a single transfer function.

Here is an example of thisreduction:

Reduced Form:

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Block Diagram Reduction

Subsystems are represented in block diagrams as blocks, eachrepresenting a transfer function. In this unit we will consider howto combine the blocks corresponding to individual subsystems sothat we can represent a whole system as a single block, andtherefore a single transfer function. Here is an example of thisreduction:

Reduced Form:

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Block Diagram Reduction

Subsystems are represented in block diagrams as blocks, eachrepresenting a transfer function. In this unit we will consider howto combine the blocks corresponding to individual subsystems sothat we can represent a whole system as a single block, andtherefore a single transfer function. Here is an example of thisreduction:

Reduced Form:

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Block Diagram Reduction

Subsystems are represented in block diagrams as blocks, eachrepresenting a transfer function. In this unit we will consider howto combine the blocks corresponding to individual subsystems sothat we can represent a whole system as a single block, andtherefore a single transfer function. Here is an example of thisreduction:

Reduced Form:

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Block Diagram Reduction

Subsystems are represented in block diagrams as blocks, eachrepresenting a transfer function. In this unit we will consider howto combine the blocks corresponding to individual subsystems sothat we can represent a whole system as a single block, andtherefore a single transfer function. Here is an example of thisreduction:

Reduced Form:

ENGI 5821 Unit 4: Block Diagram Reduction

First we summarize the elements of block diagrams:

We now consider the forms in which blocks are typically connectedand how these forms can be reduced to single blocks.

First we summarize the elements of block diagrams:

We now consider the forms in which blocks are typically connectedand how these forms can be reduced to single blocks.

First we summarize the elements of block diagrams:

We now consider the forms in which blocks are typically connectedand how these forms can be reduced to single blocks.

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Cascade Form

When multiple subsystems are connected such that the output ofone subsystem serves as the input to the next, these subsystemsare said to be in cascade form.

The algebraic form of the final output clearly shows the equivalentsystem TF—the product of the cascaded subsystem TF’s.

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Cascade Form

When multiple subsystems are connected such that the output ofone subsystem serves as the input to the next, these subsystemsare said to be in cascade form.

The algebraic form of the final output clearly shows the equivalentsystem TF—the product of the cascaded subsystem TF’s.

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Cascade Form

When multiple subsystems are connected such that the output ofone subsystem serves as the input to the next, these subsystemsare said to be in cascade form.

The algebraic form of the final output clearly shows the equivalentsystem TF—the product of the cascaded subsystem TF’s.

ENGI 5821 Unit 4: Block Diagram Reduction

When reducing subsystems in cascade form we make theassumption that adjacent subsystems do not load each other.

That is, a subsystem’s output remains the same no matter whatthe output is connected to. If another subsystem connected to theoutput modifies that output, we say that it loads the first system.

Consider interconnecting the circuits (a) and (b) below:

The overall TF is not the product of the individual TF’s!

When reducing subsystems in cascade form we make theassumption that adjacent subsystems do not load each other.That is, a subsystem’s output remains the same no matter whatthe output is connected to.

If another subsystem connected to theoutput modifies that output, we say that it loads the first system.

Consider interconnecting the circuits (a) and (b) below:

The overall TF is not the product of the individual TF’s!

When reducing subsystems in cascade form we make theassumption that adjacent subsystems do not load each other.That is, a subsystem’s output remains the same no matter whatthe output is connected to. If another subsystem connected to theoutput modifies that output, we say that it loads the first system.

Consider interconnecting the circuits (a) and (b) below:

The overall TF is not the product of the individual TF’s!

When reducing subsystems in cascade form we make theassumption that adjacent subsystems do not load each other.That is, a subsystem’s output remains the same no matter whatthe output is connected to. If another subsystem connected to theoutput modifies that output, we say that it loads the first system.

Consider interconnecting the circuits (a) and (b) below:

The overall TF is not the product of the individual TF’s!

When reducing subsystems in cascade form we make theassumption that adjacent subsystems do not load each other.That is, a subsystem’s output remains the same no matter whatthe output is connected to. If another subsystem connected to theoutput modifies that output, we say that it loads the first system.

Consider interconnecting the circuits (a) and (b) below:

The overall TF is not the product of the individual TF’s!

When reducing subsystems in cascade form we make theassumption that adjacent subsystems do not load each other.That is, a subsystem’s output remains the same no matter whatthe output is connected to. If another subsystem connected to theoutput modifies that output, we say that it loads the first system.

Consider interconnecting the circuits (a) and (b) below:

The overall TF is not the product of the individual TF’s!

We can prevent loading by inserting an amplifier.

This amplifiershould have a high input impedance so it does not load its source,and low output impedance so it appears as a pure voltage sourceto the subsystem it feeds into.

If no actual gain is desired then K = 1 and the “amplifier” isreferred to as a buffer.

We can prevent loading by inserting an amplifier. This amplifiershould have a high input impedance so it does not load its source,and low output impedance so it appears as a pure voltage sourceto the subsystem it feeds into.

If no actual gain is desired then K = 1 and the “amplifier” isreferred to as a buffer.

We can prevent loading by inserting an amplifier. This amplifiershould have a high input impedance so it does not load its source,and low output impedance so it appears as a pure voltage sourceto the subsystem it feeds into.

If no actual gain is desired then K = 1 and the “amplifier” isreferred to as a buffer.

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Parallel Form

Parallel subsystems have a common input and their outputs aresummed together.

The equivalent TF is the sum of parallel TF’s (with matched signsat summing junction).

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Parallel Form

Parallel subsystems have a common input and their outputs aresummed together.

The equivalent TF is the sum of parallel TF’s (with matched signsat summing junction).

ENGI 5821 Unit 4: Block Diagram Reduction

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

Parallel Form

Parallel subsystems have a common input and their outputs aresummed together.

The equivalent TF is the sum of parallel TF’s (with matched signsat summing junction).

ENGI 5821 Unit 4: Block Diagram Reduction

Feedback Form

Systems with feedback typically have the following form:

Noticing the cascade form within the feedforward and feedbackpaths we can simplify:

Feedback Form

Systems with feedback typically have the following form:

Noticing the cascade form within the feedforward and feedbackpaths we can simplify:

Feedback Form

Systems with feedback typically have the following form:

Noticing the cascade form within the feedforward and feedbackpaths we can simplify:

Feedback Form

Systems with feedback typically have the following form:

Noticing the cascade form within the feedforward and feedbackpaths we can simplify:

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)

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)

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)

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 system’s block diagram may require some modification beforethe reductions discussed above can be applied.

We may need to move blocks either to the left or right of asumming junction:

Moving Blocks

A system’s block diagram may require some modification beforethe reductions discussed above can be applied.

We may need to move blocks either to the left or right of asumming junction:

Moving Blocks

A system’s block diagram may require some modification beforethe reductions discussed above can be applied.

We may need to move blocks either to the left or right of asumming junction:

Or we may need to move blocks to the left or right of a pickoffpoint:

Or we may need to move blocks to the left or right of a pickoffpoint:

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

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

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

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

Block Diagram ReductionSignal-Flow Graphs

Cascade FormParallel FormFeedback FormMoving BlocksExample

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 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:

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:

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:

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

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

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

Push G1 to the right past the summing junction to create aparallel form in the feedback path

Reduce parallel form involving 1/G2 and unity

Push G1 to the right past the summing junction to create aparallel form in the feedback path

Reduce parallel form involving 1/G2 and unity

Push G1 to the right past the summing junction to create aparallel form in the feedback path

Reduce parallel form involving 1/G2 and unity

Push G1 to the right past the summing junction to create aparallel form in the feedback path

Reduce parallel form involving 1/G2 and unity

Push G1 to the right past the summing junction to create aparallel form in the feedback path

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 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

Reduce feedback form on left

Signal-Flow Graphs

Signal-flow graphs are an alternative to block diagrams.

Theyconsist of branches which represent systems (a) and nodes whichrepresent signals (b). Multiple branches converging on a nodeimplies 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)

Signal-Flow Graphs

Signal-flow graphs are an alternative to block diagrams. Theyconsist of branches which represent systems (a) and nodes whichrepresent signals (b).

Multiple branches converging on a nodeimplies 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)

Signal-Flow Graphs

Signal-flow graphs are an alternative to block diagrams. Theyconsist of branches which represent systems (a) and nodes whichrepresent signals (b). Multiple branches converging on a nodeimplies 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)

Signal-Flow Graphs

Signal-flow graphs are an alternative to block diagrams. Theyconsist of branches which represent systems (a) and nodes whichrepresent signals (b). Multiple branches converging on a nodeimplies 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)

Signal-Flow Graphs

Signal-flow graphs are an alternative to block diagrams. Theyconsist of branches which represent systems (a) and nodes whichrepresent signals (b). Multiple branches converging on a nodeimplies 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)

Signal-Flow Graphs

Signal-flow graphs are an alternative to block diagrams. Theyconsist of branches which represent systems (a) and nodes whichrepresent signals (b). Multiple branches converging on a nodeimplies 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)

Signal-Flow Graphs

Signal-flow graphs are an alternative to block diagrams. Theyconsist of branches which represent systems (a) and nodes whichrepresent signals (b). Multiple branches converging on a nodeimplies 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 intosignal-flow graphs:

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

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

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

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