Block Diagram Reduction V2

ME 413 Systems Dynamics & Control Chapter 10: Time‐Domain Analysis and Design of Control Systems

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10.1 INTRODUCTION

Block Diagram: Pictorial representation of functions performed by each component of a system and that of flow of signals.

( )C s( )R s

( ) ( ) ( )C s G s R s=

( )G s

Figure 10‐1. Single block diagram representation.

Components for Linear Time Invariant System(LTIS):

Figure 10‐2. Components for Linear Time Invariant Systems (LTIS).

Chapter 10: Time‐Domain Analysis and Design of

Control Systems: Block Diagram Reduction

A. Bazoune


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Terminology:

( )C s( )R s ( )G s1 ( )G s2

( )H s

( )Disturbance U s

( )b s± ( ) ( ) ( )E s R s b s= ± ( )m s

Figure 10‐3. Block Diagram Components.

1. Plant: A physical object to be controlled. The Plant ( )G s2 , is the controlled system, of which a

particular quantity or condition is to be controlled.

2. Feedback Control System (Closed‐loop Control System): A system which compares output to some reference input and keeps output as close as possible to this reference.

3. Open‐loop Control System: Output of the system is not feedback to the system.

4. Control Element ( )G s1

, also called the controller, are the components required to generate the

appropriate control signal ( )M s applied to the plant.

5. Feedback Element ( )H s is the component required to establish the functional relationship between

the primary feedback signal ( )B s and the controlled output ( )C s .

6. Reference Input ( )R s is an external signal applied to a feedback control system in order to

command a specified action of the plant. It often represents ideal plant output behavior.

7. The Controlled Output ( )C s is that quantity or condition of the plant which is controlled.

8. Actuating Signal ( )E s , also called the error or control action, is the algebraic sum consisting of the

reference input ( )R s plus or minus (usually minus) the primary feedback ( )B s .

9. Manipulated Variable ( )M s (control signal) is that quantity or condition which the control

elements ( )G s1

apply to the plant ( )G s2

.

10. Disturbance ( )U s is an undesired input signal which affects the value of the controlled output

( )C s . It may enter the plant by summation with ( )M s , or via an intermediate point, as shown in

the block diagram of the figure above.

11. Forward Path is the transmission path from the actuating signal ( )E s to the output ( )C s .


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12. Feedback Path is the transmission path from the output ( )C s to the feedback signal ( )B s .

13. Summing Point: A circle with a cross is the symbol that indicates a summing point. The ( )+ or ( )−

sign at each arrowhead indicates whether that signal is to be added or subtracted.

14. Branch Point: A branch point is a point from which the signal from a block goes concurrently to other blocks or summing points.

Definitions

• ( )G s ≡Direct transfer function = Forward transfer function.

• ( )H s ≡ Feedback transfer function.

• ( ) ( )G s H s ≡ Open‐loop transfer function.

• ( ) ( )C s R s ≡ Closed‐loop transfer function = Control ratio

• ( ) ( )C s E s ≡ Feed‐forward transfer function.

( )C s( )R s ( )G s

( )H s

( )B sOutputInput

( )E s

Figure 10‐4 Block diagram of a closed‐loop system with a feedback element.

10.2 BLOCK DIAGRAMS AND THEIR SIMPLIFICATION

Cascade (Series) Connections

Figure 10‐5 Cascade (Series) Connection.


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Parallel Connections

Figure 10‐5 Parallel Connection.

Closed Loop Transfer Function (Feedback Connections)

( )C s( )R s ( )G s

( )H s

( )B s

( )E s

Figure 10.4 (Repeated) Feedback connection

For the system shown in Figure 10‐4, the output ( )C s and input ( )R s are related as follows:

( ) ( ) ( )=C s G s E s

where

( ) ( ) ( ) ( ) ( ) ( )= − = −E s R s B s R s H s C s

Eliminating ( )E s from these equations gives

( ) ( ) ( ) ( ) ( )[ ]= −C s G s R s H s C s

This can be written in the form

( ) ( )[ ] ( ) ( ) ( )+ =G s H s C s G s R s1

or

( )( )

( )( ) ( )

C s G s

sR s G H s+=1

The Characteristic equation of the system is defined as an equation obtained by setting the denominator polynomial of the transfer function to zero. The Characteristic equation for the above system is

( ) ( )1+G s H s =0 .


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Block Diagram Algebra for Summing Junctions

( )= +

C=G +R±X

GR±GX

( )= +

C=GR±X

G R±X G

Figure 10‐6 Summing junctions.

Block Diagram Algebra for Branch Point

Figure 10‐7 Summing junctions.

Block Diagram Reduction Rules

In many practical situations, the block diagram of a Single Input‐Single Output (SISO), feedback control system may involve several feedback loops and summing points. In principle, the block diagram of (SISO) closed loop system, no matter how complicated it is, it can be reduced to the standard single loop form shown in Figure 10‐4. The basic approach to simplify a block diagram can be summarized in Table 1:


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TABLE 10‐1 Block Diagram Reduction Rules

1. Combine all cascade blocks

2. Combine all parallel blocks

3. Eliminate all minor (interior) feedback loops

4. Shift summing points to left

5. Shift takeoff points to the right

6. Repeat Steps 1 to 5 until the canonical form is obtained

TABLE 10‐2. Some Basic Rules with Block Diagram Transformation

G1u

2u

y

1/G

1u

1y G u

u yG

=

=

y Gu=

Gu

uy

( )2 1 2e G u u= −

Gu

y

y

G

G1u

2uy G1u

2u

y

G

uy

y

G

u

y

1/G

Gu

G

2uy 1 2y Gu u= −

u ( )1 2y G G u= −1G y21/G2G

( )Y GG X= 1 21G Y2GX

( )Y G G X= ±1 21G

2G

XY±

1 2G GX Y

1 2±G GX Y

█ Example 1: A feedback system is transformed into a unity feedback system

( )R s( ) ( )G s H s

( )C s( )1 H s

( )R s( )G s

( )C s

( )H s

=±

⋅=±

=GH

GHHGH

GRC

11

1Closed‐loop Transfer function


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█ Example 2:

Reduce the following block diagrams


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█ Example 3:

█ Example 4

G1 and G2 are in series

H1 and H2 and H3 are in parallel

G1 is in series with the feedback configuration.

█ Example 5: The main problem here is the feed‐forward of V3(s). Solution is to move this

pickoff point forward.

( )⎡ ⎤⎢ ⎥⎣ ⎦

3 21

3 2 1 2 3

G GC(s) = GR(s) 1 + G G H - H + H


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█ Example 6:


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█ Example 7

Use block diagram reduction to simplify the block diagram below into a single block relating ( )Y s to ( )R s .

█ Solution

█ Example 8

Use block diagram algebra to solve the previous example.


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█ Solution

Multiple‐Inputs cases

In feedback control system, we often encounter multiple inputs (or even multiple output cases). For a linear system, we can apply the superposition principle to solve this type of problems, i.e. to treat each input one at a time while setting all other inputs to zeros, and then algebraically add all the outputs as follows:

TABLE 10‐3: Procedure For reducing Multiple Input Blocks

1 Set all inputs except one equal to zero

2 Transform the block diagram to solvable form.

3 Find the output response due to the chosen input action alone

4 Repeat Steps 1 to 3 for each of the remaining inputs.

5 Algebraically sum all the output responses found in Steps 1 to 5

█ Example 9 : We shall determine the output C of the following system:

( )R s

( )D s

( )1G s ( )2G s ( )C s


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█ Solution

Using the superposition principle, the procedure is illustrated in the following steps:

Step1: Put ( ) 0D s ≡ as shown in Figure (a).

Step2: The block diagrams reduce to the block shown in Figure. b

Step 3: The output RC due to input ( )R s is

shown in Figure (c) and is given by the relationship

RGG

GGCR ⋅

+=

21

21

1

Step 4: Put ( ) 0R s ≡ as shown in Figure (d).

Step 5: Put ‐1 into a block, representing the negative feedback effect. (Figure d)

Step 6: Rearrange the block diagrams as shown in Figure (e).

Step 7: Let the ‐1 block be absorbed into the, summing point as shown in Figure (f).

Step 8: By Equation (7.3), the output UC

due to input U is :

UGG

GCU ⋅

+=

21

2

1

Step 9: The total output is C:

[ ]

1 2 2

1 2 1 2

21

1 2

1 1

1

R UG G GC C C R U

G G G GG G R UG G

= + = ⋅ + ⋅+ +

= ⋅ ++

( )R s( )1G s ( )2G s

( )C s

Figure (a)

( )R s( ) ( )1 2G s G s ( )C s

Figure (b)

( )R s ( ) ( )( ) ( )

1 2

1 21+G s G sG s G s

( )C s

Figure (c)

( )1G s ( )2G s( )DC s

1−

( )D s

Figure (d)

( )2G s( )D s ( )DC s

1− ( )1G s

Figure (e)

( )2G s( )D s ( )DC s

( )1G s

Figure (f)

█ Example 10:


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Block Diagram Reduction V2

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