Block Diagram Algebra 5a1o16

ME2142/ME2142E Control Systems

Block Diagram Algebra

ME2142/TM3142 Control Systems

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Block Diagram Representation

A block diagram is a graphical tool can help us to visualize the model of a system and evaluate the mathematical relationships between their elements, using their transfer functions. The Transfer Function Block Input

R(s)

G(s)

C(s)

Output

System

G ( s) =

C (s ) R( s )

The transfer function G(s) is § defined only for a linear time-invariant system and not for nonlinear systems. § Is a property of the system and is independent of the input to the system. § Commutative § Associative

G1G 2 = G2 G1 G1 + G2 = G2 + G1

ME2142/TM3142 Control Systems

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Block Diagram Elements

The Summing Point

X Signed inputs Y

+

+

+X+Y- Z

-

output Z

§

Any number of inputs. Only one output

ME2142/TM3142 Control Systems

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Block Diagram Algebra

When manipulating block diagrams, the original relationships, or equations, relating the various variables must remain the same.

Blocks in series or cascaded blocks X

G1

§

Y

G2

Z

X

G1 G2

Z

When blocks are connected in series, there must be no loading effect.

ME2142/TM3142 Control Systems

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Block Diagram Algebra

Blocks in parallel Y=G1X+G2X

Y=G1X+G2X

G2

X X

G1

+

+

ME2142/TM3142 Control Systems

Y

G1 + G2

Y

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Block Diagram Algebra

X

+

Z

G

X

G

+

Z

+ +

Y

X

Z=G(X+Y)

G

Y

G

X

Y

X

X

Z=GX+GY

Y

G 1/G

Z=G(X+Y/G) X

G

X

Z

+

+

G

Z

+

+ Y

1/G

Y

Z=GX+Y

ME2142/TM3142 Control Systems

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Closed-Loop System

R

E

+

G

C

B

H

R is called the reference input C is the output or controlled variable B is the E = (R – B) is the error C =G E B = GH E

is called the feedforward transfer function

is called the open-loop transfer function

ME2142/TM3142 Control Systems

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Closed-Loop System

R

E

+

C

G

B

C R

C = GE = G(R – B) = G(R – HC) C(1 + GH) = GR C G = R 1 + GH

H

is the closed-loop transfer function

Also E =

E R

C G

and

E 1C 1 = = R G R 1 + GH

is called the error transfer function

ME2142/TM3142 Control Systems

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Closed-Loop Control System D R

E

+

Gc

M +

+ Gp

C

B

H

Gc is the controller transfer function Gp is the plant transfer function M is the manipulated variable D is the external disturbance C = Gc G p is the feedforward transfer function E B = Gc G p H E

is the open-loop transfer function

ME2142/TM3142 Control Systems

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Closed-Loop Control System D R

E

+

Gc

M +

+ Gp

C

B

H

Gc G p C G = = R 1 + GH 1 + Gc G p H Assuming R = 0, we can re-draw D

+

Gp -

C

Gp C G = = D 1 + GH 1 + G p Gc H

GcH

ME2142/TM3142 Control Systems

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Block Diagram Manipulation Example: Determine C(s)/R(s) a R

+

+ -

Da

D

F

+

E +

G

+

+

C

-

H I

ME2142/TM3142 Control Systems

When manipulating blocks, must ensure C(s) does not change, so that C(s)/R(s) remains same.

11

Block Diagram Manipulation Example: Determine C(s)/R(s)

We wish to move this signal to before block F

Assume names of signals as shown

b = Fa-Fc+Da

ME2142/TM3142 Control Systems

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Block Diagram Manipulation Example: Determine C(s)/R(s)

To move signal b to after Block G

Assume names of signals as shown

b = Fa-Fc+Da

ME2142/TM3142 Control Systems

b = Fa-Fc+(D/F)aF

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Block Diagram Manipulation Example: Determine C(s)/R(s) a R

+

E

(D/F)a

D/F

a +

-

Da

D

+

+

F

+

G

+

C

R

b

+

-

+

+ -

F

Eb

E G

+

+

C

-

H

H

I

I

Gb

C = Gb + Eb Eb

E/G R

+

1+D/F -

+

FG

Gb

+

+

C

R

+

1+D/F -

-

+

FG

1+E/G

-

H

H

I

I

C = Gb + (E/G)Gb

ME2142/TM3142 Control Systems

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C

Block Diagram Manipulation Example

R

+

1+D/F -

G 1 + GH +

FG

1+E/G

C

-

R

+

1+D/F -

FG 1 + FGH

1+E/G

H I

R

+ -

 D  FG  E  1 +  1 +   F  1 + FGH  G 

I

C

R

 D  FG  E   1 +   1 +   F  1 + FGH  G  D FG  E  1 + 1 +  1 +  I  F  1 + FGH  G 

C

I

ME2142/TM3142 Control Systems

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C

End

ME2142/TM3142 Control Systems

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