Reduction of Multiple Subsystem

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Robotics Research Lab CHAPTER V Representation and Reduction of Multiple Subsystems

Transcript of Reduction of Multiple Subsystem

Robotics Research Lab

CHAPTER V

Representation and Reduction of Multiple Subsystems

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Why Subsystems Reduction?

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Aircraft Subsystem- Attitude Control System

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Why Subsystems Reduction?

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Space Shuttle Subsystem- Pitch Control System

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Block Diagram Representation

( )R soutput

( )C s

error( )E s

input input( )G s

output

signal signal

signal system

Summing junction

( )R s

( )R s

( )R s

( )R s

Pickoff point

1( )R s

3( )R s

( ) 1 2 3( ) ( ) ( )C s R s R s R s= + −

2( )R s

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Cascade Form

( )1G s( )R s

( )2G s( )2X s

( )3G s( )3X s

( ) ( ) ( )( ) ( ) ( ) ( )

3 3

3 2 1

C s G s X s

G s G s G s R s

=

=

( ) ( ) ( )2 1X s G s R s= ( ) ( ) ( )( ) ( ) ( )

3 2 2

2 1

X s G s X s

G s G s R s

=

=

( ) ( ) ( )1 2 3G s G s G s( )R s ( )C s

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Parallel Form

( )1G s

( )R s

( ) ( ) ( )1 1X s G s R s=

( )2G s( ) ( ) ( )2 2X s G s R s=

( )3G s( ) ( ) ( )3 3X s G s R s=

( ) ( ) ( ) ( ) ( )1 2 3C s G s G s G s R s⎡ ⎤= + +⎣ ⎦

( ) ( ) ( )1 2 3G s G s G s+ +( )R s ( )C s

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Feedback Form

( )1G s( )R s

( )2G s( )E s

( )3G s

( )2H s ( )1H s

inputtranducer controller plant

error

feedback outputtranducer

( )C soutput

negativefeedback

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Feedback Form

( )R s( )G s

( )H s

( )C s

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

2 3 1

2 1 1

where

G s G s G s G s

H s H s H s G s

=

=

( )( ) ( )1

G sG s H s+

( )R s ( )C s

Closed-loop transfer function

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Block Diagram Algebra – Linear Case

i) Summing junctions

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Block Diagram Algebra – Linear Case

ii) Pickoff point

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Analysis and Design of Feedback Systems

( )R s

( )K

s s a+( )C s

( ) ( )

( ) ( )( ) ( )

1,2

2

2

1,2

0,

42 2

KG s s as s a

C s K KT sR s s s a K s as K

a K as j

= = −+

= =+ + + +

−= − ±

K( )

1s s a+

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ex) in p.223

Open-loop :

Closed-loop :

( )R s

( )5K

s s +( )C s

( ) ( ) 1,2where 0, 5 25KG s s Ks s a

= = − =+

( )

( )

2

2

1,2

5255 25

5 5 32 2

KT ss s K

T ss s

s j

=+ +

=+ +

= − ±

25 5 / sec0.50.726 sec

% . 16.3031.6 sec

n

p

s

rad

T

O ST

ωζ

= =

==

==

Analysis and Design of Feedback Systems

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Requirement: 10% overshoot

( ) 2 5KT s

s s K=

+ +

0.591ζ→ =

2 5 4.23 / sec

17.892

n

n

n

rad

K

K

ζωω

ω

=

=

=

∴ =

Analysis and Design of Feedback Systems

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Block Diagram Representation vs. Signal-flow Graph Representation

BlockSignal

Summing junctionsPickoff points

Branches (system)Nodes (signal)

vs.

cascade

parallel

feedback

( )R s ( )1X s ( )2X s ( )C s

( )1G s ( )2G s ( )3G s

( )C s( )2G s

( )R s

( )1G s

( )3G s

( )1X s

( )2X s

( )3X s

1

1

1

( )R s ( )E s ( )G s ( )C s1

( )H s−

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Signal-flow Graphs of State Equation

In p.232

1 2

2 3

3 1 2 3

1

24 26 9 24

x xx xx x x x ry x

=

=

= − − − +

=

r1s24

1s

1s 1 y

3x 2x 1x

9−

26−

24−

# of state variables

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Signal-flow Graph Representation- Cascade Representation

( )R s24

( )C s12s +

13s +

14s +

( )3x s ( )2x s ( )1x s

( )( ) ( )( ) ( )

( ) ( ) ( )

11 2

2

1 1 2

1 44

4

X sX s s X s

X s s

x t x t x t

= → + =+

→ = − +

( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

22 2 3

3

33 3

1 33

24 2 242

X sx t x t x t

X s s

X sx t x t r t

R s s

= → = − ++

= → = − ++

State variable assignment

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Realization

[ ]

4 1 0 00 3 1 00 0 2 24

1 0 0

x x r

y x

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

=

( )R s1s24 ( )Y s( )3x s

2−

1

3− 4−

( )2x s ( )1x s1s

1s1 1

Signal-flow Graph Representation- Cascade Representation

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( )( ) ( )( )( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

242 3 4

12 24 122 3 4

12 24 122 3 4

C sR s s s s

s s s

C s R s R s R ss s s

=+ + +

= − ++ + +

= − ++ + +

122s +

( )R s

( )1X s

( )2X s

( )3X s

( )C s243s +

124s +

[ ]( )

1 1

2 2

3 3

2 123 244 12

1 1 1

x x rx x rx x ry x

C t

= − +

= − +

= − +

= −

=[ ]

1 1

2 2

3 3

ˆ ˆ2

ˆ ˆ3

ˆ ˆ4ˆ12 24 12

x x r

x x r

x x ry x

= − +

= − +

= − +

= −

Signal-flow Graph Representation- Parallel Representation

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Realization

( )R s

1s24

3−

1−

1s

2−

1s

4−

12

12

1

1

( )C s

2 0 0 120 3 0 240 0 4 12

x x r−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − + −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

diagonal, decoupled

Signal-flow Graph Representation- Parallel Representation

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ex) ( )( )

( )( ) ( )

( )( ) ( ) ( ) ( )

2

2

3

1 2

2 1 11 21

C s sR s s s

C sR s s ss

+=

+ +

= − ++ ++

( )1

1s +

( )R s

( )1X s

( )3X s

( )C s12

( )1

2s +

( )2

1s +( )2X s

1 1 2

2 2

3 3

2x x xx x rx x r

= − +

= − +

= − +1 2 3

1( )2

y c t x x x= = − +

Signal-flow Graph Representation- Jordan Canonical Representation

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Realization

( )R s

1s

1−

( )C s

1s

1−

1( )2X s ( )1X s

1s

2−

( )3X s

12

−1

1

1

2

1 1 0 00 1 0 20 0 2 1

x x r−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

Block diagonal

11 12

y x⎡ ⎤= −⎢ ⎥⎣ ⎦

Jordan canonical form- multiple poles

Signal-flow Graph Representation- Jordan Canonical Representation

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Signal-flow Graphs for Canonical Representation

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Signal-flow Graphs for Canonical Representation

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Using similarity transformations, we can obtain

i) diagonalized (parallel) form/ Jordan canonical form

ii) controllable form / observable form

iii) decomposition form ( )co co co co

Transformation of State Equations

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Similarity Transformation

0

2x

1x

1z

2z

θ

x

1xU

2xU

2zU1zU

( ) ( )( ) ( )

1 2

1 2

1 1 2

2 1 2

1 2 1 2

1 2

11 2

2

11 2

2

11 21

12 22

11 11 21 12 22

2

1 11 2 12 1 21 2 22

11 12 1

21 22 2

x x

z z

z x x

z x x

x x x x

x x

xx x U x U

x

zx z U z U

z

U p U p U

U p U p U

xz p U p U p U p U

x

z p z p U z p z p U

p p zp p z

⎡ ⎤= + = ⎢ ⎥

⎣ ⎦⎡ ⎤

= + = ⎢ ⎥⎣ ⎦

= +

= +

⎡ ⎤= + + +⎢ ⎥

⎣ ⎦= + +

⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

1

x Pzz P z−

=

=

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Transformation of State Equations

1 1

(1)

(2)

x Ax Buy Cx Du

x Pz

Pz APz Buy CPx Du

z P APz P Buy CPz Du

− −

= += +

=

= += +

= += +

Linear Time-invariant system – LTI system

Constant matrix (nonsingular)

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Transformation of State Equations

( ) ( )( ) ( )

( )( )

( ) ( )

( ) ( )

11 1

11

11 1

1 1 1 1

11 1

From (2)

Y sT s CP sI P AP P B D

U s

CP P sI P AP B D

C P sI P AP P B D

sI P AP P sP P A

T s C P sP P A B D

−− −

−−

−− −

− − − −

−− −

= = − +

⎡ ⎤= − +⎣ ⎦

⎡ ⎤= − +⎣ ⎦

− = −

⎡ ⎤= − +⎣ ⎦

( ) 1 C sI A B D−= − +

Transfer function is the same !!

( ) ( )( ) ( ) 1From (1)

Y sT s C sI A B D

U s−

= = − +

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Same characteristic polynomial !!

i.e., Same roots !!

i.e., Same eigenvalues !!

( ) ( )( )

( )( )

1 1 1

1

1

det det

det

det det det

det

sI P AP sP P P AP

P sI A P

P sI A P

sI A

− − −

− = −

⎡ ⎤= −⎣ ⎦= −

= −

Transformation of State Equations

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Transformation of State Equations

Geometrical Meaning of Eigenvector and Eigenvalue

Def: The eigenvectors of the matrix A are all vectors,

which under the transformation A become multiples of themselves.

Def: The eigenvalues of the matrix A are the values of that

satisfying for

or

i i iAx xλ=

i i iAx xλ=

0,ix ≠

0.ix ≠

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Example

ex) pp. 293-

Block diagramreduction for theantenna azimuthposition controlsystem:a. original;b. pushing inputpotentiometer tothe right past thesumming junction;c. showing equivalentforward transferfunction;d. final closed-loop transfer function

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Signal-flow graph for the antenna azimuth position control system

Example