Reduction of Multiple Subsystem
-
Upload
nicklingatong -
Category
Documents
-
view
278 -
download
4
Transcript of Reduction of Multiple Subsystem
6
Robotics Research Lab
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
7
Robotics Research Lab
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
8
Robotics Research Lab
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
9
Robotics Research Lab
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
10
Robotics Research Lab
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
13
Robotics Research Lab
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+
14
Robotics Research Lab
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
15
Robotics Research Lab
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
16
Robotics Research Lab
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−
17
Robotics Research Lab
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
18
Robotics Research Lab
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
19
Robotics Research Lab
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
20
Robotics Research Lab
( )( ) ( )( )( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
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
21
Robotics Research Lab
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
22
Robotics Research Lab
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
23
Robotics Research Lab
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
26
Robotics Research Lab
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
27
Robotics Research Lab
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−
=
=
28
Robotics Research Lab
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)
29
Robotics Research Lab
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−
= = − +
30
Robotics Research Lab
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
31
Robotics Research Lab
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 ≠
iλ
0.ix ≠
32
Robotics Research Lab
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