k() s g() l() n...−controllability / observability concept. −performance criteria / nominal...
TRANSCRIPT
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Nano System Control Lab.Chosun University 235
Summary
K ( )N s G ( )N s 0L ( )s
Plant Output
1max 0 min( L ( ) I ) I G ( )K ( ) ,N Nj j j
1max 0 min(L ( ) I ) I G ( )K ( ) ,N Nj j j
or
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Nano System Control Lab.Chosun University 236
K ( )N s G ( )N s1L ( )s
Plant Iutput
1max 1 min( L ( ) I ) I K ( )G ( ) ,N Nj j j
1max 0 min(L ( ) I ) I K ( )G ( ) ,N Nj j j
or
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Nano System Control Lab.Chosun University 237
Quiz #2 : 3-problem, 1-bonus
Some True – False questions
− Multivariable Transmission Zeros : GEP solving
− Controllability / Observability Concept.
− Performance Criteria / Nominal Stability
Understand MIMO Nyquist Criterion
• SV & SVD
• perf. Spec. via s-plane
− Bonus – Simple Eigenstructure Assignment.
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Nano System Control Lab.Chosun University 238
1 2U [u ,u ] u i
1 2V [ v , v ] vi
− right eigenvector ofHA A
HA A− right eigenvector of
1u A vi ii
H H T H H T
H
A U V , A V U , A U V1 1v A u , u A vi i i i
i i
1.
2.min min max( I A ) (A) 1 (A) 1 ,
1 1min min
max
1( I A ) (A ) 1 1(A)
min max
1 1 ,( I A ) ( A ) 1
max1
min max
( A )1( I A ) (A ) 1
3.max min1.0 , 0.0000474 '' ''depends on untis
Hw #5 solution
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Nano System Control Lab.Chosun University 239
4.
45
45
1 1 1
2,3 1 1.02j
1 2 2B , u 0 ,b b 12
1 0 0C
0 0 1cc
1q 1,2,3
0ii
1 1p ( I A ) Bq ( I A ) b ,i i i Q 1 1 1
1
1
A bS
c 0
q 1i
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Nano System Control Lab.Chosun University 240
Design of MIMO Feedback System
Standard form
K( )sr ( )s e( )s u ( )s
d( )s
y( )sL( )sG( )s
n ( )s
Error e ( ) r ( ) y ( )t t t t
e ( ) S( ) r ( ) d ( ) C( ) n ( )t s s s s s s
Error bound
max maxL ( ) I e ( ) ,j all
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Nano System Control Lab.Chosun University 241
Sensitivity
disturbance
Sensor noise
max S( )j
min S( )j
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Nano System Control Lab.Chosun University 242
Closed Loop
HighLoop gain
Sensornoise
max C( )j max
1 ( )e j
min C( )j
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Nano System Control Lab.Chosun University 243
Issue
T( ) , C( ) , S( )s s s Time Domain
T( )sr( )s e( )sy( )s
y T( ) es
1x A x Bu y C(sI A ) Bey C x
1T( ) C(sI A ) Bs
u e
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Nano System Control Lab.Chosun University 244
1y C ( ) rs
e r y Cx r
1x [A BC]x Br y C(s I A BC) B ry C x
11C ( ) C(s I A BC) Bs
e ( ) S( ) r ( )t s s s
1x [A BC]x Br C(s I A BC) B I re C x I r e
1S( ) C(s I A BC) B Is
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Nano System Control Lab.Chosun University 245
• can analyze given MIMO design
• can’t design compensator
− Nyquist diagram does not help !
− Eigenstructure assignment ; Standard form
• How about applying SISO methods ?
1r
rm
1u
u m
MIMOG( )sm m
1y
y m
11K
K mm
K ( )s
So far , Discussion ;
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Nano System Control Lab.Chosun University 246
• Dumb way
- Set Design
- Set Design
(1) Must decide which control for which output.
For TITO
1 11 1 12 2y ( ) g ( ) u ( ) g ( ) u ( )s s s s s
2 21 1 22 2y ( ) g ( ) u ( ) g ( ) u ( )s s s s s
12g ( ) 0s
21g ( ) 0s 11K ( )s
22K ( )sWatch out for
stability of TITO
• Closing the loop one at a time.
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Nano System Control Lab.Chosun University 247
• Sensible way
Close Loop #2, first
2r ( )s e( )s 2u ( )s
2y ( )s22K ( )s 22g ( )s
21g ( )s
1u ( )s
− Ignore Design 21g ( )s 22K ( )s
2u ( ) ;s 2 22 2u K e 22 2 22 21 1 22 2K r K (g u g u )
22 22 2 22 2 22 21 1(1 K g )u K r K g u
22 22 212 2 1
22 22 22 22
K K gu ( ) r u1 K g 1 K g
s 2 2 2 21 1
h ( ) r h ( )g ( ) us s s
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Nano System Control Lab.Chosun University 248
Impact on closed – loop #1
1 11 1 12 2y g ( ) u g ( ) us s 11 1 12 2 2 2 21 1g ( ) u g (h r h g u )s
11 12 2 21 1 12 2 2[g g h g ]u g h r
First Loop
1r ( )s 1e ( )s 1u ( )s
1y ( )s11K ( )s 11f ( )s
12 2g h
2r ( )s
Design for
− Good command following
− Good dist. rej. of
11K ( )s
12 2g h
11 1 12 2 2f u g h r
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Nano System Control Lab.Chosun University 249
1K
2K
1r
2r
1u
2u
1y
2y
G
d irec t con tro l
Interaction of control Loops and the Relative Gain Array (RGA) (E. Bristol, 1966)
− Tool for Static loop interaction analysis
• Cannot design each loop separately
• MIMO Loop should be stable
Which is the “best” coupling 1 1 1 22 2 2 1
y u y uor
y u y u
ind irec t con tro l
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Nano System Control Lab.Chosun University 250
RGA analysis (Relative Gain Array)
Step 1 : static gain between
(other loop gain)
1 1y & u
1u
2u 0
1y
2y2
1
1 u 0
yu
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Nano System Control Lab.Chosun University 251
Step 2 : static gain between
(other loop closed )
1 2y & u
1u
2u
1y
2y 02r2K
2
1
1 y 0
yu
2
2
1 1 u 011
1 1 y 0
( y u )( y u )
1 1( y & u )forRelative gain
Relative gain Array :11 12
21 22
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Nano System Control Lab.Chosun University 252
e.g. (=exempli gratia)
1 11 1 12 2y g u g u
2 21 1 22 2y g u g u 1
2 22 21 1u g g u 2( y 0)for
11 11 12 22 21 1y g g g g u
11 11 2211 1
11 12 22 21 11 22 12 21
g g gg g g g g g g g
11 12 1
21
1
g g gg
G
g g
n
n nn
u 0;
y 0;
( y u )( y u )
k
k
i j k jij
i j k i
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Nano System Control Lab.Chosun University 253
Note :
u 0;( y u ) g [G]ki j k j ij ij
1y 0;
y 0;
1 ( u y ) [G ]( y u ) k
k
j i k i jii j k i
T 1G (0) (G (0) )ij ij ij 0RGA evaluated at s
TITO case
11 2211
11 22 12 21
g gg g g g
, y G u
• For Static Interaction (s=0)
11 12 21: 1 g g 0 '' ''complete decoupling1 00 1
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Nano System Control Lab.Chosun University 254
• Properties of RGA
1.1 1
1n n
ij ijj i
11 0
11 1
11 22g g 0
11 22g g 0
11 22 12 21g g g g
'' ''bad decoupling
11 2211
11 22 12 21
g gg g g g
'' ''holding effect 11 121 , 0
11 0 '' ''wrong direction
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Nano System Control Lab.Chosun University 255
e.g.
1 11 0.1 1G
0.2 0.80.5 1 1
s s
s s
T 111 11 111G (0) (G (0) ) 0.8
1.25
0.8 0.20.2 0.8
1 1 1 2
2 2 2 1
y u y uy u y u
or
21 1 u ( ) 0 11[ y ( ) u ( ) ] g ( )ss s s
2
11 1 y ( ) 0 11 12 22 21[ y ( ) u ( ) ] g g g gss s
This condition is impossible for holding
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Nano System Control Lab.Chosun University 256
Property
2. RGA is invariant under input or output scaling
RGA (G(0) ) If
1 2RGA ( G (0) )S S
Then
3. If G is diagonal or triangular
Then I
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Nano System Control Lab.Chosun University 257
Relative Gain
SISO yMIMO y
i
i
Normal measure output responseof decoupling output response
0 cross coupled
1 completely decoupled
with all the output = 0
0y 0 r
0
11 12
21 22
g gG
g g
;
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Nano System Control Lab.Chosun University 258
Nominal Stability & RGA
r
yk Is
K( )s G( )s
T ( ) G ( )K ( )s s s
Thm 1. Iff det[T (0) ] 0
Then is integrally stabilizableT( )s
RGA[G (0) ]
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Nano System Control Lab.Chosun University 259
Thm 2. If some of at least one of the following is true.
(a) MIMO C.L. is unstable.
(b) Loop is unstable with all the other loops open.
(c) MIMO C.L. is unstable even with loops open.
0,ii
ithi
If for all0,ii 1, ,i n
Then MIMO C.L. system is stabilizable.
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Nano System Control Lab.Chosun University 260
Sensitivity to Modeling Error
be actual static gain matrix.
be nominal static gain matrix.
G (0)A
G (0)N
condition
1
G (0) G (0) 1 1G (0) [G ]G G
A N
N NN N
max min[ ] [ ] [ ]N N NG G G
1* min 2max ,
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Nano System Control Lab.Chosun University 261
Regulator Design
Nominal Plant Dynamic
x ( ) A x ( ) Bu ( ) : y ( ) C x( )t t t t t
Feedback Law
u ( ) G x( ) 0t t : r ( ) 0t
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Nano System Control Lab.Chosun University 262
o
B 1(sI A) G
C
u ( )s
Gyy( )t
B 1( sI A )
Gr
y( )t
x r
u
x ( )t
y( )t
G x ( )t
y( )x ( )
x ( )r
tt
t
yu ( ) G y G xr rt
o
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Nano System Control Lab.Chosun University 263
• Linear Quadratic Regulator (LQR) Problem ( R. E. Kalman )
− How do we obtain the value of ?G
Optimal Control Problem
T T
0[ x ( )Q x( ) u ( )R u( ) ]t t t t dt
: minimize
x ( ) A x( ) Bu( )t t t
- Symmetric & Positive definite, real & positive Eigenvalues.
Optimal Control - Find that minimizes
constrained to x ( ) A x( ) Bu( )t t t Ju ( )t
Q, R
J
Q = min. state size , R = min. control size
J
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Nano System Control Lab.Chosun University 264
• LQR Solution
− There exists a unique solution which can be realized in
feedback form, assuming full state feedback.
Feedback Law u ( ) G x ( )t t
LQR1 TG R B K
Where is matrix
− symmetric & positive semidefinite
− solution of CARE ( Control Algebraic Riccati Equation )
K n n
T 1 T0 K A A K Q K BR B K
Re (A BG ) 0i A, B : GivenQ, R : Design
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Nano System Control Lab.Chosun University 265
• LQR Variant 1
− Cross - Coupled cost functional
T T T
0[ x Q x 2 x Su u R u ]J dt
control u G x TQ S
MS R
1 T TG R [S B K ]
is solution of CAREK
T 1 T T0 K A A K Q [K B S]R [B K S ]
Guarantee : Re[ (A BG )] 0i : Nominal stability
, : positive semi-definite
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Nano System Control Lab.Chosun University 266
• LQR Variant 2
− Exponential weighting
T T
0[ x Q x u R u ]J ate dt
0a j
0
a
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Nano System Control Lab.Chosun University 267
Control : u G x 1 TG R B K
where is solution of CARE K
T 1 T0 K A I A I K Q K B R B Ka a
• LQR problem
x A x Bu
Control : u G x 1 TG R B K
T T[ x Q x u R u ]J fi
t
tdt
is a solution of CARE K
T 1 T0 Q KA A K KBR B K
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Nano System Control Lab.Chosun University 268
• Optimal Control – LQR Problem
minimize V L( x , x , )fi
t
tt dt
L( x , x , )t is continuous to second partial
ˆ ˆV x ( ) V x ( ) x x xt t for
Find forV x ( )t
V V(x x ) V(x)
22
2x( ) x( )
V 1 VV(x) x x V(x)x 2! xt t
21V V ;2!
V
first ordervariation of
x( )
VVx t
x
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Nano System Control Lab.Chosun University 269
• Differential Calculus
x x 0 0as t
x 0dx dxtdt dt
.cond for extremum
2 2
2 2max 0 , min 0d x d xif ifdt dt
• Variational Calculus
V 0 x 0as
x( )
VV V xx t
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Nano System Control Lab.Chosun University 270
x( )
V 0x t
.cond for extremum
2
2x( )
Vmax 0x t
if
2
2x( )
Vmin 0x t
if
V L( x , x , )fi
t
tt dt
V L( x x , x x , ) L ( x , x , )fi
t
tt t dt
L LV x xx x
f
i
t
tdt
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Nano System Control Lab.Chosun University 271
ˆx x x
ˆ( x ) x x xd d ddt dt dt
( x ) xd dt
Second term of V
L Lx x xx x x
ff f
i ii
tt t
t tt
d Ldt dtdt
L L LV x xx x x
ff
ii
tt
tt
d dtdt
Boundary Condition
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Nano System Control Lab.Chosun University 272
Necessary condition
L L 0x x
ddt
Euler equation
comment
If L L( x , u )
①
②
and are independent x , u Apply Euler eq. directly
and are dynamically constrained by x , u
x ( x , u , )f t
− change to incorporate the constraint
Lagrange Multiplier
L
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Nano System Control Lab.Chosun University 273
Example : maximize area of rectangle with fixed parameter
b
l
,a l b 2 2p l b constraint
( 2 2 )c l b l b p
2 0c bl
2 0c lb
2b 2l ,,
2 2 0
4 4
8
c l b p
pp
① ② ③
from ① , ② b l
③ ① ②, ,4 4p pl b
, ,
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Nano System Control Lab.Chosun University 274
Method of Pontryagin
min J L( x , u , )fi
t
tt dt
subject to constraint x ( x , u , ) 0f t
create TJ* L( x , u , ) p ( ) ( x )fi
t
tt t f dt
p ( )t Vector of Lagrange multiplier 1n
Design Parameter
p ( ) K x ( )t t 1n ( ) ( 1)n n n
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Nano System Control Lab.Chosun University 275
• Resulting Euler Eq.
T Tx : [L p ( x ) ] L p ( x ) 0x x
f ft
①
define TH L p f • Hamiltonian control
• State function of Pontryagin
TH p x 0x xddt
② TH px
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Nano System Control Lab.Chosun University 276
T Tu : L p ( x ) L p ( x ) 0u u
df fdt
⑤
⑥
③
④H 0u
LQR
T TL x Q x u R u , A x Buf
T T TH x Q x u R u p (A x Bu )
②⑤
④⑤
T T 1 Tu R p B 0 : u R B p
0
Tp Q x A p T T Tx Q p A p :
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Nano System Control Lab.Chosun University 277
p ( ) K x ( )t t
Tp ( ) K x K x Q x A pt
TK (A x Bu ) K x Q x A p
1 T TK A x K B( R B p) K x Q x A p
T 1 TK x Q x A K x K A x K BR B K x
T 1 TK Q A K K A K BR B K
'' . ''general eq of CARE
K : const. gain matrix ; K 0
p K x
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Nano System Control Lab.Chosun University 278
Example
1) 1st Order
x x ua 2 2
0
x uJx um m
dt
'' ''Bryson Method
2
1Qxm
21R
um
when max. des. value for xm x
max. des. value for um u
CARE
T 1 T0 K A A K Q K BR B K
2 22
10 2 K K uxm m
a
,
,
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Nano System Control Lab.Chosun University 279
2 2ux
2
( )K
u
mm
m
a a
positive
Finally 1 TG R B K
2 2ux( )m ma a
Closed – Loop system
2 2uxx (A BG ) x ( ) xm ma
x ( ) x (0) tt e
If unstable open-loop poles 0 ,a
2G a '' ''Expensive Control
0
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Nano System Control Lab.Chosun University 280
2) 2nd Order
1 2x x x ua a
2 2 2
0
x x uJx x um m m
dt
'' ''Bryson Method
21
2
1 0x
Q10
x
m
mn
n n
21
2
1 0u
R10
u
m
mm
m m
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Nano System Control Lab.Chosun University 281
Normalizing
2 2 2 2
0u J x x um P R dt
where 2ux
mP
m
2ux
mR
m
After Some Algebra
x xx
u G Gx
2x 2 2G Pa a
2 2x 1 1 2 2G 2( )R Pa a a a
,
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Nano System Control Lab.Chosun University 282
Closed – Loop : x (A BG ) x
20 0x 2 x x 0
12 4
0 2( )Pa 2
1 22
2
21 22
R
P
a aa
If is fixed , R P 2
2
If is fixed , P R
( )a ( )c( )b
,
0 0 0
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Nano System Control Lab.Chosun University 283
Summary
x A x Bu State Dynamics
T T
0J ( x Q x u R u )dt
LQR Solution
u G x State feedback
-1 TG R B K : K T 1 T0 K A A K Q K BR B K
; cost function
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Nano System Control Lab.Chosun University 284
o
B1( I A)s
G( )s
1LQT ( ) G (s I A) Bs
Loop TFM
Sensitivity TFM
Closed - Loop TFM
1
LQ LQS ( ) I T ( )s s
1
LQ LQ LQC ( ) I T ( ) T ( )s s s
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Nano System Control Lab.Chosun University 285
o
B G( )s
E
Stability Robustness
1
max min LQ[E ( ) ] [ I T ( ) ]j j
or
max LQ max[C ( ) ] 1 [E ( ) ]j j
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Nano System Control Lab.Chosun University 286
KFDE ( Kalman Freq. Domain Equality )
From T T 1 T0 K A A K N N K BR B K
After extensive algebra manipulate
T TLQ LQI T ( ) R I T ( ) R N ( )B N ( )Bs s s s KFDE
R I , 0
T TLQ LQ1KFDE : I T ( ) I T ( ) I N ( )B N ( )Bs s s s
Freq. Domain s j
2LQ1I T ( ) 1 N ( )Bi ij j
− Important for synthesis
design parameter , N
T; N N Q
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Nano System Control Lab.Chosun University 287
If 1
Then , 21 11 N ( )B N ( )Bi ij j
LQ 1T ( ) N ( )Bi ij j
Small , large ( loop gain )
good C.F. , Disturbance Reduction.
LQT ( )i j
For Low Frequency Approximation
1( ) (s I A )s
s 0 1(0) A
At DC ( 0)
1LQ 1T (0) N A Bi i
: cheap control
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Nano System Control Lab.Chosun University 288
For identical SV at DC
1NA B I
1T 1 TN B A B B
1T TN B B B A or
LQ 1T (0)i
1
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Nano System Control Lab.Chosun University 289
KFDE
2LQ1I T ( ) 1 N ( )Bi ij j
For 1
LQ1T ( ) N ( )Bi ij j
(1) Low Frequency
T 1 TLQ
1T ( ) N (B B) B Ai j
(2) High Frequency
1( ) I Aj j
For 1( ) Ijj
,
,
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Nano System Control Lab.Chosun University 290
KFDE
LQ1T ( ) N ( )Bi ij j
1 N Bi
20dbdec
maxc
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Nano System Control Lab.Chosun University 291
Design
want T 1 TNB I N (B B) B
To estimate maxc
max maxT ( ) 1cj
LQ1T ( )i j
(0 )db
max1 11 c
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Nano System Control Lab.Chosun University 292
Summary
(1) Cheap control ( 0)
− Control weighting is negligible compared to state weighting
High gain , High bandwidth
Main Result
If is minimum phaseN ( )Bs
Then 0
lim G W N
T1G B K
TW W I orthonorm alsquare m atrix
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Nano System Control Lab.Chosun University 293
(2) Expensive control
− Control weighting is huge compared to state weighting
Low gain , Low bandwidth
Main Result
If is stable
Then lim G 0
A
( )
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Nano System Control Lab.Chosun University 294
Disturbance rejection in LQR Design
System : x A x Bu ; y C x
T T T T T
0 0J y y u u x C C x u udt dt
1LQS ( ) I G ( )Bs s
As 0 1
LQS ( ) I G ( )Bs s
1I WC ( )Bs
0 LQ; : S ( ) 0in cheap control s
0 0
W C
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Nano System Control Lab.Chosun University 295
Ex )
o
u( )t
y( )t
u( )t
d( )t
x x ( )tB
L
A
C
G
CL Dynamics
x (A BG) x Ld( ) : y C x ( )t t
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Nano System Control Lab.Chosun University 296
y( ) S d( )ds s
1S ( ) C(s I A BG ) Ld s
For good distur. rejection
max S ( ) 1d s
As ( cheap control ) 0
1 1S ( ) C( ( ) BG) Ld s s
1 1C( ( ) B G) Ls
1C(BWC) L
0
W C
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Nano System Control Lab.Chosun University 297
• LQ – Servo − How to adapt LQR based design to do command following
u G x r
• LQR − Full state Feedback
C
B G( )so
d ( )s r ( )s
y( )s e( )s
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Nano System Control Lab.Chosun University 298
If output is part of the states
B ( )su ( )s
y p
x r
x ( )t entire state
System
x A x Bu
y C x ( )p p t
x ( ) D x ( )r Pt t
( )C Ip m m m n mO
( ) ( ) ( )D Ip n m m n m n mO
,
,
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Nano System Control Lab.Chosun University 299
• Mod. LQR
y( )to G y B ( )s
Gr
y px r
u ( ) G y G xy p r rt
x A BG C BG D xy p r p CL :
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Nano System Control Lab.Chosun University 300
• LQ - Servo ( Definition )
e( )tr ( )tG y B ( )s
Gr
y px r
d
y
Loop TFM , for LQ - Servo T ( )s
1T ( ) C (s I A BG D ) BGp r p ys
1LQT ( ) G (s I A ) Bs
LQT ( ) T ( )s s
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Nano System Control Lab.Chosun University 301
LTR ( Loop Transfer Recovery ) Method
K ( )s G ( )sr ( )s
e( )s u ( )s y( )s
compensator plant
H C( )sr ( )s
e( )s
y( )s
( )with Filter Loop Tagret Loop
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Nano System Control Lab.Chosun University 302
MBC ( Model – Based Compensator )
Plant : 1G ( ) C(s I A ) B C ( )Bs s
MBC : 1K ( ) G (s I A BG HC) Hs
Where Control gain matrixG u G z
x : x A x Bu
z : z A BG HC z H e
u G z
: y C x
: y re
H Filter gain matrix n m
m n
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Nano System Control Lab.Chosun University 303
Block Diagram
r ( )sI H ( )s
C
B
G B ( )s C
z ( )su( )s
y( )s
K ( )s
G ( )s
xx
x x zc
2 1n
A BG BG 0: x x r
0 A HC Hc c
y C 0 xcAc
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Nano System Control Lab.Chosun University 304
det ( I A ) det ( I A BG ) det ( I A HC)c
triangular matrix
※ Nominal stability
Re (A BG ) 0i all i
Re (A HC) 0i all i
Loop TFM
11T ( ) G ( )K ( ) C ( )BG ( ) BG HC Hs s s s s
C ( )Hs
I
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Nano System Control Lab.Chosun University 305
Step. 1.
Fix Filter matrix H s.t.
Re [A HC] 0 ,i i
C ( )Hs
Step. 2.
Apply cheap control of LQR
Re [A BG ] 0i
T
0lim G W C ; W W I
•
•
•
• Desired Loop TFM
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Nano System Control Lab.Chosun University 306
C ( )Hs
Main Result ( LTR )
As 0 , T( ) C ( )Hs s : Target Loop
r ( )s y ( )s
r ( )s y ( )s
Hdetermine
K ( )s G ( )s
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Nano System Control Lab.Chosun University 307
What is happening ?
The LTR causes the MBC based design to cancel
the dynamics of the plant , and substitute
the Filter ( Target ) Loop dynamics.
− example of inverting the plant in a stable manner.
Observ.
(1) Some poles of must cancel zeros of
the rest of the poles must be very “fast”.
(2) Zeros of must be zeros of Target loop,
(3) and has same poles.
G ( )s
G ( )sK ( )s
K ( )s C ( )H.s
C ( )Hs C ( )Bs
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Nano System Control Lab.Chosun University 308
References
'' '' , ,Dynamics of physical system Cannon HcGraw Hill
'' '' ,Linear system Fundamentals Reid
'' ; '' , &system Dynamics A unified Approach Karnopp Rosenberg