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Nano System Control Lab.Chosun University 170
SVD (Singular Value Decomposition)
1
2
0 0 0 00 0 0 00 0 0 00 0 0 0
0 0 0 0 0
n m K
1 2
; n m real matrixwhere 0K
H
H
S.V.D A UΣVU AV
Where
H -1
H -1matrix V
U : n n unitary matrix U =U
V : m m unitary =V
max SV
min SV
Nano System Control Lab.Chosun University 171
Where ; left singular vector of is right eigenvector of associated with
H 2: AA i i i
H1 2 m i jV v :v : v ; v v ij
H 2: A A i i i
HAA
i Hiλ (A A)
H1 2 i jU u , u u ; u un ij : Kronecker delta
Hλ (AA )i
Aiu
Where right singular vector of is right eigenvector of associated with
iv ; AHA A
i
Column vectors of U(V) are orthonormal
Nano System Control Lab.Chosun University 172
Geometrical Interpretation
(1) is a real matrix, exists n n -1AAn
2
n2
y Ax, x R x x x
y R y y y
T
T
Spectral norm of A
2
22 2X 0 X 1
2
AXA max max AX
X
Singular value relation
2
2 2
max 2 2X 1
min 2 1X 12
(A) max AX A
1(A) min AXA
: spectral norm
Nano System Control Lab.Chosun University 173
Visualization (2D)A
Unit Circle
0
1 1
2 2
Vector OA=v Vector OA =uVector OB=v Vector OB =u
maxH
min
1 2 1 2
0SVD : A U V ,
0
U u u , V v v
AB
B A
1y
2x
1x
2y
B
00
Nano System Control Lab.Chosun University 174
max 1
min 2
Length of vector OA σ σLength of vector OB σ σ
.
.
max SV, min SV, Range of gain
Nano System Control Lab.Chosun University 175
1) Singular values “size” of A
HA A 1, 2, Ki i i
2) SVD “Direction” of A,B
H
H
A U VU AV
max min, for
G(s)u(t) y(t)
MIMO Freq. Response
u(t) uy(t) yy G(jω)u
j t
j t
ee
Singular value
2
2
max max2 2u 1
min min2 2u 1
G(jω) max G jω u y jω
G(jω) min G jω u y jω
Nano System Control Lab.Chosun University 176
max maxy (jω)
min miny (jω) Singular value of y exists in this region.
S.V. Facts
(1) -1 -1max min
min max
1 1(A ) , (A )(A) (A)
(2) max max max(A) 1 (I+A) 1 (A)
(3) max max max(A+B) (A) (B)
(4) max max max(AB) (A) (B)
Nano System Control Lab.Chosun University 177
Feedback Performance Specs. In Frequency Domain
Use of SV to establish MIMO Performance specs.
Proof. Attributes
Review
r(s)
d(s)
u(s) y (s )
n(s)
e(s)K(s) G(s)
-Command Following-Disturbance Rejection-Insensitivity to Sensor noise-Robustness
Sinusoidal
Nano System Control Lab.Chosun University 178
1
1
t
Loop TFM : T(s) G(s)K(s)
Sens TFM : S(s) 1 T(s)
Closed-Loop TFM : C(s) 1 T(s) T(s)
e (s) S(s) r(s) d(s) C(s)n(s) e(s) r(s) y(s)
conflict S(s) C(s) I
1) Command Following : d(s) n(s) 0
tr(t) r e (t) ej t j te e : error vectore
Relation
2 2 2
max 2
S(jω)S(jω) r
S(jω) r
e re
True error
Nano System Control Lab.Chosun University 179
Define;
range of frequency that r has energyr
We have good command following by making max S(jω) 1, r
Interpretation
max2 2,maxr 1 S(jω)e
The worst error at is attained when points along the right singular vector associated with ,
rmax 1V
min2,minS(jω)e
The best error r VK
In General
min max2S(jω) S(jω)e
min
Nano System Control Lab.Chosun University 180
Good Command Following in terms of T(s)=G(s)K(s)
max S(jω) 1 ; r
1
maxmin
1(I+T) 1I+T
min min min1 T I+T 1, T(jω) 1
Overall Loop Gain
To visualize good C.F
max (S)
min (S)
dBLoop TFM
max (T)
min (T)ω
1
r
1
dB
11
Nano System Control Lab.Chosun University 181
2) Disturbance Rejection : r(s)=n(s) 0
Sinusoidal Disturbance
td(t) d e ej t j te e
Relation
max2 2
e S(jω) de S(jω) d
We have good disturbance rejection by
max
min
making S(jω) 1
also, T(jω) 1
; ω d
Define :d Range of frequency that d has energy
Interpretation ; 2d 1
Same on the C.F.
Nano System Control Lab.Chosun University 182
To visualize good C.F. & good D.R. P r d
dBdB
max (T)
min (T)ω
max (S)
min (S)
ω
P
1
:margin of design.
Quantity Relation
Let : 0 1
If max S(jω) , ω P
Then, min11 T(jω) ; ω P
and min max1 C(jω) C(jω) 1
11
C(jω) I “Plant Inversion”
Nano System Control Lab.Chosun University 183
Visualization
dB
P max S( jω)
Sensitivity TFM
dB
P
min T(jω)1
Loop TFM
dB max C(jω)
min C(jω)
1
1 Closed-Loop TFM
Nano System Control Lab.Chosun University 184
Example of the FP (FP : Feedback Performance)
(Temp. regulation in a leaky pipe line)
(1) Problem ; monitor & control temperature.
(2) Model of the system
• Dynamics of medium Flow in the pipeConstitutive relation of the medium
• Dynamics of leak• Dynamic of a heater (time-lag)
ideal gas
Assumptions what? Why? Validity, MIMO, TITO
(3) Solution-Further assumption, linearization
Application of tools & concepts
(4) Evaluation of your solution & problem
heater leak (pressure drop)
FlowPV=nRT
Nano System Control Lab.Chosun University 185
Quantitative Relations
max
min
min max
: 0 1; S(jω) 1 , ω
;1(1) 1 T(jω) ;
(2) 1 C(jω) C(jω) 1
p
p
Let
If
Then
Proof of (1)
1max max
min min
max
min
min min
1 1S(jω) I+T(jω)(I+T(jω)) 1 T(jω)
Given , S(jω)1
1 T(jω)1 1 11 T(jω) , or T(jω) 1 1
Nano System Control Lab.Chosun University 186
Proof of (2)
max
1
-1 -1 -1
-1 -1 -1max max max
min
min
(1 ) C(jω)
C(jω) I+T(jω) T(jω)
C (jω) T (jω) I+T(jω) I+T (jω)1 C (jω) I+T (jω) 1 T (jω)C(jω)
11T(jω)
①
Since min1T(jω) from
(1)
min
min min
min
1T(jω) 1
1 1 11 1C(jω) T(jω) 1 1
C(jω) 1
Nano System Control Lab.Chosun University 187
②
max
max max max
max max
max
C(jω) 1C(jω) I S(jω)
C(jω I S(jω) 1 S(jω)
1 S(jω) , since S(jω)
C(jω) 1
Summary for good C.F & D.R
pω
• Large Loop Gain
• Small Sensitivity
• Flat CL response
min T(jω) 1
max S(jω) 1
min maxC(jω) C(jω) I
for
Constraint
Nano System Control Lab.Chosun University 188
3) Insensitivity to S.N. (Sinusoidal Noise) ; r(s) d(s) 0
n( ) n ( ) ej t j ttt e e t e
Relation
max2 2
e C(jω)nC(jω)e n
Define range of frequency of noise with singnificant energyn
We can desensitize the system from S.N. by making for all n
Large separatebetween the two
( , )p n max C(jω) 1 for all n
suppose max C(jω) ; 0 1
① Then, min T(jω) 1 ;(low loopgain)1
② max1 1 S(jω) ; high sensitivity
Nano System Control Lab.Chosun University 189
Proof: ① -1 -1min min
max max
1 1C I+T 1(C) (T)
maxmax
maxmax
1 1Since (C) ,(C)
1 11 (T)(T) 1
Proof ②1
max max minmin min max
maxmax
1 1 1(S) (I+T) 1 (T)(I+T) 1 (T) (S)
11 1 (S)1 (S)
Nano System Control Lab.Chosun University 190
Design Implication
We need wide frequency separation between sets.
andp r d n
P
n
ω
dB
min S(jω)
1
max S(jω)
Nano System Control Lab.Chosun University 191
dB
1
1 P
P max C(jω)
min C(jω)
n
ω
dB
1
P
n
1 min T(jω)
max T(jω)
ω
Nano System Control Lab.Chosun University 192
Direction information in SV plot
SVDA
m m
m m (A)i
-1AComplex matrix , exists
Real diagonal matrix of
1 maxH
min
0 00 0 (A) (A A)0 0
i i
m
H HA U V : Σ=U AV
• right singular vector , orthonormal• left singular vector , orthonormal
iv ;iu ;
miv C
miu C
Nano System Control Lab.Chosun University 193
For
AX Y Y=AX
SVD HY U V X
Suppose ;H
i iX=v y=U V v
Since
H Hi
0 00
0V v , V v1
0 0
0 0
i i
rowthi
Nano System Control Lab.Chosun University 194
1 i
0
y u0
0
ii m iu u u
Directional Information
Max : 1 max , 1 max max 2
1 max max 2
v V , V 1
u U , U 1
If maxX V
max max max 2
max2
Y= U ; U 1
Y (A) max. .Amp
Min : min ,m m min min 2
m min min 2
v V , V 1
u U , U 1
Nano System Control Lab.Chosun University 195
If minX V
min min min2Y σ U Y σ (A) min. .Amp
Input
minV
Unit sphere
min (A)
maxU
minU
maxy
miny
max (A)
“ can be less than 1”max miny , y
.
maxV
output
Nano System Control Lab.Chosun University 196
Analytical Expression
Real Amp.Complex sinusoid
x(t) A j te
Re x(t) A cosωt
Im x(t) Asinωt
Complex Amp.complex sinusoid
x(t) B j te B=B j
je
Re x(t) B cos (ωt )
Im x(t) B sin (ωt )
2 2
1 ( )
B
tan x( ) B j tt e
Complex vectors 1
x( ) x , xj ti
n
x
t e x
x
Nano System Control Lab.Chosun University 197
Summary of SVD
System : Y(s)=G(s)X(s)
Pick ω , calculate G(jω) SVD
• Max - direction
Find max max max, V , U
Set max max
( )max
x ( ) V a , U b
y b
j ji i ii i
j ti i
t e e
e
• Min-minimum direction
Find min min min, V , U0
0
& : 90 difference*
& : 90 difference
min min
( )min
x ( ) V C , U d
y ( ) d
j ii i ii i
j ti i
t e e
t e
Nano System Control Lab.Chosun University 198
F-404 turbofan engine control (GE)
1 11
2 22
3 3
x xu
x * x *u
x x
1
2
3
: LP roto speed: HP roto speed
: Turbine temper
xx
x
1
2
u : Fuel Flowu : Nozzle Area
11
32
1
xyy
xy
G(s) C(sI A) B at ω 0.1 rad/sec0.863 0.11 4.609 0.47
G(j0.1)0.9347 0.25 0.88 0.32
j jj j
G(0) :steadystate
max
min
4.846 (13.7 )0.757 ( 2.4 )
DBDB
G(s)K(s)
Nano System Control Lab.Chosun University 199
Open loop TR function
1 rad/sec
2.4
13.70
max
min
( )G j
From Matrix X
0
0
max 0.22
180
min 0.23
0.22 0 0.22V
0.975 0.009 0.98
0.98V0.22
j
j
j
j
j ej e
ee
*
6.15
max 19.38
0
min 166.6
0.94U
0.22
0.23U
0.97
j
j
j
j
ee
ee
( )i max iy b j te
0.1
,
,
,( )
i min iy d ; y = Gxj te
Nano System Control Lab.Chosun University 200
For max ;
01
max 02
u 0.22sin (0.1 0) 4.55sin(0.1 6.15 )x y
u 0.98sin (0.1 0.22) 1.115sin(0.1 19.38 )t t
t t
Input to study Max. amplitude
Phases are differentFor min
min min min 0
0.174sin(0.1 0)y U
0.7341sin(0.1 166.6 )t
t
Min amplitude
0
min 0
0.98sin(0.1 180 )x u V
0.22sin(0.1 0.23 )tt
max max max maxx V , y U
Nano System Control Lab.Chosun University 201
SVD problem
x
F
v
0x
Car suspension
0M F K( ) b( v)x x x x
0If 0x
0 1 0 0 FK b 1 b v- -M M M M
x xx x
1
2 2
1 0 0 0 Fy
v 0 1 0 1 v
1 b- -1 M MG(s) C(sI A) B D b K s Ks s -s -M M M M
x xx x
M
b k
Nano System Control Lab.Chosun University 202
For n0.5 , ω 10, M 1
At ω 10.01 0.001 0.1 0.01
G(j1)0.001 0.01 0.99 0.1
kg
j jj j
max min
max min max min
U. .V SVD G(j1)1.002, 0.01
U= U U V= V V
If input u (not U) is chosen as
(1) max max max maxu V , then y Uj t j te e
(2)min min min minu V , then y Uj t j te e
1 1 1 1max min
2 2 2 2
a sin(ωt ) b sin (ωt )U U
a sin(ωt ) b sin (ωt )
,
,
Nano System Control Lab.Chosun University 204
Stability of MIMO Feedback system Nominal stability
• Time domain - locations of poles• Frequency domain- Nyquist criterion
For
1T(s) C(sI A) B T(s)r(s) e(s) y(s)
1CL: C(s) I+T(s) T(s)
For u(t) r(t) y(t) r(t) Cx(t)
x (A BC)x Bry=Cx
“Close” Loop TF1C(s) C(sI A BC) B
Stability Re (A BC) 0 for alli i
Nano System Control Lab.Chosun University 205
Key relation
0 (s) det(sI A) ; T(s)L
(s) det(sI A+BC) ; C(s)CL
0* (s) ( ) det( I T(s))CL L m m m ms
Proof ;
1m
1m
1m
m
(s) det (sI A BC) det(sI A) det(I C(sI A) B)
det (sI A) (I C(sI A) B)
det (sI A) I (sI A) BC
(s) I +T(s)
CL
OL
Nano System Control Lab.Chosun University 206
Complex variable theory facts
• = Complex scalar variable ; “Lives” on s-plane• = Scalar-valued analytic function : maps
s-plane to another complex plane
(s)fs
(s)f Analytic in R
(1) Derivative exist at each point of R
(2) Unique in R
Nano System Control Lab.Chosun University 207
Ex)If
s+1(s)= , s 1 j2s
f
then2+j2 6 2(s)= j1+j2 5 5
f
jω
..(s)f
Im
Re
The principle of the Argument
A theorem in complex variable theory
Suppose
C = closed contour in the s-plane.
(s)f = complex-valued scalar function
(1) Analytic in C(2) z zeros inside C(3) p poles inside C
Nano System Control Lab.Chosun University 208
Then, The image of C under mapping (s)f
(A) Generates a closed contour in the C.P.(B) encircle the origin 0, z-p times in a clockwise direction.
Im
Re
o
o
j
Notation
N( , (s), ) #a f C of clockwise encirclement of point by the image of under mapping (s)f
a C
Nano System Control Lab.Chosun University 209
The principle of Argument
N(0, (s),C) z pf
z p 3
Suppose , 1 2(s) (s) (s)f f f
1 1,z p 2 2,z p
Then, 1 2N(0, (s), C) N(0, (s), C) N(0, (s),C)f f f
Nano System Control Lab.Chosun University 210
S-plane
Scalar mapping plane to plane det I T(s)
Im
Re
MIMO Nyquist contour,
R
jω
Semi circle with radius
Define RD ;
R0
RD
Nano System Control Lab.Chosun University 211
The MIMO CL system is stable iff
R u R uN 0, det I T(s) , D P , or N 1, 1 det I T(s) , D P
uP # of unstable poles of T(s) (open-loop)
Proof)
Recall (s) det I+T(s)CL OL
and R R RN 0, (s), D N 0, (s), D N 0, det I T(s) ,DCL OL
MIMO Nyquist Criterion
T(s)=G(s)K(s)e(s)r(s) y(s)
T(s)
0 0 0(z p ) ?
Nano System Control Lab.Chosun University 212
Let g(s) det I+T(s)
2 2
kT(s)s(s 2 s )n n
A B
C
1
j
s-plane
Nano System Control Lab.Chosun University 213
( )g j g(j ) je
( )
. A ; s j 0g(jω) g(jω) j
sege
. B ; s Re , RT(s) 0
g(s) det I+T(s) 1
jseg
Re
Im
1
B .
A
0
C
.
. C ; s j 0seg
Nano System Control Lab.Chosun University 214
* ( ) G(s) K(s)t s
( j )K(jω)
G(jω)K(jω) ( j ) G(jω)
t
t
For SISO case
u R
R R
T(s) ( ) det I+T(s) 1 ( )
-P N(0, det I T(s) , D
N 0,1 (s), D N 1, (s), D (open- loop circlement)
t s t s
t t
G(jω) = Unstable plant
T(s): desired
1mPositive phase
margin
1
1 1G( jω)
Positive gain margin
Nano System Control Lab.Chosun University 215
MIMO Case
0
det I+G(jω)
det I+T(jω) : desired T jω G(jω)K(jω)
1
Nano System Control Lab.Chosun University 216
MIMO Nyquist Criterion
• SISO 1C(s) 1 ( ) ( )t s t s
1R R RN 0, ( ), D N 0, 1 ( ) , D N 0, ( ), DC s t s t s
C C C R 0 0
C R 0 0
Z P N 0, 1 ( ), D Z P
N 1, ( ), D Z P
t s
t s
We want CP 0
C 0 0 C R
0 C R
Z Z +P =N 1, ( ), D
P =N 1, ( ), D
t s
t s
: Nyquist criterion for SISO (ccw)
cZ Pc 0 0Z Pccw
Nano System Control Lab.Chosun University 217
• MIMO
1
1 21 2 1 0
11 1 0
C(s) I+T(s) T(s)
(s)det I+T(s)
( ) det sI A BC s s s s
only eros poles of CL(s) det(sI A) s s s
eros poles of OL
CL OL
n n nCL n n
n nOL n
s
z
z
R R R
SC UC
SO UO
N 0, (s), D N 0, (s), D N 0, det I+T(s) , D
#of Z( ) Z Z#of Z( ) Z Z
CL OL
CL
OL
nn
Stable Unstable open loop
Nano System Control Lab.Chosun University 218
UC UO R
UC UO R
Z =Z N 0, det I T(s) , D
P =P N 0, det I T(s) , D
Counter clockwise
0
det I T(s)
Stable
: roots of characteristic eq.
: poles of closed loopC(s) T(s)
UC
R UO
C R UO
We want "P 0"
N 0, det I+T(s) , D P
N 0, det I+T(s) , D P
Nano System Control Lab.Chosun University 219
Stability Robustness : SISO
Central theme- to provide guarantee that compensator designed on the basis of nominal plant model, will not yield unstable feedback system
Assume
Question: then,
Ng (s) ,NK (s)
Ng (s)Ne (s) Ny (s)NK (s) Nu (s)
stable
Is this stable?
Ag (s)e(s) y(s)NK (s)
u(s)
Nano System Control Lab.Chosun University 220
A Ng (s) g(s) g (s)
Source of g(s)
• Low order approximation of high-order system.
Directional Uncertainty
At (jω) >1g( jω) ?
1
Nt (jω)At (jω)
g( jω) has 180 uncertaintyo
• actuator of sensor dynamics.
• “Fast” mechanical dynamics : bending, torsional.
• “Small” time delay.
Nano System Control Lab.Chosun University 221
SISO Model Error (cascade Representation)
; Error reflected at the plant output
A 0 N N 1, g (s) (s) g (s) g (s) (s)l l
0 (s)lu(s) y(s)
Ng (s)
1(s)lu(s) y(s)
Ng (s)
For SISO
11ω
2ωAt (jω)
Nt (jω)
; modelling error0l
0
Nano System Control Lab.Chosun University 222
Test 1 (For Robustness)
N N
N N
N N
Vector sum-1 d (ω) t (jω)
d (ω) 1 t (jω)
d (ω) 1 t jω
" make this resonable big"
Test 2 NN
N
t (s)C (s)1+t (s)
Assume NC (s), (s)l are stable.
Then, NN
1 1If ( jω) 1 or C (jω)C ( jω) ( jω) 1
ll
The actual Loop is still stable Sufficient condition
Nt (jω)
1
Nd (ω)0
A N; t (s) (s)t (s)l
Nano System Control Lab.Chosun University 223
Proof)
A N
A N N
A N
N
t (s) (s)t (s)t (s) t (s) (s) 1 t (s)
t (s) t (s)(s) 1t (s)
ll
l
A N Nt ( jω) t ( jω) 1 t ( jω)
“The inverse is true”
1
N1 t (jω)
Nt (jω)
A Nt (jω) t (jω)
At (jω)
0
N
N N
1 t ( jω) 1( jω) 1t (jω) C ( jω)
l
divided by at the both sidesNt ( jω)
Nano System Control Lab.Chosun University 224
Alternate model
Percent deviation
m
m
m
(s) 1 e (s) (s)(s) e (s) (s)
(s) 1 e (s)
A N
N
g gg g
l
u(s) y(s)Ng (s)
me (s)
12
unstable region N
1C (jω)
me (jω)
Nano System Control Lab.Chosun University 225
N
1C (jω)
me (jω)stable
NC (jω)
Robustnessboundary
m
1e (jω)
Nano System Control Lab.Chosun University 226
Design Implication
Need ; (1) Ng (s)
(2) Some upper bound of m maxe (jω) e (ω)
Conservatism.“sufficient condition”
maxe
max
1e
NC (jω)minimize the bandwidth NC (jω)
Nano System Control Lab.Chosun University 227
Robustness Condition for SISO case
mN
Nm
1e (jω)C (jω)
1C (jω)e (jω)
; “MIMO Nyquist criterionNec. Sufficient condition”
NC (jω)
1me (jω)
Bode plot
m maxe (jω) e (ω)
Nano System Control Lab.Chosun University 228
MIMO Stability Robustness
Assume : Nominal Feedback Loop is stable
Need : Actual Feedback Loop
Nominal plant ; NG (s)Actual plant ; AG (s)
• Plant Output Error
• Plant input Error
oL (s)u(s) y(s)
NG (s)
A 0 NG (s)=L (s)G (s)
NG (s)y(s)
IL (s)
A N IG (s)=G (s)L (s)
u(s)
0 IL (s) L (s) for MIMO
Nano System Control Lab.Chosun University 229
Robustness Test (MIMO Case)
For a stable nominal loop,
L(s)NK (s) NG (s)
The actual loop is stable if
(1) L(s) is stable
and
-1max min N
N N N
-1max min N
1
L(jω) I I T (jω) , ω
T (jω) G (jω)K (jω)or
L(jω) I C (jω)
C I+T T
Nano System Control Lab.Chosun University 230
NT (s)
mE (s)
mL(s)= I+E (s)
C
Alternate Test
1max N max m
max m
1C (jω) E (jω)E (jω)
max NC (jω)
1max mE (jω)
Nano System Control Lab.Chosun University 231
Robustness Test proof;
Step ;
(1) Assume is stable
(2) Find “Smallest” model error in “worst” direction that will cause actual Feedback System to be on the verge of instability
(3) Express (2) on Nyquist plot.
(4) Use a SV fact to obtain a sufficient condition for stability
NC (s)
Nano System Control Lab.Chosun University 232
SV factmax minIf ( B ) ( A )
Then , det(A+B) 0n n n n
0. 1
N
N u
det I+T (jω)
N 0, det I T (jω) ,D PR
Adet I+T ( jω)
i
A
at ω ω*det I T ( jω) 0
For stability ;
Adet I T (jω) 0, ω
Set
-1m N
-1 -1 -1 -1m N m N N N N N
-1m N N N
B E ( ) , A=I+T ( )
0 det E +I+T det E T T +T T +T
det E T +T +I det T
s s
-1Nsince det(T ) 0,
Nano System Control Lab.Chosun University 233
A m NT I+E T
m N N m Ndet E T +T +I det I+(I+E )T 0
AT (s)
NT
mE
* Sufficient condition for stability Robustness
-1max m min N
-1min N
max N
E (jω) I+T (jω)
C (jω)
1C (jω)