balandin kogan lmi toolbox
TRANSCRIPT
-
7/24/2019 Balandin Kogan LMI Toolbox
1/135
-
7/24/2019 Balandin Kogan LMI Toolbox
2/135
-
7/24/2019 Balandin Kogan LMI Toolbox
3/135
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
4/135
H
-
7/24/2019 Balandin Kogan LMI Toolbox
5/135
ATX+ XA= Q
Q
ATX+ XA
-
7/24/2019 Balandin Kogan LMI Toolbox
6/135
V(x) =xTXx
X =XT >0
H
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
7/135
H
-
7/24/2019 Balandin Kogan LMI Toolbox
8/135
-
7/24/2019 Balandin Kogan LMI Toolbox
9/135
-
7/24/2019 Balandin Kogan LMI Toolbox
10/135
-
7/24/2019 Balandin Kogan LMI Toolbox
11/135
x= (x1, , xm)
F(x) =F0+ x1F1+ + xmFm>0 ,
F0, F1, , Fm n n Fi = F
Ti Rnn, i = 0, 1, , m > 0
uTF(x)u >0 u Rn, u = 0 .
F(x)
min(F(x))> 0 (
)
F(x)> 0
F(x)
F(x)> 0
V
Sn = {M| M=MT Rnn}
F(x) =F0+ L(x) ,
F0 Sn L(x) V Sn x V
{e1, , em} V
L(x) =m
j=1
xjFj ,
x=m
j=1
xjej , Fj =L(ej) , j= 1, , m ,
F(x)> 0
ATX+ XA+ Q >0 ,
-
7/24/2019 Balandin Kogan LMI Toolbox
12/135
A, Q Rnn
X Rnn
Q
V
Sn
(n
n)
Rm m= n(n+1)/2 {E1, , Em}
X=m
j=1 xjEj
Q +m
j=1
xj(ATEj+ EjA)> 0 ,
A=
0 1
2 3
, Q=
0 0
0 0
,
X=x1
1 0
0 0
+ x2
0 1
1 0
+ x3
0 0
0 1
=
x1 x2
x2 x3
x1
0 1
1 0
+ x2
4 3
3 2
+ x3
0 2
2 6
>0 .
>
F(x)< 0
F(x)> G(x)
F(x)
G(x)
F(x)> 0
F(x) G(x)> 0
x
F= {x| F(x)> 0}
x1, x2 F [0, 1] F(x)
F(x1+ (1 )x2) =F(x1) + (1 )F(x2)> 0 .
F1(x)> 0, , Fk(x)> 0 .
F(x) =
F1(x) 0 00 F2(x) 0
0 0 Fk(x)
>0 .
F(x)
F1(x), , Fk(x)
-
7/24/2019 Balandin Kogan LMI Toolbox
13/135
F(x)
Fi(x), i = 1, , k x
F(x)> 0
F(x)> 0
Ax= b
F(x)> 0x= Ay + b
y
F(x)> 0
x M,
M
Rn
M =x0+ M0 = {x0+ m| m M0} ,x0 Rn M0 Rn {e1, , ek} M0
F(x) =F0+ L(x)
0< F(x) =F0+ L(x0+k
j=1 xjej) =F0+ L(x0) +k
j=1 xjL(ej) == F0+ x1F1+ + xkFk= F(x) ,
F0 =F0+L(x0) Fj =L(ej) x= (x1, , xk) x Rn
F(x) > 0
x
x
x= x0+k
j=1 xjej
M
M= M11 M12
MT12 M22
M11 = M
T11
xTM x= xT1 M11x1+ 2xT1 M12x2+ x
T2 M22x2=
= (x1+ M111 M12x2)
TM11(x1+ M111 M12x2) + x
T2(M22 MT12M111 M12)x2 ,
x=
(x1, x2) M
M > 0
M11 > 0 S = M22 MT12M111 M12 > 0
S
M11 M
-
7/24/2019 Balandin Kogan LMI Toolbox
14/135
F :V Sn
F(x) =
F11(x) F12(x)
FT12(x) F22(x)
,
F22(x) F(x)> 0
F22(x)> 0 , F11(x) F12(x)F122 (x)FT12(x)> 0 .
x
x
H
ATX+ XA+ XBR1BTX+ Q 0 .
X
X > Y1 > 0
X I
I Y
>0
X
Y
-
7/24/2019 Balandin Kogan LMI Toolbox
15/135
x
F(x)> 0
f :
V R
opt= inf F(x)>0
f(x) .
x
F(x)> 0
f(x)
opt+
R
F(x) G(x)> 0 ,
F(x)> 0 ,H(x)> 0 ,
F(x)
G(x)
H(x) Sn
x= Ax ,
A Rnn
A
X=XT >0
ATX+ XA 0 .
-
7/24/2019 Balandin Kogan LMI Toolbox
16/135
xt+1=Axt
A
X=XT >0
ATXA X 0 .
x= Ax+ Bv , x(0) = 0 ,z= C x+ Dv ,
L2 v(t)
v = (0
|v(t)|2 dt)1/2 < ,
= supv=0
z
v
.
ATX+ XA XB C T
BTX I DTC D I
0
f(x) =max(F(x))
max
(F(x)) =1/2max
(FT(x)F(x))
max(FT(x)F(x))< I FT(x)F(x)> 0
I FT(x)
F(x) I
>0.
x=
x
, F(x) =
I FT(x)
F(x) I
, f(x) = ,
F(x)
F(x)
f(x)
F(x)> 0
-
7/24/2019 Balandin Kogan LMI Toolbox
17/135
x= Ax .
0
J(v) =
0
(xTQx 2vTv)d t
Q = QT 0
Q = QT 0
v=2FTXx ,
X=XT 0
ATX+ XA+ 2XF FTX+ Q= 0 ,
A+ 2F FTX
ATX+ XA+ 2HTH 0 ,(A + 2F H)TX1+ X1(A +
2F H)< 0
X1 = X
T1 > 0 X = X
T 0
FTX=H ,
-
7/24/2019 Balandin Kogan LMI Toolbox
18/135
-
7/24/2019 Balandin Kogan LMI Toolbox
19/135
+ PTTQ + QTP < 0
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
20/135
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
21/135
= (ij) i, j = 1, 2, 3, 4
WP WQ
WP =
I 0
0 I0 00 0
, WQ= 0 0
I 00 I0 0
,
11 12T12 22
-
7/24/2019 Balandin Kogan LMI Toolbox
22/135
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
23/135
= T Rnn
P Rln
Q Rkn
P =rP < n Q= rQ< n
WTPWP 0 ,
S=1I QR[ PTR (PRPTR )1PR]QTR ,
+
K=QTLPL
+ PTR KTQR+ Q
TRKPR0
+ PTR KTQR+ Q
TRKPR+
1PTR KTKPR(1KPR+ QR)(1KPR+ QR)
T .
[1
K+ QRPTR (PRP
TR )
1
](PRPTR )[
1
K+ QRPTR (PRP
TR )
1
]T
0
>0
1I > QR(1 + PTR PR)
1QTR .
1 + PTR PR QTRQR=PTR PR > 0 ,
> 0
L= S1/2
L(PRPT
R )1/2
,
L 0
QTRQR > 0 L
L < 1
Z
-
7/24/2019 Balandin Kogan LMI Toolbox
25/135
i+ PTi
TQi+ QTiPi< 0 , i= 1, . . . , N ,
i (nini) Pi Qi
(l ni) (k ni)
(k l)
WTPiiWPi
-
7/24/2019 Balandin Kogan LMI Toolbox
26/135
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
27/135
i iPT
P 0 = maxi i
i PTP
-
7/24/2019 Balandin Kogan LMI Toolbox
28/135
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
29/135
f(x)
F ={x : F(x) > 0}
fr(x) =rf(x) + (x) ,
r >0
(x) =
log det F1(x), x F, x F
.
n
xn fr(x) r =rn
rn xn x
F(x) < 0
t
F(x) tI 0
tmin 0
tmin 0
,
t
-
7/24/2019 Balandin Kogan LMI Toolbox
30/135
,
x
c
cTx
cTx 0 , y11y22 y212>0 ,
Y
Z
Y =
90, 9732 30, 3244
90, 9732
, Z= (181, 9464 15, 1622) ,
= (2, 3125 0, 9375) .
u= 2, 3125 0, 9375 .
WBT
BTWBT = 0 .
WBT =
10
.
(1 0) 2y12 y11+ y22
2y12
10
= 2y12 < 0 .
-
7/24/2019 Balandin Kogan LMI Toolbox
36/135
Y >0
y12
-
7/24/2019 Balandin Kogan LMI Toolbox
37/135
N
x= Aix+ Biu, i= 1, . . . , N .
u = x
N
N
Vi(x) = xTXix Xi = XTi > 0
(Ai+ Bi)TXi+ Xi(Ai+ Bi)< 0, i= 1, . . . , N .
N
Xi=X, i= 1, . . . , N
X1
Y =X1
Z= Y
Y ATi + AiY + ZTBTi + BiZ 0
Z
=ZY1
i+ QTiPi+ P
Ti
TQi < 0, i= 1, . . . , N .
Bi=B, i= 1, . . . , N
i+ QT + TQ
-
7/24/2019 Balandin Kogan LMI Toolbox
38/135
i = A
TiX+XAi, Q = B
TX
WTQiWQ < 0, i =
1, . . . , N
WQ
Q
WQ = X1WBT
WTBT(Y ATi + AiY)WBT 0
Y ATi +AiY+ Y
TBT+BY 0
V
-
7/24/2019 Balandin Kogan LMI Toolbox
39/135
A0=
A 0nxk
0knx 0kk
,
B0 =
0nxk B
Ik 0knu
, C0=
0knx Ik
C 0nyk
,
k
k= 0
x= A0x+ B0u ,y=C0x ,
A0 B0 C0 u = y
k
AT0 X+ XA0+ CT0
TBT0X+ XB0C0 < 0
X1
Y =X1
Y AT0 + A0Y + Y CT0
TBT0 + B0C0Y 0 .
Z= C0Y
Y AT0 + A0Y + ZTBT0 + B0Z 0
Y
Z
(Y, Z)
(nu + k)(nx + k) (nu + k)(ny+ k)
(nx > ny)
(Y, Z)
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
40/135
WBT
0X WC0 B
T0X C0
WBT
0X = X
1WBT0
WBT
0
BT0
k nx (nx+ k)
(nx+ k) X=XT
WTC0(AT0 X+ XA0)WC0 0 ,
WTBT0
(X1AT0 + A0X1)WBT
0
-
7/24/2019 Balandin Kogan LMI Toolbox
41/135
X
33
WBT0 =
1
00
, WC0 =
0
10
.
X
X=
0.7549 0.0001 0.6737 1.5655 1.0186
4.1051
, Y =
1.6050 0.2044 0.3141 0.7878 0.2290 0.3520
,
=
0.1819 0.1197
0.5487 1.5601
.
xr = 0.1819xr+ 0.1197y ,u= 0.5487xr 1.5601y ,
Ac =
0 1 0
0.5601 0 0.54870.1197 0 0.1819
1,2= 0.0582 0.7410i , 3= 0.0655 ,
x1=x3 ,x2=x4 ,x3= 2x1 x2+ u ,x4= 2x1+ 2x2 ,y1 = x1 ,y2 = x2 .
-
7/24/2019 Balandin Kogan LMI Toolbox
42/135
Ar, Br = (B
(1)r B
(2)r ), Cr, Dr = (D
(1)r D
(2)r )
A0=
0 0 1 0 00 0 0 1 0
2 1 0 0 02 2 0 0 00 0 0 0 0
, B0=
0 00 0
0 1
0 0
1 0
, C0=
0 0 0 0 1
1 0 0 0 0
0 1 0 0 0
,
X
5 5
WBT0
=
1 0 0
0 1 0
0 0 0
0 0 1
0 0 0
, WC0 =
0 0
0 0
1 0
0 1
0 0
.
X
X=
3.4330
5.3960
0.0027
0.7563 2.1220
13.1102 0.8556 1.5000 8.2465 0.4706 0.9072 1.0650 3.2732 3.0418
7.0358
,
Y =
3.0358 2.7483 0.1135 0.3296 2.4310 3.2966 0.4087 1.2210 3.5011 4.9399 1.4611 0.3288 1.8501 1.9104 4.2886
,
=
8.4770 3.8599 11.3614
22.7817 21.4799 40.8038
.
xr = 8.4770xr 3.8599y1+ 11.3614y2 ,u= 22.7817xr 21.4799y1+ 40.8038y2 ,
-
7/24/2019 Balandin Kogan LMI Toolbox
43/135
1= 7.0975, 2,3= 0.2964 2.6608i, 4= 0.7098, 5= 0.0770,
B0 =
0nxk B
Ik 0knu
, C0 =
0knx Ik
C 0nyk
WBT
0
WC0 B
T0WBT
0
=0
C0WC0 = 0
WBT0
= WBT
0
, WC0 = WC0
.
A0=
A 0nxk
0knx 0kk
X
Y
X= X11 X12
XT12 X22
, Y = Y11 Y12
YT12 Y22
.
WTC(ATX11+ X11A)WC0
X11 > 0
Y11>0
X11 Y11
X
Y
XY =I
X11Y11+ X12YT12 =I ,
X11Y12+ X12Y22 = 0 ,
XT12Y11+ X22YT12 = 0 ,
XT12Y12+ X22YT22 =I .
I X11Y11=X12YT12 .
-
7/24/2019 Balandin Kogan LMI Toolbox
44/135
k
k
(I
X11Y11)
k .
X11 Y111 =X12X122 XT12 .
X11 Y111 0 ,
X11 II Y11
0 .
X11 > 0 Y11 > 0
X >0 Y >0
X11 Y11
X11 Y11
X
X11Y111
X11 Y111 = (U1U2) 0
0 0 UT1
UT2
,
U1 Rnr U2 Rn(nr) = (1, , r) > 0
S
0r(kr)
0(kr)r Ikr
ST , S= (U1U2)
Ir 0r(kr)
0(nr)r 0(nr)(kr)
X12 = S , X22= (11 ,
, 1r , 1,
, 1) .
X
(nx nx)
X11 = X
T11 > 0 Y11 = Y
T11 > 0
k = nx
-
7/24/2019 Balandin Kogan LMI Toolbox
45/135
k=nx (nx nx)
X11 = X
T11 > 0 Y11 = Y
T11 > 0
WTC(ATX11+ X11A)WC0
X=
X11 V
V V
, V =X11 Y111 .
x1=x3 ,
x2=x4 ,x3= 2x1 x2+ u ,x4= 2x1+ 2x2 ,y = x1 .
A(i,j)r , i, j= 1, , 4 B(i)r , i=
1, , 4
C(j)r , j = 1, , 4 Dr
A0
B0
C0
X
8 8
WBT =
1 0 0
0 1 0
0 0 0
0 0 1
, WC=
0 0 0
1 0 0
0 1 0
0 0 1
,
X11 Y11 4 4 X12 X22
-
7/24/2019 Balandin Kogan LMI Toolbox
46/135
X11=
1871.7 165.9 329.8 51.7 544.1 44.2 340.5 1434.8 522.1
445.7
,
X12=
0.8125 0.5733 0.0660 0.08190.1152 0.0103 0.8299 0.54580.5323 0.7412 0.2679 0.30900.2076 0.3491 0.4849 0.7745
,
X22=
0.0005 0 0 0
0 0.0007 0 0
0 0 0.0013 0
0 0 0 0.0917
,
= 104
6.0558 6.3583 1.1637 58.602 16132.05.6231 5.9042 1.0805 54.416 14979.01.0674 1.1207 0.2052 10.329 2843.00.0180 0.0189 0.0035 0.175 48.00.0054 0.0056 0.0010 0.052 14.0
.
1= 1813.1 , 2 = 5.4 , 3,4= 0.4 0.9i ,5,6= 0.3 0.6i , 7= 1.1, 8= 1.2 ,
=
,Q,P, Xmin
.
-
7/24/2019 Balandin Kogan LMI Toolbox
47/135
=
2.2561 4.4298 6.7194 1091.4 691.673.2879
3.3957
7.2738 982.0
686.88
0.4228 0.3313 0.1676 185.2 360.800.1479 0.1586 0.0361 4.5 381.110.0023 0.0029 0.0066 0.9 2.16
,
1 = 1.5847 , 2= 1.1216 , 3,4 = 0.7534 2.3388i ,5,6 = 0.5314 1.9128 , 7= 0.2723, 8= 0.2206 .
Ar = 0 Br = 0 Cr = 0 =Dr
(nxnx)
X = XT > 0
Y = YT > 0
WTC(ATX+ XA)WC0 ,
WTBT(Y A
T
+ AY)WBT 0 .
x= Ax+ Bu ,y= C x ,
x Rnx u Rnu y Rny
xr =Axr+ Bu+ L(Cxr y) ,u= K xr ,
xr Rnx K L
e= xxr
(x, e)
xe
= A + BK BK0 A + LC
xe
.
-
7/24/2019 Balandin Kogan LMI Toolbox
48/135
A+BK
A+ LC
(A, B)
(AT, CT)
Y1AT + AY1+ Z
T1B
T + BZ1 < 0 ,
Y2A + ATY2+ Z
T2C+ C
TZ2 0 Z1 Y2 = Y
T2 > 0 Z2
K=Z1Y11 , L= Y
12 Z
T2 .
C
ny < nx nz = nx ny
z=F z+ T Bu+ Qy ,
z
Rnz
y
u
F T Q
T A F T =QC .
e= z T x
e= F e .
F
z
T x
y
C=
Iny 0nynz
.
A
B
A=
A11 A12
A21 A22
, B=
B1
B2
,
A11 (ny ny) B1 (ny nu)
F T
Q
F =A22+ LA12, T = (L Inz), Q= A21+ LA11 (A22+ LA12)L ,
-
7/24/2019 Balandin Kogan LMI Toolbox
49/135
L
F
(AT22, A
T12)
xr = (A22+ LA12)xr+ (B2+ LB1)u+ [A21+ LA11 (A22+ LA12)L]y ,u= K1xr+ K2y ,
K1 K2
xr =T x+ e
x= (A + BK)x BK1e ,e= F e ,
K= (K2+ K1L K1) .
K
A+BK
L
K1 K2
Y1AT + AY1+ Z
T1B
T + BZ1 < 0 ,
Y2A22+ AT22Y2+ Z
T2A12+ A
T12Z2 0 Z1 Y2 = Y
T2 > 0 Z2
K= (K2+ K1L K1) =Z1Y11 , L= Y
12 Z
T2 .
x1=x3 ,x2=x4 ,x3= 2x1 x2+ u ,x4= 2x1+ 2x2 ,y = x1 .
C= (1 0 0 0) .
-
7/24/2019 Balandin Kogan LMI Toolbox
50/135
A=
0 | 0 1 0 0 | 0 0 12 | 1 0 0
2 | 2 0 0
, B=
0
0
1
0
.
Y1=
102.195 71.981 27.278 3.674 73.319 2.119 24.064
203.296 25.975 41.835
,
Z1=335.704 96.619 5.270 75.512
,
Y2=
46.007 6.289 27.506 339.231 81.770
44.784
,
Z2 =175.690 61.945 6.288
,
K=18.180 20.162 4.044 10.707
,
LT =
10.372 2.786 11.598
.
xr =
0 10.372 1
1 2.786 02 11.598 0
xr+
0
1
0
u+
17.298
4.610
9.567
y ,
u= (20.162 4.044 10.707)xr 362.730y ,
1= 0.903 , 2,3= 0.941 2.405i ,4,5=
1.457
2.862i , 6,7 =
0.564
0.510i ,
-
7/24/2019 Balandin Kogan LMI Toolbox
51/135
xt+1
=Axt+ Bu
t ,
yt =C xt ,
xt Rnx ut Rnu yt Rny
k
x(r)t+1 = Arx
(r)t + Bryt ,
ut = Crx(r)t + Dryt ,
x(r)t Rk
xt+1=Acxt , Ac=
A + BDrC BCr
BrC Ar
,
xt = (xt, x
(r)t )
Vt(xt) = x
TtXxt X
T = X > 0
Vt+1 Vt= xTt+1Xxt+1 xTtXxt = xTt(ATcXAc X)xt < 0 .
ATcXAc X
-
7/24/2019 Balandin Kogan LMI Toolbox
52/135
=
X1 A0AT0 X
, P = (0 C0) , Q=
BT0 0
.
WP =
0 I
WC0 0
, WQ=
WBT0 0
0 I
,
(nx +k)
(nx+ k) X=XT >0
0 I
WC0 0
T X1 A0
AT0 X
0 I
WC0 0
-
7/24/2019 Balandin Kogan LMI Toolbox
53/135
A0 B0 C0
X
Y
X= X11 X12
XT12 X22 , Y =
Y11 Y12
YT12 Y22
WTC(ATX11A X11)WC 0
X11 II Y11
0 ,
(I X11Y11) k ,
Ar =
0
Br = 0 Cr = 0 = Dr
= X
1 A
AT X , P= (0 C) , Q= BT 0 .
WTC(ATXA X)WC
-
7/24/2019 Balandin Kogan LMI Toolbox
54/135
X1 >0
WTBT(AX1AT X1)WBT
-
7/24/2019 Balandin Kogan LMI Toolbox
55/135
LMI
D
x = Ax
D
A
D
A
D
D
D
A
X
AX+ XAT 0 .
(m m)
f(z) =+ z+ zT ,
= T Rmm
Rmm
D= {z C : f(z)< 0}
-
7/24/2019 Balandin Kogan LMI Toolbox
56/135
LMI
f(z)
LMI
x =
(z)
y =
(z)
LMI
z D f(z) =f(z)< 0 ,
LMI
LMI
f(z)
(m m)
M(A, X) = X+ (AX) + T (AX)T ,
Mij(A, X) =ijX+ ijAX+ jiXAT , i, j= 1, , m .
M(A, X)
f(z)
(X,AX,XAT) (1, z,z) .
D LMI
A
D
X=XT
M(A, X)< 0 , X >0 .
v
A
vA= v
(I v)M(A, X)(I v) = (vXv)f() .
X >0
f()< 0
D
A=
(1, , n) D
M(A, X)
A
X=X
M(A, X) = X+ (AX) + T (AX) .
M(A, I) =UT
(f(1), , f(n))U ,
U
f(i) < 0, i = 1,
, n
M(A, X) < 0 X = I A
-
7/24/2019 Balandin Kogan LMI Toolbox
57/135
LMI
D1 ={z : z
-
7/24/2019 Balandin Kogan LMI Toolbox
58/135
LMI
LMI
LMI
D1
D2 LMI D1 D2 f1 f2
f= (f1, f2)
M(A, X) =
(M1(A, X), M2(A, X)) .
A
LMI
D1 D2
X
A
D1 D2
X=XT >0
M1(A, X)< 0 M2(A, X)< 0
LMI
x= Ax+ Bu ,
x Rnx
u Rnu
u= x ,
D
LMI
X = XT > 0
M(A+B, X)< 0
Z= X
M(A, X) + BZ+ T ZTBT
-
7/24/2019 Balandin Kogan LMI Toolbox
59/135
LMI
D2
0.1X X+ AX+ BZX+ XAT + ZTBT 0.1X 0 ,
A=
0 0 1 00 0 0 12 1 0 0
2 2 0 0
, B=
0010
.
= (10.0067, 9.5078, 4.0023, 6.0056) ,
1,2 = 0.9890 0.0111i , 3 = 1.0104 , 4= 1.0140 .
-
7/24/2019 Balandin Kogan LMI Toolbox
60/135
-
7/24/2019 Balandin Kogan LMI Toolbox
61/135
H2
H2
2
-
7/24/2019 Balandin Kogan LMI Toolbox
62/135
H2
RL2 H(s)
[HT(j)H(j)] d < .
H2
H2= { 12
[HT(j)H(j)] d}1/2 .
H(s) =C(sI
A)1B .
x= Ax+ Bv , x(0) = 0z=C x ,
x Rnx
v Rnv
z Rnz
H2
v(i)(t) =(t) ei
(t)
ei
i
(nv nv)
i =1, 2, . . . , nv z(i)
Z(i)(j)
Z(i)(j) =H(j)ei
H22 = 1
2
[HT(j)H(j)] d=
= 1
2
nvi=1
eTiHT(j)H(j)ei d = 1
2
nvi=1
Z(i)T(j)Z(i)(j) d =
= nvi=1
0
z(i)Tz(i) dt .
-
7/24/2019 Balandin Kogan LMI Toolbox
63/135
H2
H2
z(i)(t) =CeAtBei, t > 0 (K
TSK) =
(SK KT)
K
S
H22=nvi=1
0
eTiBTeA
TtCTCeAtBei dt=
=nvi=1
0
(eATtCTCeAtBeie
TiB
T) dt=
=
0
(eA
TtCTCeAtnvi=1
BeieTiB
T) dt=
=
0
(eAT
tCTCeAtBBT) dt=
= (BT
0
eATtCTCeAt dtB) =
(C
0
eAtBBTeATt dtCT) .
A
P =Po=
0
eATt
CT
CeAt
dt
ATP+ P A + CTC= 0 ,
P=Pr =
0
eAtBBTeATt dt
AP+ P AT + BBT = 0 .
H2
H22 = (BTPoB) ,
Po
H22 = (CPrCT) ,
Pr
v
E[v(t)vT(t + )] =()I
J= limt
E|z(t)|2 ,
-
7/24/2019 Balandin Kogan LMI Toolbox
64/135
E
J= limt
E (zzT) =
= limt
E[
t0
(t, )v() dt
0
vT()T(t, ) d] =
= limt
[
t0
(t, )
t0
E[v()vT()]T(t, ) d d=
= limt
[
t0
(t, )T(t, ) d],
(, ) =C eA()B
= t
J= limt
[C
t0
eABBTeAT dCT] =
[C
0
eABBTeAT dCT] = H22 .
H2
H2
A
H2<
X=XT >0
S=ST
ATX+ XA XBBTX I
0 ,
(S)< ;
Y =YT >0
R= RT
AY + Y AT Y CT
CY I
0 ,
(R)< .
(BTPoB)0
ATX+ XA+ CTC
-
7/24/2019 Balandin Kogan LMI Toolbox
65/135
X > Po (B
TPoB) (BTXB)0
(A, B)
(A, C)
u=
(DTD)1BTP x ,
P =PT 0
ATP+ P A P B(DTD)1BTP+ CTC= 0 ,
min J = xT0 P x0
2
P
x= Ax+ x0(t) + Bu , x(0) = 0 ,z=C x+ Du ,
-
7/24/2019 Balandin Kogan LMI Toolbox
66/135
(t)
x= (A + B)x+ x0(t) , x(0) = 0 ,z= (C+ D)x .
H2 J
H2
(C+ D)[sI (A + B)]1x02 < |x0| , x0= 0 .
Y =YT >0
R= RT
(A + B)Y + Y(A + B)T Y(C+ D)T
(C+ D)Y |x0|I 0 ,
R < |x0| .
R
Y |x0|Y R |x0|R
(A + B)Y + Y(A + B)T Y(C+ D)T(C+ D)Y I
0 , R < .
Z = Y
xT0 Y1x0
|x0|2 < , x0= 0 ,
Y II I
>0 ,
= ZY1
Y = YT > 0
Z
AY + Y AT + BZ+ ZTBT Y CT + ZTDT
CY + DZ I
0 .
-
7/24/2019 Balandin Kogan LMI Toolbox
67/135
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
68/135
x1 = x2 ,
x2 = 20x1 x2+ u ,
z1=x1 ,z2=u ,
A=
0 1
20
, B =
0
1
,
C= 1 0
0 0 , D=
0
1 .
0= 10 = 0.1
Y =
0.4914 0.0593
0.0593 49.1460
, Z= (0 2.0354) ,
= (0.005 0.0414)
= 2.0354
P =
4.1428 0.0050
0.0050 0.0414
, = (DTD)1BTP= (0.005 0.0414).
x= 1 1
0 0
x+ 0
1
u ,
z= (0 1) x+ u .
u= (0 1) x ,
P = 0 0
0 1
.
-
7/24/2019 Balandin Kogan LMI Toolbox
69/135
u= (4 3) x .
P =
8 4
4 3
.
x= Ax+ B1v+ B2u , x(0) = 0 ,z= C x+ Du ,
v
u= x ,
>0
J = limt
E|z(t)|2 dt < 2 .
x= (A + B2)x+ B1v ,z= (C+ D)x .
H2
J < 2
(A + B2)Y + Y(A + B2)T Y(C+ D)T
(C+ D)Y I
0 ,
R < .
Z= Y
=Z Y1
Y =YT >0
Z
AY + Y AT + B2Z+ ZTBT2 Y CT + ZTDT
CY + DZ I
0 , R <
Y
Z
R
-
7/24/2019 Balandin Kogan LMI Toolbox
70/135
x= Ax+ Bu , x(0) =x0 ,z= C1x+ Du ,y=C2x ,
y Rny
k
xr =Arxr+ Bry , xr(0) = 0 ,u= Crxr+ Dry ,
xr
Rk
J=
0
|z(t)|2 dt < 2|x0|2 , x0= 0 .
xc = Acxc+ x0(t) , xc(0) = 0 ,z= Ccxc ,
xc= (x, xr)
Ac= A + BDrC2 BCr
BrC2 Ar
, x0 = x00
,Cc= (C1+ DDrC2 DCr) .
(t)
z
Hc(s) =Cc(sI Ac)1x0
Hc
2<
|x0
|,
x0
= 0 .
AcY + Y ATc Y CTc
CcY |x0|I
0 ,
R < |x0|
Y = YT > 0
R = RT
R
Y
|x0|Y R
|x0|R AcY + Y ATc Y CTc
CcY I 0 , R < .
-
7/24/2019 Balandin Kogan LMI Toolbox
71/135
Y
I
0
(I 0) I
>0 .
=
Ar Br
Cr Dr
Ac=A0+ BC , Cc=C0+ DC ,
A0=
A 0nxk
0knx 0kk
, B =
0nxk B
Ik 0knu
,
C = 0knx Ik
C2 0nyk
, C0= (C1 0nzk) ,
D = (0nzk D) .
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
72/135
WP =
Y1 0
0 I
WG .
k
(nx+ k) (nx+ k)
X=XT >0 Y =YT >0
WTG
AT0 X+ XA0 CT0
C0 I
WG0 .
Y
X
Y
XY =I
A0 C0 X Y
X=
X11 X12
XT12 X22
, Y =
Y11 Y12
YT12 Y22
.
G= 0knx Ik 0knz
C2 0nyk 0nynz
, Q= 0knx Ik 0knzBT 0nuk D
T
,
WG WQ
WG=
WC2 0
0 0
0 I
, WQ=
W(1)Q
0
W(2)Q
,
W
(1)Q W
(2)Q
BTW(1)Q + DTW(2)Q = 0 .
-
7/24/2019 Balandin Kogan LMI Toolbox
73/135
WC2 0
0 00 I
T
ATX11+ X11A ATX12 C
T1
0 0 I
WC2 0
0 00 I
,
W(1)Q
0
W(2)Q
T
Y11AT + AY11 AY12 Y11C
T1
0 YT12CT1
I
W(1)Q
0
W(2)Q
,
X11 Y11 X
Y
WC2 00 I
T ATX11+ X11A CT1
C1 I
WC2 0
0 I
0
X11 I
I Y11
0 ,
(I X11Y11) k
X > 0
Y > 0
X11 Y11
X11 < I .
Y11 Y12 I
YT12 Y22 0
I 0 I
>0 ,
Y22>0 , Y11 Y12Y122 YT12 I
I I
>0 .
-
7/24/2019 Balandin Kogan LMI Toolbox
74/135
Y > 0
X11 Y11
X111 I
I I
>0 ,
X11 < I
k
(nx nx) X11 = X
T11 > 0 Y11 = Y
T11 > 0
(k= nx)
(nx nx) X11 = XT11 > 0
Y11=Y
T11 >0
WC2 00 I
T ATX11+ X11A CT1
C1 I
WC2 0
0 I
-
7/24/2019 Balandin Kogan LMI Toolbox
75/135
A, B, D
C1 =
1 0
0 0
, C2= (1 0) .
0 = 10 = 0.1
= 2.0354
X11 =
2.0354 0.0011
0.0011 1.5211
, Y11=
0.4914 0.0593
0.0593 49.1470
,
xr = 57.8622 0.07020.0671 0.1424 xr+
0.1201
98.6673 y ,
u= (0.0255 0.0420)xr 0.0050y .
-
7/24/2019 Balandin Kogan LMI Toolbox
76/135
-
7/24/2019 Balandin Kogan LMI Toolbox
77/135
x= Ax+ Bvz= C x+ Dv , x(0) = 0 ,
x Rnx
v Rnv
z Rnz
L2
v(t)
v
= (
0 |
v(t)
|2 dt)1/2 0
zv < , v, v = 0 .
supv=0
zv = sup(,) H(j ) = H ,
H(s) =D + C(sIA)1B
v
z j =
1
H = maxi
i(H) ,
-
7/24/2019 Balandin Kogan LMI Toolbox
78/135
i(H) =1/2i (HH
) i
H
H(s) supRe s0 H(s) <
H<
H(j )HT(j)< 2I , (, ) .
I HT(j )
H(j ) 2I
0 , (, ) , |v| = 0
L(x, v)
x Cnx
v Cnv
(A, B)
V(x) =
xTXx
X = X
V L(x, v)< 0 , x,v, |x| + |v| = 0
2
xX(Ax+ Bv) L(x, v)< 0 , x,v, |x| + |v| = 0 .
L(x, v) = (xT, vT) L x
v
, L=
L11 L12
LT12 L22
,
ATX+ XA L11 XB L12BTX LT12 L22
-
7/24/2019 Balandin Kogan LMI Toolbox
79/135
(j I A)1B
I
T
L11 L12
L
T
12 L22
(j I A)1B
I
>0 .
L(x, v) =2vv (Cx+ Dv)(Cx+ Dv) ,
2I HT(j )H(j )> 0 , (, ) ,
2
xX(Ax+ Bv) + (Cx+ Dv)(Cx+ Dv) 2vv
-
7/24/2019 Balandin Kogan LMI Toolbox
80/135
X
= inf Fc(X,)
-
7/24/2019 Balandin Kogan LMI Toolbox
81/135
H
xc=Acxc+ Bcv ,z= Ccxc+ Dcv ,
Ac=A + B2 , Bc=B1 , Cc= C1+ D12, Dc=D11 .
v
z
Hc(s) =Dc+ Cc(sI Ac)1Bc
Hc< .
ATcX+ XAc XBc CTc
BTcX I DTcCc Dc I
0
X1 0 0
0 I 0
0 0 I
Y(A + B2)
T + (A + B2)Y B1 Y(C1+ D12)T
BT1 I DT11(C1+ D12)Y D11 I
-
7/24/2019 Balandin Kogan LMI Toolbox
82/135
H
x1 = x2 ,
x2 = 20x1 x2+ v+ u ,
z1 = x1 ,z2 = u ,
A=
0 1
20
, B1=B2 =
0
1
,
C1 = 1 0
0 0 , D11 =
0
0 , D12=
0
1 .
0 = 10 =
0.1
= 0.7071 = (0.005, 0.1)
H
u= 0.005x1 0.1x2
=0.7071
H
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
83/135
H
WP =
0 0
I 0
0 I
, WQ=
W(1)Q 0
0 I
W(2)Q 0
,
W(1)Q W
(2)Q
BT2W(1)Q + D
T12W
(2)Q = 0 .
I DT11
D11 I
max(DT11D11)
N2 | 0 0 | I
T
Y AT + AY Y C T1 | B1C1Y I | D11
BT1 DT11 | I
N2 | 0 0 | I
0
Y
H k
xr =Arxr+ Bry ,u= Crxr+ Dry ,
xr Rk k = 0
u= Dry
xc=Acxc+ Bcv ,z= Ccxc+ Dcv ,
xc= (x, xr)
Ac =
A + B2DrC2 B2Cr
BrC2 Ar
, Bc =
B1+ B2DrD21
BrD21
,
Cc= (C1+ D12DrC2 D12Cr) , Dc=D11+ D12DrD21 .
-
7/24/2019 Balandin Kogan LMI Toolbox
84/135
v
z
Hc(s) =Dc+ Cc(sI Ac)1Bc
Hc< .
ATcX+ XAc XBc CTc
BTcX I DTcCc Dc I
0
=
Ar Br
Cr Dr
Ac= A0+ BC , Bc= B0+ BD21 ,Cc=C0+ D12C , Dc= D11+ D12D21 ,
A0= A 0nxk
0knx 0kk
,
B = 0nxk B2
Ik 0knu
, C =
0knx Ik
C2 0nyk
,
B0=
B1
0knv
, C0= (C1 0nzk) ,
D12 = (0nzk D12) ,
D21 =
0knv
D21
.
+ PTTQ + QTP
-
7/24/2019 Balandin Kogan LMI Toolbox
85/135
H
WTP
AT0 X+ XA0 XB0 CT0
BT0X I DT11C0 D11 I
WP
-
7/24/2019 Balandin Kogan LMI Toolbox
86/135
X
Y
WTP
AT0 X+ XA0 XB0 C
T0
BT0X
I DT11
C0 D11 I
WP
-
7/24/2019 Balandin Kogan LMI Toolbox
87/135
H
= 0.708
A0 B0 C0
X Y
X=
X11 X12
XT12 X22
, Y =
Y11 Y12
YT12 Y22
.
P = 0knx Ik 0knv 0knz
C2 0nyk D21 0nynz ,
R=
0knx Ik 0knv 0knz
BT2 0nuk 0nunv DT12
,
WP WR
WP =
W(1)P 0
0 0
W(2)P 0
0 I
, WR=
W(1)R 0
0 0
0 I
W(2)R 0
,
W
(1)P W
(2)P W
(1)R W
(2)R
C2W(1)P + D21W
(2)P = 0 , B
T2W
(1)R + D
T12W
(2)R = 0 .
W(1)
P 00 0
W(2)P 0
0 I
T
AT
X11+ X11A AT
X12 X11B1 CT1
0 XT12B1 0
I DT11 I
W(1)
P 00 0
W(2)P 0
0 I
,
W(1)R 0
0 0
0 IW(2)R 0
T
Y11AT + AY11 AY12 B1 Y11C
T1
0 0 YT12CT1
I DT
11
I
W(1)R 0
0 0
0 IW(2)R 0
,
-
7/24/2019 Balandin Kogan LMI Toolbox
88/135
X11 Y11 X Y
N1 | 0 0 | I
T
ATX11+ X11A X11B1
| CT1
BT1X11 I | DT11 C1 D11 | I
N1 | 0 0 | I
0
X11 I
I Y11
0 ,
(I
X11Y11)
k
X >0
Y >0
X11 Y11 H
k
(nxnx) X11 = XT11 > 0 Y11 = YT11 > 0
D11 = 0
v
>0
H (k = nx)
H
-
7/24/2019 Balandin Kogan LMI Toolbox
89/135
x(1)r = 0.0592x(1)r 0.8358x(2)r 24.76y ,
x(2)r
= 1.0667x(1)r
0.8604x(2)r
+ 20.24y ,
u= 0.2586x(1)r + 0.1836x(2)r 0.0389y
= 0.7072
xt+1= Axt+ Bvt ,zt=C xt+ Dvt ,
xt Rnx vt Rnv zt Rnz A
l2
vt
v = (
t=0|vt|2)1/2 < .
= supv=0
zv .
= inf{ :zv < , v,v = 0} .
>0
zv < , v, v = 0 .
supv=0
zv = sup[0,2) H(e
j ) = H ,
H(q) = D+C(qI A)1B
v
z
j=
1
H = maxi
i(H) ,
-
7/24/2019 Balandin Kogan LMI Toolbox
90/135
i i H H(q)
sup|q|1 H(q) <
H
<
HT(ej )H(ej)< 2I , [0, 2) .
H(ej ), [0, 2)
Vt+1 Vt L(xt, vt)< 0 , xt, vt, |xt| + |vt| = 0
L(x, v)
x Cnx
v Cnv
Vt =
V(xt) =xTtX xt X
X=X
(Axt+ Bvt)
X(Axt+ Bvt) xT
tX xt L(xt, vt)< 0 , xt, vt, |xt| + |vt| = 0
L[(ej I A)1Bvt, vt]> 0 , [0, 2) , |vt| = 0 .
L(x, v) = (xT, vT) L x
v
, L=
L11 L12
LT12 L22
,
ATXA X L11 ATXB L12
BTXA LT12 BTXB L22
0 .
L(x, v) =2vv (Cx+ Dv)(Cx+ Dv)
-
7/24/2019 Balandin Kogan LMI Toolbox
91/135
2I HT(ej )H(ej )> 0 , [0, 2) .
(Axt+ Bvt)X(Axt+ Bvt) xTtX xt+ (Cxt+ Dvt)(Cxt+ Dvt) 2vt vt < 0 .
X
X
1
xt (ATXA X+ 1CTC)xt+ 2 xt (ATXB+ 1CTD)vt+
+vt (1DTD I+ BTXB)vt < 0 .
xt vt
ATXA X+ 1CTC ATXB+ 1CTD(ATXB+ 1CTD)T I+ BTXB+ 1DTD
-
7/24/2019 Balandin Kogan LMI Toolbox
92/135
H
xt+1=Axt+ B1vt+ B2ut ,
zt =C1xt+ D11vt+ D12ut ,yt =C2xt+ D21vt ,
xt Rnx vt Rnv ut Rnu zt Rnz
yt Rny
k
x(r)t+1 = Arx
(r)t + Bryt ,
ut = Crx(r)t + Dryt ,
x(r)t Rk
zv < , v, v = 0
x(c)t+1=Acx(c)t + Bcvt ,
zt=Ccx(c)t + Dcvt ,
Ac=
A + B2DrC2 B2Cr
BrC2 Ar
, Bc =
B1+ B2DrD21
BrD21
,
Cc= (C1+ D12DrC2 D12Cr) , Dc=D11+ D12DrD21 .
=
Ar BrCr Dr
Ac= A0+ BC , Bc= B0+ BD21 ,Cc=C0+ D12C , Dc= D11+ D12D21 ,
A0=
A 0nxk
0knx 0kk
,
B = 0nxk B2
Ik 0knu
, C =
0knx Ik
C2 0nyk
,
B0=
B1
0knv
, C0= (C1 0nzk) ,
D12 = (0nzk D12) , D21 = 0knv
D21
.
-
7/24/2019 Balandin Kogan LMI Toolbox
93/135
H
vt zt
Hc(q) =Dc+ Cc(qI Ac)1Bc
Hc< .
ATcXAc X ATcXBc CTcBTcX Ac I+ BTcX Bc DTc
Cc Dc I
0
H
X1 Ac Bc 0ATc X 0 CTcBTc 0 I DTc
0 Cc Dc I
0 ,
X 0 CTc0 I DTc
Cc Dc I
+
ATc
BTc
0
X(Ac Bc 0)< 0 .
Ac
ATcXAc X ATcXBc CTcBTcX Ac I+ BTcX Bc DTc
Cc Dc I
-
7/24/2019 Balandin Kogan LMI Toolbox
94/135
+ PTTQ + QTP 0
WTP
X1 A0 B0 0AT0 X 0 CT0BT0 0 I DT11
0 C0 D11 I
WP
-
7/24/2019 Balandin Kogan LMI Toolbox
95/135
H
X
Y
WTP
Y A0 B0 0
A
T
0 X 0 CT
0
BT0 0 I DT110 C0 D11 I
WP
-
7/24/2019 Balandin Kogan LMI Toolbox
96/135
WTP
Y11 Y12 A 0 B1 0
Y22 0 0 0 0
X11 X12 0 CT1 X22 0 0 I DT11 I
WP
-
7/24/2019 Balandin Kogan LMI Toolbox
97/135
H
WP =
N1 | 0 |
0 | I
,
N1= (W
(1)P , W
(2)P ) (C2 D21)
WTQ
AY11AT Y11 AY11CT1 | B1 I+ C1Y11CT1 | D11 | | I
WQ 0 Y11 = Y
T11 > 0
X11 I
I Y11
0 ,
(I X11Y11) k ,
-
7/24/2019 Balandin Kogan LMI Toolbox
98/135
-
7/24/2019 Balandin Kogan LMI Toolbox
99/135
x1 = x2 ,x2 = x1+ u ,y=x1
u= y
X = XT > 0
Y = YT > 0
WTC(ATX+ XA)WC
-
7/24/2019 Balandin Kogan LMI Toolbox
100/135
WBT =
1
0
, WC=
0
1
,
x120, Li(X, Y)< 0, i= 1, 2, 3} ,
L3(X, Y) =
X II
Y
.
L3(X, Y)< 0
X > Y1
A1 min = 0
X
Y
A
A1
X Y1 < I
A1
A1
-
7/24/2019 Balandin Kogan LMI Toolbox
101/135
min= min{: (X, Y , G1, G2)< I, X >0,Y >0 , Li(X, Y)< 0, i= 1, 2, 3
},
(X, Y , G1, G2) = (I G1)
X I
I Y
IG1
+
+(G2 I)
X I
I Y
G2
I
,
Gi=GTi, i= 1, 2
A2
A1
(X, Y , G1, G2)< I
(X, Y , G1, G2) = (G1+ Y1)Y(G1+ Y1) + (G2+ X1)X(G2+ X1)+
+(X Y1) + (Y X1) 0
L3(X, Y)< 0 X > Y
1
min= 0
X Y
A
G1= Y1 G2 = X1
j= 0
G1=G
(j)1 G2 = G
(j)2
A2 j+1, Xj, Yj
G
(j+1)1 =Y1j G(j+1)2 =X1j
j=j + 1
G(0)1 G(0)2 j
limj
j = 0 , limj
Xj =X , limj
Yj =Y .
(X, Y , G1, G2)
= ((Xj+1, Yj+1, G(j+1)1 , G
(j+1)2 )) ((Xj , Yj, G(j)1 , G(j)2 ))
= 1+ 2 =
[((Xj+1, Yj+1, G(j+1)1 , G
(j+1)2 )) ((Xj , Yj, G(j+1)1 , G(j+1)2 ))]+
+[((Xj, Yj, G(j+1)1 , G
(j+1)2 )) ((Xj, Yj, G(j)1 , G(j)2 ))] .
(j + 1)
X = Xj+1 Y = Yj+1
-
7/24/2019 Balandin Kogan LMI Toolbox
102/135
(Xj, Yj, G(j+1)1 , G
(j+1)2 ) (Xj, Yj, G(j)1 , G(j)2 ) =
= (G(j+1)1 + Y1
j )Yj(G(j+1)1 + Y
1j ) + (G
(j+1)2 + X
1j )Xj(G
(j+1)2 + X
1j )
(G(j)1 + Y1j )Yj(G(j)1 + Y1j ) (G(j)2 + X1j )Xj(G(j)2 + X1j ) .
G(j+1)1 = Y1j G(j+1)2 = X1j
(Xj, Yj, G(j+1)1 , G
(j+1)2 ) (Xj, Yj, G(j)1 , G(j)2 ) =
(Y1j Y1j1)Yj(Y1j Y1j1) (X1j X1j1)Xj(X1j X1j1) 0 .
A B 0
(A) (B)
0
j
= 0 XY = I
X Y A > 0
A
G
(0)1 G
(0)2
j < |j+1 j| <
= u
y = +
A=
0 11 0
, B=
01
, C= (1 1) .
Li(X, Y) < 0, i = 1, 2 G(0)1 G(0)2
G(0)1 =
0.8709 0.17950.1795 0.7873
, G
(0)2 =
0.8842 0.62630.6263 0.9803
.
104
= 103
= 8 106
X= 0.8425 0.0028
0.0028 0.6975 , Y =
1.1869 0.00470.0047 1.4337
= 2.2263
-
7/24/2019 Balandin Kogan LMI Toolbox
103/135
= u
y =
A=
0 11 0
, B=
01
, C= (1 0) .
xr R1 Li(X, Y)0 ;
X11>0 , X22 XT12X111X12 > 0 ;X22>0 , X11 X12X122XT12 > 0 .
X=
X11 X12
XT12 X22
,
X11 X22
X11>0 X 0
X22 XT12X111X12 0 .
X22>0 X 0 X11 X12X122XT12 0 .
X=
X11 X12 X13
XT12 X22 X23
XT13 XT23 X33
.
X22>0 X >0
X11 X13XT13 X33
X12
XT23
X122 (XT12 X23)> 0 .
X
x =
(x1, x2, x3)
xTXx= (x2+ X122 X
T12x1+ X
122X23x3)
TX22(x2+ X122 X
T12x1+ X
122X23x3)+
+xT1(X11 X12X122XT12)x1+ xT3(X33 XT23X122X23)x3++2xT1(X13 X12X122X23)x3 .
X22 >0 X >0
X11 X12X122XT12 X13 X12X122X23
XT
13 XT
23X1
22XT
12 X33 XT
23X1
22X23
>0 ,
-
7/24/2019 Balandin Kogan LMI Toolbox
109/135
A
n n
R
r r
X
Y
n r r n
A + XRY
(A + XRY)1 =A1
A1X(R1 + Y A1X)1Y A1 .
A
(m n)
r
A= (U1 U2)
0
0 0
V1
V2
=U1V1 ,
=
(1, , r) , 1 r >0 ,
i = 1/2i (AA
)
A U = (U1 U2) V = (V1 V2)
UU=I
VV =I
R(A) =
(U1) , N(A) = (U2) ,R(A) =
(V1) , N(A) = (V2) ,
R()
N()
()
X
11= XT
11> 0
Y
11= YT
11> 0
(n
n)
X12 X22 = X
T22 (n k) (k k)
X=
X11 X12
XT12 X22
>0 ,
X11 X12
XT12 X22
1
=
Y11 Y12
YT12 Y22
Y12 Y22
X11 I
I Y11
0 , (I
X11Y11)
k .
X11Y11+ X12YT12 =I ,
X11Y12+ X12Y22 = 0 ,
XT12Y11+ X22YT12 = 0 ,
XT12Y12+ X22YT22 =I .
I X11Y11=X12YT12 .
-
7/24/2019 Balandin Kogan LMI Toolbox
110/135
k
k
(I
X11Y11)
k .
Y11 = X111 (I X12YT12)
XT12X111 + (X22 XT12X111 X12)YT12 = 0 .
Y12
Y12XT12X
111 + Y12(X22
XT12X
111X12)Y
T12 = 0 .
Y12XT12=I Y11X11 .
(I Y11X11)X111 + Y12(X22 XT12X111X12)YT12 = 0 .
X111 Y11 = Y12(X22 XT12X111 X12)YT12 .
X >0
X22 XT12X111 X12>0 .
X111 Y11 0 ,
X11 I
I Y11
0 .
X11 Y11
(I X11Y11) =r k .
X11
Y12 Y22
Y111 =X11 X12X122 XT12 ,
X11 Y111 =X12X122 XT12 .
-
7/24/2019 Balandin Kogan LMI Toolbox
111/135
X12 n k X22 = XT22 > 0 k k
X11
Y111 =
(I
X11Y11)Y
111
r
(X11 Y111 ) =r .
X11 Y111
X11 Y111 = (U1U2) 0
0 0
UT1
UT2
,
U1 Rnr
U2 Rn(nr)
=
(1, , r)> 0
S
0r(kr)
0(kr)r Ikr
ST , S= (U1U2)
Ir 0r(kr)
0(nr)r 0(nr)(kr)
,
X12 = S , X22= (11 , , 1r , 1, , 1) .
Y11
Y12
Y22
X22 > 0
X11 X12X122 XT12=Y111 >0 ,
X >0
Q= QT Rnn
A Rnm
A max[D1(Q22
QT12Q111Q12)D
T]
(C2 C1)
xT(Q AAT)x = xTQx < 0
x
AT
x= 0
-
7/24/2019 Balandin Kogan LMI Toolbox
113/135
AX=C ,
A
C
(m n)
(m q)
X
(n q)
X0 X0= C=rC
X0 = V C ,
V
rC
C
C
C= (C1 C2) , C2=C1D
D
X= (X1 X2) X1 RnrC
X1 AX1 =C1 C1 X2 = X1D AX2 = C2
X0 = (X1 X2)
AX0 = C C X0 C = X0 = rC
X0 C X0 =V C
V
AXB=C
X
AY =C , ZB=C
Y
Z
-
7/24/2019 Balandin Kogan LMI Toolbox
114/135
X
Y =
XB Z=AX
Y, Z
Y0
rC Y0=V C
C=AY0=AV C=AV ZB
X=V Z
-
7/24/2019 Balandin Kogan LMI Toolbox
115/135
Ax= b ,
A
(mn)
b Rm
x Rn
m= n =
A
N(A) =N(AT) = {0}
x= A1b
A
N(A)
N(AT)
m > n
A= n
A
b
R(A)
AT
AT
A
x= (ATA)1ATb .
b R(A)
x=
min Ax b2 .
b = bR(A)+bN(AT) bR(A) R(A) bN(AT) N(AT)
bN(AT)R(A)
Ax bR(A) bN(AT)2 = Ax bR(A)2 + bN(AT)2 .
min Ax b2 = bN(AT)2 ,
Ax= bR(A) x= (ATA)1ATbR(A)
(ATA)1ATbR(A) = (A
TA)1ATb
x= (ATA)1ATb ,
-
7/24/2019 Balandin Kogan LMI Toolbox
116/135
m < n
A = m
A
x
x+xN(A) x N(A)
x=
min{Ax=b}
x .
Rn
Rn = R(AT) N(A) ,
N(A)
x = xR(AT) +xN(A)
AxR(AT)=b xR(AT) R(AT) Rm xR(AT) = A
T
AAT= b
AAT = (AAT)1b
xR(AT)= AT(AAT)1b
x= AT(AAT)1b .
A = r < min(m, n)
A
A= (U1 U2)
0
0 0
VT1
VT2
=U1V
T1 ,
=
(1, , r) , 1 r >0 ,R(A) =
(U1) , N(AT) = (U2) ,R(AT) =
(V1) , N(A) = (V2) ,
UT1UT2
(U1 U2) =I ,
VT1
VT2
(V1 V2) =I .
Rn
Rm
Rn = R(AT) N(A) , Rm = R(A) N(AT)
x= xR(AT)+ xN(A) , b= bR(A)+ bN(AT) .
Ax b2 = Ax bR(A)2 + bN(AT)2 == AxN(A)+ AxR(AT) bR(A)2 + bN(AT)2 =
= AxR(AT) bR(A)2 + bN(AT)2 ,
Ax= bR(A) ,
-
7/24/2019 Balandin Kogan LMI Toolbox
117/135
x
R(AT)
bR(A) =U11 1 Rr bN(AT) =U22 2 Rmr
b = U11 + U22 U
T1U1 = I U
T1U2 = 0 1 = U
T1b
bR(A) = U1U
T1b x
R(AT)
x = V1
Rr
AV1=U1UT1b .
A= U1V
T1 U
T1 =
1UT1b
x= V11UT1b .
A+ =V11UT1
A
A A+
AA+A= A , A+AA+ =A+ , (AA+)T =AA+ , (A+A)T =A+A .
m= n =
A A+ =A1 ;m > n,
A= n A+ = (ATA)1AT ;
m < n,
A= m
A+ =AT(AAT)1 .
x= A+b .
-
7/24/2019 Balandin Kogan LMI Toolbox
118/135
-
7/24/2019 Balandin Kogan LMI Toolbox
119/135
x= Ax+ Bu , x Rnx
, u Rnu
(A, B)
t0
eABBTeAT d
t >0
(B AB . . . Anx1B) =nx;
(sI A B) =nx , s C;
nx
A + B
A
e
Cnx
eA= e
C
eB= 0
(A, B)
(sI A B) =nx , s 0;
A + B
eA= e
eB= 0
-
7/24/2019 Balandin Kogan LMI Toolbox
120/135
x= Ax ,y = Cx , x Rnx , y Rny
(A, C)
t0
eATCTCeA d
t >0
(CT ATCT . . . (AT)nx1CT) =nx;
(sI AT CT) =nx , s C;
nx A+ C
A
e
Cnx
Ae= e
C
Ce = 0
(A, C)
(sI AT CT) =nx , s 0;
A + C
Ae= e
Ce = 0
-
7/24/2019 Balandin Kogan LMI Toolbox
121/135
ATX+ XA+ CTC= 0 .
A
X=XT 0
(i)
(A, C)
X=XT >0
(A, C)
A
(i)
(ii)
A
X=
0
eATtCTCeAt dt
X
AT(X X) + (X X)A= 0 .
A
X=X
(A, C)
x0= 0 Xx0 = 0
xT0 Xx0 =
0
xT0 eATtCTCeAtx0 dt=
0
yTy dt ,
y
x= Ax ,y= C x
-
7/24/2019 Balandin Kogan LMI Toolbox
122/135
x(0) =x0 y(t) 0
(iii)
X0 = XT0 0
A
Ax0 = x0 x0
= 0
0 0 =x0A
TX0x0+ x0X0Ax0+ x
0C
TCx0= 2( )x0X0x0+ x
0C
TCx0 .
Cx0 = 0
Ax0 =x0 x0= 0 0
(A, C)
AT
XA X+ CT
C= 0 .
A
X=XT 0
(i)
(A, C)
X=XT >0
(A, C)
A
(i)
(ii)
A
X=t=0
(At)TCTCAt
X
AT(X X)A (X X) = 0 ,
A
X=X
(A, C)
x0= 0 Xx0 = 0
xT0 Xx0=t=0
xT0(At)TCTCAtx0 =
t=0
yTt yt ,
yt
xt+1=Axt ,yt=C xt
-
7/24/2019 Balandin Kogan LMI Toolbox
123/135
x0 yt 0
(iii)
X0 = XT0 0
A
Ax0 = x0 x0
= 0
|| 1 0 =x0A
TX0Ax0 x0X0x0+ x0CTCx0= (||2 1)x0X0x0+ x0CTCx0 .
Cx0 = 0
Ax0 = x0 x0= 0 || 1
(A, C)
-
7/24/2019 Balandin Kogan LMI Toolbox
124/135
-
7/24/2019 Balandin Kogan LMI Toolbox
125/135
(m n)
A
(p q)
B
(mp nq)
A B=
a11B a12B a1nBa21B a22B a2nB
am1B am2B amnB
.
AC
BD
(A B)(C D) =AC BD ;
(A B)T =AT BT
A
B
A B= A B , (A B) =A B ;
A
B
A B
(A B)1 =A1 B1 .
-
7/24/2019 Balandin Kogan LMI Toolbox
126/135
-
7/24/2019 Balandin Kogan LMI Toolbox
127/135
S
F(x)< 0 , x = 0
x Rnx
Gi(x) 0 , i= 1, . . . , m ,
F(x)
Gi(x)
S(x) =F(x) 1G1(x) . . . mGm(x)
S(x)< 0 , x = 0
i 0 S
m = 1
x0 G1(x0) < 0
1 > 0
F(x) 1G1(x)< 0 , x = 0 .
S
1 = 1
11 F(x) < 0 11 F(x)< 0
m 2
S
i 0, i = 1, . . . , m
-
7/24/2019 Balandin Kogan LMI Toolbox
128/135
-
7/24/2019 Balandin Kogan LMI Toolbox
129/135
(A, B)
X
2
xX(Ax+ Bv) L(x, v)< 0 , x,v, |x| + |v| = 0
L(x, v)
x, v
v= 0
L[(j I A)1
Bv,v]> 0 , (, ) .
L(x, v) = (xT, vT) L x
v
, L=
L11 L12
LT12 L22
,
ATX+ XA L11 XB L12
BTX LT12 L22
0 .
(A, B)
X
(Ax+ Bv)X(Ax+ Bv) xTXx L(x, v)< 0 , x,v, |x| + |v| = 0
L(x, v)
x, v
v= 0
L[(ej I A)1Bv,v]> 0 , [0, 2) .
-
7/24/2019 Balandin Kogan LMI Toolbox
130/135
L(x, v) = (xT, vT) L x
v
, L=
L11 L12
LT12 L22
,
ATXA X L11 ATXB L12
BTXA LT12 BTXB L22
0 .
-
7/24/2019 Balandin Kogan LMI Toolbox
131/135
H
H
H
-
7/24/2019 Balandin Kogan LMI Toolbox
132/135
H
H
D
S
-
7/24/2019 Balandin Kogan LMI Toolbox
133/135
S
H
H
-
7/24/2019 Balandin Kogan LMI Toolbox
134/135
H
H2 H
H
H
-
7/24/2019 Balandin Kogan LMI Toolbox
135/135