Balandin Kogan LMI Toolbox

7/24/2019 Balandin Kogan LMI Toolbox

1/135


2/135


3/135

+ PTTQ + QTP


4/135

H


5/135

ATX+ XA= Q

Q

ATX+ XA


6/135

V(x) =xTXx

X =XT >0

H

+ PTTQ + QTP


7/135

H


8/135


9/135


10/135


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 ,


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)


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


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


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 .


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


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 ,


18/135


19/135

+ PTTQ + QTP < 0

+ PTTQ + QTP


20/135

+ PTTQ + QTP


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


22/135

+ PTTQ + QTP


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


25/135

i+ PTi

TQi+ QTiPi< 0 , i= 1, . . . , N ,

i (nini) Pi Qi

(l ni) (k ni)

(k l)

WTPiiWPi


26/135

+ PTTQ + QTP


27/135

i iPT

P 0 = maxi i

i PTP


28/135

+ PTTQ + QTP


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


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 .


36/135

Y >0

y12


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


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


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


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


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 .


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 ,


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 .


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


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


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

.


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

.


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 ,


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) .


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 ,


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


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


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


54/135

X1 >0

WTBT(AX1AT X1)WBT


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}


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


57/135

LMI

D1 ={z : z


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


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 .


60/135


61/135

H2

H2

2


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 .


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 ,


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


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 ,


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 .


67/135

+ PTTQ + QTP


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

.


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


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 < .


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


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 .


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 .


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


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 .


76/135


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) ,


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


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


80/135

X

= inf Fc(X,)


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


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


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 .


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


85/135

H

WTP

AT0 X+ XA0 XB0 CT0

BT0X I DT11C0 D11 I

WP


86/135

X

Y

WTP

AT0 X+ XA0 XB0 C

T0

BT0X

I DT11

C0 D11 I

WP


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

,


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) ,


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 .


v

, L=

L11 L12

LT12 L22

,

ATXA X L11 ATXB L12

BTXA LT12 BTXB L22

0 .

L(x, v) =2vv (Cx+ Dv)(Cx+ Dv)


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


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

.


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


94/135

+ PTTQ + QTP 0

WTP

X1 A0 B0 0AT0 X 0 CT0BT0 0 I DT11

0 C0 D11 I

WP


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


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


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 ,


98/135


99/135

x1 = x2 ,x2 = x1+ u ,y=x1

u= y

X = XT > 0

Y = YT > 0

WTC(ATX+ XA)WC


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


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


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


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 ,


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 .


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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 .


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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


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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


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X

Y =

XB Z=AX

Y, Z

Y0

rC Y0=V C

C=AY0=AV C=AV ZB

X=V Z


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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 ,


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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) ,


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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 .


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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


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


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


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


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)


124/135


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 .


126/135


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


128/135


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 , (, ) .


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) .


130/135


v

, L=

L11 L12

LT12 L22

,

ATXA X L11 ATXB L12

BTXA LT12 BTXB L22

0 .


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H

H

H


132/135

H

H

D

S


133/135

S

H

H


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H

H2 H

H

H


135/135

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