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(Small-gain theorem in nonlinear system analysis and controller design)

(!) . . . . . . . . . . . .

1- Case study

. . . *. . . html . html PDF «» ! PDF word copy-paste ! word html . html

**

!

. * . ** .

[1,2,4] -- --- . .

x1 + x2 + ... + xn

p p

x x

p

(

p 1/ p

)

: x n p ·

; 1 p <

max xi

i

. : A p ·

A

p

m ax A x

x

p

=1

p

. i A

2

= m ax i ( A )

i

: A - ---

1- Norm

p u(t) . · :

u u

p

p u (t ) dt - Sup u (t )

t

1/ p

; 1 p <

. u (t ) = e- t 1(t) - [1,2] -- L <+ x0 f : m n · : x0 x

f ( x) - f ( x0 ) 2 L x - x0

2

x = f ( x) M ·

x(0) M x(t ) M , t

: h : n ·

h h ( x ) = x1 h x2 h xn

n : f : n n h : ·

L f h := h . f L0f h = h ( x ) , Lkf h = L f ( Lkf -1 h )

. V ( x ) = L f V V : n x = f ( x ) -

x : a f ( x) b f(0)= 0 [a,b] f : · x

. [-2,2] f ( x) = 2 x sin x -

2- Lipschitz 3- Invariant set 4- Lie derivative 5- Sector

. (0,+) f ( x ) = x - [1,2]

L p m --

u: [0, ) Rm · . Lm

m . p Lm L p ·

L p m {u (t ) u : [0, ) m , u

p

< }

[0,) f L . p . Lp 1 p (a > 0) f(t) = e

-at

-

. p f : . L1 . Lp 1< p g(t) = (t +1)-1 -

0

1 dt = t +1

[17,1] KL K -- (0) = 0 . : [0, c) R+ · (. R+) . K . K lim ( t ) = c = · t (r,s) s KL : [0, c) ×R+ R+ · s (r,s) r K r

lim (r,s) = 0

s

. K 2 K 2 (r ) = r 2 1 (r ) = tan -1 (r ) -

-s -k . KL k> 0 2 (r, s) = e r 1 (r , s) =

r - 1 + k rs

6- Lebesgue measurable space

[20] --

. . . -- [1] ---

x = 0 x = f ( x, t )

t0 c > 0 K · x = 0 x(t ) ( x(t0 ) ) : t0 [0, t ) x(t0 ) < c . t0 c > 0 KL · x(t ) ( x(t0 ) , t - t0 ) : t0 [0, t ) x(t0 ) < c . [1] - --- y x Lp Lp y x · . Lp

7- Interconnected systems

. 8- Uniform stability 9- Uniformly asymptotically stable 10- Input-output stability

.

p x Lp Lp ·

y

p

p x

p

+ bp

: bp

. L BIBO - [14,15] --- ---- BIBO . [13]

. BIBO

x u x = f ( x , u ) ·

( x (t0 ) ) x(t0) . : KL K ( u ) u L

x (t ) ( x (t 0 ) , t - t 0 ) + ( u )

. . . x (t ) < ( x (t0 ) , t - t0 ) + ( u ) · . = = · . ISS ISS ----

V: Rn R . x Rn x = f ( x , u ) ·

K K

ISS

: x

11- Input-output stable with finite gain 12- Input-to-state stability 13- Sontag 14- Locally input-to-state stable, or: (, )-ISS 15- Gain function 16- Input-to-state practically stable, or: (, )-ISpS

1) ( x ) V ( x ) ( x ) 2) x ( u ) V . f ( x , u ) - ( x ) x

ISS - [2,14] .

. u = 0 x = - x3 + u -

4 2 : ( ) = 3 2 ( x) = 1 x V ( x) = ( x) = ( x) = x 2

1 2

x (2 u )

13

V . f ( x, u ) = - x 4 + ux - 1 x 4 2 x

. V(x) = ½ x2 ISS ISS ... . [14-16]

-- [6] ---

. A · . Q ·

Q = QH , Q H (Q * )T : Q >0 Q ·

x 0 : xH Q x > 0

: ( ) -

17- Linear matrix inequalities (LMI) 18- Hurwitz 19- Hermitian 20- Positive definite

A12 A <0 (i ) A = 11 T A12 A22 T (ii ) A22 < 0 & A11 - A12 A22 -1 A12 < 0

. U HU = U U H = I U · U A = U V H Am×n . V

= [ d

0]

d = diag { 1 , 2 ,..., n }

or = d 0

1 2 n 0

: G(s) -

G ( s)

= sup 1 (G ( j ))

. || G || < + G H . G H G

·

[6,7] --- :

F ( x ) = F0 + xi Fi > 0

i =1 m

. Fi F0 x m . MATLAB Robust Control Toolbox P > 0 LMI G

AT P + PA PB min > 0 BT P - I C D

(A,B,C,D) G(s) :

CT DT < 0 - I

21- Schur complement 22- Unitary

: LMI

% [A,B,C,D]: State-space realization n = size(A,1) setlmis([]) % specify structure and size of P P = lmivar(1,[n 1]); gamma = lmivar(1,[1 1]); % HinfLMI = newlmi % LMI # 1 lmiterm([HinfLMI 1 1 P],1,A,'s') % AP + PA' lmiterm([HinfLMI 1 2 P],1,B) % lmiterm([HinfLMI 1 3 0],C') % lmiterm([HinfLMI 2 2 gamma],-1,1) % lmiterm([HinfLMI 2 3 0],D') % lmiterm([HinfLMI 3 3 gamma],-1,1) % Ppos = newlmi; % LMI # 2 lmiterm([Ppos 1 1 P],-1,1) % P > 0 LMIsys = getlmis; c = mat2dec(LMIsys,zeros(n),1); options = [1e-5,0,0,0,0]; % Relative accuracy of solution [normhinf, xopt] = mincx(LMIsys,c,options); %---------------------------------

. . . . -- . - «» «» ­ (!) .

1- Small gain theorem 2- Nyquist 3- Circle criterion 4- Popov criterion 5- Describing function method 6- Passivity theorem

. . [3,8,20] --- : G2 G1

x1 = A1 x1 + B1e1 y1 = C1 x1 G1 ( s ) = C1 ( sI - A1 ) -1 B1 x2 = A2 x2 + B2e2 y2 = C2 x2 G2 ( s ) = C2 ( sI - A2 ) -1 B2

: .

-

x1 = A1 x1 + B1C2 x2 + B11u1 x2 = A2 x2 + B2C1 x1 + B22u2

G1 × G2

( I - G2G1 )-1 G2 ( I - G1G2 )-1 H -1 ( I - G1G2 ) -1 G1 ( I - G2G1 )

< 1 G1 , G2 H ·

: .

[4,5,6,10] ---

7- Internally stable

I-G1G2 G2

1 r G1

< r G1H

. G1H

[1,11] ---

. L G2 G1 G2: Lq Lm G1: Lm Lq (-- ) e1 , y2 Lm , e2 , y1 Lq u1 Lm , u2 Lq :

y1 1 e1 + 1 , y2 2 e2 + 2

: u2 Lm u1 Lm 12 <1 - ·

e1 e2 1 1 - 1 2 1 1 - 1 2 ( u1 + 2 u 2 + 2 + 2 1 ) ( u 2 + 1 u1 + 1 + 1 2 )

- ·

e1 u1 + (G2 e2 ) u1 + 2 e2 + 2 u1 + 2 ( u 2 + 1 e1 + 1 ) + 2

= 1 2 e1 + ( u1 + 2 u2 + 2 + 2 1 )

e1 1 1 - 1 2 ( u1 + 2 u 2 + 2 + 2 1 )

: 12 <1

[9,25] ---

. x = (x1 , x2 ) , F = ( f1 , f 2 ) , x = F(x)

x2 . ISS x2 x1 K 2 1 . ISS x1

8- Proper, i.e. G(s) 0 as s 9- Well defined

s 0 : ( 1 1 2 1 )( s) s : K 2 1 ( f g (.) f ( g (.)) ) . . n [20,28]

[17,18,29,33] . [32,45,50]

[26,31,35] . [34] . . * ! -- . -- G2 G1 . . G1 G2 . Robust Control MATLAB .

[8] .

[24] .

! .. * 10- Nominal plant 11- Perturbations 12- Uncertainty 13- Robust control 14- Robustness

.

[46] . Biomathematics

RNA . . [23,27,30,32,33,41-49] --

[12] .

(s) . ||||< 2 LTI .

-

1 2 , W ( s) = s -1 s + 10 P( s ) = (1 + W ( s ) ( s ) ) Po ( s) Po ( s ) =

. P . K(s) = k :

H - ( s ) = - Po K W -2k = 1 + Po K ( s + 10)( s - 1 + k )

15- Drosophila

H -(s) .

2 : . k >1 .

H - ( j ) < 1 ; : 2

H - ( j ) =

4k 2 2k 1 max H - ( j ) = < 2 2 2 ( + 10)( + ( k - 1) ) 10( k - 1) 2

.k > 5/3 : k >1 - . . [25-50] . . . !

16- Feedback linearization 17- Backstepping

[17] -- «» . . - . (ISpS) - . . - . x Rn |x| u(t) u :

x1 = f1 ( x1 , u11 , u12 ) ; u11 = x2 x2 = f 2 ( x2 , u21 , u22 ) ; u21 = x1

ni mi fi(0)= 0 xi , ui 2 ( xi , ui1 , ui 2 ) fi

«» (ISS)

. i=1,2

- .. *

i ( i , i ) - ISS ui2 ui1

x u (i = 1, 2) i x , i u K i , i KL

for x1 (0) 1 , u12 1u , x2 1x : t 0 ; x1 (t ) 1 ( x1 (0) , t ) + 1x ( x2 ) + 1u ( u12 )

for x2 (0) 2 , u22 2u , x1 2 x : t 0 ; x2 (t ) 2 ( x2 (0) , t ) + 2 x ( x1 ) + 2u ( u22 )

. u = 12 x = 1 u22 x2 : , , d1 , d 2 > 0 -

(1 + ) 1x (1 + ) 2 x ( s) s + d1 , 0 s x x (1 + ) 2 (1 + ) 1 ( s) s + d 2

and l := d1 + d 2

u

x

< m in { , 1 x , 2 x , 1 , 2 }

. ( , ) - ISpS u x (,)

r1 ( s ) := (1 + -1 ) 1u ( s ) + 1x (1 + )(1 + -1 ) 2u ( s ) , r2 ( s ) := (1 + -1 ) 2u ( s ) + 2 x (1 + )(1 + -1 ) 1u ( s ) , r ( s ) := r1 ( s ) + r2 ( s)

a ( u ) u x (0) t 0 KL K

x (t ) ( x (0) , t ) + ( r + a ) ( u ) + l ,

: x (t )

t 0

.

--- :

x = - x 3 + 1 y 3/ 2 2 2 y = -k y + y + x + u ; k > 0 (1) (2)

: V1 ( x) = 1 x 2 () 2

3 3/ 2 V1 ( x ) - 1 x 4 - 1 x ( x - y ) 2 2

: x

y

V1 ( x) - 1 x 4 2

ISS y () ---

(1x = 1 = +) . 1 ( s ) = s

: V2 ( y ) = 1 y 2 () 2

2 ( y ) -k y - y (k y - x + u ) , V2 4 4

y

k 2

:

V2 ( y ) - 1 k y 2 4

k 4 y x + u 2 k

: (u , x) ()

if

1 y (0) 2 = 1 k , u 2u = 16 k 2 , 2 1 x 2 x = 16 k 2

y (t ) 2 ( y (0), t ) + 2 ( x ) + 2 ( u )

1(s) . 2 ( s) =

4 k

4s KL 2 k

.

2 a2 = (1 + ) , a1 =

2 (1 + )3/ 2 k

(1 + ) 1 (1 + ) 2 ( s ) = a1 s s + 1 a12 4 , 2 1 (1 + ) 2 (1 + ) 1 ( s ) = a2 s s + 4 a2

s 0 ( = +)

k > 0

k k2 k min { , 1 x , 2 x , 1 , 2 } = min , = , a1 a 2 2 16 2

l= d1 + d 2

a12 + a2 2 a12 (1 + )3 = =2 4 2 k

l < k/2 k . ( , ) - ISpS ---

k k2 k . min , = k = 0.5 2 16 2

:

. k = 8 . :

. [18] . ISS -- «» . ISS ISS «» . ISS . :

V1 -1 (V1 ) + 1 (V2 ) V2 - 2 (V2 ) + 2 (V1 )

(1)

K 2 , 1 2 , 1 2 , 1 , 2 , 1

.

1- Monotone systems [42] 2- Integral input-to-state stable

- V2 V1 . () ISS () K . . . - K - () : (R2 0)

W1 = -1 (W1 ) + 1 (W2 ) f1 (W1 , W2 ) W2 = - 2 (W2 ) + 2 (W1 ) f 2 (W1 , W2 ) (2)

. () 2 0 () [0, Tmax ) ---

W2 (0) > V2 (0) . W (0) = [W1 (0) W2 (0) ]

T

W1 (0) > V1 (0)

t [0, Tmax ) : W1 (t ) > V1 (t ) & W2 (t ) > V2 (t )

-- . ()

2 : 0

+- := {[ w1 , w2 ] 2 0 f1 ( w1 , w2 ) 0 & f 2 ( w1 , w2 ) 0} -- := {[ w1 , w2 ] 2 0 f1 ( w1 , w2 ) 0 & f 2 ( w1 , w2 ) 0} -+ := {[ w1 , w2 ] 2 0 f1 ( w1 , w2 ) 0 & f 2 ( w1 , w2 ) 0}

2 1 . 1 := -- +- , 2 := -+ -- : ( w1 , w2 )

3- Positively invariant set

-- +- -+ 2 K 2 1 = 0 [25]

K . . () -- ---

-- : W(t) W (0) ()

t +

lim W (t ) = 0

: () ( ) ---

-- +- -+ = 2 0 and -- +- -+ = {0}

. 2 0 () L- < L+ + -

w+ w L -

lim sup 1-1 (1 ( w)) = L+

lim 2 -1 ( 2 ( w)) = +

L- < L+ + -

w+ w L -

lim sup 2 -1 ( 2 ( w)) = L+

lim 1-1 (1 ( w)) = +

-

w+ w+

lim 1-1 (1 ( w)) = + lim 2 -1 ( 2 ( w)) = +

--- :

2 x1 x2 x1 = - x + 1 + ( x + 1)( x + 1) 1 1 2 x = - 2 x2 + x 1 2 x2 + 1

. . ( x1 , x2 ) +

2

V1 ( x1 ) = x1 , V2 ( x2 ) = x2

: ()

1 ( s ) = 2 ( s ) =

2s s , 1 ( s ) = , 2 (s) = s s +1 s +1

:

2w 1- w 2w 2 -1 ( 2 ( w)) = 1+ w

1-1 (1 ( w)) =

. L- = 1 , L+ = 2 . --- :

. 2 0 -- [19] [*] : .

x = ax - yz y = -by + xz z = -cx + xy a , b, c > 0

(1)

) c=5, b=12, a=4.5 . ( ) c=5, b=12, a=0.4 ( :

1- Two-scroll attractor

- bc - bc bc bc 0 E0 = 0 , E1 = - ac , E2 = ac , E3 = - ac , E4 = ac ab - ab - ab ab 0

[**] . . u=k x () E0 k <-a --- .

x = ax - yz + u y = -by + xz z = -cz + xy (2)

: -

V ( x, y , z ) = x 2 + 1 y 2 + 1 z 2 2 2 V = L f V = 2( a + k ) x 2 - by 2 - cz 2 ( x , y , z ) 0, k < -a V < 0

. E0

() . !

^ ^ ^ : Ei = ( x, y, z )T

^ x = x - x ^ y = y - y z = z - z ^

: u1 , u2 , u3

x = ax - zy - yz - yz + u1 ^ ^ ^ ^ y = zx - by + xz + xz + u 2 ^ ^ z = yx + xy - cz + xy + u3

:

u1 k1 u = 0 2 u3 0 0 k2 0 0 x y 0 1 - xy

(. x , y , z x, y, z )

:

^ ^ x = ( a + k1 ) x - zy - yz - zy ^ ^ y = zx + ( -b + k 2 ) y + xz + xz z = yx + xy - cz ^ ^ (3)

: () E=(0,0,0)T ---

( k < -a -

1

ac + bc c

)

2

,

k2

( <b-

ac + bc c

)

2

: () -

2- Abuse of notation!

^ ^ x = (a + k1 ) x - zy - yz - zy (state x, y , input z ) ^ ^ y = zx + (-b + k2 ) y + xz + xz ^ ^ z = yx + xy - cz (state z , input x, y )

(4) (5)

2 . ISS () V ( z ) = 1 z 2

^ ^ V ( z ) = -cz 2 + z ( yx + xy ) -cz 2 + z choose: 1 ( r ) = z 1 (

(

ac x + bc y

)

ac + bc 1 r , 0< < c c - 2 c - ( x, y ) ) x , y z V ( z ) - z 2 ac + bc

: K

(r ) = (r ) = 1 r 2 , (r ) = r 2 2

:

1 (r ) =

ac + bc c r , 0 < < , 1 K c - 2

V ( x, y ) = 1 ( x 2 + y 2 ) x, y z () 2 .

^ ^ V ( x, y ) = ( a + k1 ) x 2 + ( -b + k 2 ) y 2 - yxz + xyz ( a + k1 ) x 2 + ( -b + k 2 ) y 2 + z

(

ac x + bc y

)

^ ^ ^ (note that: x = bc , y = ac , z = ab ) c - 2 choose: 2 (r ) = r ab + bc

( x, y ) 2 ( z ) z

(

ac x + bc y

)

(

ac + bc c - 2 ac + bc c - 2

x2 + y 2

(

ac x + bc y

)

) (x

2

2

+ y2 )

:

( x, y ) a + k + V 1 ( x, y ) 2 ( z )

(

ac + bc c - 2

)

2

x 2 + -b + k + 2

(

ac + bc c - 2

)

2

y2

( k < -a -

1

ac + bc c

)

2

,

k2

( <b- ( -b+

ac + bc c

)

2

: (, )

( a+

ac + bc c - 2

)

2

+ k1 < - ,

ac + bc c - 2

)

2

+ k2 < - ,

>0

V ( x, y ) - ( x 2 + y 2 )

V(x,y) K . ISS

(r ) = (r ) = 1 r 2 , (r ) = r 2 2

2 (r ) =

c - 2 r , 2 K ac + bc

1 ( 2 ( r )) =

c - 2 r < r : r > 0 c -

. ISS () E0 .

[*] W-B. Liu. G-R. Chen, A new chaotic system and its generation, International Journal of Bifurcation & Chaos 13(1) (2003) 261-7. [**] MT. Yassen, Controlling chaos for the dynamical system of coupled dynamos, Chaos, Solitons & Fractals 13 (2002) 341-52.

[20] -- ... - . . . (LMI) . . --- : ()

x1 = A1 x1 + A12 x2 + B11u1 x2 = A2 x2 + A21 x1 + B22u2 A A= 1 A21

A12 , A2

B B = 11 B22

K 0

(1)

Acl K = 1 0 K2 .

A + B11K1 Acl = A + BK = 1 A21 A2 + B22 K 2 A12

) . A21 = B2 C1 A12 = B1 C2 A21 A12 : (.

H1 ( s ) = C1 ( sI - A1 ) -1 B1 , H 2 ( s ) = C2 ( sI - A2 ) -1 B2

A ()

. (A2, B2, C2) (A1, B1, C1)

1- Decentralized control .. *

K2 K1 -

C1 ( sI1 - A1 - B11 K1 )-1 B1

× C2 ( sI 2 - A2 - B22 K 2 )-1 B2

<1

(2)

. () A - . - .

LMI A21 = B2 C1 A12 = B1 C2

. () . A G ( s) = D + C ( sI - A)-1 B ([*] ) --- G ( s) < 1

AX + XAT BT X = X T > 0 such that CX B -I D XC T DT < 0 -I

:

0 A12 = B10C2 A = 1 A21

A

A12 --- A2

A21 = B2 C1 A12 = B1 C2 A21 = B20C10

C1 ( sI1 - A1 ) -1 B1

× C2 ( sI 2 - A2 ) -1 B2

<1

P , P2 , X , Y (all > 0) such that 1 P A1T + A1 P + B10 XB10T 1 1 0 C1 P 1

2- Bounded real lemma

0 0 P AT + A2 P2 + B2 YB2 T PC10T 1 <0 & 2 2 0 -Y C2 P2 0 P2C2 T <0 -X

0 B1 = B10 X 1/ 2 , B2 = B2 Y 1/ 2 ,

C1 = Y -1/ 2C10 ,

0 C2 = X -1/ 2C2 -

:

P A1T + A1 P + B1 B1T 1 1 C1 P 1

T T P2 A2 + A2 P2 + B2 B2 PC1T 1 <0 & - I1 C2 P2 T P2C2 <0 -I2

: --

C1 ( sI1 - A1 ) -1 B1

<1

&

C2 ( sI 2 - A2 ) -1 B2

<1

. . [20] . A21 A12 - - C1 ( sI1 - A1 ) -1 B1 × C2 ( sI 2 - A2 ) -1 B2 - . LMI A21 = B2 C1 A12 = B1 C2 K2 K1 A21 = B2 C1 A12 = B1 C2 () --- : ()

P , P2 , X 1 , X 2 (all > 0) and Y1 , Y2 such that 1

T P A1T + A1 P + B10 X 1 B10T + B11Y1 + Y1T B11 PC10T 1 1 1 <0 & C10 P -X2 1 T T 0 0 0 P2 A2 + A2 P2 + B2 X 2 B2 T + B22Y2 + Y2T B22 P2C2 T <0 0 C2 P2 - X1 K1 = Y1 P -1 , K 2 = Y2 P2 -1 1

. LMI -- -- . MATLAB LMI --- n . n

/ : .

A1 B C A(n) = 2 1 BnC1 B1C2 A2 BnC2 B1Cn B2Cn An

( Ai , Bi , Ci ); i = 1...n n

Gi ( s ) = Ci ( sI - Ai ) -1 Bi .

: . G1,2 G2 G1

G1, 2 ( s ) = C (2) ( sI - A(2) ) B (2)

-1

= G1 ( I - G2G1 ) -1 + G2 ( I - G1G2 ) -1 + G2G1 ( I - G2G1 )-1 + G1G2 ( I - G1G2 ) -1

:

G1, n -1 ( s ) = C (n - 1) ( sI - A(n - 1) ) B (n - 1) = G1, n- 2 ( I - Gn -1G1, n - 2 ) -1 +

-1

Gn -1 ( I - G1, n - 2Gn -1 ) -1 + Gn -1G1, n - 2 ( I - Gn-1G1, n - 2 ) -1 + G1, n - 2Gn-1 ( I - G1, n- 2Gn -1 ) -1

. A(n)

:

A(n) G1, n -1 × Gn

< 1 A(n-1) An ---

. ... A(n-2) A(n-1) A(n) -- . A(1)=A1 . N : N

xi = Ai xi + Bi vi , yi = Ci xi , i = 1,..., N

: /

vi = H ij yi + ui , i = 1,..., N

j =1 N

:

x = A( N ) x + B N u , y = C N x

B N = diag [ B1 ,..., BN ] ,

C N = diag [C1 ,..., C N ]

A1 A1N , A( N ) = AN 1 AN

Ai Aij = Bi H ij C j

i= j i j

(Ai , Bi) [**] : A cl(N) (i=1,...,N) Ki

A1N A1 + B1 K1 Acl ( N ) = AN 1 AN + BN K N

: Ki -- . A1 + B1 K1 K1 - K2 1 = H 21C1 ( sI - A1 - B1 K1 ) -1 B1 H12

C2 ( sI - A2 - B2 K 2 ) -1 B2

-

< 1/ 1

. A cl(2) 1,2 = C (2) ( sI - Acl (2) ) B(2)

-1

-

C (2) = [ H 31C1

A + B K H 32C2 ] , Acl (2) = 1 1 1 B2 H 21C1

B1 H12C1 B1 H13 , B(2) = B H A2 + B2 K 2 2 23

C3 ( sI - A3 - B3 K 3 ) -1 B3

< 1/ 1,2 K3

. A cl(3)

A cl(N),..., A cl(5), A cl(4) K4 , K5 ,..., KN -

.

() . --- . A A2 A1 () «» A cl K21 K12 B21 B12

A1 Acl = A21 + B21 K 21 A12 + B12 K12 A2

R2 R1 (. C( X ) X ) :

C( R1 ) = C( B12 ) + C( A12 ) C( R2 ) = C( B21 ) + C( A21 )

S2 S1 K21 K12

B12 K12 + A12 = R1S2 , B21 K 21 + A21 = R2 S1

(3)

:

: Q2 Q1

T T Q1 R1 = 1 , Q2 R2 = 2 : T1 , T2 are nonsingular 0 0

: Q2 Q1 ()

B12 A12 T1 B21 A21 T2 K12 + = S2 , K 21 + = S1 0 0 0 0 0 0 B Q1 B12 = 12 , Q1 A12 = .... : ) 0

(.

S1 = T2 -1 B21 K 21 + T2 -1 A21 , S 2 = T1-1 B12 K12 + T1-1 A12 :

B21 K21+ A21 =B2 C1 B12 K12 + A12 =B1 C2 K21 K12 ---

C1 ( sI - A1 )-1 B1

× C2 ( sI - A2 )-1 B2

< 1

:

P , P2 , X 1 , X 2 (all > 0) and Y1 , Y2 such that 1 P A1T + A1 P + R1 X 1 R1T 1 1 -1 -1 T2 A21 P + T2 B21 Y1 1

T T P A 21 T2 -T + Y1T B21 T2 -T 1 <0 & -X2

T T T T P2 A2 + A2 P2 + R2 X 2 R2 P2 A12 T1-T + Y2T B12 T1-T -1 <0 -1 - X1 T1 A12 P2 + T1 B12 Y2 -1 -1 K12 = Y2 P2 , K 21 = Y1 P 1

.

:

1 0 1 0 0 0 0 0 , A = 0 , B = B = 0 A1 = 0 1 0 1 2 11 22 -23 -24 -10 -15 -18 -15 1 0 7 -3 0.5 0 0.5 1 2 0 , A = B 0C 0 = -1 0 -1 0 1 0 A12 = B10C2 = 21 2 1 1 0 2 2 1 1 0.5 1 0.5 1

. A = 1 A21

A

A12 A2

: --

K1 = [ -153.83 -185.36 -35.09] , K 2 = [ -994.1 1879.1 -109]

:

35.5 A12 = B1C2 = -38.65 -74.76 -2.8 A21 = B2C1 = -11.187 67.1 0 0 0.197 0.394 4.87 1.67 3.13 0.206 9.73 26.4 0 0.089 -0.089 0 0.142 0.019 -0.085 52.8

C1 ( sI - A1 - B11 K1 ) -1 B1 A + B11 K1 Acl = 1 A21

× C2 ( sI - A2 - B22 K 2 ) -1 B2

<1

is stable A2 + B22 K 2 A12

-- . B12 = [3 0 1] , B21 = [ 0.5 0.4 1]

T T

K12 = [ -2.46 -4.635 -0.28] , K 21 = [ -4.16 -0.336 3.116 ] :

: B21 K21 + A21 B12 K12 + A12

6.8 A12 + B12 K12 = B1C2 = -6.84 -51.6 -10.2 A21 + B21 K 21 = B2C1 = 2.1 68.87

0 -8.8 2.1 0.0456 0.09 0.078 10.4 0 0.06 0.096 17.9 0.7 9 E - 4 8 E - 4 6 E - 4 7.5 2.7 -0.076 0 0.076 0.15 -3.56 2.17 0.042 -0.023 15 5.43 0.01 0.0067 0.002

C1 ( sI - A1 ) -1 B1

× C2 ( sI - A2 ) -1 B2

<1

A1 Acl = A21 + B21 K 21

A12 + B12 K12 is stable A2

[*] M. Green. D.J.N. Limebeer, Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ, 1995. [**] M.E. Sezer. D.D. Siljak, On structurally fixed modes, Proceedings of IEEE International Symposium on Circuits and Systems (1981) 558-65.

-- [21] . :

z = q( y, z, t ) x1 = x2 + 1 ( y, z , t ) ... x = x + ( y, z , t ) ,... i +1 i i xn = u + n ( y, z , t )

y = x1

(1) xi , u , y ; z n0

q i ( ) z : .

* pi 1 i n

(i)

i ( y, z , t ) pi* i1 ( y ) + pi* i 2 ( z ) ( y, z , t ) × n 0 × +

.i2(0) = 0 i2 i1 . V0 ISPS ISpS y z (ii) : V0

d 0 0,

0 , 0 class K ; such that

( y, z , t ) × n 0 × +

V0 ( z )q ( y, z , t ) - 0 ( z ) + 0 ( y ) + d 0 , z

. i1 (0) 0 . (i)

z d = 0 y = 0 z ISpS -

. y

= v( , y ) , u = µ ( , y ) (2)

1- Smooth 2- Proper

. ()-() . . () --- . :

i = i +1 + Li +1 y - Li (1 + L1 y ) , 1 i n - 2 n-1 = u - Ln -1 (1 + L1 y )

. A (1 i n - 1) Li

- L1 -L 2 A= - Ln - 2 - Ln -1 1 0 0 0 0 1 0 0 0 0 1 0

i =

xi +1 - i - L i y p

*

: 1 i n-1

* , p* := max {1, pi*+1 , L i p1 ; 1 i n - 1}

* ( y, z , t ) = [ 2 - L11

= A + 1 * ( y, z , t ) p*

3 - L2 1 n - Ln -11 ]

T

:

(3)

(z y ) i xi +1 . . i + Li y : P

3- Regulation

PA + AT P = -2 I

T : () V = P V

1 V = -2 T + 2 T P * * ( y, z , t ) p

2 T 2

(4)

2

n -1 n -1 2 - + 2 P ( i +1,1 ( y ) + 11 ( y ) ) + 2 P ( i +1,2 ( z ) + 12 ( z ) ) i =1 i =1

. (y,z) ISpS V

n -1 d := ( i +1,1 ( y ) + 11 ( y ) ) i =1

y 2

n -1 d := ( i +1,2 ( z ) + 12 ( z ) ) i =1

z

2

z () . . . ( y, 1 ,..., n -1 )

z = q ( y, z , t ) , = A + * ( y, z , t ) / p* y = + L y + p* + ( y, z, t ) 1 1 1 1 i = i +1 + Li +1 y - Li (1 + L1 y ) ; 1 i n - 2 n -1 = u - Ln -1 (1 + L1 y )

(5)

--- - 1 () ( ,y) - . . u (1 i n - 1), i

:

= A + 1 * ( y, z , t ) p*

y = 1 + L1 y + p*1 + 1 ( y, z , t )

: V1

V1 = 1 1 ^ V ( ) + ( y 2 ) + ( p - p)2 2 2 0 1

K 0 , > 0

^ . p .

: () (i)

V1 - 1 2P

2

0

+

T

0

( dy + dz ) + ( y 2 ) y 1 + L1 y + p *1 + 1 ( y , z , t ) (6)

^ ^ + ( p - p) p /

y2 ( y 2 ) . +

. : 1 i1

0 0 dy y 21 ( y ) + d , ( d := dy y =0

)

(7)

: 11 1 > 0 (i)

y p*1 + 1

2 + 1 1 1 2 1 + p y 2 11 ( y ) + 0 + 20 41 4

2

+ 12 ( z ) + 1 11 (0)2

(8)

: () () ()

2 ( y 2 ) y 2 1 + L + 2 P ( y ) + V1 1 0 ( y 2 ) 1 y

2 + 1 1 p 11 ( y ) + 0 + 41 4

2

2P 2 1 1 2 0 ^ ^ + p ( p - p ) + 12 ( z ) + - (dz + d ) + 1 11 (0) 2 0 20

:

^ 1 := - p + ( y 2 ) y 2 11 +

2 2

20 + 1 1 + 4 41

2P 2 + 1 1 ^ 1 := - y v1 ( y ) - L1 y - y 1 - p y 11 + 0 + 2 0 ( y ) 4 41 ^ w2 := 1 - 1 ( y, p ) (9)

.v1(0) > 0 v1 > 0 :

1 1 2 ^ ^ ^ ^ V - - y 2 v1 ( y 2 ) + y w2 - p ( p - p ) + ( p - p )( p - 1 ) 20

+ 12 ( z ) +

2

2P

2 0 (dz + d ) + 1 11 (0) 2

0

() -

- wk +1 k y

: 21

( p * 1 + 1 ) 1 2

k +1

0

+ y 2 + p wk +12 21 ( y , 1 ,..., k -1 )

2 2

+ 12 ( z ) + 1 11 (0) 2

:

k +1 := k + wk +12 21 k +1 := -ck +1 wk +1 - wk - L k +1 y + Lk (1 + L1 y ) +

+

j =1 k -1

k y

(1 + L1 y ) +

k

k ^ p

k +1

k j

^ ( j +1 + L j +1 y - L j (1 + L1 y ) ) - p - w j

j =2

j -1 wk +1 21 ^ p

wk + 2 := k +1 - k +1

(10)

. w1 = 0 ck+1 > 0

2 Vk +1 = Vk ( , y , w2 ,..., wk ) + 1 wk +1 2

[21] .

() w2 , 1 , 1 : n . wi +1 , i , i ()

u = n + Ln y = n + Ln y

^ . p = n

--- . v1 : v1

( y 2 ) y 2 v1 ( y 2 ) - n + 1 ( y 2 )

K 1 i 2 :

n 12 ( z ) +

2

2P

2

0

dz 1 ( z )

2

: [21]

2 Vn - cVn + 1 ( z ) + 2

1 c := min n , 2 , 2ci , ; 2 i n 2 max ( P )

2 :=

2

p +

2

2P

2 0 d + n 1 11 (0) 2

0

: 0 < 3 < c

2 2 2 1 ( -1 (V0 ) 2 ) , Vn max c - 3 c - 3

. Vn - 3 Vn

: K

( z ) V0 ( z ) ( z )

: (ii)

V0 ( z ) q ( y, z , t ) - 0 ( z ) + ( -1 0 ( y ) ) + d 0 z

(11)

. K :

-1 0 ( y ) 1 ( y 2 ) + 4 ; 4 > 0 4

-1 0 ( y ) 1 Vn ( , y , w2 ,..., wn ) + 4 2

: ()

V0 ( z ) q( y, z , t ) - 0 ( z ) + (Vn ) + d 0 + (2 4 ) z

: 6 > 0 0 < 5 < 1

2d + 2 ( 4 ) 2 (Vn ) , 0 -1 0 if V0 ( z ) max 0 -1 1- 5 1- 5 - ( z ) V0 5 0

K [25]

:

( s) <

1- 5 0 2 c - 3 -1 1-1 2 s , s > 0

--- (ii) (i) --- . ()-() . y pi* d 0 = i1 (0) = i 2 (0) = 0 (ii) (i) ---

s 0

lim+ sup

0 ( s)

s2

< + ,

i 2 (s)2 lim sup < + , 1 i n s 0 0 (s)

+

()

lim ( x(t ) + z (t ) + (t ) ) = 0

t

:

i.e. regulation property

. [21]

[22] -- [22] . . . --- .

1- Electrohydraulic velocity servosystem

: - - - - - - - - . -

x1 = x2 = 1 {- Bm x1 + qm x2 - qm C f Ps sgn x1} Jt 2e V0

-

:

1 ( Ps - x2 sgn x3 ) - qm x1 - Cim x2 + Cd W x3 1 K x3 = - x3 + r u Tr Kq

y = x1

: : x1 : x2 : x3 :

J t = 0.03 Kg m 2 qm = 7.96 × 10 -7 m3 / rad Bm = 1.1× 10-3 Nms C f = 0.104 V0 = 1.2 ×10-4 m3 e = 1.391×109 Pa Cd = 0.61 Cim = 1.69 × 10-11 m3 / Pa. s Ps = 107 Pa = 850 Kg / m3 Tr = 0.01 s K r = 1.4 × 10 -4 m3 / s.V K q = 1.66 m 2 / s W = 8 × 10-3 m . :

x1 N = arbitrary x2 N = x3 N = 1 {Bm x1N + qm Ps C f } qm qm x1 N + Cim x2 N Cd W

( Ps - x2 N ) /

Kq

x 3 N x3 Kr x1 > 0 .

uN =

. x3 > 0 . :

x = f ( x) + g ( x) u y = h( x ) -27.6 - 3.667 × 10-2 x1 + 2.65 × 10 -5 x2 f ( x) = -1.845 × 107 x1 - 391.8 x2 + 1.22 × 1010 x3 107 - x2 -100 x3 g ( x) = ( 0 0 8.43 × 10 -3 ) ,

T

h( x) = x1

--- :

u= 1 Lg L

2 f

{- L h( x )

3 f

h( x) + v} ,

f = ( f1 , f 2 , f3 )T

1 v = -k0 ( x1 - yd ) - k1 ( f1 - yd ) - k2 (- Bm f1 + qm f 2 ) - d + d y y Jt

k0 , k1, k2 > 0 yd : yd = x1N = 200 rad/s . k1 k2 > k0

k0 = 5000, k1 = 5150, k2 = 151

:

u= - 1 10 - x2

7 7

( 413.65 - 45 x - 9.63 × 10

1

-4

x2 + 7.227 × 1011 x32 )

x3 (1.094 ×109 x1 + 23228 x2 ) + 40413x3 10 - x2

. x(0) = 0

--- :

u = uN -

Kq Kr

Tr ( k1 - 1 / Tr )( x3 - x3 N )

k1 = 1s-1 yd = x1N = 200 rad/s :

u = 5928.6 x3 + 0.6942

. x(0) = 0

-- . . .

[1] :

z = A z + Bu v = Cz

. C A-1 B = - I A . u = v = 0 :

x = f ( x) + g ( x) v z = A z + B u v = Cz

. u v d2 d1 x d2 d1 x (d1 , d2) u (. ) < * (. || . ||2 || . || ) d1 2 , d 2 2 < + d1 , d 2 L2

) . x = f ( x ) + g ( x )(u + d )

: ( = 0 , d1 + d 2 = d

x d + ; , > 0

[1] : e1

u .[ f ( x ) + g ( x )(u + e1 ) ] c1 || x || + c2 || e1 || ; c1 , c2 0

.

= z + A-1 B (u + d 2 ) :

y1 = u , y2 = -C , v1 = d1 + d 2 = d , v2 = d 2

: H1

x = f ( x) + g ( x)(u + e1 ) y1 = u .[ f ( x) + g ( x)(u + e1 ) ]

. *

: H2

1 -1 = A + A Be2 y2 = -C

: A

e At k e - a t

f = k A -1 B . C / a

2 = k C . 0 / 2a y2 f e2 + 2

;

f , 2 < +

y1 c1 x + c2 e1 : H1

y1 1 e1 + 1 : 1 = c1 + c2 , 1 = c1

(- )

1 2 = 1 f < 1

:

x

1- 1 f

{d

+ f d 2 + f 1 + 2 } +

( x d + + 2 : 0 ) --- :

x d + u .[ f ( x ) + g ( x )(u + e1 ) ] c1 x + c2 e1 ; e1 or : u + 8.43 × 10 -3 u e1 c1 x + c2 e1 x3

L2 . x = +

.

*. - . . . «» «» «» .«» - n [28] . !

. *

[1] H.K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, 1996. [2] M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1993. [3] K. Zhou. J.C. Doyle. K. Glover, Robust and Optimal Control, PrenticeHall, Englewood Cliffs, NJ, 1996. [4] J. Doyle. B. Francis. A. Tannenbaum, Feedback Control Theory, Macmillan, 1990, pp 4-50. [5] C. Scherer, Theory of Robust Control, Delft University of Tech, 2001. [6] S. Scogestad. I. Postlethwaite, Multivariable Feedback Control, John Wiley & Sons, NY, 2001. [7] S. Boyd. L.E. Ghaoui. E. Feron. V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. [8] D-W. Gu. P.Hr. Petkov. M.M. Konstantinov, Robust Control Design with MATLAB, Springer, London, 2005, pp 21-44. [9] R.A. Freeman. P.V. Kokotovic, Robust Nonlinear Control Design, Birkhauser, Boston, 1996, pp 21-43. [10] P. Albertos. A. Sala, Multivariable Control Systems: An Engineering Approach, Springer, London, 2004, pp 109-10. [11] R.S.S. Pena. M. Sznaier, Robust Systems: Theory and Applications, John Wiley & Sons, NY, 1998, pp 248-50. [12] Massachusetts Institute of Technology, Lecture Notes on Dynamic Systems, Fall 2003, Rec 9. [13] E.D. Sontag, Smooth Stabilization Implies Coprime Factorization, IEEE Transactions on Automatic Control. 34 (4) (1989) 435-443 [14] E.D. Sontag. Further facts about input to state stabilization, IEEE Transactions on Automatic Control 35 (4) (1990) 473-6.

[15] E.D. Sontag. L. Grune. F.R. Wirth, Asymptotic stability equals exponential stability, and ISS equals finite energy gain ­if you twist your eyes, Systems & Control Letters 38 (1999) 127-34. [16] D.J. Hill. P.J. Moylan, Stability results for nonlinear feedback systems, Automatica 13 (1977) 377-82. [17] M.P. Tzamtzi. S.G. Tzafestas, A small gain theorem for locally input to state stable interconnected systems, Journal of the Franklin Institute 336 (1999) 893-901. [18] D. Angeli. A. Astolfi, A tight small gain theorem for not necessarily ISS systems, Systems & Control Letters 56 (2007) 87-91. [19] D-B. Liu. X-S. Yang, Global stabilization of the new chaotic system based on small-gain theorem, Chaos, Solitons and Fractals (2006) doi: 10.1016/j.chaos.2006.06.086. [20] Z. Duan. L. Huang. L. Wang. J. Wang, Some applications of small gain theorem to interconnected systems, Systems & Control Letters 52 (2004) 263-73. [21] Z-P. Jiang, A combined backstepping and small-gain approach to adaptive output feedback control, Automatica 35 (1999) 1131-9. [22] M. Jovanovi, Nonlinear control of an electrohydraulic velocity servosystem, University of California, Santa Barbara, CA. [23] A.R. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Transactions on Automatic Control 41 (9) (1996) 1256-70. [24] W.M. Griggs. B.D.O. Anderson. A. Lanzon, A mixed small gain and passivity theorem in the frequency domain, Systems & Control Letters 56 (2007) 596-602. [25] Z-P. Jiang. I.M.Y. Mareels. Y. Wang, A Liapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems, Automatica. J. IFAC 32 (8) (1996) 1211-65. [26] D.S. Laila. D. Nesi, Lyapunov based small-gain theorem for parameterized discrete-time interconnected ISS systems, Proceeding of IEEE Conference on Decision and Control (2002) 2292-7. [27] W.M. Haddad. V.S. Chellaboina. D.S. Bernstein, An implicit small gain condition and an upper bound for the real structured singular value, Systems & Control Letters 29 (1997) 197-205. [28] S. Dashkovskiy. B. Rüffer. F.R. Wirth, An ISS small gain theorem for general networks, Fachbereich 3- Mathematik und Informatik (2005) Report 05-05.

[29] A.R. Teel, Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Transactions on Automatic Control 43 (7) (1998) 960-4. [30] R.S. Chandra. R. D'Andrea, Necessity of the small gain theorem for multidimentional systems, IEEE Conference on Decision and Control, 2002. [31] B. Ingalls, Discussion on: `The non-uniform in time small gain theorem for a wide class of control systems with outputs', Proceeding of IEEE Conference on Decision and Control, 2003. [32] E.D. Sontag. B. Ingalls. A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections, Journal of the Franklin Institute 339 (2002) 211-29. [33] Z-P. Jiang. A.R. Teel. L. Praly. Small-gain theorem for ISS systems and applications, Mathematics of Control Signals and Systems 7 (1994) 95120. [34] R.S. Chandra. R. D'Andrea, A scaled small gain theorem with applications to spatially interconnected systems, Proceeding of IEEE Conference on Decision and Control (2003) 2859-64. [35] Z-P. Jiang. Y. Lin. Y. Wang, Nonlinear small-gain theorems for discrete-time feedback systems and applications, Automatica 40 (2007) 2129-36. [36] Z-P. Jiang. I. Mareels. A small-gain control method for nonlinear cascaded systems with dynamic uncertainties, IEEE Transactions on Automatic Control 42 (1997) 292-308. [37] Z-J. Wu. X-J. Xie. S-Y. Zhang, Adaptive backstepping controller design using stochastic small-gain theorem, Automatica 43 (2007) 60820. [38] W.M. Haddad. D.S. Bernstein, Robust stabilization with positive real uncertainty: beyond the small gain theorem, Systems & Control Letters 17 (1991) 191-208. [39] S. Battilotti, Robust output feedback stabilization via a small gain theorem, International Journal of Robust and Nonlinear Control 8 (1998) 211-29. [40] W.M. Haddad. V.S. Chellaboina. J.R. Corrado. D.S. Bernstein, Robust fixed-structure controller synthesis using the implicit small-gain bound, Journal of the Franklin Institute 337 (2000) 85-96. [41] P.D. Leenheer. D. Angeli. E.D. Sontag, On predator-prey systems and small gain theorems, Mathematical Biosciences and Engineering 2 (1) (2005) 25-42.

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