`&gt; &gt;         «  »    (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 + ... + xnp px xp (p 1/ p):      x   n  p  ·  ; 1  p &lt; max xii.           :      A  p  ·Ap m ax A xxp=1p.    i  A2= m ax  i ( A )i:    A  -    ---1- Normp      u(t)  .         · :     u up  p    u (t ) dt   -   Sup u (t )t1/ p; 1  p &lt;.   u (t ) = e- t 1(t)    -  [1,2]   --       L &lt;+      x0  f :  m   n  · :  x0       x  f ( x) - f ( x0 ) 2  L x - x02      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 , up&lt; }[0,)    f         L       .   p     .  Lp  1 p      (a &gt; 0) f(t) = e-at - .  p    f  : . L1   . Lp  1&lt; p      g(t) = (t +1)-1  -  01 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) = 0s . K    2   K    2 (r ) = r 2  1 (r ) = tan -1 (r )  - -s -k . KL   k&gt; 0   2 (r, s) = e r  1 (r , s) =r  -  1 + k rs6- Lebesgue measurable space[20]    --                          .       .               .       -- [1]   --- x = 0    x = f ( x, t )          t0   c &gt; 0    K      ·   x = 0    x(t )   ( x(t0 ) ) :  t0  [0, t )  x(t0 ) &lt; c .         t0   c &gt; 0    KL      ·      x(t )   ( x(t0 ) , t - t0 ) :  t0  [0, t )  x(t0 ) &lt; c .   [1]  -  ---   y   x  Lp      Lp  y   x     · . Lp7- Interconnected systems        .               8- Uniform stability 9- Uniformly asymptotically stable 10- Input-output stability      .   p     x  Lp           Lp    ·ypp xp+ bp:        bp . L   BIBO   -  [14,15]     ---  ----        BIBO                   .  [13].           BIBO         x   u   x = f ( x , u )    ·    ( x (t0 )   ) x(t0)        .   :     KL     K     ( u   )  u  Lx (t )   ( x (t 0 ) , t - t 0 ) +  ( u  )        .           .   .         x (t ) &lt;  ( 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   21 2x  (2 u )13V . 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 &gt;0     Q  · x  0 : xH Q x &gt; 0:    ( ) -  17- Linear matrix inequalities (LMI) 18- Hurwitz 19- Hermitian 20- Positive definiteA12  A &lt;0 (i ) A =  11 T A12 A22    T (ii ) A22 &lt; 0 &amp; A11 - A12 A22 -1 A12 &lt; 0. U HU = U U H = I     U  ·  U   A = U  V H      Am×n          .                 V = [ d0] d = diag { 1 ,  2 ,...,  n }  or  =  d  0 1   2     n  0:     G(s)  - G ( s)= sup  1 (G ( j )). || G || &lt; +   G  H    .   G  H  G ·[6,7]   --- :         F ( x ) = F0 +  xi Fi &gt; 0i =1 m   .       Fi  F0   x  m     .  MATLAB Robust Control Toolbox       P &gt; 0   LMI   G AT P + PA PB  min  &gt; 0   BT P - I  C D   (A,B,C,D)   G(s)  :CT   DT  &lt; 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 &gt; 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 )&lt; 1   G1 , G2  H   ·:   . [4,5,6,10]    ---7- Internally stableI-G1G2   G2 1 r    G1&lt; 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 &lt;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 &lt;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)            . ||||&lt; 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 &gt;1         .H - ( j ) &lt; 1 ;  :       2H  - ( j ) =4k 2 2k 1  max H  - ( j ) = &lt; 2 2 2  ( + 10)( + ( k - 1) ) 10( k - 1) 2.k &gt; 5/3 :   k &gt;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 &gt; 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 2u x &lt; 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 &gt; 0  (1) (2):          V1 ( x) = 1 x 2   ()   23 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  ()     22  ( y )  -k y - y (k y - x + u ) , V2 4 4yk 2: V2 ( y )  - 1 k y 2 4k 4  y  x + u    2 k:      (u , x)    ()    if1 y (0)   2 = 1 k , u   2u = 16 k 2 , 2 1 x   2 x = 16 k 2y (t )   2 ( y (0), t ) +  2 ( x ) +  2 ( u )    1(s)     .  2 ( s) =4 k4s   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     &gt; 0      k k2  k min { ,  1 x ,  2 x ,  1 ,  2 } = min  ,  = , a1  a 2  2 16  2  l= d1 + d 2a12 + a2 2 a12 (1 +  )3 =  =2 4 2 k          l &lt; 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) &gt; V2 (0)    .  W (0) = [W1 (0) W2 (0) ]  T W1 (0) &gt; V1 (0)  t  [0, Tmax ) : W1 (t ) &gt; V1 (t ) &amp; W2 (t ) &gt; V2 (t )           --      .    ()     2 :         0       +- := {[ w1 , w2 ]   2 0  f1 ( w1 , w2 )  0 &amp; f 2 ( w1 , w2 )  0}   -- := {[ w1 , w2 ]   2 0  f1 ( w1 , w2 )  0 &amp; f 2 ( w1 , w2 )  0}   -+ := {[ w1 , w2 ]   2 0  f1 ( w1 , w2 )  0 &amp; 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- &lt; L+  +   -w+ w L -lim sup 1-1 (1 ( w)) = L+lim  2 -1 ( 2 ( w)) = +     L- &lt; 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 )   +   2V1 ( 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+ w1-1 (1 ( w)) =    .   L- = 1 , L+ = 2        .           --- :           .     2 0                        -- [19]               [*]       :       .            x = ax - yz    y = -by + xz  z = -cx + xy  a , b, c &gt; 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 &lt;-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 &lt; -a  V &lt; 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 &lt; -a -1ac + bc c)2,k2( &lt;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&lt; &lt; 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 &lt;  &lt; , 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 - 2x2 + y 2(ac x + bc y)) (x22+ 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 &lt; -a -1ac + bc c)2,k2( &lt;b- ( -b+ac + bc c)     2:            (, )( a+ac + bc c - 2)2+ k1 &lt; - ,ac + bc c - 2)2+ k2 &lt; - , &gt;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 &lt; r : r &gt; 0  c - .  ISS                    ()   E0         .      [*] W-B. Liu. G-R. Chen, A new chaotic system and its generation, International Journal of Bifurcation &amp; Chaos 13(1) (2003) 261-7. [**] MT. Yassen, Controlling chaos for the dynamical system of coupled dynamos, Chaos, Solitons &amp; Fractals 13 (2002) 341-52.[20]         --     ... -                  .                     .           .                             (LMI)         .      .      --- :          ()   x1 = A1 x1 + A12 x2 + B11u1    x2 = A2 x2 + A21 x1 + B22u2 A A= 1  A21A12  , 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&lt;1(2) .        ()        A           -       .    -       .           LMI      A21 = B2 C1  A12 = B1 C2       .    ()        .  A  G ( s) = D + C ( sI - A)-1 B   ([*]   ) ---      G ( s)  &lt; 1 AX + XAT  BT  X = X T &gt; 0 such that   CX  B -I D XC T   DT  &lt; 0 -I  :          0 A12 = B10C2      A =  1  A21AA12     ---  A2       A21 = B2 C1  A12 = B1 C2    A21 = B20C10 C1 ( sI1 - A1 ) -1 B1× C2 ( sI 2 - A2 ) -1 B2&lt;1      P , P2 , X , Y (all &gt; 0) such that 1  P A1T + A1 P + B10 XB10T 1 1  0 C1 P  12- Bounded real lemma0 0  P AT + A2 P2 + B2 YB2 T PC10T  1 &lt;0 &amp;  2 2  0 -Y  C2 P2  0 P2C2 T  &lt;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  1T T  P2 A2 + A2 P2 + B2 B2 PC1T  1 &lt;0 &amp;  - I1  C2 P2  T P2C2  &lt;0 -I2 :              --      C1 ( sI1 - A1 ) -1 B1&lt;1&amp;C2 ( sI 2 - A2 ) -1 B2&lt;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 &gt; 0) and Y1 , Y2 such that 1T  P A1T + A1 P + B10 X 1 B10T + B11Y1 + Y1T B11 PC10T  1 1 1  &lt;0 &amp; C10 P -X2   1 T T 0 0 0  P2 A2 + A2 P2 + B2 X 2 B2 T + B22Y2 + Y2T B22 P2C2 T   &lt;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 +-1Gn -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&lt; 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,..., Nj =1 N:         x = A( N ) x + B N u , y = C N xB N = diag [ B1 ,..., BN ] ,C N = diag [C1 ,..., C N ] A1  A1N      , A( N ) =    AN 1  AN    Ai Aij =   Bi H ij C ji= 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 H12C2 ( sI - A2 - B2 K 2 ) -1 B2-&lt; 1/  1.   A cl(2)           1,2 = C (2) ( sI - Acl (2) ) B(2)-1 -C (2) = [ H 31C1A + B K H 32C2 ] , Acl (2) =  1 1 1  B2 H 21C1B1 H12C1   B1 H13   , B(2) =  B H  A2 + B2 K 2   2 23 C3 ( sI - A3 - B3 K 3 ) -1 B3&lt; 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&lt; 1     :    P , P2 , X 1 , X 2 (all &gt; 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 &lt;0 &amp; -X2  T T T T  P2 A2 + A2 P2 + R2 X 2 R2 P2 A12 T1-T + Y2T B12 T1-T   -1 &lt;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  A21AA12         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&lt;1 is stable A2 + B22 K 2   A12--     . B12 = [3 0 1] , B21 = [ 0.5 0.4 1]   T TK12 = [ -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&lt;1A1   Acl =   A21 + B21 K 21A12 + 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 IT :  ()      V =  P   V    1  V = -2 T  + 2 T P * * ( y, z , t ) p2 T 2(4)2n -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 z2  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 ,  &gt; 0 ^ .          p .  :  ()   (i)      V1  - 1 2P20 +T0( 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 &gt; 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  22P 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) &gt; 0       v1     &gt; 0  :  1 1 2   ^ ^ ^ ^ V -  -   y 2 v1 ( y 2 ) +   y w2 -  p ( p - p ) + ( p - p )( p -  1 ) 20 +  12 ( z ) +22P2 0 (dz + d ) + 1  11 (0) 20             ()   - - wk +1  k y:     21 ( p * 1 + 1 )  1 2k +10 +   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 jj =2 j -1   wk +1  21 ^ p wk + 2 :=  k +1 -  k +1(10). w1 = 0  ck+1 &gt; 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 ) +22P20dz  1 ( z )2:        [21] 2  Vn  - cVn + 1 ( z ) +  2  1   c := min  n , 2 , 2ci ,   ; 2  i  n   2  max ( P )    2 :=2p +22P2 0 d + n  1  11 (0) 20:   0 &lt;  3 &lt; 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 &gt; 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 &gt; 0  0 &lt;  5 &lt; 1       2d + 2 ( 4 )   2    (Vn ) ,   0 -1  0 if V0 ( z )  max   0 -1  1- 5 1- 5        -  ( z )  V0 5 0K               [25]   :         ( s) &lt;1- 5 0 2   c - 3  -1   1-1    2   s  , s &gt; 0    ---             (ii)  (i)   ---         .    ()-()     .           y    pi*       d 0 =  i1 (0) =  i 2 (0) = 0   (ii)  (i)   --- s 0lim+ sup 0 ( s)s2&lt; + ,i 2 (s)2 lim sup &lt; + ,  1  i  n s 0  0 (s)+            ()     lim ( x(t ) + z (t ) +  (t ) ) = 0t :            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 ) / Kqx 3 N       x3         Kr    x1 &gt; 0  .              uN =     . x3 &gt; 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 ) ,Th( x) = x1    --- :             u= 1 Lg L2 f{- L h( x )3 fh( x) + v} ,f = ( f1 , f 2 , f3 )T1   v = -k0 ( x1 - yd ) - k1 ( f1 - yd ) - k2  (- Bm f1 + qm f 2 ) - d  + d y y Jt         k0 , k1, k2 &gt; 0      yd  :    yd = x1N = 200 rad/s     .  k1 k2 &gt; k0k0 = 5000, k1 = 5150, k2 = 151:  u= - 1 10 - x27 7( 413.65 - 45 x - 9.63 × 101-4x2 + 7.227 × 1011 x32 )x3 (1.094 ×109 x1 + 23228 x2 ) + 40413x3 10 - x2.      x(0) = 0     --- :      u = uN -Kq KrTr ( 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      (.     )   &lt;     * (. || . ||2  || . ||    ) d1 2 , d 2 2 &lt; +   d1 , d 2  L2          ) .    x = f ( x ) + g ( x )(u + d ) :             (  = 0 , d1 + d 2 = dx  d +  ; , &gt; 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 . 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