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Андриевский Б.Р., Фрадков А.Л. Избранные главы теории автоматического управления

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t nal { ª®­¥ç­®¥ §­ 祭¨¥ t.

y0 { ¢¥ªâ®à-á⮫¡¥æ ­ ç «ì­ëå §­ 祭¨©.

tol { âà¥¡ã¥¬ë© ¤®¯ãáª. ­ 祭¨¥ ¯® 㬮«ç ­¨î: tol = 10;6.

trace { ¯à¨ ­¥­ã«¥¢®¬ §­ 祭¨¨ ª ¦¤ë© è £ ¢ë¢®¤¨âáï ­ ¯¥ç âì. ® 㬮«ç ­¨î trace = 0.

:

T - ¯®«ã祭­ë© ¢¥ªâ®à-áâப §­ 祭¨© ¢à¥¬¥­¨ ( à£ã¬¥­- â ¨­â¥£à¨à®¢ ­¨ï).

Y - ¯®«ã祭­®¥ à¥è¥­¨¥, ¯® ¢¥ªâ®à-á⮫¡æã ¤«ï ª ¦¤®£® ¬®¬¥­â ¢à¥¬¥­¨.

c = poly(x)

POLY { å à ªâ¥à¨áâ¨ç¥áª¨© ¬­®£®ç«¥­.

᫨ A { n n-¬ âà¨æ , poly(A) ï¥âáï ¢¥ªâ®à®¬-áâப®© á n+1 í«¥¬¥­â ¬¨, ïî騬¨áï ª®íää¨æ¨¥­â ¬¨ å à ªâ¥- à¨áâ¨ç¥áª®£® ¬­®£®ç«¥­ det( In ; A): ᫨ V { ¢¥ªâ®à, â® poly(V) ï¥âáï ¢¥ªâ®à®¬, í«¥¬¥­âë ª®â®à®£® ¥áâì ª®íää¨- 樥­âë ¬­®£®ç«¥­ á § ¤ ­­ë¬¨ ¢ V ª®à­ï¬¨. «ï ¢¥ªâ®à®¢

ROOTS ¨ POLY { ¢§ ¨¬­® ®¡à â­ë¥ ä㭪樨.

y = polyval(c, x)

POLYVAL { ¢ëç¨á«¥­¨¥ §­ 祭¨ï ¯®«¨­®¬ .

᫨ V { ¢¥ªâ®à, í«¥¬¥­âë ª®â®à®£® ¥áâì ª®íää¨æ¨¥­âë ¬­®£®ç«¥­ , â® polyval(V, s) ï¥âáï §­ 祭¨¥¬ ¯®«¨­®¬ , ¢ëç¨á«¥­­ë¬ ¢ â®çª¥ s. ᫨ S { ¬ âà¨æ ¨«¨ ¢¥ªâ®à, â® ¯®«¨­®¬ ¢ëç¨á«ï¥âáï ¯à¨ ¢á¥å §­ 祭¨ïå S.

¬. POLYVALM ¤«ï ¢ëç¨á«¥­¨ï ¬ âà¨ç­®£® ¬­®£®ç«¥­ .

[coe s, poles, k] = residue(u, v, k)

RESIDUE { à §«®¦¥­¨¥ ­ ¯à®á⥩訥 ¤à®¡¨ ¨«¨ ¢ëç¨á«¥- ­¨¥ ¢ëç¥â®¢.

[R, P, K] = residue(B, A) ¢ëç¨á«ï¥â ¢ëç¥âë, ¯®«îá ¨ æ¥-

«ãî ç áâì (â.¥. à §«®¦¥­¨¥ ­

¯à®á⥩訥) ®â­®è¥­¨ï ¤¢ãå

¬­®£®ç«¥­®¢ B ¨ A:

 

 

 

 

 

 

B(s)

=

r1

+

r2

 

+ +

rn

+ k(s):

A(s)

s ; p1

s ; p2

s ; pn

¥ªâ®àë B ¨ A § ¤ îâ ª®íää¨æ¨¥­âë ¬­®£®ç«¥­®¢ ¯® ã¡ë- ¢ ­¨î á⥯¥­¥© s. ëç¥âë ᮤ¥à¦ âáï ¢ ¢¥ªâ®à-á⮫¡æ¥ R, ¯®«îá à ᯮ« £ îâáï ¢ ¢¥ªâ®à-á⮫¡æ¥ P, 楫 ï ç áâì { ¢ ¢¥ªâ®à-áâப¥ K.

422

[B, A] = residue(R, P, K) ¯à¥®¡à §ã¥â ¯à¥¤áâ ¢«¥­¨¥ ¢ ¢¨¤¥ ¯à®á⥩è¨å ¤à®¡¥© ®¡à â­®, ¢ B/A-ä®à¬ã.

r = roots(c)

ROOTS { ¢ëç¨á«ï¥â ª®à­¨ ¯®«¨­®¬ .

roots(C) ¢лз¨б«п¥в ª®а­¨ ¯®«¨­®¬ , ª®ндд¨ж¨¥­вл ª®в®а®- £® п¢«повбп н«¥¬¥­в ¬¨ ¢¥ªв®а C. б«¨ C ¨¬¥¥в N+1 ª®¬-

¯®­¥­âë, ¯®«¨­®¬ ¨¬¥¥â ¢¨¤ C1XN + : : : + CN X + CN+1:¬. â ª¦¥ ROOTS1 ¨ POLY.

y = table1(tab, x0)

TABLE1 { ¯®¨áª ¯® â ¡«¨æ¥.

Y = table1(TAB, X0) ¯® â ¡«¨æ¥ áâப TAB ­ 室¨â «¨­- ¥©­®-¨­â¥à¯®«¨à®¢ ­­®¥ §­ 祭¨¥, ®¡à é ïáì á® §­ 祭¨¥¬ X0 ª ¯¥à¢®¬ã á⮫¡æã TAB. ¥à¢ë© á⮫¡¥æ â ¡«¨æë ¤®«¦¥­ ¡ëâì ¬®­®â®­­® ¢®§à áâ î騬. ᫨ §­ 祭¨¥ X0 ¢ë室¨â

§£à ­¨æë ¯¥à¢®£® á⮫¡æ TAB. X0 ¬®¦¥â ¡ëâì ¢¥ªâ®à®¬.¬. â ª¦¥ TABLE2.

à®£à ¬¬ë ¨áá«¥¤®¢ ­¨ï á¨á⥬ ã¯à ¢«¥­¨ï

X = are(F, G, H)

ARE { ¥è¥­¨¥ «£¥¡à ¨ç¥áª®£® ãà ¢­¥­¨ï ¨ªª â¨

X = are(F, G, H) ¢®§¢à é ¥â à¥è¥­¨¥ (¥á«¨ ®­® ¨¬¥¥âáï) «£¥¡à ¨ç¥áª®£® ãà ¢­¥­¨ï ¨ªª â¨:

FT X + XF ; XGX + H = 0

£¤¥ ¬ âà¨æë G=GT 0, H=HT :

[Ab, Bb, Cb]=balreal(A, B, C)

BALREAL { á¡ « ­á¨à®¢ ­­ ï ॠ«¨§ æ¨ï ãà ¢­¥­¨© á®- áâ®ï­¨ï ¨ ¯®­¨¦¥­¨¥ ¯®à浪 ¬®¤¥«¨. 1

[Ab, Bb, Cb] = balreal(A, B, C) ¢®§¢à é ¥â á¡ « ­á¨à®¢ ­- ­ãî ॠ«¨§ æ¨î ãà ¢­¥­¨© á®áâ®ï­¨ï á¨á⥬ë (A B C):

[Ab, Bb, Cb, G, T] = balreal(A, B, C) â ª¦¥ ¢®§¢à é -

¥â ¢¥ªâ®à G, ᮤ¥à¦ 騩 ¤¨ £®­ «ì­ë¥ í«¥¬¥­âë £à ¬¨ - ­ á¡ « ­á¨à®¢ ­­®© ॠ«¨§ 樨 ¨ ¬ âà¨æã T ¯à¥®¡à §®¢ - ­¨ï ¯®¤®¡¨ï, ¨á¯®«ì§®¢ ­­ãî ¯à¨ ¯à¥®¡à §®¢ ­¨¨ (A B C) ª (Ab Bb Cb): ᫨ ­®à¬ «¨§ æ¨ï á¨áâ¥¬ë ¢ë¯®«­¥­ ãᯥ譮,

1 ®¢®àïâ, çâ® ª¢ ¤à â­ ï ¬ âà¨æ á¡ « ­á¨à®¢ ­ , ¥á«¨ ã ­¥¥, ­ - ᪮«ìª® ¢®§¬®¦­®, ­®à¬ë ¯® áâப ¬ ¨ á⮫¡æ ¬ ᮢ¯ ¤ îâ [53].

423

¬ «ë¥ í«¥¬¥­âë £à ¬¨ ­ G ¯®ª §ë¢ îâ í«¥¬¥­âë, ª®â®àë¥ ¬®£ãâ ¡ëâì ¨áª«îç¥­ë ¯à¨ ¯®­¨¦¥­¨¨ ¯®à浪 ¬®¤¥«¨.

[mag, phase] = bode(a, b, c, d, iu, w)

BODE { ¬¯«¨â㤭® ¨ ä §®-ç áâ®â­ë¥ å à ªâ¥à¨á⨪¨ (¤¨ - £à ¬¬ë ®¤¥) «¨­¥©­®© á¨áâ¥¬ë ­¥¯à¥à뢭®£® ¢à¥¬¥­¨.

[MAG, PHASE] = bode(A, B, C, D, iu, W) ¢ëç¨á«ï¥â ç áâ®â-

­ãî å à ªâ¥à¨á⨪ã á¨á⥬ë

 

x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t)

(C.1)

®â i-£® ¢å®¤ ¯à¨ s = |!. ¥ªâ®à W ¤®«¦¥­ ᮤ¥à¦ âì §­ ç¥- ­¨ï ç áâ®â (¢ à ¤¨ ­ å), ¤«ï ª®â®àëå âॡã¥âáï ¢ëç¨á«¨âì ¤¨ £à ¬¬ë ®¤¥. BODE ¢ëç¨á«ï¥â ¬ âà¨æë MAG ¨ PHASE (¢ £à ¤ãá å), ç¨á«® á⮫¡æ®¢ ª®â®àëå à ¢­® à §¬¥à­®á⨠¢ë- 室 y, á length(W) áâப ¬¨.

[MAG, PHASE] = bode(NUM, DEN, W) ¢ëç¨á«ï¥â ç áâ®â- ­ãî å à ªâ¥à¨á⨪ã á¨á⥬ë, § ¤ ­­®© ¯¥à¥¤ â®ç­®© äã­ª- 樥©

G(s) =

NUM(s)

 

(C.2)

DEN (s)

 

 

 

£¤¥ NUM ¨ DEN ᮤ¥à¦ â ª®íää¨æ¨¥­âë ¯®«¨­®¬®¢ ¯® ã¡ë- ¢ î騬 á⥯¥­ï¬.

¬. LOGSPACE ¤«ï ¯®«ã祭¨ï ç áâ®â ¢ «®£ à¨ä¬¨ç¥áª®¬

¬áèâ ¡¥.

Co=ctrb(A, B)

CTRB { ä®à¬¨à®¢ ­¨¥ ¬ âà¨æë ã¯à ¢«ï¥¬®áâ¨. ctrb(A, B) ¢®§¢à é ¥â ¬ âà¨æã ã¯à ¢«ï¥¬®á⨠Co = [ B, AB, A2B, : : : , An;1B ].

[Abar, Bbar, Cbar, T, K] = ctrbf(A, B, C)

CTRBF { âà¥ã£®«ì­ ï ä®à¬ ã¯à ¢«ï¥¬®áâ¨.

[Abar, Bbar, Cbar, T, K] = ctrbf(A, B, C) ¢®§¢à é ¥â à §- ¡¨¥­¨¥ ­ ¯®¤¯à®áâà ­á⢠ã¯à ¢«ï¥¬ëå ¨ ­¥ã¯à ¢«ï¥¬ëå á®áâ®ï­¨©.

[Abar, Bbar, Cbar, T, K] = ctrbf(A, B, C, TOL) ¨á¯®«ì§ã¥â â®ç­®áâì TOL.

᫨ ¬ âà¨æ ã¯à ¢«ï¥¬®á⨠Co(A B) ¨¬¥¥â rank r n, â® ¨¬¥¥âáï ¯à¥®¡à §®¢ ­¨¥ ¯®¤®¡¨ï T â ª®¥, çâ®

424

Abar = T AT 0 Bbar = T B Cbar = CT 0 (T0 = T ;1) ¨ ¯à¥®¡à §®-

¢ ­­ ï á¨á⥬

¨¬¥¥â ¢¨¤

 

 

 

 

Anc

0

0

 

 

 

Abar = A21

Ac

Bbar = Bc

Cbar = [ Cnc Cc ]

£¤¥ ¯ à

(Ac Bc) ã¯à ¢«ï¥¬ ï ¨ Cc(sI

;

Ac);1Bc

 

C(sI

; A);1B:

 

 

[P, G] = c2d(a, b, t)

 

 

 

 

C2D { ¯à¥®¡à §®¢ ­¨¥ ãà ¢­¥­¨© á®áâ®ï­¨ï ­¥¯à¥à뢭®© á¨áâ¥¬ë ª ¤¨áªà¥â­®© ä®à¬¥.

[P, G] = c2d(A, B, T) ¯à¥®¡à §ã¥â á¨á⥬㠭¥¯à¥à뢭®£® ¢à¥¬¥­¨

x(t) = Ax(t) + Bu(t)

(C.3)

ª ãà ¢­¥­¨ï¬ á®áâ®ï­¨ï ¤¨áªà¥â­®© á¨á⥬ë

 

x[n + 1] = P x[n] + G u[n]

(C.4)

¯à¨ íªáâà ¯®«ïâ®à¥ ­ã«¥¢®£® ¯®à浪 ¢å®¤­®£® ᨣ­ «

¨ ¨­-

â¥à¢ «¥ ¤¨áªà¥â­®á⨠(¢ë¡®à®ç­®¬ ¨­â¥à¢ «¥, sample time) T.

[Wn, Z] = damp(A)

DAMP { ᮡá⢥­­ë¥ ç áâ®âë ¨ ª®íää¨æ¨¥­âë ¤¥¬¯ä¨à®- ¢ ­¨ï.

[Wn, Z] = damp(A) ¢®§¢à é ¥â ¢¥ªâ®àë Wn ¨ Z, ᮤ¥à¦ - 騥 ᮡá⢥­­ë¥ ç áâ®âë ¨ ª®íää¨æ¨¥­âë ¤¥¬¯ä¨à®¢ ­¨ï à£ã¬¥­â A. ¥à¥¬¥­­ ï A ¬®¦¥â ¨¬¥âì ®¤¨­ ¨§ á«¥¤ãîé¨å ä®à¬ ⮢:

1)¥á«¨ A ª¢ ¤à â­ ï, â® ¯à¥¤¯®« £ ¥âáï, çâ® íâ® ¬ âà¨æ "A" ãà ¢­¥­¨© á®áâ®ï­¨ï\

2)¥á«¨ A { ¢¥ªâ®à-áâப , â® ¯à¥¤¯®« £ ¥âáï, çâ® íâ® ¢¥ª- â®à ª®íää¨æ¨¥­â®¢ å à ªâ¥à¨áâ¨ç¥áª®£® ¬­®£®ç«¥­ á¨áâ¥- ¬ë\

3)¥á«¨ A { ¢¥ªâ®à-á⮫¡¥æ, â® ¯à¥¤¯®« £ ¥âáï, çâ® íâ® ¢¥ª- â®à ª®à­¥© å à ªâ¥à¨áâ¨ç¥áª®£® ¬­®£®ç«¥­ \

® ¢á¥å á«ãç ïå, DAMP ¢®§¢à é ¥â ᮡá⢥­­ë¥ ç áâ®âë

¨ª®íää¨æ¨¥­âë ¤¥¬¯ä¨à®¢ ­¨ï á¨á⥬ë.

G = dgram(A, B)

DGRAM { £à ¬¨ ­ë ã¯à ¢«ï¥¬®á⨠¨ ­ ¡«î¤ ¥¬®á⨠¤¨á- ªà¥â­®© á¨á⥬ë.

425

­¨ï u = ;Kx ¬¨­¨¬¨§¨àã¥â äã­ªæ¨î ¯®â¥àì J = ¯à¨ ãà ¢­¥­¨¨ á¢ï§¨
x[n + 1] = Ax[n] + Bu[n]:
426
xT Qx + uT Ru
[K, S] = dlqr(A, B, Q, R)

dgram(A, B) ¢®§¢à é ¥â £à ¬¨ ­ ã¯à ¢«ï¥¬®á⨠¤¨áªà¥â- ­®© á¨á⥬ë.

dgram(A', C') ¢®§¢à é ¥â £à ¬¨ ­ ­ ¡«î¤ ¥¬®á⨠¤¨áªà¥â- ­®© á¨á⥬ë

¬. â ª¦¥ GRAM.

[L, M, P] = dlqe(A, G, C, Q, R)

DLQE { ᨭ⥧ ¤¨áªà¥â­®£® «¨­¥©­®£® «£®à¨â¬ ®æ¥­¨¢ - ­¨ï ¯® ª¢ ¤à â¨ç­®¬ã ªà¨â¥à¨î.

«ï ¤¨áªà¥â­®© á¨á⥬ë

x[n + 1] = Ax[n] + Bu[n] + Gw[n] ; ãà ¢­¥­¨ï á®áâ®ï­¨ï

z[n] = Cx[n] + Du[n] + v[n] ; ãà ¢­¥­¨ï ¨§¬¥à¥­¨© á ¬®¬¥­â ¬¨ ¢®§¬ã饭¨© ¨ è㬮¢ ¨§¬¥à¥­¨ï:

Efwg = Efvg = 0 EfwwT g = Q EfvvT g = R

äã­ªæ¨ï dlqe(A, G, C, Q, R) ¢®§¢à é ¥â ¬ âà¨æã ª®íää¨æ¨¥­- ⮢ L ¤¨áªà¥â­®£® áâ 樮­ à­®£® 䨫ìâà «¬ ­ :

{ ãà ¢­¥­¨¥ á®áâ®ï­¨ï

x[n + 1] = Ax [n] + Bu[n] { ãà ¢­¥­¨¥ ¨§¬¥à¥­¨©

x [n] = x[n] + L(z[n] ; Hx[n] ; Du[n]):

¨«ìâà «¬ ­ ¢ëà ¡ âë¢ ¥â ®¯â¨¬ «ì­ãî ¯® «¨­¥©­®¬ã ª¢ ¤à â¨ç­®¬ã ªà¨â¥à¨î ( ) ®æ¥­ªã x á®áâ®ï­¨ï x:

[L, M, P] = dlqe(A, G, C, Q, R) ¢®§¢à é ¥â ¬ âà¨æã ª®íä- ä¨æ¨¥­â®¢ ¯¥à¥¤ ç¨ L, à¥è¥­¨¥ «£¥¡à ¨ç¥áª®£® ãà ¢­¥­¨ï¨ªª ⨠M ¨ ª®¢ ਠ樮­­ãî ¬ âà¨æ㠮訡®ª ®æ¥­¨¢ ­¨ï ¯® १ã«ìâ â ¬ ¨§¬¥à¥­¨© P = Ef(x ; x)(x ; x)T g:

DLQR { ᨭ⥧ «¨­¥©­®£® ¤¨áªà¥â­®£® ॣã«ïâ®à ¯® ª¢ - ¤à â¨ç­®¬ã ªà¨â¥à¨î.

[K, S] = dlqr(A, B, Q, R) ¢ëç¨á«ï¥â ®¯â¨¬ «ì­ãî ¬ âà¨æã ª®íää¨æ¨¥­â®¢ ®¡à â­®© á¢ï§¨ K â ªãî, çâ® § ª®­ ã¯à ¢«¥-

P

AXAT + C = X:

ª¦¥ ¢®§¢à é ¥âáï ãáâ ­®¢¨¢è¥¥áï à¥è¥­¨¥ S ¯à¨á®¥¤¨­¥­- ­®£® ¤¨áªà¥â­®£® ¬ âà¨ç­®£® ãà ¢­¥­¨ï ¨ªª â¨

S ; AT SA + AT SB;1 (R + BT SB)BST A ; Q = 0:

X = dlyap(A, C)

DLYAP { à¥è¥­¨¥ ¤¨áªà¥â­®£® ãà ¢­¥­¨ï ï¯ã­®¢ .

X = dlyap(A, C) ­ 室¨â à¥è¥­¨¥ ¤¨áªà¥â­®£® ãà ¢­¥­¨ïï¯ã­®¢

¬. â ª¦¥ LYAP.

[Ab, Bb, Cb, Db] = dmodred(A, B, C, D, ELIM)

DMODRED { ¯®­¨¦¥­¨¥ ¯®à浪 ¤¨áªà¥â­®© ¬®¤¥«¨.

[Ab, Bb, Cb, Db] = dmodred(A, B, C, D, ELIM) 㬥­ìè ¥â ¯®- à冷ª ¬®¤¥«¨, ¯à¥­¥¡à¥£ ï ª®¬¯®­¥­â ¬¨ ¢¥ªâ®à á®áâ®ï­¨ï, ®¯¨á ­­ë¬¨ ¢ ¢¥ªâ®à¥ ELIM. ¥ªâ®à á®áâ®ï­¨ï à §¤¥«ï¥âáï ­ X1, ª®â®àë© á®åà ­ï¥âáï, ¨ X2, ª®â®àë© ®¯ã᪠¥âáï,

A11

A12

B1

C = [ C1 C2 ]

A = A21

A22

B = B2

x[n + 1] = Ax[n] + Bu[n] y[n] = Cx[n] + Du[n]:

X2[n+1] ¯а¨а ¢­¨¢ ¥вбп X2[n], ¨ ¯®«гз¥­­л¥ га ¢­¥­¨п а §а¥и овбп ®в­®б¨в¥«м­® X1. ®«гз¥­­ п б¨бв¥¬ ¨¬¥¥в ­ LENGTH(ELIM) ¬¥­миго а §¬¥а­®бвм ¢¥ªв®а б®бв®п­¨п ¨ ¬®¦¥в а бб¬ ва¨¢ вмбп ª ª б¨бв¥¬ , г ª®в®а®© б®бв®п­¨п ELIM ¨§¬¥­повбп ¬£­®¢¥­­®.

¬. â ª¦¥ DBALREAL, BALREAL ¨ MODRED

[a, b] = d2c(phi, gamma, t)

D2C { ¯à¥®¡à §®¢ ­¨¥ ãà ¢­¥­¨© á®áâ®ï­¨ï ¨§ ¤¨áªà¥â­®-

£® ¢à¥¬¥­¨ ¢ ­¥¯à¥à뢭®¥.

[A, B] = d2c(P, G, T) ¯à¥®¡à §ã¥â ¤¨áªà¥â­ãî á¨á⥬ã (C.4) ª ­¥¯à¥à뢭®¬ã ¢¨¤ã ãà ¢­¥­¨© á®áâ®ï­¨ï (C.3) ¯à¨ íªáâà - ¯®«ïâ®à¥ ­ã«¥¢®£® ¯®à浪 ¢å®¤­®£® ᨣ­ « ¨ ¨­â¥à¢ «¥ ¤¨áªà¥â­®á⨠T.

[mag, phase] = dbode(a, b, c, d, iu, w)

DBODE { ¬¯«¨â㤭®- ¨ ä §®-ç áâ®â­ë¥ å à ªâ¥à¨á⨪¨ (¤¨ £à ¬¬ë ®¤¥) «¨­¥©­®© á¨áâ¥¬ë ¤¨áªà¥â­®£® ¢à¥¬¥­¨.

427

[MAG, PHASE] = dbode(A, B, C, D, iu, W) ¢ëç¨á«ï¥â ç áâ®â- ­ãî å à ªâ¥à¨á⨪ã á¨á⥬ë (C.1) ®â i-£® ¢å®¤ ¯à¨ z = e|!:¥ªâ®à W ¤®«¦¥­ ᮤ¥à¦ âì §­ 祭¨ï ç áâ®â (¢ à ¤¨ ­ å), ¤«ï ª®â®àëå âॡã¥âáï ¢ëç¨á«¨âì ¤¨ £à ¬¬ë ®¤¥. ¡ëç­® ! : DBODE ¢ëç¨á«ï¥â ¬ âà¨æë MAG ¨ PHASE (¢ £à ¤ã- á å), ç¨á«® á⮫¡æ®¢ ª®â®àëå à ¢­® à §¬¥à­®á⨠¢ë室 y ¨ length(W) áâப.

[MAG, PHASE] = dbode(NUM, DEN, W) ¢ëç¨á«ï¥â ç áâ®â- ­ãî å à ªâ¥à¨á⨪ã á¨á⥬ë, § ¤ ­­®© ¯¥à¥¤ â®ç­®© äã­ª-

樥©

G(z) =

NUM(z)

 

(C.5)

DEN (z)

 

 

 

£¤¥ NUM ¨ DEN ᮤ¥à¦ â ª®íää¨æ¨¥­âë ¯®«¨­®¬®¢ ¯® ã¡ë- ¢ î騬 á⥯¥­ï¬.

[y, x] = dimpulse(a, b, c, d, iu, n)

DIMPULSE { äã­ªæ¨ï ¢¥á (¨¬¯ã«ìá­ ï äã­ªæ¨ï) ¤¨áªà¥â- ­®© «¨­¥©­®© á¨á⥬ë.

Y = dimpulse(A, B, C, D, iu, n) ¢ëç¨á«ï¥â ॠªæ¨î á¨á⥬ë x[n + 1] = Ax[n] + Bu[n] y[n] = Cx[n] + Du[n] (C.6)

­ ®¤¨­®ç­ë© ¯à®¡­ë© ᨣ­ « ( - à®­¥ª¥à ), ¯à¨«®¦¥­­ë© ª

i-¬ã ¢å®¤ã. ¥«®¥ n § ¤ ¥â ç¨á«® â®ç¥ª, ¤«ï ª®â®àëå á«¥¤ã¥â ¯®«ãç¨âì äã­ªæ¨î ¢¥á . DIMPULSE ¢ëç¨á«ï¥â ¬ âà¨æã Y , ç¨á«® á⮫¡æ®¢ ª®â®à®© à ¢­® à §¬¥à­®á⨠¢ë室 y ¨ n áâப.

[Y, X] = dimpulse(A, B, C, D, iu, n) â ª¦¥ ¢®§¢à é ¥â ¯à®æ¥áá ¨§¬¥­¥­¨ï á®áâ®ï­¨ï á¨á⥬ë.

Y = dimpulse(NUM, DEN, n) ¢ëç¨á«ï¥â äã­ªæ¨î ¢¥á ¯® ®¯¨á ­¨î á¨áâ¥¬ë ¢ ¢¨¤¥ ¯¥à¥¤ â®ç­®© ä㭪樨 (C.5), £¤¥ NUM , DEN ᮤ¥à¦ â à ᯮ«®¦¥­­ë¥ ¯® ã¡ë¢ ­¨î ª®íää¨- 樥­âë ¯®«¨­®¬®¢.

[y, x] = dlsim(a, b, c, d, u, x0)

DLSIM { ¬®¤¥«¨à®¢ ­¨¥ ¤¨áªà¥â­®© «¨­¥©­®© á¨á⥬ë

Y = dlsim(A, B, C, D, U) ¢ëç¨á«ï¥â ॠªæ¨î á¨á⥬ë(C.6) ­ ¢å®¤­ãî ¯®á«¥¤®¢ ⥫쭮áâì U. âà¨æ U ¤®«¦­ ᮤ¥à- ¦ âì â ª®¥ ª®«¨ç¥á⢮ á⮫¡æ®¢, ª ª®¢ à §¬¥à­®áâì ¢å®¤ u. ¦¤ ï áâப U ᮮ⢥âáâ¢ã¥â ­®¢®¬ã ¬®¬¥­â㠢६¥­¨. DLSIM ä®à¬¨àã¥â ¬ âà¨æã Y, ç¨á«® á⮫¡æ®¢ ª®â®à®© à ¢­® ª®«¨ç¥áâ¢ã §­ 祭¨© ¢ë室 y ¨ ç¨á«® LENGTH(U) áâப.

428

[Y, X] = dlsim(A, B, C, D, U) â ª¦¥ ¢®§¢à é ¥â ¯à®æ¥áá ¨§- ¬¥­¥­¨ï á®áâ®ï­¨ï á¨á⥬ë.

dlsim(A, B, C, D, U, X0) ¬®¦­® ¨á¯®«ì§®¢ âì, ¥á«¨ § ¤ ­ë (­¥­ã«¥¢ë¥) ­ ç «ì­ë¥ ãá«®¢¨ï.

Y = dlsim(NUM, DEN, U) ¢ëç¨á«ï¥â ¯¥à¥å®¤­ë© ¯à®æ¥áá ¢ «¨­¥©­®© ¤¨áªà¥â­®© á¨á⥬¥, § ¤ ­­®© ¯¥à¥¤ â®ç­®© äã­ª- 樥© (C.5), £¤¥ NUM ¨ DEN ᮤ¥à¦ â à ᯮ«®¦¥­­ë¥ ¯® ã¡ë- ¢ ­¨î ª®íää¨æ¨¥­âë ¯®«¨­®¬®¢.

dlsim(NUM, DEN, U) íª¢¨¢ «¥­â­® lter(NUM, DEN, U).

[y, x] = dstep(a, b, c, d, iu, n)

DSTEP { ¯¥à¥å®¤­ ï äã­ªæ¨ï ¤¨áªà¥â­®© «¨­¥©­®© á¨áâ¥- ¬ë.

Y = dstep(A, B, C, D, iu, n) ¢ëç¨á«ï¥â ¯¥à¥å®¤­ãî äã­ªæ¨î á¨á⥬ë (C.6) ¯® i-¬ã ¢å®¤ã. ¥«®¥ n ¯®ª §ë¢ ¥â âॡ㥬®¥ ç¨á«® â®ç¥ª (è £®¢). DSTEP ¢ëç¨á«ï¥â ¬ âà¨æã Y, ç¨á«® á⮫¡æ®¢ ª®â®à®© à ¢­® à §¬¥à­®á⨠¢ë室 y, ç¨á«® áâப à ¢­® n.

[Y, X] = dstep(A, B, C, D, iu, n) â ª¦¥ ¢®§¢à é ¥â ¯à®æ¥áá

¨§¬¥­¥­¨ï ¢¥ªâ®à á®áâ®ï­¨ï á¨á⥬ë.

Y = dstep(NUM, DEN, n) ¢ëç¨á«ï¥â ¯¥à¥å®¤­ë© ¯à®æ¥áá ¯® ¯¥à¥¤ â®ç­®© ä㭪樨 (C.5), £¤¥ NUM ¨ DEN ᮤ¥à¦ â ª®íä- ä¨æ¨¥­âë ç¨á«¨â¥«ï ¨ §­ ¬¥­ â¥«ï ¯¥à¥¤ â®ç­®© ä㭪樨, à ᯮ«®¦¥­­ë¥ ¢ ¯®à浪¥ ã¡ë¢ ­¨ï á⥯¥­¥©.

[y, x] = impulse(a, b, c, d, iu, t)

IMPULSE { äã­ªæ¨ï ¢¥á (¨¬¯ã«ìá­ ï äã­ªæ¨ï) ­¥¯à¥àë¢- ­®© «¨­¥©­®© á¨á⥬ë.

Y = impulse(A, B, C, D, iu, T) ¢ëç¨á«ï¥â äã­ªæ¨î ¢¥á á¨-

á⥬ë (C.1) ¯® ®â­®è¥­¨î ª i-¬ã ¢å®¤ã. ¥ªâ®à T ¤®«¦¥­ á®- ¤¥à¦ âì ॣã«ïà­ãî ¯®á«¥¤®¢ ⥫쭮áâì ¬®¬¥­â®¢ ¢à¥¬¥­¨, ª®â®à ï § ¤ ¥â ¢à¥¬¥­­ãî ®áì ¤«ï ä㭪樨 ¢¥á . IMPULSE ä®à¬¨àã¥â ¬ âà¨æã Y ᮤ¥à¦ éãî á⮫쪮 á⮫¡æ®¢, ª ª®¢ à §¬¥à­®áâì ¢¥ªâ®à y ¨ LENGTH(T) áâப.

[Y, X] = impulse(A, B, C, D, iu, T) â ª¦¥ ¢®§¢à é ¥â ¯à®æ¥áá

¨§¬¥­¥­¨ï á®áâ®ï­¨ï á¨á⥬ë.

Y = impulse(NUM, DEN, T) ¢ëç¨á«ï¥â äã­ªæ¨î ¢¥á ¯® ®¯¨- á ­¨î á¨áâ¥¬ë ¢ ¢¨¤¥ ¯¥à¥¤ â®ç­®© ä㭪樨 (C.2), £¤¥ NUM ¨ DEN ᮤ¥à¦ â ª®íää¨æ¨¥­âë ¬­®£®ç«¥­®¢ ¯® ã¡ë¢ î騬 á⥯¥­ï¬.

429

[L, P] = lqe(A, G, C, Q, R)

LQE { ᨭ⥧ «¨­¥©­®£® ¨¤¥­â¨ä¨ª â®à á®áâ®ï­¨ï ¯® ª¢ - ¤à â¨ç­®¬ã ªà¨â¥à¨î.

«ï ­¥¯à¥à뢭®© á¨á⥬ë:

x(t) = Ax(t) + Bu(t) + Gw(t)

- ãà ¢­¥­¨¥ á®áâ®ï­¨ï

z(t) = Cx(t) + Du(t) + v(t)

- ãà ¢­¥­¨¥ ¨§¬¥à¥­¨©

á ª®¢ ਠ樮­­ë¬¨ ¬ âà¨æ ¬¨ ¢®§¬ã饭¨© ¨ è㬮¢ ¨§¬¥à¥- ­¨ï:

Efw(t)g = Efv(t)g = 0 Efw(t)wT (t)g = QEfv(t)vT (t)g = R

äã­ªæ¨ï

lqe(A, G, C, Q, R) ¢®§¢à é ¥â ¬ âà¨æã ª®íää¨æ¨¥­â®¢ ¯¥- । ç¨ L â ªãî, çâ® áâ 樮­ à­ë© 䨫ìâà «¬ ­

x^(t) = Ax^(t) + Bu(t) + L(z(t) ; Hx^(t) ; Du(t))

¢ëà ¡ âë¢ ¥â «¨­¥©­ãî, ®¯â¨¬ «ì­ãî ¯® ª¢ ¤à â¨ç­®¬ã ªà¨â¥à¨î ®æ¥­ªã x(t).

[L, P] = lqe(A, G, C, Q, R) { ¢®§¢à é ¥â ¬ âà¨æã ª®íää¨- 樥­â®¢ ¯¥à¥¤ ç¨ L ¨ à¥è¥­¨¥ «£¥¡à ¨ç¥áª®£® ãà ¢­¥­¨ï¨ªª ⨠P, ª®â®à®¥ ï¥âáï ¬ âà¨æ¥© ª®¢ ਠ権 ®è¨¡ª¨ ®æ¥­¨¢ ­¨ï.

[K, S] = lqr(A, B, Q, R, N)

LQR { ᨭ⥧ «¨­¥©­®£® ॣã«ïâ®à ¯® ª¢ ¤à â¨ç­®¬ã ªà¨â¥à¨î ¤«ï ­¥¯à¥à뢭®© á¨á⥬ë.

[K, S] = lqr(A, B, Q, R) ¢ëç¨á«ï¥â ®¯â¨¬ «ì­ãî ¬ âà¨æã ª®-

íää¨æ¨¥­â®¢ ®¡à â­®© á¢ï§¨ K â ªãî, çâ® § ª®­ ã¯à ¢«¥­¨ï u(t) = ;Kx(t) ¬¨­¨¬¨§¨àã¥â äã­ªæ¨î ¯®â¥àì

J = Z0

1

(xT (t)Qx(t) + uT (t)Ru(t))dt

(C.7)

¯à¨ á¢ï§¨ x(t) = Ax(t) + Bu(t):

 

 

஬¥ ⮣®, ¢®§¢à é ¥âáï ¬ âà¨æ

S { à¥è¥­¨¥ ¯à¨á®¥¤¨-

­¥­­®£® ãà ¢­¥­¨ï ¨ªª ⨠SA + AT

S ; SBR;1BT

S + Q = 0

[K, S] = lqr(A, B, Q, R, N) ¢ª«îç ¥â ¯¥à¥ªà¥áâ­ãî á®áâ ¢«ï-

îéãî 2xT (t)Nu(t), ª®â®à ï á¢ï§ë¢ ¥â u ¨ x ¢ 楫¥¢®© äã­ª- 樨.

430

а®£а ¬¬ ®бгй¥бв¢«п¥в а §«®¦¥­¨¥ ¯® б®¡бв¢¥­­л¬ ¢¥ª- в®а ¬. ®¦­® ¨б¯®«м§®¢ вм ­ «®£¨з­го ¯а®ж¥¤гаг [K, S] = lqr2(A, B, Q, R, N), ¨б¯®«м§гойго «£®а¨в¬ га ¤«п а¥- и¥­¨п «£¥¡а ¨з¥бª®£® га ¢­¥­¨п ¨ªª в¨ ¨ з¨б«¥­­® ¡®«¥¥ ¤®бв®¢¥а­ , з¥¬ LQR.

[K, S] = lqry(A, B, Q, R, N)

LQRY { ᨭ⥧ «¨­¥©­®£® ॣã«ïâ®à ª¢ ¤à â¨ç­®¬ã ªà¨- â¥à¨î á ã¯à ¢«¥­¨¥¬ ¯® ¢ë室㠨 ª¢ ¤à â¨ç­®¬ã ªà¨â¥à¨î ¤«ï ­¥¯à¥à뢭ëå á¨á⥬.

[K, S] = lqry(A, B, C, D, Q, R) ¢ëç¨á«ï¥â ®¯â¨¬ «ì­ãî ¬ -

âà¨æã ®¡à â­®© á¢ï§¨ K â ªãî, çâ® § ª®­ ã¯à ¢«¥­¨ï u(t) = ;Ky(t) ¬¨­¨¬¨§¨àã¥â 楫¥¢ãî äã­ªæ¨î

J = Z0

1

(yT (t)Qy(t) + uT (t)Ru(t))dt

(C.8)

¯à¨ á¢ï§¨

 

 

 

 

x(t) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t):

 

஬¥ ⮣®, ¢®§¢à é ¥âáï ¬ âà¨æ

S { à¥è¥­¨¥ ¯à¨á®¥¤¨­¥­-

­®£® ãà ¢­¥­¨ï ¨ªª ⨠SA + AT S ; SBR;1BT S + Q = 0

[K, S] = lqry(A, B, Q, R, N) ¢ª«оз ¥в ¯¥а¥ªа¥бв­го б®- бв ¢«пойго 2yT (t)Nu(t), ª®â®à ï á¢ï§ë¢ ¥â u ¨ y ¢ 楫¥¢®© ä㭪樨.

[y, x] = lsim(a, b, c, d, u, t, x0)

LSIM { ¬®¤¥«¨à®¢ ­¨¥ «¨­¥©­®© á¨áâ¥¬ë ­¥¯à¥à뢭®£® ¢à¥¬¥­¨ ¯à¨ ¯à®¨§¢®«ì­®¬ ¢å®¤¥.

lsim(A, B, C, D, U, T) ¢ëç¨á«ï¥â ¨ ¢ë¢®¤¨â ॠªæ¨î á¨- á⥬ë (C.1) ­ ¢å®¤­®© ¯à®æ¥áá U. âà¨æ U ¤®«¦­ ¨¬¥âì ç¨á«® á⮫¡æ®¢, ᮢ¯ ¤ î饥 á à §¬¥à­®áâìî ¢¥ªâ®à ¢å®¤

u. ¦¤ ï áâப U ᮮ⢥âáâ¢ã¥â ­®¢®¬ã ¬®¬¥­â㠢६¥­¨,

¨ U ¤®«¦­ ¨¬¥âì length(T) áâப.

Y=lsim(A, B, C, D, U, T) ¢ëç¨á«ï¥â (¡¥§ ¢ë¢®¤ ) ¬ âà¨æã Y, ç¨á«® á⮫¡æ®¢ ª®â®à®© à ¢­® à §¬¥à­®á⨠¢ë室 y ¨ length(T) áâப.

[Y, X] = lsim(A, B, C, D, U, T) â ª¦¥ ¢®§¢à é ¥â ¯à®æ¥áá

¨§¬¥­¥­¨ï á®áâ®ï­¨ï á¨á⥬ë.

lsim(A, B, C, D, U, T, X0) ¬®¦­® ¨á¯®«ì§®¢ âì, ¥á«¨ § ¤ ­ë ­¥­ã«¥¢ë¥ ­ ç «ì­ë¥ ãá«®¢¨ï.

431