Андриевский Б.Р., Фрадков А.Л. Избранные главы теории автоматического управления
.pdft nal { ª®¥ç®¥ § 票¥ t.
y0 { ¢¥ªâ®à-á⮫¡¥æ ç «ìëå § 票©.
tol { âà¥¡ã¥¬ë© ¤®¯ãáª. 票¥ ¯® 㬮«ç ¨î: tol = 10;6.
trace { ¯à¨ ¥ã«¥¢®¬ § 票¨ ª ¦¤ë© è £ ¢ë¢®¤¨âáï ¯¥ç âì. ® 㬮«ç ¨î trace = 0.
:
T - ¯®«ãç¥ë© ¢¥ªâ®à-áâப § 票© ¢à¥¬¥¨ ( à£ã¬¥- ⠨⥣à¨à®¢ ¨ï).
Y - ¯®«ã祮¥ 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 § ¤ 묨 ¢ V ª®àﬨ. «ï ¢¥ªâ®à®¢
ROOTS ¨ POLY { ¢§ ¨¬® ®¡à âë¥ äãªæ¨¨.
y = polyval(c, x)
POLYVAL { ¢ëç¨á«¥¨¥ § ç¥¨ï ¯®«¨®¬ .
᫨ V { ¢¥ªâ®à, í«¥¬¥âë ª®â®à®£® ¥áâì ª®íää¨æ¨¥âë ¬®£®ç«¥ , â® polyval(V, s) ï¥âáï § 票¥¬ ¯®«¨®¬ , ¢ëç¨á«¥ë¬ ¢ â®çª¥ s. ᫨ S { ¬ âà¨æ ¨«¨ ¢¥ªâ®à, â® ¯®«¨®¬ ¢ëç¨á«ï¥âáï ¯à¨ ¢á¥å § 票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 |
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¬®£®ç«¥®¢ B ¨ A: |
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B(s) |
= |
r1 |
+ |
r2 |
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+ + |
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®¢ ®¥ § 票¥, ®¡à é ïáì á® § 票¥¬ X0 ª ¯¥à¢®¬ã á⮫¡æã TAB. ¥à¢ë© á⮫¡¥æ â ¡«¨æë ¤®«¦¥ ¡ëâì ¬®®â®® ¢®§à áâ î騬. ᫨ § 票¥ 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¯®«¥ ãᯥè®,
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㤮 ¨ ä §®-ç áâ®âë¥ å à ªâ¥à¨á⨪¨ (¤¨ - £à ¬¬ë ®¤¥) «¨¥©®© á¨áâ¥¬ë ¥¯à¥à뢮£® ¢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 |
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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) |
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(C.2) |
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DEN (s) |
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£¤¥ NUM ¨ DEN ᮤ¥à¦ â ª®íää¨æ¨¥âë ¯®«¨®¬®¢ ¯® ã¡ë- ¢ î騬 á⥯¥ï¬.
¬. LOGSPACE ¤«ï ¯®«ã票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 ¢¨¤ |
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Anc |
0 |
0 |
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Abar = A21 |
Ac |
Bbar = Bc |
Cbar = [ Cnc Cc ] |
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£¤¥ ¯ à |
(Ac Bc) ã¯à ¢«ï¥¬ ï ¨ Cc(sI |
; |
Ac);1Bc |
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C(sI |
; A);1B: |
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[P, G] = c2d(a, b, t) |
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C2D { ¯à¥®¡à §®¢ ¨¥ ãà ¢¥¨© á®áâ®ï¨ï ¥¯à¥à뢮© á¨áâ¥¬ë ª ¤¨áªà¥â®© ä®à¬¥.
[P, G] = c2d(A, B, T) ¯à¥®¡à §ã¥â á¨á⥬㠥¯à¥à뢮£® ¢à¥¬¥¨
x(t) = Ax(t) + Bu(t) |
(C.3) |
ª ãà ¢¥¨ï¬ á®áâ®ï¨ï ¤¨áªà¥â®© á¨á⥬ë |
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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 { ᮡáâ¢¥ë¥ ç áâ®âë ¨ ª®íää¨æ¨¥âë ¤¥¬¯ä¨à®- ¢ ¨ï.
[Wn, Z] = damp(A) ¢®§¢à é ¥â ¢¥ªâ®àë Wn ¨ Z, ᮤ¥à¦ - 騥 ᮡáâ¢¥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 ᮡáâ¢¥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
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 { á¨â¥§ ¤¨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饨© ¨ 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 ண® 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 { á¨â¥§ «¨¥©®£® ¤¨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 ⮩ á¢ï§¨ K â ªãî, çâ® § ª® ã¯à ¢«¥-
P
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 묨 ¢ ¢¥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¥à뢮¥.
[A, B] = d2c(P, G, T) ¯à¥®¡à §ã¥â ¤¨áªà¥âãî á¨á⥬ã (C.4) ª ¥¯à¥à뢮¬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㤮- ¨ 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 § 票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) |
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(C.5) |
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DEN (z) |
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£¤¥ 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 « ( - ஥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®© äãªæ¨¨ (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«¥¤®¢ ⥫ì®áâì 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 § 票© ¢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®© äãªæ¨¨ (C.5), £¤¥ NUM ¨ DEN ᮤ¥à¦ â ª®íä- ä¨æ¨¥âë ç¨á«¨â¥«ï ¨ § ¬¥ â¥«ï ¯¥à¥¤ â®ç®© äãªæ¨¨, 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«¥¤®¢ ⥫ì®áâì ¬®¬¥â®¢ ¢à¥¬¥¨, ª®â®à ï § ¤ ¥â ¢à¥¬¥ãî ®áì ¤«ï äãªæ¨¨ ¢¥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®© äãªæ¨¨ (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 { á¨â¥§ «¨¥©®£® ¨¤¥â¨ä¨ª â®à á®áâ®ï¨ï ¯® ª¢ - ¤à â¨ç®¬ã ªà¨â¥à¨î.
«ï ¥¯à¥à뢮© 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묨 ¬ âà¨æ ¬¨ ¢®§¬ã饨© ¨ 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© ä¨«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⮢ ¯¥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 { á¨â¥§ «¨¥©®£® ॣã«ïâ®à ¯® ª¢ ¤à â¨ç®¬ã ªà¨â¥à¨î ¤«ï ¥¯à¥à뢮© á¨á⥬ë.
[K, S] = lqr(A, B, Q, R) ¢ëç¨á«ï¥â ®¯â¨¬ «ìãî ¬ âà¨æã ª®-
íää¨æ¨¥â®¢ ®¡à ⮩ 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) |
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¯à¨ á¢ï§¨ x(t) = Ax(t) + Bu(t): |
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஬¥ ⮣®, ¢®§¢à é ¥âáï ¬ âà¨æ |
S { à¥è¥¨¥ ¯à¨á®¥¤¨- |
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¥®£® ãà ¢¥¨ï ¨ªª ⨠SA + AT |
S ; SBR;1BT |
S + Q = 0 |
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[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 { á¨â¥§ «¨¥©®£® ॣ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¥àë¢ëå á¨á⥬.
[K, S] = lqry(A, B, C, D, Q, R) ¢ëç¨á«ï¥â ®¯â¨¬ «ìãî ¬ -
âà¨æã ®¡à ⮩ 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) |
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¯à¨ á¢ï§¨ |
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x(t) = Ax(t) + Bu(t) |
y(t) = Cx(t) + Du(t): |
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஬¥ ⮣®, ¢®§¢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¥¢®© äãªæ¨¨.
[y, x] = lsim(a, b, c, d, u, t, x0)
LSIM { ¬®¤¥«¨à®¢ ¨¥ «¨¥©®© á¨áâ¥¬ë ¥¯à¥à뢮£® ¢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.
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