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2 MATLAB/Simulink による現代制御入門サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行当時のものです.

4 i MATLAB/Simulink MATLAB/Simulink MATLAB/Simulink MATLAB/Simulink M Simulink ( 4. MATLAB/Simulink 5. (LMI) MATLAB/Simulink

5 ii OS Windows 7 (64bit) MATLAB MATLAB Version 7.11 (R21b) Simulink Version 7.6 (R21b) Control System Toolbox Version 9. (R21b) (.1) Symbolic Math Toolbox Version 5.5 (R21b) (.2) Robust Control Toolbox Version 3.5 (R21b) 9 MATLAB SeDuMi Version 1.3 (2145) SDPT3 Version 4. YALMIP Version 3 (R2193) (.3) 23 (.1) R28b Symbolic Math Toolbox Maple ( MuPAD R28a R28b Symbolic Math Toolbox (.2) 8 9 LMI LMILAB (.3) B.7

6 iii MATLAB n ()

7 iv MATLAB/Simulink MATLAB Simulink MATLAB/Simulink MATLAB/Simulink

8 v MATLAB/Simulink ( 1) ( 2) MATLAB/Simulink (LQ ) ( ) MATLAB/Simulink ( ) LMI LMI LMI BMI LMI LMI

9 vi LMI MATLAB YALMIP A 197 A A A A A B MATLAB/Simulink 211 B.1 MATLAB B.2 M M B B B.5 Simulink B.6 Symbolic Math Toolbox B.7 LMI LMI C 229 C C C C.4 MATLAB

10 1 1 PID () () 1.1 classical control theory PID control PID ( 1.1) 2 pole stability ( 1.1) u(t) =f 1 (t) 2 y(t) =z 2 (t) 1, 2 P : M 1 z 1 (t) =u(t) k ( z 1 (t) y(t) ) M 2 ÿ(t) = f k (t) {}}{ k ( z 1 (t) y(t) ) + μ ( ż 1 (t) ẏ(t) ) f μ (t) (1.1) {}}{ μ ( ż 1 (t) ẏ(t) ) P : y(s) =P (s)u(s) (1.2) ( 1.1) PID Proportional ()Derivative ()Integral ()

11 2 1 f k (t) [N] f μ (t) [N] M 1 [kg] 1 M 2 [kg] 2 k [kg/s 2 ] μ [kg/s] u(t) =f 1 (t) 2 u(s) =L [ u(t) ], y(s) =L [ y(t) ] u(t), y(t) transfer function ( 1.2) P (s) :=y(s)/u(s) u(s) y(s) M 1 =.5, M 2 =1,k =2,μ =1y() =, ẏ() =, z 1 () =, ż 1 () = (1.1) z 1 (s) P : y(s) =P (s)u(s), P(s) = 2s +4 s 2 (s 2 +3s +6) (1.3) 1.2 P D ( PD ) y ref (t) y(t) e(t) :=y ref (t) y(t) ( 1.3) P D {}}{ K : u(t) = k P e(t) k D ẏ(t) K : u(s) =k P e(s) k D sy(s) (1.4) ( 2 y(t) =z 2 (t) y ref (t) ) P D k P, k D () 1.2 P D ( 1.2) A.1 (p.197) P (s) f(t) f(s) =L [ f(t) ] ( 1.3) A B A := B

12 ( 1.1) P D k P, k D 1 y ref (s) y(s) T (s) :=y(s)/y ref (s) P (s)k P T (s) = 1+P (s) ( ) = N T (s) k D s + k P D T (s) { N T (s) =2k P s +4k P D T (s) =s 4 +3s 3 + ( 6+2k D ) s 2 + ( 2k P +4k D ) s +4kP (1.5) 2 1/T (s) ) 4k P +2k P s 1 + k D s + 3 s 2 + k P 2k P 4k P + ( ) ( ) 2k P +4k D s + 6+2kD s 2 +3s 3 + s 4 4k P +2k P s 4k D s + ( 6+2k D ) s 2 +3s 3 + s 4 4k D s +2k D s 2 6s 2 +3s 3 + s 4 6s 2 +3s 3.. 1/T (s) 1 T (s) =1+ k D k P s + 3 2k P s 2 + (1.6) 1/T (s) s =(1.6) 3 P D k P, k D 2 G M (s) 2 G M (s) = ω 2 n s 2 +2ζω n s + ωn 2, ω n >, ζ > (1.7) y M (s) =G M (s)y ref (s) ω n ζ 1.3 (1.7) G M (s) 1 G M (s) =1+ 2ζ ω n s + 1 ωn 2 s 2 (1.8) (1.8) (1.6) 2 (1.4) k P, k D k P = 3ω2 n 2, k D =3ζω n (1.9)

13 4 1 P D (1.4), (1.9) 1.4 ζ =.7, ω n =.5, 2 { y ref (t<) (t) = 1 (t ) 1.4 P D ω n G M (s) T (s) ω n T (s) D T (s) 4 P D (1.4) k P, k D 2 T (s) 4 (D T (s) =) (a) ( <ζ<1) (b) (ζ =1) (ζ >1) y M (s) =G M (s)y ref (s) (a) ω n =.5, ζ =.7 (b) ω n =2,ζ = P D P D 2 (y(t) =z 2 (t), ẏ(t) =ż 2 (t)) (1.4) 1 (z 1 (t), ż 1 (t)) modern control theory 2 1

14 ( 1.1) 2 y ref (t) =yc ref (y(t) =z 2 (t) =yc ref ) 1 z 1 (t) yc ref 1 P D 2 P D {}}{ K : u(t) = k P1 e 1 (t) k D1 ż 1 (t)+ k P2 e 2 (t) k D2 ż 2 (t) (1.1) e 1 (t) =y ref (t) z 1 (t), e 2 (t) =y ref (t) z 2 (t) x 1 (t) z 1 (t) x 2 (t) x(t) = x 3 (t) = ż 1 (t) z 2 (t) x 4 (t) ż 2 (t) } } () k = [ k 1 k 2 k 3 k 4 ] = [ kp1 k D1 k P2 k D2 ], h = kp1 + k P2 state variable (1.1) x(t) kx(t) y ref (t) hy ref (t) state feedback K : u(t) =kx(t)+hy ref (t) (1.11) ( 1.5 ) 4 (1.11) y ref (s) y(s) T (s) y ref (s) y(s) T (s) :=y(s)/y ref (s) T (s) = N T (s) D T (s) N T (s) =2 ( ) ( ) k P1 + k P2 s +4 kp1 + k P2 D T (s) =s 4 + ( ) 3+2k D1 s 3 +2 ( ) 3+k P1 + k D1 + k D2 s 2 +2 { k P1 + k P2 +2 ( )} ( ) k D1 + k D2 s +4 kp1 + k P2 (1.12) k P1, k D1, k P2, k D2 T (s) (D T (s) =) P D (1.4) 1.5

15 6 1 pole placement method optimal regulator theory (4.3 ) ( 8 ) k, h J = { 6 ( y(t) y ref c ) 2 + u(t) 2} dt (1.11) k = [ ], h =7.75 ( 1.6 (a)) 1.1 P D ( 1.6 (b)) (a) (1.11) 1.6 (b) P D (1.4), (1.9) (ω n =2,ζ =.7) x(t) u(t) y(t) x(t) x(t) state space representation { P : ẋ(t) =Ax(t)+bu(t) y(t) =cx(t) state equation (1.13) (1.13) 1 output equation ( 1.4) ( 1.4) (p.15)

16 u(t) y(t) single-input single-output system 1 (1 1 SISO ) PID 1 multiple-input multiple-output system ( 1.5) (MIMO ) ( 1.8) 1.8 1, 2 f 1 (t), f 2 (t) f 1 (t) 2 f 2 (t) 1 z 1 (t) 2 z 2 (t) 2 u(t) y(t) [ ] [ ] [ ] [ ] u 1 (t) f 1 (t) y 1 (t) z 1 (t) u(t) = =, y(t) = = (1.14) u 2 (t) f 2 (t) y 2 (t) z 2 (t) ( 1.5) p q p q

17 p p ẋ(t) =Ax(t)+Bu(t), x() = x P : y(t) =Cx(t) η(t) =x(t) (5.1) x(t) R n u(t) R p y(t) R p η(t) R n 5.1 K : u(t) = {}}{ Kx(t)+Hy ref (t) (5.2) y ref (t) =y ref c following control (5.2) P D

18 (p.15) 1.1 (p.2) 2 M 1 =.5, M 2 =1,k =2,μ =1 { ẋ(t) =Ax(t)+bu(t) P : y(t) =cx(t) x 1 (t) z 1 (t) x 2 (t) x(t) = x 3 (t) = ż 1 (t) z 2 (t), u(t) =f 1(t), x 4 (t) ż 2 (t) A = 1, b = 2, c = [ 1 ] (5.3) (5.3) y(t) =z 2 (t) { y ref (t<) (t) = (5.4) yc ref (t ) y(t) =y ref c K : u(t) =kx(t)+hy ref (t) (5.5) ) x(t), u(t) x = [ ] x 1 x 2 x 3 x T 4, u ( 2 y ref c (5.3) y ref (t) =yc ref, x(t) =x, u(t) =u ẋ = { = Ax + bu = yc ref = cx [ ][ A b c x u ] [ ] = 1 y ref c (5.6)

19 94 5 x, u [ ] M A b = c =4 = M A b := c (5.6) x, u [ ] [ ] 1 [ ] x A b = yc ref u c 1 x yc ref x 2 = x 3 = yc ref = y ref c x u (5.7) (5.7) 2 y ref c (u =) 1 y ref c (x 3 = yc ref ) 1 (x 1 = yc ref ) (5.5) y(t) y ref (t) =yc ref x, u x(t) := x(t) x, ũ(t) :=u(t) u (5.3), (5.6) x(t) =ẋ(t) =Ax(t)+bu(t) =A ( x(t)+x ) + b (ũ(t)+u ) = A x(t)+bũ(t)+ax + bu }{{ = A x(t)+ bũ(t) } e(t) =y ref c = cx(t) =y ref c c ( x(t)+x ) = c x(t)+yc ref cx = c x(t) { x(t) =A x(t)+bũ(t) e(t) = c x(t) (5.8) x, u x(t), ũ(t) (5.8) A cl := A + bk λ = α + jβ α t x(t) =e A clt x() e(t) =y ref c y ref (t) =yc ref (5.7) K : ũ(t) =k x(t) (5.9) y(t) = c x(t) (y(t) y ref (t) =y ref c ) (5.9) K : u(t) =k ( x(t) x ) + u = kx(t)+ [ k 1 ][ x u ]

20 = kx(t)+hyc ref, h = [ k 1 ][ ] 1 [ ] A b c 1 (5.1) (5.5) y ref (t) =yc ref (4.3.3 ) A cl := A+bk λ 2 ± 2j, 2 ± j k (5.1) h 1 2 ± 2j, 2 ± j (4.53) (p.81) Δ(λ) δ, δ 1, δ 2, δ 3 Δ(λ) :=(λ +2 2j)(λ +2+2j)(λ +2 j)(λ +2+j) = λ 4 +8λ 3 +29λ 2 +52λ + 4 (5.11) = δ 3 =8, δ 2 =29, δ 1 =52, δ =4 (4.56) (p.82) Δ A Δ A = A 4 + δ 3 A 3 + δ 2 A δ 1 A + δ I = (5.12) 2 (4.9) (p.71) V c V c = [ b Ab A 2 b A 3 b ] = (5.13) (4.57) (p.82) (5.5) k, h V 1 c {}}{ e {}}{ [ ] k = [ = ] (5.14) [ ] h = 1 } 2 2 {{ 2 2 } 4 4 = 1 (5.15) [ ] 4 k [ ] 1 [ ] A b c 1 Δ A

21 96 5 k, h (5.14), (5.15) (5.5) yc ref = y(t) y ref (t) =yc ref u(t) y(t) p p (5.1) Point! () p p (5.1) M = A B C O [ = M := ] A B (5.16) C O ( 5.1) A cl := A + BK K K : u(t) =Kx(t)+Hy ref (t), y ref (t) =y ref c H = [ K I ][ ] 1 [ ] A B O C O I ) e := lim t e(t) = (5.17) steady-state error ( e(t) :=y ref (t) y(t) 5.1 { [ ] [ ] ẋ(t) =Ax(t)+bu(t) 1 P :, A =, b =, c = [ 1 ] (5.18) y(t) =cx(t) y(t) y ref (t) =y ref c ( 5.1) (5.16) (5.1)

22 K : u(t) =kx(t)+hy ref (t) (5.19) [ ] A b (1) M := y(t) =yc ref c x, u (2) A + bk 2 ± 2j k h h = [ k 1 ][ ] 1 [ ] A b c 1 (5.2) (5.16) (5.16) invariant zero Point! p p p p (5.1) s M(s) [ =, M(s) := ( si A ) ] B C O (5.21) (s =) M() = A B C O = (5.22) (5.16) (5.16) M() p p (5.1) { [ ] [ ] P : ẋ(t) =Ax(t)+bu(t) 2 1, A =, b = y(t) =cx(t) 3 2, c = [ 2 2 ] (5.23) (5.21)

24 ( ) () 8.1 (LQ ) p n { ẋ(t) =Ax(t)+Bu(t), x() = x P : η(t) =x(t) (8.1) 4.3 K : u(t) =Kx(t) (8.2) x(t) = [ x 1 (t) x n (t) ]T R n u(t) = [ u 1 (t) u p (t) ]T R p η(t) = [ η 1 (t) η n (t) ]T R n 4.3 (1) (p.75) K u(t)

25 156 8 u i (t) u i (t) 4.3 (1) x 2 (t) K K ( 4.7 (p.83) ) optimal regulator (p.16) M =1,k = 1, μ =1 [ ] [ ] 1 ẋ(t) =Ax(t)+bu(t), A =, b = (8.3) u(t) =f(t), y(t) =z(t), x(t) = [ x 1 (t) x 2 (t) ]T = [ z(t) ż(t) ] T (8.3) K : u(t) =kx(t), k = [ 5 1 ] (8.4) (t x(t) ) k A cl := A + bk 1 ± 2j x = [ 1 ] T x 1 (t), x 2 (t) u(t) 2 J x1 = x 1 (t) 2 dt >, J x2 = x 2 (t) 2 dt >, J u = u(t) 2 dt > 8.1 x i (t) J xi J xi x i (t) x i (t) u(t) u(t) J u x i (t) J xi (J xi x i (t) )

26 8.1 (LQ ) x 1 (t), x 2 (t), u(t) 2 J u(t) J u (J u u(t) ) 8.1 x() = x = [ 1 ]T 2 J x1 =.45, J x2 =1.25, J u =17.5 J x1, J x2, J u J x1 <J x2 J u J x1 + J x2 + J u J u J x1, J x2 J u J x1, J x2, J u J = q 1 J x1 + q 2 J x2 + rj u ( = q1 x 1 (t) 2 + q 2 x 2 (t) 2 + ru(t) 2) dt (8.5) q i, r> q 1 = 1, q 2 =, r =1 (8.5) J J = = 45 q 1 J x1 q 2 J x2 r J u = 467.5

27 158 8 x 1 (t) J x1 u(t) J u q 1 =.1, q 2 =, r =1 (8.5) J J = =.45 q 1 J x1 q 2 J x2 r J u = u(t) J u x 1 (t) J x1 8.1 (8.5) p n (8.1) n p ( n p J = q i J xi + r j J uj = q i x i (t) 2 + r j u j (t) )dt 2 (8.6) i=1 j=1 i=1 j=1 J xi, J uj x i (t), u j (t) 2 J xi = x i (t) 2 dt, J uj = u j (t) 2 dt q i, r j > ( 8.1) Point! (8.6) q i, r j > q i x i (t) (J xi ) r j > u j (t) (J uj ) (8.6) (8.2) (8.6) q i, r j q i, r j (8.6) ( 8.1) r j = u j (t) r j >

28 J = 8.1 (LQ ) 159 Q x(t) {}}{ x(t) {}} T { [ x1 (t) x n (t) ] q 1 x 1 (t)..... q n x n (t) + [ u 1 (t) u p (t) ] r 1 u 1 (t).... u(t) T r p u p (t) } {{ } R } {{ } u(t) dt (8.7) Q = Q T, R = R T quadratic form > 2 Q = Q T, R = R T > 2 (8.8) () linear quadratic (LQ) optimal control LQ Point! (LQ ) n (8.1) (i) Q = Q T > (ii) (Q o, A) ( 8.2) Q = Q T = Q T o Q o Q = Q T R = R T > J = ( x(t) T Qx(t)+u(t) T Ru(t) ) dt (8.8) (8.2) ( 8.3) Point! (LQ ) (8.1) (8.8) (8.8) (8.2) K = K opt ( 8.2) (Q o, A) (7.24) (p.149) ( 8.3) A.4 (p.26)

29 16 8 K opt := R 1 B T P opt (8.9) P = P opt Riccati equation ( 8.4) PA+ A T P PBR 1 B T P + Q = O (8.1) P = P T > (8.8) J min = x T P opt x (8.11) A.4 (p.26) (8.9) K opt (8.1) (A.58) (p.28) P opt A cl + A T clp opt = Q cl, { A cl := A + BK opt Q cl := K T optrk opt + Q (8.12) (i) (ii) A cl := A + BK opt (t x(t) ) (p.156) 1 (8.3) J = ( x(t) T Qx(t)+ru(t) 2) dt (8.13) Q, r [ ] [ ] 3 q 1 (1) Q = >, r =1> (2) Q = (q 1 > ), r =1> 6 [ ] PA+ A T P 1 r PbbT P + Q = O, P = P T p 11 p 12 = > (8.14) p 12 p 22 P (8.13) ( 8.4) Riccati

30 () ( () ) 1997 () MATLAB/Simulink () Scilab () MATLAB/Simulink c (12 71) FAX () // PrintedinJapan ISBN

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63> 単純適応制御 SAC サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/091961 このサンプルページの内容は, 初版 1 刷発行当時のものです. 1 2 3 4 5 9 10 12 14 15 A B F 6 8 11 13 E 7 C D URL http://www.morikita.co.jp/support