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2 構造物のシステム制御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 刷発行時のものです.

4 i The deeper we research, the more we find there is to know, and as long as human life exists I believe that it will always be so. Albert Einstein,

5 ii 2 ( ) ( ) ( ) 1 2 ( ) 3 ( ) 4 ( ) 5 6 LQ H ( ) 7

6 iii 8 MATLAB 1) )

7 iv ( ) ( ) MATLAB ( ) ( ) ( ) ( )

8 v PBH (Popov-Belevitch-Hautus)

9 vi LQ H H H H H /H

10 vii MOESP z(t) z(t) A A A A

11 viii A A A A A A A A

12 ( ) ( (Kármán) ( ) ( ) Theodore von Kármán with Lee Edson: The Wind and Beyond, The Theodore von Kármán; Pioneer in Aviation and Pathfinder in Space, Chap. 26, ). 1.1, 1) (Theodore von Kármán ) (Kármán vortex street 32 ) ( 1995). ( á [ a :])

13 2 1 ( ) (flexible structures) 1 3 dynamic force 3 (CalTech) 1930

14 (systems control) ( [dynamical system] ) (automatic control) 18 (James Watt ) ( govenor) (automation) R. E. (R.E.Kalman) (modern control theory) (classical control theory),.

15 ( ) ( ) ( ) ( ) (system modeling)

16 1.3 5 (system identification problem) ( ) (systems analysis) ( ) (system synthesis) (dynamics) mẍ(t)+cẋ(t)+kx(t) =f(t) x(t) m, c, k f(t) L( ) =md 2 ( )/dt 2 + cd( )/dt + k( ) f 1 (t), f 2 (t) x 1 (t), x 2 (t), L[ x 1 ]=f 1, L[ x 2 ]=f 2, L[ α 1 x 1 + α 2 x 2 ]=m d2 dt 2 (α 1x 1 + α 2 x 2 )+c d dt (α 1x 1 + α 2 x 2 ) +k(α 1 x 1 + α 2 x 2 ) = α 1 {mẍ 1 (t)+cẋ 1 (t)+kx 1 (t)} + α 2 {mẍ 2 (t)+cẋ 2 (t)+kx 2 (t)} = α 1 f 1 (t)+α 2 f 2 (t) α 1, α 2 L[ ] L[ α 1 x 1 + α 2 x 2 ]=α 1 L[ x 1 ]+α 2 L[ x 2 ] L (linear operator) (linear differential equation) f(t) =α 1 f 1 (t) +α 2 f 2 (t) x(t) =

17 6 1 α 1 x 1 (t)+α 2 x 2 (t) (superposition) (linear system) [ M ] {ẍ(t)} +[D ] {ẋ(t)} +[K ] {x(t)} = {0} { } [ ] M ẍ(t)+dẋ(t)+kx(t) =0 I I n n m O n m m = n O n 0 0 a T, A T T A =[a ij ] What we currently call mathematical system theory is (if I may say so) a marvelous beginning, a real promise that the new science is doable, just like when Galileo and Newton showed that physics was (then) doable. ( ) R. E. Kalman, ) Never lose a holy curiosity. Albert Einstein 1) (Rudolf Emil Kalman )

18 1.3 7 A det A ( n i=1 a ii) tr A A A =diag{a 11,a 22,,a nn } A := B A B B := A (4.1) bis (4.1) const. I.C. B.C. (Q.E.D.)

19 8 2 My theme: 1. Get the physics right. 2. After that, it is all mathematics. : R. E. Kalman: The Evolution of System Theory; My Memories and Hopes, Plenary Lecture at 16th IFAC World Congress, Prague, July 4, ). ( ) 2 ( ) x(t) u(t), 1) (Rudolf Emil Kalman ) 7.7.2, A. Einstein and L. Infeld: The Evolution of Physics (The Cambridge Univ. Press 1961) ( A. /L )

20 2.1 9 mẍ(t)+kx(t) =u(t) (2.1) x(t) 2 x(0) = x 0 ẋ(0) = ẋ 0 x(t) =x 0 cos ωt + ẋ0 ω sin ωt + 1 mω t 0 u(τ)sinω(t τ) dτ (2.2) ω = k/m 2.1 u(t) 1 (2.2) x(t) x 0 ẋ 0 2 x(t) ẋ(t) 2 x(t) x(t) = x(t) (2.3) ẋ(t) x 1 (t), x 2 (t) x 1 (t) =x(t), x 2 (t) =ẋ(t) ẋ 1 (t) =x 2 (t) (2.1) ẋ 2 (t) = k m x 1(t)+ 1 m u(t) ẋ 1 (t) =x 2 (t) ẋ 2 (t) = k m x 1(t)+ 1 m u(t) ẋ 1 (t) = 0 1 x 1 (t) ẋ 2 (t) k m u(t) x 2 (t) m

21 10 2 ẋ(t) =Ax(t)+bu(t) (2.4) A = 0 1 k, b = 0 m 0 1 m (2.3) x(t) (2.4) x(t) 2 (2.1) (x 1 (t),x 2 (t)) 1 2 (2 ) m i w i (t) (reference axis) z i (t) m i, d i, k i 2.2 3

23 110 6 Control Theory is supposed to deal with physical systems, and not merely with mathematical objects such as a differential equation or a transfer function We must therefore pay careful attention to the relationship between physical systems and their representation via differential equations, transfer functions, etc. ( ) R E Kalman, ( ) ( ) ( ) 6.1 ( )

24 ( ) 2.4 1) (disturbance) 6.1 ( a ) (b) (c) (random) (a) ( ) 6.1 1) 1989 / 4m 33 m 10 ( ), ( 2010)

25 112 6 (c) ( ) ( ) ( ) LQ H 6.2 Σ: ẋ(t) =Ax(t)+Bu(t) y(t) =Cx(t) (6.1) x(t) R n y(t) R m u(t) R l (6.1) x(t) u(t) =Fx(t) A + BF F x(t) x(t) ˆx(t) u(t) =F ˆx(t) (6.2) ˆx(t) 4.5 ˆx(t) ˆx(t) =Dz(t)+Hy(t) ż(t) =Âz(t)+ ˆBu(t)+Ky(t) (6.3) (z(t) R n m ) F F (6.2) (6.1) (6.3) 1

26 ẋ(t) =Ax(t)+BF ˆx(t) = Ax(t)+BF[ Dz(t)+Hy(t)] =(A + BFHC)x(t)+ BFDz(t) (y(t) = Cx(t)) (6.3) ż(t) =Âz(t)+ ˆBF ˆx(t)+Ky(t) = Âz(t)+MBF[ Dz(t)+Hy(t)]+Ky(t) =(Â + MBFD)z(t)+(MBFH + K)Cx(t) ˆB = MB ((4.33c) ) 2 ẋ(t) A + BFHC BFD = x(t) (6.4) ż(t) (MBFH + K)C Â + MBFD z(t) ((4.30) ) η(t) =z(t) Mx(t) (6.5) x(t) = I 0 x(t) η(t) M I z(t) (6.4) x(t) η(t) ẋ(t) = I 0 ẋ(t) η(t) M I ż(t) = I 0 A + BFHC BFD x(t) M I (MBFH + K)C Â + MBFD z(t) = A + BFHC BFD x(t) KC MA Â z(t) = A + BFHC BFD I 0 x(t) KC MA Â M I η(t) A + BF(HC + DM) = BFD x(t) KC MA+ ÂM Â η(t)

27 114 6 (4.33a) (4.33b) (4.5 ) ẋ(t) = A + BF BFD x(t) (6.6) η(t) 0 Â η(t) si (A + BF) BFD 0= 0 si Â = si (A + BF) si Â (6.7) ( A.8 (A.13c) ) A + BF Â (separation property of the observer-regurator design procedure) {γ i i =1, 2,,n m} F {μ i i =1, 2,,n} ˆx(t) (6.2) {γ i } {μ i } 6.3 LQ x(t) y(t) x(t) ˆx(t) ẋ(t) =Ax(t)+Bu(t), x(0) = x 0 (6.8)

28 6.3 LQ 115 (x(t) R n u(t) R l ), 1) J(u) = T 0 [ x T (t)mx(t)+u T (t)nu(t)]dt (6.9) {u(t), 0 t T } M N 1 x T (t)mx(t) x(t) 0 2 u T (t)nu(t) (6.9) (performance index) x(t) u(t) 2 2 (quadratic cost functional) [0,T ] [0, ) (6.9) {u(t)} (optimal control problem) (optimal regulator problem) (Richard Bellman ) 2) u(t) u(t) =Fx(t) x(t) (6.9) x(t) 2 x(t) 2 x T (t)π(t)x(t) Π(t) n n d dt [ xt (t)π(t)x(t)]=ẋ T (t)π(t)x(t)+x T (t) Π(t)x(t)+x T (t)π(t)ẋ(t) =[Ax(t)+Bu(t)] T Π(t)x(t) + x T (t) Π(t)x(t)+x T (t)π(t)[ Ax(t)+Bu(t)] = x T (t)[ Π(t)+Π(t)A + A T Π(t)]x(t)+x T (t)π(t)bu(t) + u T (t)b T Π(t)x(t) 0= [ x T (t)mx(t)+u T (t)nu(t)]+x T (t)mx(t)+u T (t)nu(t) = [ x T (t)mx(t)+u T (t)nu(t)] + x T (t)[ Π(t)+Π(t)A + A T Π(t)+M ] x(t) 1) (functional) {u} J(u) 2) 7.8

30 179 8 Build your models from data, not from assumptions. ( ) R. E. Kalman, Discovery and Invention: The Newtonian Revolution in System Technology, ), (M,D,K) ( ) (exogenous input) (subspace[-based] system identification method) ( ) QR (SVD) ( ) 1) (Issac Newton, ) ( Philosophiae Naturalis Principia Mathematica, ) 2 (1713) (Scholium Generale) Hypotheses non fingo. ( ) I do not frame hypotheses. hypotheses ((Sire je n ai pas eu besoin de cette hypothèse.)) ( )

31 180 8 ( ) Σ D ( ) u(k) y(k) Σ D x(k +1)=Ax(k)+Bu(k) Σ D y(k) =Cx(k)+Du(k), k =0, 1, 2, (8.1) (2.6 (2.77) F, G A, B ) u(k) R l y(k) R m x(k) R n n (A, B, C, D) (8.1) (system identification problem) {u(k),y(k)} n (A, B, C, D) ( [to identify] ) 1) Σ D {u(k)} (white Gaussian random sequence) E{[ u(k) E{u(k)} ][ u(l) E{u(l)} ] T } = Q k δ kl Q k δ kl 1 ) to estimate = to calculate approximately (the amount, extent, magnitude, etc.); to identify = recognize as being a specified person or thing, associate (onself) closely in feeling or interest (with a person, idea, etc.) ; ( ),

32 ) k (= t k ) l (= t l ) u(k) u(l) ( ) (E{u(k)} =0) {u(k)} 7.3 {w(t)} t k {w(t k )} w (h) (t) =u(k), t k t<t k + h (h >0) {u(k)} E{u(k)u T (l)} = Qδ kl h 0 u(k) =w (h) (t) h=0 = w(t k ) ξ(k) = tk +h t k w (h) (t) dt E{ξ(k)} =0 E{ξ(k)ξ T (l)} = = = tk +h tl +h t k t l tk +h tl +h t k t l tk +h t k Qdt= Qh E{w (h) (t)[ w (h) (τ)] T } dτ dt E{u(k)u T (l)} dτ dt h 0 {ξ(k)} h 0 Q Q/h {ξ(k)} Q/h Q (Dirac delta) (h 0) w (h) (t) 8.1 {u(k),y(k)} k=0,1,,5 1) (Leopold Kronecker ) God made the integers; all the rest is the work of man. ( ( ) ) (Kronecker s delta): δ kl = { 1 (k = l) 0 (k l)

33 2013 Printed in Japan ISBN

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63> 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. http://www.morikita.co.jp/books/mid/9241 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. i MATLAB/Simulink MATLAB/Simulink 1. 1 2. 3. MATLAB/Simulink