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2 倒立振子で学ぶ制御工学 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.

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4 i (PID ) MATLAB/Simulink book_ip_page.html ()

5 ii I () II () //

6 iii MATLAB/Simulink MATLAB/Simulink ( mfiles.zip ) mfiles p1c2 2.. p1c8 8 p2c1 1 p2c2 2 p2c3 3 (3.2 ) AutoGenU_InvPend 3 (3.3 ) page.html (Windows R2013a R2016a ) ip_toolbox_1.0.2.zip ip_toolbox_1.0.2 iptools odqlab_2.1.3 cdip_sample adip_sample / M Simulink ODQ Toolbox/Lab ( 2.3 ) C hoge ip_toolbox_1.0.2 iptools odqlab_2.1.3 >> addpath( C:Y=hogeY=ip_toolbox_1.0.2Y=iptools ) >> addpath( C:Y=hogeY=ip_toolbox_1.0.2Y=odqlab_2.1.3 ) iptools 3.4 (p. 56) ODQ Toolbox/Lab 2.3 (p. 173) 1 (p. 135) 2.3 (p. 173) SeDuMi YALMIP

7 iv f(s) =Lˆf(t) f(t) =L 1ˆf(s) f(z) =Zˆf[k] f(t) Laplace Laplace t s f[k] :=f(kt s) (k =0, 1,...) Z R R n n R m n m n C C n n C m n m n I (I n) (n n ) 0(0 m n) (m n ) M M M 1 M M + (m n ) M M 1 2 M M =(M 1 2 ) M 1 2 M M rank(m) M () tr(m) M diag{a 1,..., a n} M 0 M (M ) M 0 M (M ) M 0 M (M ) M 0 M (M ) M N M N Kronecker x x Euclid x x M F M Frobenius M M j (j 2 = 1) Reˆλ λ = α + jβ α Imˆλ λ = α + jβ β min f f () max f f () inf f f (infimum) () sup f f (supremum) () subject to

8 v I 1 1 () D/A DC PID () PID P PD PI PID PID P P D I PD PID ()

9 vi MATLAB/Simulink () () () ()

10 vii 8 () Z H II LMI () LMI LMI () ()

11 viii

12 I I PID

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14 3 1 () ( 1.1) 1) 2) 3) (Segway) 4) (1.3 (a)) ibot 5) 6) (1.3 (b)) 1.2

15 4 1 (a) 1.3 (b) () LEGO MINDSTORMS NXT/EV3 7) 11) 12) 17) Lagrange () PID

16 ) 15) 13) 12), 13) 14), 15) 13) 20) 1.4 ( ) 1.4 Furuta Pendulum 12) 14), 16) 1.5 LEGO MINDSTORMS NXT/EV3 LEGO PowerFunctions XL 18) mindsensors.com GlideWheel-M 19) 10), 11) 1.5 LEGO MINDSTORMS 21) 1.6 Pendulum Robot ( ) Pendubot ), 14)

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18 37 3 () () () () 1) 4) 5) 8) 2) 4),9),10) (p. 8) Newton-Euler Lagrange 5) 7) () (a) Newton-Euler Newton-Euler 5) 7) Newton-Euler 3.1 F (t) =M z(t) (3.1) T (t) =J θ(t) (3.2)

19 F (t) [N] T (t) [N m] ( 1) M [kg] ( 2) J [kg m 2 ] ( 3) z(t) [m] θ(t) [rad] m c [kg] μ c [kg/s] m p [kg] μ p [kg m 2 /s] J p [kg m 2 ] l p [m] g [m/s 2 ] H(t) V (t) 1), 3), 6) H(t) V (t) f c (t) μ c ż(t) m c z(t) =f c (t) μ c ż(t) H(t) (3.3) ( 4) { { m p Ẍ 2 (t) =H(t) m p Ÿ 2 (t) =V (t) m p g, X 2 (t) =z(t)+l p sin θ(t) Y 2 (t) = l p cos θ(t) (3.4) μ p θ(t) ( 1) ( 2) M ( 3) 2l [m] m [kg] J =(4/3)ml 2 [kg m 2 ] ( 4)

20 J p θ(t) = μp θ(t)+v (t)sinθ(t) lp H(t)cosθ(t) l p (3.5) (3.3) (3.5) H(t), V (t) (m c + m p ) z(t)+m p l p cos θ(t) θ(t) = μ c ż(t)+m p l p θ(t) 2 sin θ(t)+f c (t) (3.6) m p l p cos θ(t) z(t)+(j p + m p lp) θ(t) 2 = μ p θ(t)+mp gl p sin θ(t) (3.7) (b) Lagrange Lagrange 5), 7), 8) 3.3 Maple, Mathematica, MATLAB/Symbolic Math Toolbox 11) W (t) U(t) D(t) W (t) = {}}{ 1 2 m cż(t) 2 + U(t) =m p gy 2 (t) }{{} {}}{ 1 2 m pẋ2(t) 2 +, D(t) = 1 2 μ cż(t) 2 } {{ } {}}{ 1 2 m pẏ2(t) μ p θ(t) 2 }{{} {}}{ 1 2 J p θ(t) 2 (3.8) (3.4) { Ẋ2 (t) =ż(t)+l p θ(t)cosθ(t) Ẏ 2 (t) = l p θ(t)sinθ(t) (3.9) q(t) = [ q 1 (t) q 2 (t) ] = [ z(t) θ(t) ] f(t) = [ f 1 (t) f 2 (t) ] = [ f c (t) 0 ] 3.3 (k z, k θ ) (h(t)) (μ z, μ θ ) Mż(t)2 2 J θ(t) kzz(t)2 2 k θθ(t) 2 Mgh(t) 1 2 μzż(t)2 1 2 μ θ θ(t) 2

21 40 3 Lagrangian L(t) =W (t) U(t) D(t) Lagrange Lagrange ( ) d L(t) L(t) dt q i (t) q i (t) + D(t) q i (t) = f i(t) (i =1, 2) (3.10) (3.6), (3.7) MATLAB/Symbolic Math Toolbox Lagrange (3.10) (3.6), (3.7) M M p1c311_cdip_lagrange.m 1 clear; format compact 2 3 syms m_c mu_c real m c, μ c 4 syms J_p m_p mu_p g l_p real J p, m p, μ p, g, l p 5 syms z th dz dth ddz ddth fc real z, θ, ż, θ, z, θ, f c 6 7 q = [ z th ] ; q =[q 1 q 2 ] =[z θ ] 8 dq = [ dz dth ] ; q =[ q 1 q 2 ] =[ż θ ] 9 ddq = [ ddz ddth ] ; q =[ q 1 q 2 ] =[ z θ ] 10 f = [ fc 0 ] ; f =[f 1 f 2 ] =[f c 0 ] 11 % X2 = q(1) + l_p*sin(q(2)); X 2 = z + l p sin θ 13 Y2 = l_p*cos(q(2)); Y 2 = l p cos θ dx2 = diff(x2,q(1))*dq(1)... Ẋ 2 = X2 z ż + X2 θ θ 16 + diff(x2,q(2))*dq(2); 17 dy2 = diff(y2,q(1))*dq(1)... Ẏ 2 = Y2 z ż + Y2 θ θ 18 + diff(y2,q(2))*dq(2); 19 % W = (1/2)*m_c*dq(1)^2... W = 1 2 mcż mpẋ (1/2)*m_p*dX2^ (1/2)*m_p*dY2^ mpẏ Jp θ (1/2)*J_p*dq(2)^2; 24 U = m_p*g*y2; U = m pgy 2 25 D = (1/2)*mu_c*dq(1)^ (1/2)*mu_p*dq(2)^2; 27 D = 1 2 μcż μp θ 2 28 L = W - U; L = W U 29 % N = length(q); N =2q 31 for i = 1:N 32 dlq(i) = diff(l,dq(i)); L qi = L q i 33 (i =1,,N) 34 temp = 0; 35 for j = 1:N «d L NX Lqi 36 temp = temp + diff(dlq(i),dq(j))*ddq(j)... = q j + Lq i dt q i q 37 + diff(dlq(i),q(j))*dq(j); j=1 j q j 38 end 39 ddlq(i) = temp; 40 «d L 41 eq(i) = ddlq(i) - diff(l,q(i))... L + D f i dt q i q i q i 42 + diff(d,dq(i)) - f(i); 43 end q j «

22 eq = simplify(eq ) eq M p1c311_cdip_lagrange.m eq = - l_p*m_p*sin(th)*dth^2 - fc + dz*mu_c + ddz*(m_c + m_p) + ddth*l_p*m_p*cos(th) (3.6) J_p*ddth + dth*mu_p + ddth*l_p^2*m_p + ddz*l_p*m_p*cos(th) - g*l_p*m_p*sin(th) (3.7) f c (t) v(t) (a) DC 3.2 DC di v a (t) =R a i a (t)+l a (t) a + e b (t), dt e b (t) =k b θm (t) (3.11) J m θm (t) =τ m (t) τ L (t) μ m θm (t), τ m (t) =k t i a (t) (3.12) 8), 12) DC 3.4 L a di a (t)/dt 0 (3.11), (3.12) DC J m θm (t) = μ m θm (t)+ k t R a v a (t) τ L (t), μ m = μ m + k tk b R a (3.13) 3.2 DC 3.4 DC v a(t) i a(t) R a L a e b (t) k b θ m(t) τ m(t) J m μ m τ L (t) k t (b) DC 1.12 (p. 8) DC DC

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24 II I LMI II LMI 1 2 3

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26 135 1 LMI (LMI: linear matrix inequality) LMI LMI LMI ( ) 1), 2) y(s) =P(s)u(s), P(s) = ω 2 n s 2 +2ζω n s + ω 2 n (1.1) ω n > 0 0 <ζ<1 2 u(t) =1(t 0) y(t) % 3/(ζω n ) T s 3/(ζω n ) <T s

27 136 1 LMI (0 0 <ζ<1) ζω n > 3 T s (1.2) ζ> 1 2 (1.3) <ζ<1 λ = ζω n ± jω n 1 ζ 2 λ λ A(α) := { λ C Re[λ] < α } (1.4) A(α) α- ( 1.3) α (1.2) 5%α =3/T s θ ζ >cos θ tan θ> Im[λ] Re[λ] = 1 ζ 2 λ S(k) := { λ C Im[λ] <k Re[λ], Re[λ] < 0, k =tanθ>0 } (1.5) S(k) k ( 1.4) (1.3) ζ >cos θ =1/ 2 k =1 ζ

28 A(α)α- 1.4 S(k) (k k =tanθ) 1.5 B(β) ( 1.5) λ B(β) := { λ C Re[λ] > β } (1.6) ẋ(t) =Ax(t)+Bu(t), x(0) = x 0 (1.7) y(t) =Cx(t) (1.8) x(t) R n u(t) R m y(t) R p (A, B, C) A λ (1.7) (1.4) (1.5) (LMI: linear matrix inequality) ( 1) (1.4) LMI 3), 4) 1.1 A(α) (α-) LMI A R n n α R (i) (ii) (i) (ii) A λ λ A(α) ( 1.3) LMI n P 0 A P + PA+2αP 0 (1.9) A λ α- LMI P 0 (1.9) P ( 1) LMI ( ) LMI

29 138 1 LMI LMI P A α- 1.1 LMI A P, α LMI A A(α) (α-) [ ] [ ] 0 1 ξ 1 ξ 2 A =, P =, α =1 (T s = 3) (1.10) ξ 2 1 A λ =( 9 ± 3 7j)/8 Re[λ] = 9/8 α = 1 (A A(α) ) A A(α) LMI (P 0 (1.9) ) ( 2) [ ] P 0 = ξ 1 ξ 2 0 ξ 2 1 [ ] (1.11) A P + PA+2αP 0 = 8ξ 1 +18ξ 2 9 4ξ 1 + ξ 2 9 4ξ 1 + ξ ξ 1 0 (1.12) LMI (ξ 1,ξ 2 ) LMI LMI LMI» ( 2) ξ1 ξ 2 ξ 3 P = ξ 3 =1 ξ 2 ξ 3 α-

30 (ξ 1,ξ 2 ) λ α- LMI SeDuMi (Ver ( )) LMI YALMIP (Release ) ( 3) P 0 (1.9) ((1.11), (1.12) ) LMI M M p2c112_ex1_alpha_stability.m 1 clear; format compact 2 % A = [ ]; 5 n = length(a); 6 alpha = 1; 7 ep = 1e-6; 8 % xi1 = sdpvar(1); xi2 = sdpvar(1); 10 P = [ xi1 xi2 ξ 1, ξ 2 P = 11 xi2 1 ]; 12 % LMI = [ P >= ep*eye(n) ]; 14 LMI = LMI + [ A *P + P*A + 2*alpha*P <= - ep*eye(n) ]; 15 % solvesdp(lmi) 17 % P = double(p)» A = nx(t) α =1 ε =10 6» ξ1 ξ 2 ξ 2 1 P εi ( 0) ( 4) A P + PA+2αP εi ( 0) LMI P YALMIP LMI ε>0 F (ξ) 0 F (ξ) εi ( 0) (1.13) M p2c112_ex1_alpha_stability.m SeDuMi 1.32 by AdvOL, and Jos F. Sturm, Alg = 2: xz-corrector, theta = 0.250, beta = eqs m = 2, order n = 5, dim = 9, blocks = 3 nnz(a) = 7 + 0, nnz(ada) = 4, nnz(l) = 3 it : b*y gap delta rate t/tp* t/td* feas cg cg prec 0 : 9.31E : 0.00E E E-10 ( 3) SeDuMi YALMIP johanl/yalmip/ SeDuMi YALMIP http: // ( 4) ξ1 ξ 2 P = 9 11 P = sdpvar(n,n, sy ); ( sy ξ 2 ξ 3 )

31

32 223 A Ackermann 87 A/D 169, 173 AutoGenU 220 B Butterworth 28 C Cayley-Hamilton 81 Cholesky 73 D D/A 7, 9, 11, 46, 120, 169, 173 DC 12, 41 DC 41 Dirac 65 DMC 216 D 21, 23, 25, 26 E Euclid 51, 68 F Frobenius 144 F/V 13 G GPC 216 H Hankel 181 Hankel 63 Hurwitz 71 H 13 H 195 I inf 143, 145 ITAE 28 I 21, 25, 26 I PD 32 I PD J Jordan () 88 K Kronecker 160 L Lagrange 39 Lagrangian 40 Laplace 64 Lie 84 LMI 137, 141 LPV , , PDLMI LPV 158 LMI 161 LTI Luenberger 109 Lyapunov 69 () , 89, 203, , , 72 M Möbius 127 MPC M 52 N Newton-Euler 37 Nyquist 69 O ODQ Toolbox 186 P PDLMI PD 23 PD PFC 216 PID 21, 22 PID 20, PI 25, 96 PI , 99 PWM () 12 P 22, 42, 45, 95 P 22, , 98 P 21, 25 P D 31 P D 30, R Receding Horizon Riccati 89, 101, 143, 205 Routh-Hurwitz 69 S Schur 170 Schur 144 SeDuMi 139, 186 SOS 161 sup 143, 145 T Tustin 126, 127

33 224 Y YALMIP 139, 161 Youla 113 Z Z α- 136 LMI , () Lyapunov 69 Nyquist 69 Routh-Hurwitz , , , 54 39, , 199 Lagrange 40, 54 Newton-Euler Luenberger , , 143, , 105 () , 25, 26 PI 99 P , 9, 10, 169, 173 inf 63, 89, 100, 106, , () , 146, 163 Z , 118 () , 76, 85, 100, 112, , 77 81, 82, , 80 () () PD 23 PID 25 PI 25 P 22 P D , , , 72, , 85, 108, 137 Ackermann LMI 146 PDLMI LMI 137, , 47, LMI , , 47, , , 45, , 143, 205, 214, 217 LMI 145

34 225 16, 88, 101, 143, 147, 152, 164, () 169, , , 11, 119, , sup , () , () , 71, , 71, , , 147, I 136 LMI , 120, , , 83 69, 70, 71 () , 43, 55, 92 LMI 44, , 161 LPV 126, , , , , , I PD 32 PD 23 PID 25 PI 25, 96, 99 P 22, 29, 45, 65, 95, 98 P D 30 27, , 146 Dirac , 51 27, I PD 32 PD 23 PID 25 PI 25 P 23 P D 31 21, 22, 29, 32, 93, 95, 99, 102 PI 96, 99

35 226 P 95, 98 9, 11 Dirac () , 175, 181 5, 14, 78, 87, 90, 146, 151, 156, 162, , 7, 103, 113, 124, 130, 182, 193, 207 7, , 98, , , LMI PDLMI 16, , 71 69, , , 71 70, H 201, , 66, 76, PD P D D P PID I PD , 85, 94 33, 84, , 80 24, , 71, , 71, , 68, 92, , , 12, , , , , 169, 187 Riccati , , , , , 10, 24, 50, 107, , 24, PDLMI , 201, 222

36 SVBL Carnegie Mellon University 2007/102009/

37 i

38 c FAX PrintedinJapan ISBN

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