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1 J.JSSAC (2005) Vol. 11, No. 3,4, pp Noda2005 SCILAB 1 Matlab is one of the most popular commercial software among the researchers in the area of control and system engineering. Although Matlab has no special functions as a programming language, it is equipped with many state-of-the-art packages, which makes Matlab competitive with its clones. In this paper, we focus on SCILAB, one of the clones of Matlab, which is developed by INRIA. Unlike other clones, the SCILAB has packages which are powerful enough to match those of Matlab. The SCILAB also has the function to define a new data type and its operators in the programming, which is similar to class in C++ programming language. In this paper, utilizing the function, we introduce power series arithmetic into the SCILAB, following the next three guidelines. 1. The power series are truncated at the total degree of variables. 2. Computations of the power series should be efficient. 3. The power series should be treated as if they are numbers. We developed the packages for controller designs which utilizes the power series arithmetic. With the packages, the H 2 optimal controller is obtained in the form of truncated power series of the design parameters. We present a design example of H 2 optimal controller with restrictions in its overshoot and undershoot (such design restrictions require trial and error process in a conventional controller design method). We also present a technique to detect numerical errors due to power series arithmetic. Mathematica Maple MATLAB kitamoto@yamaguchi-u.ac.jp c 2005 Japan Society for Symbolic and Algebraic Computation
2 , MATLAB LMI MATLAB Mathematica Maple Mathematica Control System Professional 2 LMI MATLAB Mathematica Maple MATLAB SCILAB SCILAB MATLAB Maple (Extended symbolic math toolbox) MATLAB Maple Extended symbolic math toolbox MATLAB Extended symbolic math toolbox Maple MATLAB 2 MATLAB 3 MATLAB SCILAB SCILAB 4 SCILAB 5 SCILAB MATLAB MATLAB CAD MATLAB 1 1
3 J.JSSAC Vol. 11, No. 3,4, Mathematica GNU Octave SCILAB MATLAB CAD LMI Simulab MATLAB 3 SCILAB 3.1 SCILAB INRIA MATLAB LAPACK MATLAB LMI MATLAB Simulab SCICOS MATLAB SCILAB SCILAB C++ C++ C 2 tlist tlist tlist([, 1,, n ], 1, 2,, n) tlist 1,, n 1 + 2i
4 , tlist([ complex, real, imag ], 1.0, 2.0) c c.real c.imag SCILAB SCILAB ps (i) (ii) (iii) (ii) ps α 0 + α 1 x + α 2 y + α 3 x 2 + α 4 xy + α 5 y 2 [α 0 α 1 α 2 α 3 α 4 α 5 ] x, y 3 p 1 = α 0 + α 1 x + α 2 y + α 3 x 2 + α 4 xy + α 5 y 2 p 2 = β 0 + β 1 x + β 2 y + β 3 x 2 + β 4 xy + β 5 y 2 p 1 + p 2 = α 0 + β 0 + (α 1 + β 1 )x + (α 2 + β 2 )y + (α 3 + β 3 )x 2 + (α 4 + β 4 )xy + (α 5 + β 5 )y 2 p 1 p 2 = α 0 β 0 + (α 0 β 1 + α 1 β 0 )x + (α 2 β 0 + α 0 β 2 )y + (α 0 β 3 + α 1 β 1 + α 3 β 0 )x 2 +(α 0 β 4 + α 1 β 2 + α 2 β 1 + α 4 β 0 )xy + (α 0 β 5 + α 2 β 2 + α 5 β 0 )y 2 (ii) (ii) ps_str
5 J.JSSAC Vol. 11, No. 3,4, ID ps ID " #! 1: id = ps_str([ x, y ],2); p1 = ps(id,[1,2,3,-1,2,-2]); p2 = ps(id,[3,4,-1,2,-1,1]); ps_str() ID x, y 2 ID id ps() ID 1 + 2x + 3y x 2 + 2xy 2y 2 p1 3 4x y + 2x 2 xy + y 2 p2 (iii) 3.3 SCILAB C++ SCILAB x y %<x >_a_<y > <x > <y > x y ps %ps_a_ps() p1, p2 ps p1+p2 %ps_a_ps(p1,p2) ps ID %ps_a_ps()
6 , function c = %ps_a_ps(a,b) if (a.id ~= b.id) then error("%ps_a_ps: ids are different!!"); end c=ps(id,a.coef+b.coef); endfunction ID ID ID %ps_a_ps() a s m r %ps_s_ps() %ps_m_ps() %ps_r_ps() ^ p == o 2 2: + a <> n - s < 1 * m > 2 / r <= 3 ^ p >= 4 t () e == o () i %ps_a_ps() p1 ps p1+1 %ps_a_s() s 1+p1 %ps_a_s() %s_a_ps() 4 SCILAB poly multpoly SCILAB psmat multpoly
7 J.JSSAC Vol. 11, No. 3,4, ps psmat ps multpoly multpoly(, ) x 2 p0 + p1*x + p2*x^2 f p0 p1 p2 ps f = multpoly( x,list(p0,p1,p2)) psmat psmat 2 2 p11 p12 m p21 p22 p11 p12 p21 p22 ps m = psmat(2,2,list(p11,p12,p21,p22)) multpoly multpoly_newton() 1: function xn = multpoly_newton(p,x0) 2: xn = x0; 3: tdeg = ps_str_data(p.coef(1).id).tdeg; 4: pd = p ; 5: i = 1; 6: while (i-1)<tdeg 7: i = 2*i; 8: xn = xn-p(xn)/pd(xn); 9: end; 10: endfunction p multpoly multpoly_newton() p x0 xn = x0 x0 xn x0 3 tdeg pd = p p pd multpoly p(xn) p xn ps ()
8 , SCILAB psmat multpoly psmat_inv() 1: function pn = psmat_inv(a) 2: if (a.row ~= a.col) then 3: error(" Matrix is not square!!"); 4: end 5: a0 = psmat_const(a); 6: e = eye(a0); 7: pn = inv(a0); 8: tdeg = ps_str_data(a(1,1).id).tdeg; 9: i = 1; 10: while (i-1)<tdeg 11: i = 2*i; 12: pn = pn+pn*(e-a*pn); 13: end; 14: endfunction 5 SCILAB Mathematica Ver CPU Pentium M 1.6GHz, 1GByte Windows Mathematica Mathematica 1 2
9 J.JSSAC Vol. 11, No. 3,4, x, y 2 α 0,0 + α 1,0 x + α 0,1 y + α 2,0 x 2 + α 1,1 xy + α 0,2 y 2 2 α 0,0 + α 0,1 y + α 0,2 y 2 + (α 1,0 + α 1,1 y + α 1,2 y 2 )x + (α 2,0 + α 2,1 y + α 2,2 y 2 )x 2 Mathematica t x i x i t t Mathematica Mathematica : Math Math Math Math Math. Mathematica ms Math. / Mathematica 3 Mathematica 1
10 , Mathematica : Math Math Math Math Mathematica < 1 > Mathematica 3 5 5: Math Math Math Math
11 J.JSSAC Vol. 11, No. 3,4, Mathematica Mathematica 4 Mathmatica 5 Mathematica SCILAB H 2 d dt x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (1) x(t), u(t), y(t) A, B, C, D t > 0 x(t), y(t) u(t) x(t) x(t) u(t) = Kx(t) ( K x(t), y(t) K K H 2 H 2 0 { x(t) T Qx(t) + u(t) T Ru(t) } dt (2) Q, R x(t) T, u(t) T x(t), u(t) u(t) (1 u(t) (2 u(t) Riccati PA+A T P PBR 1 B T P+Q = 0 P P K = B T P K u(t) = Kx(t) u(t) 6.2
12 , psmat_inv(a) psmat a [2],[7] psmat_eig(a) psmat a [2] [6],[8],[12] psmat_lyap(a,b) a,b psmat Lyapunov a*x+x*a^t = b x psmat_are(a,b,c) a,b,c psmat Riccati a^t*p+p*a-p*b*p+c = 0 p [3],[5] psmat_expm(a) psmat a [9] step(a,b,c,d,t) a,b,c,d t A = a, B = b, C = c, D = d 1 impulse(a,b,c,d,t) a,b,c,d t A = a, B = b, C = c, D = d 1 eval_y(a,b,c,d,t) a,b,c,d t A = a, B = b, C = c, D = d 1 t =t y(t) psmat_eval_y(a,b,c,d,t) psmat a,b,c,d ps t A = a, B = b, C = c, D = d 1 t =t y(t) ps [9] psmat_peaktime(a,b,c,d,t) psmat a,b,c,d t A = a, B = b, C = c, D = d 1 y(t) y peak ps t y(t) = y peak t ps y(t) t t [9] A, B, C, D A = , B = 1 0, C = [ 1 2 ], D = 0 (3) H H 2 Q, R 2 u(t) = Kx(t) Q, R Q = , R = 1 (4)
13 J.JSSAC Vol. 11, No. 3,4, Riccati PA + A T P PBR 1 B T P + Q = 0 P P = (5) K = B T P = [ ] (6) u(t) = Kx(t) = 2.12x 1 (t) 1.29x 2 (t) (7) 6 6 y(t) y( ) = y(t) t : y(t) : t 6 t = 2 1 y(t) 10% (=1.1) 0 y(t) Q 10% (8) Q u(t) Q, R z 1, z 2 Q = (1 + z 1) (1 + z 2 ) 5, R = 1 (9) z 1, z 2 z 1 0.5, z (1 + z 1 ) 5, (1 + z 2 ) Q H 2
14 , z 1, z 2 H 2 z 1, z 2 SCILAB : 1, 3 1: a = [-1 1; -3 1]; 2: b = [-1 0] ; 3: c = [1 2]; 4: d = 0; Q, R 9 1: id = ps_str([ z1, z2 ],7); 2: q = psmat(id,zeros(2,2)); 3: q1 = ps(id); 4: q1(0) = [1 1]; 5: q(1,1) = q1*q1*q1*q1*q1; 6: q2 = ps(id); 7: q2(0) = [1 0 1]; 8: q(2,2) = q2*q2*q2*q2*q2; 9: r = [1]; z 1, z Q 2 Q psmat q1 (1 + z 1 ) [, z 1, z 2, ] [1 1] (1 + z 1 ) 5 Q (1,1) (1 + z 1 ) Q (2,2) (1 + z 2 ) 5 9 R 1 H 2 1: p = psmat_are(a,b*inv(r)*b,q); 2: k = b *p; 3: at = a-b*k; 1 Riccati A T P + PA + PBR 1 B T P Q = 0 P Q z 1, z 2 p psmat z 1, z 2
15 J.JSSAC Vol. 11, No. 3,4, K K psmat u(t) = Kx(t) 3 u(t) A (= A BK) at at psmat 1: [y1,t1]=psmat_peaktime(at,b,c,d,0.1); 2: [y2,t2]=psmat_peaktime(at,b,c,d,2); 6 z 1 = 0, z 2 = 0 t = 0.1, t = 2.0, t = 4 t = psmat z 1, z 2 t1 y1 t1 = *z *z *z1^ *z 1*z *z2^ *z1^ *z1^2*z *z1*z2^ *z2^ *z1^ *z1^3* z *z1^2*z2^ *z1*z2^ *z2^ *z1^ *z1^4*z *z1^3*z2^ *z1^2*z2^ *z1*z2^ *z2^ *z1^ *z1^5*z *z1^4*z2^ *z1^3*z2^ *z1^2*z2^ *z1*z2^ *z2^ *z1^ *z1^6*z *z1^5*z2^ *z1^4*z2^ *z1^3*z2^ *z1^2*z2^ *z1*z2^ *z2^7 y1 = *z *z *z1^ * z1*z *z2^ *z1^ *z1^2*z *z1*z2^ *z2^ *z1^ *z1^3 *z *z1^2*z2^ *z1*z2^ *z2^ *z1^ *z1^4*z *z1^3*z2^ *z1^2*z2^ *z1*z2^ *z2^ *z1 ^ *z1^5*z *z1^4*z2^ *z1^3*z2^ *z1^2*z2^ *z1*z2^ *z2^ *z1^ *z1^6*z *z1^5*z2^ *z1^4*z2^ *z1^3*z2^ *z1^2*z2^ *z1*z2^ *z2^7 y1, y2 y( ) = 1
16 , : tmp = c*psmat_inv(at)*b; 2: nf = -1/tmp(1); 3: ny1 = y1*nf; 4: ny2 = y2*nf; y( ) = C(A BK) 1 B 1/( C(A BK) 1 B) y1, y2 1 C(A BK) 1 B 2 1/(C(A BK) 1 B) nf 3-5 nf y1, y2 ny1, ny2 ny1 ny2-1 z 1 z 2 8, z 2 0 z z 1 8: z 1 9: ny1 ny2-1 ny1 < % ny2-1<0.1 10% 1 z 1 = 0.3, z 2 = 0.4 ny1= ny2= K 1: q = zeros(2,2); 2: q(1,1) = (1+0.3)^5; 3: q(2,2) = (1-0.4)^5; 4: p = riccati(a,b*b,q, c ); 5: k = b *p; 1-3 Q 4 Riccati riccati() SCILAB 5
17 J.JSSAC Vol. 11, No. 3,4, z z 1 1: K 7 y( ) = % 10% z 1 0.5, z Q (1 + z 1 ) 5, (1 + z 2 ) (1 + z 1 ) 5, (1 + z 2 ) Q (1 + z 1 ) 5, (1 + z 2 ) z 1, z 2 z 1, z (1 + z 1 ) 5, (1 + z 2 ) (2 + z 1 ) 5, (2 + z 2 ) 5 z 1 < 0.5, z 2 < (2 + z 1 ) 5, (2 + z 2 ) z 1 < 0.5, z 2 < 0.5 { (1 + z1 ) 5, (1 + z 2 ) 5}, { (2 + z 1 ) 5, (1 + z 2 ) 5}, { (1 + z 1 ) 5, (2 + z 2 ) 5}, { (2 + z 1 ) 5, (2 + z 2 ) 5} (z 1, z 2 ) = (0, 0) (z 1, z 1 ) = ( 0.5, 0.5), ( 0.5, 0.5), (0.5, 0.5), (0.5, 0.5) ny1,ny2 ny1,ny2
18 , (z 1, z 2 ) = ( 0.5, 0.5) 0.5 z 1, z 2 0 ny1, ny2 ny1, ny2 (z 1, z 2 ) = ( 0.5, 0.5), (0.5, 0.5), (0.5, 0.5) (z 1, z 2 ) = ( 0.5, 0.5) 30% 0.5 < z 1 < 0, 0 z ny1, ny (1 + z 1 ) 5 1, 1 (1 + z 2 ) ( z 1 ) 5, ( z 2 ) 5 z 1 < 0.25, z 2 < (1 + z 1 ) 5, (1 + z 2 ) SCILAB MATLAB SCILAB C++ SCILAB (a) (b) (c) SCILAB Mathematica Ver Mathematica 1/5 Riccati H 2
19 J.JSSAC Vol. 11, No. 3,4, [1] Scilab Group: Scilab, ftp://ftp.inria.fr/inria/scilab/contrib/scijapanese/intro-jp.pdf [2] Kitamoto, T.: Approximate eigenvalues, eigenvectors and inverse of a matrix with polynomial entries, Jpn. J. Indus. Appl. Math., 11(1), 1994, [3] :,, J78- A(4), 1995, [4] : H 2,, J81-A(2), 1998, [5] : Riccati,, J81-A(3), 1998, [6] :,, J81-A(4), 1998, [7],, J84-A(2), 2001, [8] Kitamoto, T.: On computation of approximate eigenvalues and eigenvectors, IEICE Trans. Fundamentals, E85-A(3), 2002, [9] Kitamoto, T.: Computation of the peak of time response in the form of formal power series, IEICE Trans. Fundamentals, E86-A(12), 2003, [10] Kung, H.T., Traub, J.F, All algebraci function can be computed fast, J. ACM, 25, 1978, [11] Lipson, J.D., Newton s method: A great algebraic algorithm, Proceedings of ACM Symposium on Symbolic and Algebraic Computations, 1976, [12] Yokoyama, K., Takeshima, T.: On Hensel construction of eigenvalues and eigenvectors of matrices with polynomial entries, Proceedings of the ISSAC 93, ACM PRESS, 1993,
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