A MATLAB Toolbox for Parametric Rob TitleDesign based on symbolic computatio Design of Algorithms, Implementatio Author(s) 坂部, 啓 ; 屋並, 仁史 ; 穴井, 宏和 ; 原, 辰次 Citation 数理解析研究所講究録 (2004), 1395: 231-237 Issue Date 2004-10 URL http://hdl.handle.net/2433/25954 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University
$\dagger_{\mathrm{y}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{i}@\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{b}}$ $\mathrm{p}_{\mathrm{u}\mathrm{j}1\mathrm{t}\mathrm{s}\mathrm{u}}$ LABORATORIES $*\cdot$ UNIVERSITY $\mathrm{t}$ 1395 2004 231-237 231 A MATLAB Toolbox for Parametric Robust Control System Design based on symbolic computation KEI SAKABE HITOSHI YANAMI ( ) ( ) ALPHAOMEGA INC. HIROKAZU $\mathrm{a}_{\mathrm{n}\mathrm{a}\mathrm{i}}$ ( ) LTD. \ddagger $\mathrm{t}\mathrm{h}\mathrm{b}$ FUJITSU LABORATORIES $\mathrm{l}\mathrm{t}\mathrm{d}$ SHINJI HARA OF TOKYO \S 1 (fixed-structure controller) ( Quantffier Elimination: ) (semi-algebraic set ) GUI QEPCAD $[5, 7]$ 2 SDC(Sign Definite Condition) CAD(Cylindrical Algebraic Decomposition) (double-exponential) (single-exponential) [2] 6 $\Gamma$ -region PARADISE[6] sakabe@a2z.c0.jp.fujitsu.co.jp fanai@jp.fujitsu.com sshinjij{ara@ipc.i.u-tokyo.ac.jp XXXVI-I
$\omega_{t}$ $\gamma_{t}$ 232 SDC Maple SyNRAC $[3,4]$ MATLAB Extended symbolic math toolbox Maple MATLAB Maple SyNRAC GUI GUI GUI 3 2 SDC(Sign Definite Condition) $\mathrm{h}_{\infty}$ Sign Definite Condition( SDC) SDC SDC $f$ (x) Sign Definite 1 $f$(x): $\mathrm{r}\vdash t\mathrm{r}$ $x\in[a, b]$, $a<b$ (x) ( $f$ $[a, b]$ )Sign definite SDC 1: $\mathrm{p}\mathrm{i}$ 1 $S$ (s) $T$ (s) $S(s)= \frac{1}{1+p(s)c(s)}=\frac{s^{2}-s}{s^{2}+(k-1)s+m}$ (1) $T(s)= \frac{p(s)c(s)}{1+p(s)c(s)}=\frac{ks+m}{s^{2}+(k-1)s+m}$ (2) $\omega_{s}$ $\gamma_{s}$ 2 $S$(s) $T$ (8) $[\omega_{1},\omega 2]$ $ G _{[\cdot[u_{2}]}1 = \Delta\sup_{\omega_{1}\leq\omega\leq\omega_{2}} G(j\omega) $ (3) ( : ) $j$ XXXVI-2
$\frac{\mathrm{r}}{\mathrm{e}}\mathrm{a}2^{\mathrm{v}\wedge\cdot\cdot\vee\vee\wedge}\mathrm{o}_{\ovalbox{\tt\small REJECT}_{<}-}^{8\mathrm{o}\mathrm{d}*\mathrm{b}\cdot r\mathrm{v}}\mathrm{o}^{:}..\cdot \mathrm{f}^{\acute{\mathrm{i}}}.\cdot-j.i.\cdot.\tau_{b}^{\mu} $. 233 $\mathrm{m}$ 2: $S(s),T$(s) $ S(s) $ (o,i.1 $<\gamma_{s}$ (4) $ $ T(s) $ $ [$w_{2}$,ool $<\gamma$t(5) $n$ $G(s)=C(sI-A)^{-1}B+D$ $H=[C^{T}CA$ $\mathrm{x}(\gamma^{2}i-d^{t}d)^{-1}[-dt$ -t] C $B^{T}]$ (6) $h(s^{2})= $ si-h $ = \dot{.}\sum_{=0}^{n}h:s^{2:}$ (7) $s^{2}$, $x$ $f(x)= \Delta.\cdot\sum_{=0}^{n}h_{i}x^{:}$ (8) $n$ $G(s)=C(sI-A)^{-1}B+D$ $ G [\mathrm{t}d_{1},\omega_{2}]<\gamma$ $ G(j\omega_{1}) <\gamma$ $\omega_{2^{1}}\leq x\leq\omega_{2^{2}}$ $ G(j\omega_{2}) <\gamma$ [\sim ] $(4),(5)$ $(1),(2)$ $f(x)\neq 0$ $f_{s}(x)=x^{2}+ \frac{(2m\gamma_{s}^{2}-(k-1)^{2}\gamma_{\epsilon}^{2}+1)x+m^{2}\gamma_{s}^{2}}{-1+\gamma_{s}^{2}}$ (9) $f_{t}(x)=x^{2}+(2m-$ (k-1y $+ \frac{k^{2}}{\gamma_{t}^{2}}$ ) $x+m2(1- \frac{1}{\gamma_{t}^{2}})$ (10) SyNRAC SDC $\forall x>0,f(x)>0$ $[\omega_{1^{2}},\omega_{2^{2}}]$ $(9),(10)$ $[0, \infty]$ $z= \frac{x+\omega_{1^{2}}}{1+\overline{\omega}_{2}^{\urcorner}x}$ (11) $f_{s}(z)= \frac{(\gamma_{s}^{2}-1)\omega_{e}^{4}z^{2}+(z-\omega_{s}^{2})^{2}m^{2}\gamma_{s}^{2}+((k-1)^{2}\gamma_{s}^{2}-2m\gamma_{s}^{2}-1)(\omega_{s}^{2}z-1)\omega_{*}^{2}z}{(\gamma_{s}^{2}-1)(z-\omega_{s}^{2})^{2}}$ (12) $f_{t}(z)=(z- \omega_{2^{2}})^{2}+m^{2}(1-\frac{1}{\gamma_{t^{2}}})+$ (2m-(k $-1)^{2}+ \frac{k^{2}}{\gamma\iota^{2}}$ ) (13) $(z-\omega_{2^{2}})$ XXXVI-3
$\iota$ 0$ $\backslash$ $\rceil$ 234 $\omega_{s}=1(\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}),$ $\gamma_{s}=-20(\mathrm{d}\mathrm{b})=0.1,$ $\omega_{t}=20(\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}),$ $\gamma_{t}=-10(\mathrm{d}\mathrm{b})\simeq 0.3163$ $f_{s}$ (z), (z) $f_{s}$ $f_{t}$ 3, 4 (z), (z) $3_{\text{ }}4$ ( 5) $f_{t}$. $i:\backslash \cdot$. 1 $\tau\backslash$ $:\cdot.-$ 1 1 0 $$ $\backslash$ $8$ 2 0 0 $\mathrm{k}\uparrow 3: 4: 5: ( ) 3 GUI ( 6) ( 7) 8) ( 9) 4 6: 7: 8: 9: ( 6) 2 ( 7) XXXVI-4
$\mathrm{s}$ 235 ( 10) $\mathcal{l}$ {f(t)} $=F(s)$ (14) 10: 11: ( 11) $\mathrm{s}$ $($, $\copyright)_{\text{ }}$ XXXVI-5
$\mathrm{s}$ 238 ( ) (, 2 (, ) (\copyright,, ( ) (,0) (0) ( ) (0) - (,\copyright ) (, ) 4 CAD 3 1 SDC $\langle$ 2 GUI 3 $\backslash \text{ }$ GUI GUI 11 GUI GUI 12 GUI 13 MATLAB GUI :GUIDE GUI 14 XXXVI-6
Anai Anai, Yanami, Hara. Hara. 237 12: GUI 13: GUIDE 1 GUI 5 21 COE ( ) (JST) [1],. h., 27(6):714-716, [2] $\mathrm{h}$ [3] $\mathrm{h}$ 1991. and $\mathrm{s}$ Fixed-structure robust controller synthesis based on sign definite condition by a special quantifier elimination. In Proceedings of American Control Conference eooo, pages 1312-1316, 2000. $\mathrm{h}$ and $\mathrm{s}$ SyNRAC: a maple package for solving real algebraic constraints toward a robust parametric control toolbox. In Proceedings of SICE Annual Conference 2003 (Pukui, Japan), pages 1716-1721, 2003. [4] Hirokazu Anai and Hitoshi Yanami. SyNRAC: A maple package for solving real algebraic constraints. In Proceedings of International Workshop on Computer Algebra Systems and their Applications (CASA) ZOOS (Saint Petersburg, Russian Federation), P.M.A. Sloot et al. $(Eds.)$ : ICCS 2003, LNCS 2657 pages 828-837. Springer, 2003. [5] P. Dorato, W.Yang, and C.Abdallah. Application of quantifier elimination theory to robust multiobject feedback design. J. Symb. Comp. 11, pages 1-6, 1995. [6] T. B\"untet L. G\"uvenc D. Kaesbauer M. Kordt M. Muhler J. Ackermann, P. Blue and D. Odenthal. Robust control: The parameter space approach. Springer, 2002. [7] M. Jirstrand. Nonlinear control system design by quantifier elimination. J. of Symb. Comp. 24, pages 137-152, 1997. XXXVI-7