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#author("2023-12-14T14:24:46+09:00","default:exp","exp") #author("2023-12-21T08:52:19+09:00","default:exp","exp") [[授業]] *Advanced Automation 2023 [#b4ea11e2] [[latest lecture>#uedd99b9]] ** &color(green){[lecture #1]}; 2023.9.7 outline of the lecture, review of classical and modern control theory (1/3) [#v87a417c] - outline of this lecture -- syllabus([https://vos-lc-web01.nagaokaut.ac.jp/]) -- evaluation --- mini report #1 ... 10% --- mini exam #1 ... 10% --- mini report #2 ... 10% --- mini exam #2 ... 10% --- final report ... 60% -- [[schedule2023]] (tentative) //-- map &ref(授業/制御工学特論2017/map_v1.1.pdf); for review &ref(授業/制御工学特論2017/map_v1.1_review.pdf); -- map &ref(授業/制御工学特論2017/map_v1.1_review.pdf); - review : stabilization of SISO unstable plant by classical and modern control theory -- transfer functions / differential equations -- poles / eigenvalues -- impulse response / initial value response -- ... %-- 2023/09/07 13:21 --% s = tf('s') P = 1/(s-1) pole(P) impulse(P) pole(P) K = 2 help step Tyr = K/(s-1+K) step(Tyr) #ref(2023.09.07-1.jpg,left,noimg,whiteboard #1); #ref(2023.09.07-2.jpg,left,noimg,whiteboard #2); #ref(2023.09.07-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #2]}; 2023.9.14 review of classical and modern control theory (2/3) with introduction of Matlab/Simulink [#x512561b] + introduction of Matlab and Simulink &ref(授業/制御工学特論2015/text_fixed.pdf); Basic usage of MATLAB and Simulink used for 情報処理演習及び考究II/Consideration and Practice of Information Processing II: Advanced Course of MATLAB -- interactive system (no compilation, no variable definition) -- m file //-- example: stabilization of inverted pendulum (sorry in Japanese) //--- [[derivation of equation of motion>http://c.nagaokaut.ac.jp/~kobayasi/i/Matlab/ex/text.pdf]] //--- [[stabilization of 1-link pendulum>http://c.nagaokaut.ac.jp/~kobayasi/i/Matlab/ex/1link.html]] //--- [[stabilization of 2-link pendulum>http://c.nagaokaut.ac.jp/~kobayasi/i/Matlab/ex/2link.html]] // // + system representation: Transfer Function(TF) / State-Space Representation (SSR) // -- example: mass-spring-damper system -- definition of SSR -- from SSR to TF -- from TF to SSR: controllable canonical form + open-loop characteristic -- open-loop stability: poles and eigenvalues -- Bode plot and frequency response &ref(ex0914_1.m); &ref(mod0914_1.mdl); --- cut off frequency; DC gain; -40dB/dec; variation of c --- relation between P(jw) and steady-state response + closed-loop stability -- Nyquist stability criterion (for L(s):stable) -- Nyquist plot &ref(ex0914_2.m); &ref(mod0914_2.mdl); --- Gain Margin(GM); Phase Margin(PM) %-- 2023/09/14 13:06 --% a = 1 u=[1;2;3] ex0914_1.m ex0914_1 sqrt(k/m) sqrt(k/m)/(2*pi) #ref(2023.09.14-1.jpg,left,noimg,whiteboard #1); #ref(2023.09.14-2.jpg,left,noimg,whiteboard #2); #ref(2023.09.14-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #3]}; 2023.9.21 review of classical and modern control theory (3/3) [#b704f31a] + LQR problem -- controllability -- cost function J >= 0 -- positive (semi-)definite matrices -- solution of LQR problem -- example &ref(ex0921_1.m); &ref(mod0921_1.mdl); + ARE and quadratic equation -- scalar case (solve by hand) //-- closed loop stability ... Lyapunov criterion //-- Jmin -- matrix case &ref(授業/制御工学特論2015/lqr.pdf); ≒ &ref(授業/制御工学特論2015/proof4.pdf); (from B3「動的システムの解析と制御」) %-- 2023/09/21 13:42 --% ex0921_1 A eig(A) Uc B A*B ex0921_1 help are P eig(P) G F F = [0, 0] J ex0921_1 F J J(end) Jmin #ref(2023.09.21-1.jpg,left,noimg,whiteboard #1); #ref(2023.09.21-2.jpg,left,noimg,whiteboard #2); #ref(2023.09.21-3.jpg,left,noimg,whiteboard #3); #ref(2023.09.21-4.jpg,left,noimg,whiteboard #4); ** &color(green){[lecture #4]}; 2023.9.28 relation between LQR and H infinity control problem (1/2) [#u796f2cb] - GOAL: to learn difference in concepts between LQR problem and H infinity control problem //- review of LQR problem and the simple example + a simple example relating LQR and H infinity control problems -- For given plant G \[ G = \left[\begin{array}{c|c:c} a & 1 & b \\ \hline \sqrt{q} & 0 & 0 \\ 0 & 0 & \sqrt{r} \\ \hdashline 1 & 0 & 0 \end{array} \right] = \left\{ \begin{array}{l} \dot x = ax + bu + w\\ z = \left[ \begin{array}{c} \sqrt{q} x \\ \sqrt{r} u \end{array}\right] \\ x = x \end{array}\right. \] with zero initial state value x(0) = 0, find a state-feedback controller \[ u = -f x \] such that \begin{eqnarray} (i) &&\quad \mbox{closed loop is stable} \\ (ii) &&\quad \mbox{minimize} \left\{\begin{array}{l} \| z \|_2 \mbox{ for } w(t) = \delta(t) \quad \mbox{(LQR)} \\ \| T_{zw} \|_\infty \mbox{($H_\infty$ control problem)}\end{array}\right. \end{eqnarray} -- comparison of norms in (ii) (for a = -1, b = 1, q = 1, r = 1) \[ \begin{array}{|c||c|c|}\hline & \mbox{LQR}: f=-1+\sqrt{2} & \quad \quad H_\infty: f=1\quad\quad \\ \hline\hline J=\|z\|_2^2 & & \\ \hline \|T_{zw}\|_\infty & & \\ \hline \end{array} \] + an alternative description to LQR problem ++ J = (L2 norm of z)^2 ++ impulse resp. with zero initial value = initial value resp. with zero disturbance + definition of H infinity norm (SISO) s = tf('s'); G1 = 1/(s+1); bode(G1); norm(G1, 'inf') G2 = 1/(s^2 + 0.1*s + 1); bode(G2); norm(G2, 'inf') + definition of H infinity norm (SIMO) + solve the problem by hand + solve the problem by tool(hinfsyn) &ref(ex0928_1.m); %-- 2023/09/28 13:41 --% s = tf('s'); G1 = 1/(s+1); bode(G1); norm(G1, 'inf') format e format long e norm(G1, 'inf') G3 = s/(s+1); bode(G3); norm(G3, 'inf') G2 = 1/(s^2 + 0.1*s + 1); bode(G2); norm(G2, 'inf') grid ctrlpref bode(G2); grid #ref(2023.09.28-1.jpg,left,noimg,whiteboard #1); #ref(2023.09.28-2.jpg,left,noimg,whiteboard #2); #ref(2023.09.28-3.jpg,left,noimg,whiteboard #3); #ref(2023.09.28-4.jpg,left,noimg,whiteboard #4); ** &color(green){[lecture #5]}; 2023.10.05 relation between LQR and H infinity control problem (2/2) [#e039fec2] + cont. -- solve the problem by hand -- solve the problem by tool(hinfsyn) &ref(ex0928_1.m); + complete the table in simple example + confirm the cost function J for both controllers by simulation &ref(mod1005.mdl); -- block diagram in the simulink model -- how to approximate impulse disturbance with a step function -- (unit) impulse disturbance resp. with zero initial condition = (unit) initial condition resp. with zero disturbance + confirm the closed-loop H infinity norm for both controllers by simulation -- H infinity norm = L2 induced norm -- review: steady-state response for sinusoidal input signal; how to choose the frequency such that the output amplitude is maximized ? -- the worst-case disturbance w(t) for the simple example ? + general state-feedback case: &ref(授業/制御工学特論2015/hinf.pdf); -- includes the simple example as a special case -- LQR &ref(授業/制御工学特論2015/lqr.pdf); is included as a special case in which gamma -> infinity, w(t) = 0, B2 -> B, and non-zero x(0) are considered %-- 2023/10/05 13:06 --% ex0928_1 K help sigma clp0 norm(clp, 'inf') norm(clp0, 'inf') sqrt(2-sqrt(2)) mod1005 f = -1+sqrt(2) h = 0.01 x0 = 0 plot(t, x) zz zz(end) x0 x0 = 1 zz f f = 1 zz x0 = 0 zz h x0 h = 100 f zz sqrt(zz(end)/ww(end)) f = -1+sqrt(2) sqrt(zz(end)/ww(end)) #ref(2023.10.05-1.jpg,left,noimg,whiteboard #1); #ref(2023.10.05-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #6]}; 2023.10.12 Mixed sensitivity problem 1/3 [#w60b9fda] + outline: &ref(授業/制御工学特論2017/map_v1.1_mixedsens1.pdf); -- sensitivity function S and complementary sensitivity function T + H infinity control problem (general case) -- with generalized plant G -- including the state-feedback case + reference tracking problem -- how to translate the condition (ii) into one with H infinity norm ? -- corresponding generalized plant G ? -- introduction of weighting function for sensitivity function in (ii) + design example &ref(ex1012_1.m); &ref(ex1012_2.m); + the small gain theorem -- proof: Nyquist stability criterion //+ from performance optimization to robust stabilization %-- 2023/10/12 13:43 --% ex1012_1 P eig(P) ctrlpref ex1012_1 ex1012_2 K_hinf eig(K_hinf) #ref(2023.10.12-1.jpg,left,noimg,whiteboard #1); #ref(2023.10.12-2.jpg,left,noimg,whiteboard #2); #ref(2023.10.12-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #7]}; 2023.10.19 Mixed sensitivity problem 2/3 [#vea23cff] + outline: from point to set &ref(授業/制御工学特論2017/map_v1.1_mixedsens2.pdf); + the small gain theorem ... robust stability = H infinity norm condition + normalized uncertainty Delta + uncertainty model + simple example of plant set -- given plant P tilde --- frequency response of plant with perturbation &ref(ex1019_1.m); -- how to determine P0 and WT ? --- frequency response based procedure for P0 and WT &ref(ex1019_2.m); + robust stabilization problem and equivalent problem -- design example and simulation &ref(ex1019_3.m); &ref(mod1019.mdl); %-- 2023/10/19 13:26 --% ex1019_1 ex1019_3 ex1019_2 ex1019_3 mod1019 c c = 0.8 c = 2 c = 1.5 #ref(2023.10.19-1.jpg,left,noimg,whiteboard #1); #ref(2023.10.19-2.jpg,left,noimg,whiteboard #2); #ref(2023.10.19-3.jpg,left,noimg,whiteboard #3); #ref(2023.10.19-4.jpg,left,noimg,whiteboard #4); ** &color(green){[lecture #8]}; 2023.10.26 Mixed sensitivity problem 3/3 [#db1ca877] //- schedule (no lecture will be given on Nov.31) - review: &ref(授業/制御工学特論2017/map_v1.1_mixedsens2.pdf); (1)performance optimization and (2)robust stabilization - outline: ++ how to design controllers considering both conditions in (1) and (2) ++ gap between NP(nominal performance) and RP(robust performance) + mixed sensitivity problem => (1) and (2) : proof + generalized plant for mixed senstivity problem + design example &ref(ex1026_1.m); minimize gamma by hand + gamma iteration by bisection method &ref(ex1026_2.m); tradeoff between model robust stability and performance + intro. to RP: weak point of mixed sensitivity problem(problem of NP) &ref(ex1026_3.m); %-- 2023/10/26 13:36 --% pwd ex1026_1 K ex1026_1 ex1026_2 gam ex1026_2 gam ex1026_2 gam ex1026_2 gam ex1026_3 ex1026_2 ex1026_3 #ref(2023.10.26-1.jpg,left,noimg,whiteboard #1); #ref(2023.10.26-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #9]}; 2023.11.2 robust performance problem 1/3 [#b2bf2dca] -- [[schedule2023]] + review -- mixed sensitivity problem : N.P. but not R.P. //-- robust performance problem (R.P.) c.f. the last whiteboard, but can not be solved by tool //-- the small gain theorem + robust performance problem (R.P.), but can not be solved by tool + an equivalent robust stability (R.S.) problem to R.P. -- (i) introduction of a fictitious uncertainty Delta_p (for performance) -- (ii) for 2-by-2 uncertainty block Delta hat which includes Delta and Delta_p + definition of H infinity norm for general case (MIMO) -- definition of singular values and the maximum singular value M = [1, 1; 1i/sqrt(2), -1i/sqrt(2)] M' eig(M'*M) svd(M) -- mini report #1 &ref(report1.pdf); ... You will have a mini exam #1 related to this report + proof of ||Delta hat||_inf <= 1 + design example: &ref(ex1102_1.m); -- robust performance is achieved but large gap -- non structured uncertainty is considered ... the design problem is too conservative %-- 2023/11/02 13:54 --% M = [1, 1; 1i/sqrt(2), -1i/sqrt(2)] M' eig(M'*M) svd(M) pwd ex1102_1 #ref(2023.11.02-1.jpg,left,noimg,whiteboard #1); #ref(2023.11.02-2.jpg,left,noimg,whiteboard #2); #ref(2023.11.02-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #10]}; 2023.11.9 Robust performance problem (2/3) [#t4e095f2] + return of mini report #1 //+ review //-- robust performance but too conservative // ex1108_1 //-- robust stability problem for Delta hat and its equivalent problem(?) with Delta tilde //-- structured unertainty Delta hat and unstructured uncertainty Delta tilde + SVD: singular value decomposition -- definition -- meaning of the largest singular value (a property and proof) -- 2 norm of vectors (Euclidean norm) -- SVD for 2-by-2 real matrix &ref(ex1109_1.m); %-- 2023/11/09 13:26 --% M = [1, 1; 1i/sqrt(2), -1i/sqrt(2)] [U, S, V] = svd(M) U'*U format long e U'*U U*U' V'*V U*S*V' U*S*V'- M format short U*S*V'- M ex1109_1 rand(1,1) rand(1,3) #ref(2023.11.09-1.jpg,left,noimg,whiteboard #1); #ref(2023.11.09-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #11]}; 2023.11.16 Robust performance problem (3/3) [#d8bda72f] + review : R.S. problems for structured and unstructured uncertainty + scaled H infinity control problem + relation between three problems + how to determine structure of scaling matrix + design example &ref(ex1116_1.m); ex1102_1 gam2 = gam_opt ex1116_1 gam_opt + mini exam #1 (10 min.) %-- 2023/11/16 13:03 --% ex1102_1 gam_opt format long gam_opt gam2 = gam_opt ex1116_1 gam_opt gam2 opts = hinfsynOptions opts.AutoScale = 'off' ex1116_1 #ref(2023.11.16-1.jpg,left,noimg,whiteboard #1); #ref(2023.11.16-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #12]}; 2023.11.23 Robust performance problem (3/3) (cont.), Control system design for practical system (1/3) [#w368135b] + return of mini exam #1 + review of scaling &ref(ex1123_1.m); + mini report #2 &ref(report2.pdf); + introduction of a practical system: Speed control of two inertia system with servo motors -- experimental setup &br; &ref(授業/制御工学特論2019/setup_fixed.pdf); &br; &ref(授業/制御工学特論2019/photo.jpg,left,noimg); -- objective of control system = reference speed tracking control problem: to drive the drive-side servomotor by generating proper driving signal u (T_M) using drive-side speed y (\omega_M) such that y tracks the reference speed command r -- frequency response experiment and physical model of speed control system #ref(ex1123_2.m); &ref(servo1.dat); &ref(servo2.dat); -- room 374 @ Dept. Mech. Bldg. 2 %-- 2023/11/23 13:09 --% ex1123_1 format long eig(clp2.a) ex1123_2 #ref(2023.11.23-1.jpg,left,noimg,whiteboard #1); #ref(2023.11.23-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #13]}; 2023.11.30 &color(red){&size(20){[CANCELLED]};}; [#o399a1a7] &color(red){&size(25){Due to a hardware problem in our experimental environment, today's lecture is cancelled. I'm sorry for the inconvenience.};}; &size(25){Please check the modified schedule at [[schedule2023]]}; ** &color(green){[lecture #13]}; 2023.12.7 Control system design for practical system (2/3) [#f7d44e25] + return of mini report #2; ... You will have a mini exam #2 related to this report next week -- [[schedule2023]] + review of the experimental system //-- closed-loop system of 2-by-2 plant G and controller K //-- closed-loop gain is desired to be minimized for constant speed operation -- the same robust performance problem setup given in lecture can be used to design controllers + design example -- frequency response experiment data&br; [[servo1.dat>/:~exp/seigyokougakutokuron_2023/exp/freqresp/1/servo1.dat]]&br; -- determination of plant model(nominal plant and multiplicative uncertainty weight)&br; &ref(nominal.m);&br; &ref(weight.m);&br; &ref(weight_2.m); ... loose weight was further used to prevent high-frequency oscillation -- configuration of generalized plant and controller design by scaled H infinity control problem using one-dimensional search on the scaling d&br; &ref(cont.m);&br; &ref(cont_2.m); ... for loose weight design //-- comparison of closed-loop gain characteristics with and without control&br; //&ref(compare.m); -- result of control experiment and evaluation&br; [[result.dat>/:~exp/seigyokougakutokuron_2023/exp/design_example/1/result.dat]]&br; [[result_2.dat>/:~exp/seigyokougakutokuron_2023/exp/design_example/2/result.dat]]; ... for loose weight design&br; &ref(perf.m); + final report and remote experimental system ++design your controller(s) so that the system performance is improved compared with the design example ++Draw the following figures and explain the difference between two control systems &color(red){(your controller and the design example)};: +++bode diagram of controllers //+++gain characteristic of closed-loop system from w to z +++gain characteristic of sensitivity function +++time response of control experiment ++Why is the performance of your system improved(or unfortunately deteriorated)? --&size(30){&color(red){due date: 4th(Thu) Jan 17:00};}; --submit your report(pdf file) by e-mail to kobayasi@nagaokaut.ac.jp --You can use Japanese --maximum controller order is 20 --submit your &size(25){&color(red){controller.dat, controller_order.dat, and controller.mat};}; &color(black){&size(30){at this page:[[participant list2023>/:~exp/seigyokougakutokuron_2023]](download is also possible)};}; &size(30){&color(red){not later than 26th(Tue) Dec};}; --the system will be started until next lecture --You can send up to 10 controllers --&size(30){&color(black){[[control experimental results will be uploaded here>/:~exp/seigyokougakutokuron_2023/exp]]};}; --freqresp ... frequency response will be measured and uploaded everyday + how to improve the performance ? -- performance and sensitivity gain -- accuracy of the nominal(physical) model, weighting for robust stability -- caution: measurement noise in high frequency range due to rotational speed calculation //+ detailed explanation of m-files in the previous lecture + specifications of the experimental system ++ program sources for frequency response experiment --- [[freqresp.h>/:~exp/seigyokougakutokuron_2023/freqresp.h]] --- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2023/freqresp_module.c]] --- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2023/freqresp_app.c]] --- format of servo1.dat 1st column ... frequency (Hz) 2nd column ... gain from T_M(Nm) to Omega_M(rad/s) 3rd column ... phase (deg) from T_M to Omega_M 4th column ... gain from T_M to Omega_L 5th column ... phase (deg) from T_M to Omega_L ++ program sources for control experiment --- [[hinf.h>/:~exp/seigyokougakutokuron_2023/hinf.h]] --- [[hinf_module.c>/:~exp/seigyokougakutokuron_2023/hinf_module.c]] --- [[hinf_app.c>/:~exp/seigyokougakutokuron_2023/hinf_app.c]] --- format of result.dat 1st column: time (s) 2nd column: y (Omega_M (rad/s)) 3rd column: Omega_L (rad/s) 4th column: u (T_M (Nm)) 5th column: r (rad/s) ++ configuration of control experiment --- reference signal r is specified as described in hinf.h and hinf_module.c: r = (double)COM_ROT_SPEED;; if(t > 2){ t1 = t - 2; t1 -= ((int)(t1 / REF_PERIOD))*REF_PERIOD; t1 /= 0.2; if(t1 < 1){ r += REF_AMP * t1; }else if(t1 < 4){ r += REF_AMP; }else if(t1 < 6){ r += REF_AMP * (5 - t1); }else if(t1 < 9){ r -= REF_AMP; }else if(t1 < 10){ r -= REF_AMP * (10 - t1); } } --- control signal u is limited as specified in hinf.h and hinf_module.c: #define U_MAX (RATED_TORQ / 3.0) if(u > U_MAX) u = U_MAX; if(u < -U_MAX) u = -U_MAX; u is generated by PI control for t < 1(s). Your designed controller is started at t = 1(s). ++ calculation of rotational speed --- The rotational speed is approximately calculated by using difference for one sampling period in hinf_module.c and freqresp_module.c like: theta_rad[0] = (double)read_theta(0) / (double)Pn212 * 2.0 * M_PI; theta_rad[1] = (double)read_theta(1) / (double)Pn212 * 2.0 * M_PI; y = (theta_rad[0] - theta_rad_before[0]) / msg->sampling_period; z = (theta_rad[1] - theta_rad_before[1]) / msg->sampling_period; theta_rad_before[0] = theta_rad[0]; theta_rad_before[1] = theta_rad[1]; where the sampling period is given as 0.25 ms. %-- 2023/12/07 13:33 --% pwd cd .. nominal weight cont perf axis([9.9, 10, -10, 50]) axis([9.99, 10, -10, 50]) weight_2 pwd weight_2 cont_2 perf cd .. perf ** &color(green){[lecture #14]}; 2023.12.14 Control system design for practical system (3/3) [#uedd99b9] - web based remote experiment system //-- your password were sent by e-mail -- usage; how to upload controller's -- powered by prof. Takebe, National Institute of Technology, Nagaoka College //--- now you can login after registration - supplemental explanations -- room temperature is displayed and stored in temp.txt (Bosch Sensortec BME280 is used) //-- generating wav file [[filter.c>/:~exp/seigyokougakutokuron_2021/filter.c]] (txt2wav.c is used to convert the text file to wav file) -- c2d() is used to discretize the resultant continuous-time controller in cont.m -- You can send up to 10 controllers (don't fall into trial and error; think always about the reason) -- no strict control objective is given ( there is a freedom to define what is good performance ) - preparation of your own controller(s) by using the remote experiment system - mini exam #2 %-- 2023/12/14 13:50 --% load result.dat plot(result(:,1), result(:,2)) axis([9, 10, 10, 25]) load result2.dat plot(result(:,1), result(:,2)) plot(result2(:,1), result2(:,2)) axis([9, 10, -10, 50]) axis([9.9, 10, -10, 50]) plot(result2(:,1), result2(:,2), '.-') axis([9.9, 10, -10, 50]) nominal weight weight2 weight_2 //■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ &color(black,red){&size(20){!!! the remaining page is under construction (the contents below are from last year) !!!};}; //&color(black,red){&size(20){!!! the remaining page is under construction (the contents below are from last year) !!!};}; ** &color(green){[lecture #15]}; 2022.12.22 Control system design for practical system (cont.) [#dad360a1] ** &color(green){[lecture #15]}; 2023.12.21 Control system design for practical system (cont.) [#dad360a1] - return of mini exam #2 - [[schedule2022]] no lecture will be given next week - %%return of mini exam #2%% //- [[schedule2022]] no lecture will be given next week //--- the system will be unavailable from %%21(Fri)%% &color(red){7:30 on 22(Sat)}; to &color(red){19:00 on}; 22(Sat) due to electrical construction scheduled on 22(sat) - preparation of your own controller(s) by using the remote experiment system //**related links [#g1a68a2b] //-東ティモール工学部復興支援/support of rehabilitation for faculty of eng. National University of Timor-Leste //--[[How to control objects>/:~kobayasi/easttimor/2009/index.html]] to design, to simulate and to experiment control system by using MATLAB/Simulink with an application of Inverted Pendulum //--[[Prof. Kimura's page>http://sessyu.nagaokaut.ac.jp/~kimuralab/index.php?%C0%A9%B8%E6%B9%A9%B3%D8%C6%C3%CF%C0]]
