[[授業]] *Advanced Automation [#ye2d4ebb] ** &color(green){[lecture #1]}; 2015.9.3 outline of the lecture, review of classical and modern control theory (1/3) [#bdb8b1f6] - outline of this lecture -- syllabus -- evaluation --- mini report #1 ... 10% --- mini exam #1 ... 10% --- mini report #2 ... 10% --- mini exam #2 ... 10% --- final report ... 60% -- [[schedule2015]] (tentative) -- map #ref(map_v1.0_review.pdf); - review : stabilization of 1st-order unstable plant by classical and modern control theory -- transfer function -- differential equation -- eigenvalue and pole -- ... %-- 9/3/2015 2:09 PM --% s = tf('s') Ptf = 1/(s+1) pole(Ptf) impulse(Ptf) Pss = ss(Ptf) initial(Pss, 1) initial(Pss, 2) #ref(2015.09.03-1.jpg,left,noimg,whiteboard #1); #ref(2015.09.03-2.jpg,left,noimg,whiteboard #2); #ref(2015.09.03-3.jpg,left,noimg,whiteboard #3); #ref(2015.09.03-4.jpg,left,noimg,whiteboard #4); &ref(ex0902.m); ** &color(green){[lecture #2]}; 2015.9.10 review of classical and modern control theory (2/3) with introduction of Matlab/Simulink [#x66b415e] + introduction of Matlab and Simulink &ref(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 difinition) -- m file // + system representation: Transfer Function(TF) / State-Space Representation (SSR) // -- example: mass-spring-damper system -- difinition 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(ex0910_1.m); &ref(mod0910_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(ex0910_2.m); &ref(mod0910_2.mdl); --- Gain Margin(GM); Phase Margin(PM) %-- 9/10/2015 1:55 PM --% ex0910_1 P P.den P.den{:} P.num{:} ex0910_1 ex0910_2 #ref(2015.09.10-1.jpg,left,noimg,whiteboard #1); #ref(2015.09.10-2.jpg,left,noimg,whiteboard #2); #ref(2015.09.10-3.jpg,left,noimg,whiteboard #3); #ref(2015.09.10-4.jpg,left,noimg,whiteboard #4); ** &color(green){[]}; 2015.9.17 cancelled [#w7af2c97] ** &color(green){[]}; 2015.9.25 no lecture (lectures for Monday are given) [#ne6e400c] ** &color(green){[lecture #3]}; 2015.10.1 review of classical and modern control theory (3/3) [#b83a9a65] + LQR problem -- controllability -- cost function J >= 0 -- (semi)-positive definiteness + solution of LQR problem -- ARE and quadratic equation -- closed loop stability ... Lyapunov criterion -- Jmin &ref(lqr.pdf); ≒ &ref(proof4.pdf); (from B3「動的システムの解析と制御」) + example &ref(mod1001.mdl); A = [1, 2; 0, -1]; % unstable plant B = [0; 1]; Uc = ctrb(A,B); det(Uc) % should be nonzero C = eye(2); % dummy D = zeros(2,1); % dummy F = [0, 0]; % without control x0 = [1; 1]; % initial state Q = eye(2); R = 1; P = are(A, B/R*B', Q); eig(P) % should be positive F = R\B'*P; x0'*P*x0 %-- 10/1/2015 2:08 PM --% mod1001 A = [1, 2; 0, -1]; % unstable plant B = [0; 1]; Uc = ctrb(A,B); A B Uc det(Uc) C = eye(2); % dummy D = zeros(2,1); % dummy F = [0, 0]; % without control x0 = [1; 1]; % initial state Q = eye(2); R = 1; F P = are(A, B/R*B', Q); P eig(P) F = R\B'*P; F J x0 x0'*P*x0 A-B*F eig(A-B*F) #ref(2015.10.01-1.jpg,left,noimg,whiteboard #1); #ref(2015.10.01-2.jpg,left,noimg,whiteboard #2); #ref(2015.10.01-3.jpg,left,noimg,whiteboard #3); ... I'm sorry but all of equations are in the pdf file. #ref(2015.10.01-4.jpg,left,noimg,whiteboard #4); #ref(2015.10.01-5.jpg,left,noimg,whiteboard #5); ** &color(green){[lecture #4]}; 2015.10.8 relation between LQR and H infinity control problem (1/2) [#d821c8a6] - GOAL: to learn difference in concepts between LQR problem and H infinity control problem - review of LQR problem and the simple example + an equivalent problem + a simple example of state-feedback H infinity control problem + definition of H infinity norm (SISO) s = tf('s'); P1 = 1/(s+1); bode(P1); norm(P1, 'inf') P2 = 1/(s^2 + 0.1*s + 1); bode(P2); norm(P2, 'inf') + definition of H infinity norm (SIMO) + solve the problem by hand + solve the problem by tool(hinfsyn) &ref(ex1008.m); %-- 10/8/2015 1:48 PM --% s = tf('s'); P1 = 1/(s+1); bode(P1); norm(P1, 'inf') P2 = 1/(s^2 + 0.1*s + 1); bode(P2); norm(P2, 'inf') ex1008 #ref(2015.10.08-1.jpg,left,noimg,whiteboard #1); #ref(2015.10.08-2.jpg,left,noimg,whiteboard #2); #ref(2015.10.08-3.jpg,left,noimg,whiteboard #3); #ref(2015.10.08-4.jpg,left,noimg,whiteboard #4); #ref(2015.10.08-5.jpg,left,noimg,whiteboard #5); -Q: 最後にfを求めてどうするのか分からなかった。 -A: 閉ループ系のH∞ノルムを最小化するfを求め、LQRの最適解と比較する予定でしたが、最後まで説明できずすみません。 -Q: |Tzw|∞ は感度関数になる? -A: Tzw が感度関数になるか?という意味と思いますが、一般化プラントの設定次第でそうなります(例えば目標値信号を w、偏差を z に選ぶ場合など)。次々回、その場合を扱います。 ** &color(green){[lecture #5]}; 2015.10.15 relation between LQR and H infinity control problem (2/2) [#w4c0811d] + complete the table in simple example + behavior of hinfsyn in &ref(ex1008.m); + confirm the cost function J for both controllers by simulation &ref(mod1015.mdl); + confirm the closed-loop H infinity norm for both controllers by simulation (common mdl file is available) -- review: steady-state response (see photo 8 @ lec. #2) -- how to construct the worst-case disturbance w(t) which maximizes L2 norm of z(t) ? -- what is the worst-case disturbance in the simple example ? + general case: &ref(hinf.pdf); includes the simple example as a special case -- LQR &ref(lqr.pdf); is included as a special case where gamma -> infinity, non-zero x(0), and B2 -> B %-- 10/15/2015 1:14 PM --% ex1008 K dcgain(K) gopt ex1008 mod1015 f f = 1 x0 = 0 h = 0.1 zz zz(end) h = 1e-6 zz(end) f = -1+sqrt(2) h zz(end) x0 = 1 zz(end) f h h = 10 zz(end)/ww(end) x0 x0 = 0 zz(end)/ww(end) sqrt(zz(end)/ww(end)) h h = 100 sqrt(zz(end)/ww(end)) #ref(2015.10.15-1.jpg,left,noimg,whiteboard #1); #ref(2015.10.15-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #6]}; 2015.10.22 Mixed sensitivity problem 1/3 [#tfbf02f5] + review &ref(map_v1.0_intro1.pdf); and outline + H infinity control problem (general form) + reference tracking problem + weighting function for sensitivity function + design example &ref(ex1022_1.m); &ref(ex1022_2.m); + the small gain theorem -- proof: Nyquist stability criterion + from performance optimization to robust stabilization %-- 10/22/2015 2:06 PM --% ex1022_1 eig(P) ex1022_2 #ref(2015.10.22-1.jpg,left,noimg,whiteboard #1); #ref(2015.10.22-2.jpg,left,noimg,whiteboard #2); #ref(2015.10.22-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #7]}; 2015.10.29 Mixed sensitivity problem 2/3 [#dd1fc284] + review &ref(map_v1.0_intro2.pdf); and outline + an equivalent problem of robust stabilization for reference tracking problem + uncertainty model and normalized uncertainty Delta + robust stabilization problem and an equivalent problem + practical example of plant with perturbation &ref(ex1029_1.m); + how to determine the model &ref(ex1029_2.m); + design example and simulation &ref(ex1029_3.m); &ref(mod1029.mdl); %-- 10/29/2015 1:52 PM --% ex1029_1 ex1029_2 ex1029_3 mod1029 c c = 0.8 #ref(2015.10.29-1.jpg,left,noimg,whiteboard #1); #ref(2015.10.29-2.jpg,left,noimg,whiteboard #2); #ref(2015.10.29-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #8]}; 2015.11.5 Mixed sensitivity problem 3/3 [#oc5e65d6] + review : (1)robust stabilization and (2)performance optimization + mixed sensitivity problem : a sufficient condition for (1) and (2) -- proof by definition of H infinity norm + construction of the generalized plant + design example &ref(ex1105_1.m); + gamma iteration by bisection method &ref(ex1105_2.m); + a problem of the mixed sensitivity problem: nominal performance and robust performance &ref(ex1105_3.m); + introduction of robust performance problem %-- 11/5/2015 1:37 PM --% ex1105_1 ex1105_2 gam ex1105_2 WT ex1105_2 ex1105_3 ex1105_2 ex1105_3 #ref(2015.11.05-1.jpg,left,noimg,whiteboard #1); #ref(2015.11.05-2.jpg,left,noimg,whiteboard #2); #ref(2015.11.05-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #9]}; 2015.11.12 robust performance problem 1/3 [#tae1c2b9] -- [[schedule2015]] mini report and exam + review: robust performance problem + an equivalent robust stability problem + definition of H infinity norm for general case (MIMO) + definition of (maximum) singular value M = [j, 0; -j, 1] M' eig(M'*M) svd(M) + mini report #1 &ref(report1.pdf); -- write by hand -- due date and place of submission -> see [[schedule2015]] -- check if your answer is correct or not before submission by using Matlab -- You will have a mini exam #1 related to this report + SVD: singular value decomposition -- definition [U,S,V] = svd(M) M = [j, 0; -j, 1; 2, 3] -- unitary matrix and 2 norm of vectors -- a property of SVD: input-output interpretation -- illustrative example: rotation matrix &ref(ex1112_1.m); + H infinity norm of Delta hat %-- 11/12/2015 1:01 PM --% M = [j, 0; -j, 1] M' eig(M'*M) svd(M) M = [j, 0; -j, 1] M' eig(M'*M) svd(M) (3+sqrt(5))/2 sqrt((3+sqrt(5))/2) help svd [U,S,V] = svd(M) U'*U M = [j, 0; -j, 1; 2, 3] [U,S,V] = svd(M) V'*V ex1112_1 #ref(2015.11.12-1.jpg,left,noimg,whiteboard #1); #ref(2015.11.12-2.jpg,left,noimg,whiteboard #2); #ref(2015.11.12-3.jpg,left,noimg,whiteboard #3); #ref(2015.11.12-4.jpg,left,noimg,whiteboard #4); -Q: Σでノルムが決まる?→固有値で -A: 与えられた行列Mを、入力をベクトル a、出力をベクトル b とする入出力システム \[ b = M a \] とみなすと、出力の2ノルムの最大値は、入力の2ノルムの \[ \bar \sigma(M) \] 倍となります(それを超えるような a はない)。 質問の意図と違っている場合はまた聞いてください。 -Q: it was too fast -A: This might be caused by my less explanation in Japanese. I will improve this in the next lecture. ** &color(green){[lecture #10]}; 2015.11.19 Robust performance problem (2/3) [#la4839f6] + return of mini report #1 + review and outline: robust stability problem for Delta hat and its equivalent problem(?) + signal vector's size is not restricted in H infinity control problem and small gain theorem + H infinity norm of Delta hat + design example: robust performance is achieved &ref(ex1119_1.m); + non structured uncertainty is considered ... the design problem is too conservative %-- 11/19/2015 1:23 PM --% doc hinfsyn ex1105_2 ex1105_3 gam_opt ex1119_1 gam_opt svd([1/sqrt(2), 0; 1/sqrt(2), 0]) #ref(2015.11.19-1.jpg,left,noimg,whiteboard #1); #ref(2015.11.19-2.jpg,left,noimg,whiteboard #2); -Q: Delta tilder is more tractability than Delta hat, Delta hat is more solvability than Delta tilder, OK? -A: That is right. For problems -- (P1) robust stabilization against Delta hat -- (P2) robust stabilization against Delta tilde,&br; P1 is more solvable than P2 because of the smaller uncertain set. &br; P2 is more tractable than P1 because of the ignorance of the structure. ** &color(green){[lecture #11]}; 2015.11.26 Robust performance problem (3/3) [#d50833d3] + review -- robust performance problem with Delta hat and conservative design problem with Delta tilde -- inclusion relation between two uncertain sets + introduction of the scaled H infinity control problem + how to determine structure of scaling matrix //+ mini report #2 &ref(report2.pdf); + design example &color(red){moved to next lecture}; % less conservative design ex1105_2 ex1105_3 ex1119_1 gam_opt0 = gam_opt; K_opt0 = K_opt; #ref(ex1126_1.m); + effect of scaling matrix &color(red){moved to next lecture}; #ref(ex1126_2.m); + mini exam #1 #ref(2015.11.26-1.jpg,left,noimg,whiteboard #1); #ref(2015.11.26-2.jpg,left,noimg,whiteboard #2); -Q: 薄いマーカーが見辛い -A: すみません。次回、新しいマーカーに交換してもらいます。 ** &color(green){[lecture #12]}; 2015.12.3 Robust performance problem (3/3) (cont.), Control system design for practical system (1/3) [#k0c92c66] + return of mini exam #1; schedule of mini report #2 and exam #2 + review of the scaled H infinity control problem + comments on mu-synthesis prolem + design example &color(red){(moved from the previous lecture)}; % less conservative design ex1105_2 ex1105_3 ex1119_1 gam_opt0 = gam_opt; K_opt0 = K_opt; #ref(ex1126_1.m); + effect of scaling matrix &color(red){(moved from the previous lecture)}; #ref(ex1126_2.m); + mini report #2 &ref(report2.pdf); -- write by hand -- due date and place of submission -> see [[schedule2015]] -- check if your answer is correct or not before submission by using Matlab -- You will have a mini exam #2 related to this report + controller design for practical system: active noise control in duct -- introduction of experimental setup #ref(exp_apparatus1.jpg,left,noimg); #ref(exp_apparatus2.jpg,left,noimg); -- objective of control system: to drive control loudspeaker by generating proper driving signal u using reference microphone output y such that the error microphone's output z is attenuated against the disturbance input w -- frequency response experiment #ref(ex1203_1.m); #ref(spk1.dat); #ref(spk2.dat); %-- 12/3/2015 1:27 PM --% ex1105_2 ex1105_3 ex1119_1 gam_opt0 = gam_opt; K_opt0 = K_opt; who gam_opt0 ex1126_1 gam_opt d_opt ex1126_2 ex1203_1 #ref(2015.12.03-1.jpg,left,noimg,whiteboard #1); #ref(2015.12.03-2.jpg,left,noimg,whiteboard #2); #ref(2015.12.03-3.jpg,left,noimg,whiteboard #3); -Q: Why \[ \mbox{(i) } \hat G(0.0111) \mbox{ is stable, and} \] \[ \mbox{(ii) } \hat G(0.0103) \mbox{ is unstable,} \] hold ? -A: In the first design (i), controller K_opt0 and minimized value of gamma (gam_opt0) are obtained so that the H infinity norm of the resultant closed-loop system clp0 without connecting Delta tilde is less than 1. Therefore, the closed-loop system composed of clp0 and Delta tilde is stable according to the small gain theorem. On the other hand, in the second design (ii), controller K_opt and minimized value of gamma (gam_opt) are obtained so that the H infinity norm of the resultant closed-loop system *with scaling* (clp_d) is less than 1. Please note that there is no guarantee for the H infinity norm of the closed-loop system *without scaling* (clp_1) to be less than 1. Indeed, we confirmed that the H infinity norm of clp_1 was larger than 1 in our example, by which the closed-loop system composed of clp_1 and Delta tilde is unstable according to the small gain theorem. However, Delta hat which has diagonal structure, can be connected to clp_1 without loosing closed-loop stability. (Maybe I didn't explain the last sentence. I'm sorry for this, if this is the reason of your question.) If this answer is not sufficient for your question, please ask again. ** &color(green){[lecture #13]}; 2015.12.10 Control system design for practical system (2/3) [#o15403b0] + return of mini report #2 + review of the experimental apparatus and frequency response experiment + design example -- determination of plant model(nominal plant and additive uncertainty weight)&br; &ref(nominal.m);&br; &ref(subspace.m); ... replacement of n4sid in System Identification Toolbox (not provided in IPC)&br; &ref(weight.m); -- 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); -- comparison of closed-loop gain characteristics with and without control&br; &ref(compare.m); -- result of control experiment&br; &ref(result.dat);&br; &ref(compare_result.m); + room 157 @ Dept. Mech. Bldg.2 ex1203_1 ctrlpref ex1203_1 346/3.6 ex1203_1 nominal weight cont nominal compare compare_result #ref(2015.12.10-1.jpg,left,noimg,whiteboard #1); #ref(2015.12.10-2.jpg,left,noimg,whiteboard #2); -Q: Each of the files I wanted to clarify what they represent. -A: A detailed explanation will be given in the next lecture. ** &color(green){[lecture #14]}; 2015.12.17 Control system design for practical system (3/3) [#ja8537cc] + final report ++design your controller(s) so that the system performance is improved compared with the design example introduced in the previous lecture ++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 +++time response and frequency spectrum (PSD) of control experiment ++Why is the performance of your system improved(or unfortunately deteriorated)? --&size(30){&color(red){due date: 8th(Fri) Jan 17:00};}; --submit your report(pdf or doc) by e-mail to kobayasi@nagaokaut.ac.jp --You can use Japanese --maximum controller order is 35 --submit your &size(25){&color(red){controller.dat, controller_order.dat, and controller.mat};}; %%to kobayasi@nagaokaut.ac.jp%% &color(magenta){&size(30){at this page:[[participant list2015>/:~exp/seigyokougakutokuron_2015]](download is also possible)};}; &size(30){&color(red){not later than 25th(Fri) Dec};}; --You can send up to 5 controllers --&size(25){&color(red){[[control experimental results will be uploaded here>/:~exp/seigyokougakutokuron_2015/exp]]};}; --freqresp ... frequency response will be measured and uploaded everyday + how to improve the performance ? -- order of the nominal plant -- weighting for robust stability + detailed explanation of m-files in the previous lecture + specifications of the experimental system ++ experimental equipments -- loudspeakers: AURA SOUND NSW2-326-8A (2inch, 15W) -- pressure sensors: NAGANO KEIKI KP15 -- A/D, D/A converters: CONTEC AD12-16(PCI), DA12-4(PCI) -- PC: Dell Dimension 1100 -- OS: Linux kernel 2.4.22 / Real Time Linux 3.2-pre3 ++ program sources for frequency response experiment -- [[freqresp.h>/:~exp/seigyokougakutokuron_2015/freqresp.h]] -- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2015/freqresp_module.c]] -- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2015/freqresp_app.c]] -- format of spk1.dat (u is used instead of w for spk2.dat) ---1st column ... frequency (Hz) ---2nd column ... gain from w(V) to y(V) (signal's unit is voltage (V)) ---3rd column ... phase from w to y ---4th column ... gain from w to z ---5th column ... phase from w to z ++ program sources for control experiment -- [[hinf.h>/:~exp/seigyokougakutokuron_2015/hinf.h]] -- [[hinf_module.c>/:~exp/seigyokougakutokuron_2015/hinf_module.c]] -- [[hinf_app.c>/:~exp/seigyokougakutokuron_2015/hinf_app.c]] -- format of result.dat ---1st column: time (s) ---2nd column: z (V) ---3rd column: y (V) ---4th column: u (V) ---5th column: w (V) ++ configuration of control experiment -- disturbance signal w is specified as described in hinf.h and hinf_module.c: #define AMP 3.0 // amplitude for disturbance #define DIST_INTERVAL 5 // interval step for updating w count_dist++; if(count_dist >= DIST_INTERVAL){ w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // uniform random number in [-AMP, AMP] count_dist = 0; } da_conv(V_OFFSET + w, 0); // D/A output to noise source w is updated with 1ms period (sampling period 0.2ms times DIST_INTERVAL 5) -- control signal u is limited to [-4, 4] as specified in hinf.h and hinf_module.c: #define U_MAX 4.00 if(u > U_MAX) u = U_MAX; if(u < -U_MAX) u = -U_MAX; u is set to 0 for t < 10(s). (controller is operated for 10 <= t < 15.) -- a high pass filter with cut-off frequency are used to cut DC components in z and y as described in hinf.h and hinf_module.c // HPF(1 rad/s) to cut DC in z and y #define AF 9.9980001999866674e-01 #define BF 1.9998000133326669e-04 #define CF -1.0000000000000000e+00 #define DF 1.0000000000000000e+00 ad_conv(&yz); // A/D input // HPFs yf = CF*xf_y + DF*yz[0]; xf_y = AF*xf_y + BF*yz[0]; zf = CF*xf_z + DF*yz[1]; xf_z = AF*xf_z + BF*yz[1]; // %%[[experiment directory>/:~exp/seigyokougakutokuron_2015]]%% // + mini exam #2 %-- 12/17/2015 1:36 PM --% help bodemag #ref(2015.12.17-1.jpg,left,noimg,whiteboard #1); -Q: I wanted to make a print on the final report. -A: Do you mean you want to submit your report in hard copy? If so, I will receive it. //[[network camera>/:~exp/picture/image.jpg]] //#ref(2014.12.18-1.jpg,left,noimg,whiteboard #1); //#ref(2014.12.18-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #15]}; 2015.12.24 Control system design for practical system (cont.) [#a165727c] - preparation of your own controller(s) -- submission procedure of controllers has been changed -- wav file is available - questionnaires -- to university -- for web-based experimental environment %-- 12/24/2015 1:11 PM --% compare #ref(2015.12.24-1.jpg,left,noimg,whiteboard #1); //■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ &color(black,red){&size(20){!!! the remaining page is under construction (the contents below are from 2014) !!!};}; //&color(black,red){&size(20){!!! the remaining page is under construction (the contents below are from 2014) !!!};}; // %-- 12/25/2014 12:58 PM --% load cont.mat who K_opt who Kd who Ghat load result.dat plot(result(:,1), result(:,2)) plot(result(:,1), result(:,3)) who bode(K_opt) bode(Kd) Kd1 = Kd K_opt1 = K_opt load cont.mat bode(K_opt1, 'b', K_opt, 'r') bode(Kd1, 'b', Kd, 'r') Kd_tmp = c2d(K_opt1, 0.000001); bode(Kd1, 'b', Kd, 'r', Kd_tmp, 'm') clear all load cont.mat who bode(K_opt) K_example = K_opt; load cont.mat bode(K_example, 'b', K_opt, 'r') #ref(2014.12.25-1.jpg,left,noimg,whiteboard #1); #ref(2014.12.25-2.jpg,left,noimg,whiteboard #2); //**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]]