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#author("2024-12-19T11:55:04+09:00","default:exp","exp") #author("2024-12-19T14:16:33+09:00","default:exp","exp") [[授業]] *Advanced Automation 2024 [#l41b3bc1] [[latest lecture>#sbe12a36]] ** &color(green){[lecture #1]}; 2024.9.5 outline of the lecture, review of classical and modern control theory (1/3) [#t6acac00] - 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% -- [[schedule2024]] (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 -- ... s = tf('s') P = 1/(s-1) pole(P) impulse(P) K = 2 Tyr = K/(s-1+K) step(Tyr) -minute paper [[https://cera-e1.nagaokaut.ac.jp/ilias/]] %-- 2024/09/05 13:40 --% s = tf('s') P = 1/(s-1) pole(P) impulse(P) K = 2 Tyr = K/(s-1+K) step(Tyr) #ref(2024.09.05-1.jpg,left,noimg,whiteboard #1); #ref(2024.09.05-2.jpg,left,noimg,whiteboard #2); #ref(2024.09.05-3.jpg,left,noimg,whiteboard #3); #ref(2024.09.05-4.jpg,left,noimg,whiteboard #4); ** &color(green){[lecture #2]}; 2024.9.12 review of classical and modern control theory (2/3) with introduction of Matlab/Simulink [#w02ec80c] + 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(ex0912_1.m); &ref(mod0912_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(ex0912_2.m); &ref(mod0912_2.mdl); --- Gain Margin(GM); Phase Margin(PM) %-- 2024/09/12 13:07 --% a = 2 a + 1 t=[1 2 3] t + 2 pwd ex0912_1 sqrt(k/m) sqrt(k/m)/(2*pi) #ref(2024.09.12-1.jpg,left,noimg,whiteboard #1); #ref(2024.09.12-2.jpg,left,noimg,whiteboard #2); #ref(2024.09.12-3.jpg,left,noimg,whiteboard #3); #ref(2024.09.12-4.jpg,left,noimg,whiteboard #4); ** &color(green){[lecture #3]}; 2024.9.19 review of classical and modern control theory (3/3) [#y0d3f70a] + LQR problem -- controllability -- cost function J >= 0 -- positive (semi-)definite matrices -- solution of LQR problem -- example &ref(ex0919_1.m); &ref(mod0919_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」) %-- 2024/09/19 13:53 --% ex0919_1 Uc [B A*B] ex0919_1 F #ref(2024.09.19-1.jpg,left,noimg,whiteboard #1); #ref(2024.09.19-2.jpg,left,noimg,whiteboard #2); #ref(2024.09.19-3.jpg,left,noimg,whiteboard #3); #ref(2024.09.19-4.jpg,left,noimg,whiteboard #4); ** &color(green){[lecture #4]}; 2024.9.26 relation between LQR and H infinity control problem (1/2) [#xb3706ee] - 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 = s/(s+1); bode(G2); norm(G2, 'inf') G3 = 1/(s^2 + 0.1*s + 1); bode(G3); norm(G3, 'inf') + definition of H infinity norm (SIMO) G4 = [1/(s+2); -1/(s+2)]; norm(G4, 'inf') + solve the problem by hand + solve the problem by tool(hinfsyn) &ref(ex0926_1.m); %-- 2024/09/26 14:04 --% s = tf('s'); G1 = 1/(s+1); bode(G1); norm(G1, 'inf') G2 = s/(s+1); bode(G2); norm(G2, 'inf') bode(G1); norm(G1, 'inf') bode(G2); norm(G2, 'inf') G3 = 1/(s^2 + 0.1*s + 1); bode(G3); norm(G3, 'inf') ctrlpref bode(G3); grid on G4 = [1/(s+2); -1/(s+2)]; norm(G4, 'inf') 1/sqrt(2) #ref(2024.09.26-1.jpg,left,noimg,whiteboard #1); #ref(2024.09.26-2.jpg,left,noimg,whiteboard #2); #ref(2024.09.26-3.jpg,left,noimg,whiteboard #3); #ref(2024.09.26-4.jpg,left,noimg,whiteboard #4); ** &color(green){[lecture #5]}; 2024.10.03 relation between LQR and H infinity control problem (2/2) [#e37d16ec] + cont. -- solve the problem by hand -- solve the problem by tool(hinfsyn) &ref(ex0926_1.m); + complete the table in simple example + confirm the cost function J for both controllers by simulation &ref(mod1003.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 %-- 2024/10/03 13:17 --% ex0926_1 sqrt(2-sqrt(2)) mod1003 h = 0.01 x0 = 0 f = 1 plot(t, x) zz plot(t, zz) zz(end) x0 zz(end) x0 = 1 zz(end) f = -1+sqrt(2) zz(end) h = 100 x0 = 0 f sqrt(zz(end)/ww(end)) f = 1 sqrt(zz(end)/ww(end)) #ref(2024.10.03-1.jpg,left,noimg,whiteboard #1); #ref(2024.10.03-2.jpg,left,noimg,whiteboard #2); #ref(2024.10.03-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #6]}; 2024.10.10 Mixed sensitivity problem 1/3 [#zb524741] + 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(ex1010_1.m); &ref(ex1010_2.m); + the small gain theorem -- proof: Nyquist stability criterion //+ from performance optimization to robust stabilization %-- 2024/10/10 13:47 --% ex1010_1 pole(P) ex1010_2 K_hinf K_hinf.a eig(K_hinf.a) ex1010_2 #ref(2024.10.10-1.jpg,left,noimg,whiteboard #1); #ref(2024.10.10-2.jpg,left,noimg,whiteboard #2); #ref(2024.10.10-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #7]}; 2024.10.17 Mixed sensitivity problem 2/3 [#pc4fbfb4] + 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(ex1017_1.m); -- how to determine P0 and WT ? --- frequency response based procedure for P0 and WT &ref(ex1017_2.m); + robust stabilization problem and equivalent problem -- design example and simulation &ref(ex1017_3.m); &ref(mod1017.mdl); %-- 2024/10/17 13:36 --% ex1017_1 ex1017_2 ex1017_1 ex1017_2 ex1017_1 ex1017_2 ex1017_3 mod1017 c c = 0.8 c = 2 c = 1.5 #ref(2024.10.17-1.jpg,left,noimg,whiteboard #1); #ref(2024.10.17-2.jpg,left,noimg,whiteboard #2); #ref(2024.10.17-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #8]}; 2024.10.24 Mixed sensitivity problem 3/3 [#d736b1d3] //- 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(ex1024_1.m); minimize gamma by hand + gamma iteration by bisection method &ref(ex1024_2.m); tradeoff between model robust stability and performance + intro. to RP: weak point of mixed sensitivity problem(problem of NP) &ref(ex1024_3.m); %-- 2024/10/24 13:41 --% ex1024_1 K ex1024_1 ex1024_2 gam ex1024_2 gam ex1024_2 gam help hinfsyn ex1024_3 ex1024_2 ex1024_3 #ref(2024.10.24-1.jpg,left,noimg,whiteboard #1); #ref(2024.10.24-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #9]}; 2024.10.31 robust performance problem 1/3 [#a3a962dd] -- [[schedule2024]] + 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/sqrt(2), 1/sqrt(2); 1i, -1i] 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(ex1031_1.m); -- robust performance is achieved but large gap -- non structured uncertainty is considered ... the design problem is too conservative %-- 2024/10/31 14:05 --% M = [1/sqrt(2), 1/sqrt(2); 1i, -1i] M' eig(M'*M) svd(M) ex1031_1 #ref(2024.10.31-1.jpg,left,noimg,whiteboard #1); #ref(2024.10.31-2.jpg,left,noimg,whiteboard #2); #ref(2024.10.31-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #10]}; 2024.11.7 Robust performance problem (2/3) [#dca84207] + 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(ex1107_1.m); %-- 2024/11/07 13:24 --% M = [sqrt(2), sqrt(2); -1i/sqrt(2), 1i/sqrt(2)] 1i j i [U, Sigma, V] = svd(M) U'*U format long e U'*U V*V' ex1107_1 #ref(2024.11.07-1.jpg,left,noimg,whiteboard #1); #ref(2024.11.07-2.jpg,left,noimg,whiteboard #2); #ref(2024.11.07-3.jpg,left,noimg,whiteboard #3); ** &color(green){[lecture #11]}; 2024.11.14 Robust performance problem (3/3) [#n12bc9db] + 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(ex1114_1.m); &ref(dgam.jpg,noimg); ex1031_1 gam2 = gam_opt ex1114_1 gam_opt + mini exam #1 (10 min.) %-- 2024/11/14 13:22 --% ex1031_1 gam_opt format long e gam_opt gam2 = gam_opt ex1114_1 gam_opt gam2 #ref(2024.11.14-1.jpg,left,noimg,whiteboard #1); #ref(2024.11.14-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #12]}; 2024.11.21 Robust performance problem (3/3) (cont.), Control system design for practical system (1/3) [#te317596] + return of mini exam #1 + review of scaling &ref(ex1121_1.m); + mini report #2 &ref(report2.pdf); + introduction of a practical system: active noise control in duct -- experimental setup &br; &ref(photo1.jpg,left,noimg); &ref(photo2.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(ex1121_2.m); &ref(spk1.dat); &ref(spk2.dat); -- room 157 @ Dept. Mech. Bldg. 2 %-- 2024/11/21 13:03 --% ex1121_1 gam2 gam3 ex1121_2 ctrlpref ex1121_2 c0 = 346 L = 1.55 c0/(2*L) #ref(2024.11.21-1.jpg,left,noimg,whiteboard #1); #ref(2024.11.21-2.jpg,left,noimg,whiteboard #2); ** &color(green){[lecture #13]}; 2024.11.28 &color(red){&size(20){[CANCELLED]};}; [#c3eeff23] &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 [[schedule2024]]}; ** &color(green){[lecture #13]}; 2024.12.5 Control system design for practical system (2/3) [#wab6af0f] + return of mini report #2; ... You will have a mini exam #2 related to this report next week -- [[schedule2024]] + 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 -- frequency response data of G can be used; how to handle modeling error of G ? + design example (modeling error for Gyu is only considered for simplicity) -- frequency response experiment data&br; [[spk1.dat>/:~exp/seigyokougakutokuron_2024/exp/freqresp/1/spk1.dat]]&br; [[spk2.dat>/:~exp/seigyokougakutokuron_2024/exp/freqresp/1/spk2.dat]]&br; -- determination of plant model(nominal plant and additive uncertainty weight)&br; &ref(nominal.m);&br; &ref(n4sid_replaced.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 and evaluation&br; [[result.dat>/:~exp/seigyokougakutokuron_2024/exp/design_example/1/result.dat]]&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 +++time response of control experiment ++Why is the performance of your system improved(or unfortunately deteriorated)? --&size(30){&color(red){due date: 6th(Mon) Jan 17:00};}; --submit your report(pdf file) 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};}; &color(black){&size(30){at this page:[[participant list2024>/:~exp/seigyokougakutokuron_2024]](download is also possible)};}; &size(30){&color(red){not later than 24th(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_2024/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: FOSTEX FE87E(10W) --- 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_2024/freqresp.h]] --- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2024/freqresp_module.c]] --- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2024/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_2024/hinf.h]] --- [[hinf_module.c>/:~exp/seigyokougakutokuron_2024/hinf_module.c]] --- [[hinf_app.c>/:~exp/seigyokougakutokuron_2024/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 0.5 // amplitude for disturbance w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // uniform random number in [-AMP, AMP] da_conv(V_OFFSET + w, 0); // D/A output to noise source --- control signal u is limited to [-3, 3] as specified in hinf.h and hinf_module.c: #define U_MAX 3.00 if(u > U_MAX) u = U_MAX; if(u < -U_MAX) u = -U_MAX; u is set to 0 for t < 5(s). (controller is operated for 5 <= t < 10) %-- 2024/12/05 13:31 --% nominal pwd nominal ls data ls data/spk1.dat save data/nominal.mat G0 G_g G0_g w nominal weight cont compare #ref(2024.12.05-1.jpg,left,noimg,whiteboard #1); #ref(2024.12.05-2.jpg,left,noimg,whiteboard #1); ** &color(green){[lecture #14]}; 2024.12.12 Control system design for practical system (3/3) [#sbe12a36] - review -- experimental system --- block diagram &ref(photo3.jpg,left,noimg); --- objective: z → 0 ... virtual open end (`absorption' was not correct) -- frequency response experiment --- AMP in freqresp.h has been decreased to avoid a saturation problem //#define AMP_CH0 0.5 #define AMP_CH0 0.4 //#define AMP_CH1 0.5 #define AMP_CH1 0.4 -- design example (download and save spk1.dat and spk2.dat in data/) >> nominal >> weight >> cont >> compare (upload the controller, carry out control experiment, and download the result.dat) >> perf - 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_2024/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 %-- 2024/12/12 12:58 --% nominal weight cont compare perf #ref(2024.12.12-1.jpg,left,noimg,whiteboard #1); //■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ //&color(black,red){&size(20){!!! the remaining page is under construction (the contents below are from last year) !!!};}; ** &color(green){[lecture #15]}; 2024.12.19 Control system design for practical system (cont.) [#te3dcf93] - %%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 %-- 2024/12/19 14:18 --% pwd cd .. ls perf result plot(t, result(:,2)) plot(result(:,1), result(:,2)) size(result) plot(result(49000:,1), result(49000:,2)) plot(result(49000:end,1), result(49000:end,2)) plot(result(49000:end,1), result(49000:end,2), '.-') plot(result(49000:end,1), result(49000:end,2), '-.') plot(result(49000:end,1), result(49000:end,2), 'bo-') //**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]]
