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[[授業]]

*Advanced Automation [#we2e9bed]

** &color(green){[]}; 2014.9.4 cancelled [#n7818221]

** &color(green){[lecture #1]}; 2014.9.11 outline of the lecture, review of classical and modern control theory (1/3) [#z238c276]

- 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% 
-- map
#ref(map_v1.0_review.pdf);

- review
-- equation of motion
-- transfer function
-- state space representation
-- eigenvalue and pole

#ref(2014.09.11-1.jpg,left,noimg,whiteboard #1);
#ref(2014.09.11-2.jpg,left,noimg,whiteboard #2);
#ref(2014.09.11-3.jpg,left,noimg,whiteboard #3);
#ref(2014.09.11-4.jpg,left,noimg,whiteboard #4);
#ref(2014.09.11-5.jpg,left,noimg,whiteboard #5);
#ref(2014.09.11-6.jpg,left,noimg,whiteboard #6);


** &color(green){[lecture #2]}; 2014.9.18 CACSD introduction with review of classical and modern control theory (2/3) [#b977e2f2]

+ 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
//
+ How to define open-loop system
//
++ &color(black,cyan){TF}; 
 s = tf('s');
 T1 = 1 / (s+1);
 T2 = 1 / (s^2 + 0.1*s + 1);
++ &color(black,lightgreen){SSR};
 A = [-0.3, -1; 1, 0];
 B = [1; 0];
 C = [0, 1];
 D = 0;
 S3 = ss(A, B, C, D);
-- Bode plot
 bode(T1, 'b-', T2, 'g', S3, 'r--');
 grid on;
//
+ open-loop stability can be checked by
//
++ &color(black,cyan){poles of TF};
 roots(T2.den{:})
++ &color(black,lightgreen){eigenvalues of A-matrix in SSR};
 eig(S3.a)
++ also by simulation
#ref(mod0918_1.mdl);
//
+ closed-loop stability
//
 L = 1/(s^3+1.5*s^2+1.5*s+1); % example of open-loop system
 roots(L.den{:}) % confirm the open-loop system is stable 
//
++ graphical test by &color(black,yellow){Nyquist stability criterion}; and &color(black,yellow){Bode plot}; with &color(black,yellow){GM};(gain margin) and &color(black,yellow){PM};(phase margin)
 nyquist(L)
//
 bode(L)
++ numerical test by closed-loop system
 clp_den = L.den{:} + L.num{:};
 roots(clp_den)
++ simulation
#ref(mod0918_2.mdl);

 %-- 9/18/2014 1:06 PM --%
 t = [1 2 3]
 u = [1;2;3]
 V = [1 2 3; 4 5 6; 7 8 9]
 t'
 t'*t
 who
 k=0:0.1:10:
 k=0:0.1:10;
 y = sin(k);
 whos
 plot(x,y)
 plot(k,y)
 foo
 print -djpeg sin.jpg
 s = tf('s');
 T1 = 1/(s+1)
 T2 = 1/(s^2+0.1*s+1);
 A = [-0.3, -1; 1, 0];
 B = [1; 0];
 C = [0, 1];
 D = 0;
 S3 = ss(A, B, C, D);
 A
 B
 S3
 bode(T1, 'b-', T2, 'g', S3, 'r--');
 grid on;
 T2
 T3 = tf(S3);
 T3
 T2
 T2.num
 T2.num{:}
 T2.den{:}
 roots(T2.den{:})
 S3
 S3.a
 eig(S3.a)
 mod0918_1
 bode(T1, 'b-', T2, 'g', S3, 'r--');
 grid on;
 roots(L.den{:})
 L
 L = 1/(s^3+1.5*s^2+1.5*s+1);
 L
 roots(L.den{:})
 nyquist(L)
 bode(L)
 grid on
 nyquist(L)
 L
 clp_den = L.den{:} + L.num{:};
 clp_den
 roots(clp_den)
 mod0918_2


#ref(2014.09.18-1.jpg,left,noimg,whiteboard #1);
#ref(2014.09.18-2.jpg,left,noimg,whiteboard #2);


** &color(green){[lecture #3]}; 2014.9.25 CACSD introduction with review of classical and modern control theory (3/3) [#u1cf67cc]

+ 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(proof4.pdf); (from B%%4%%3「動的システムの解析と制御」)
+ example
&ref(mod0925.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);
 P-P' % should be zero
 eig(P) % should be positive
 F = R\B'*P;

 %-- 9/25/2014 2:17 PM --%
 A = [1, 2; 0, -1]; % unstable plant
 B = [0; 1];
 Uc = ctrb(A,B);
 Uc
 det(Uc)]
 det(Uc)
 C = eye(2); % dummy
 D = zeros(2,1); % dummy
 F = [0, 0]; % without control
 x0 = [1; 1]; % initial state
 mod0925
 Q = eye(2);
 R = 1;
 P = are(A, B/R*B', Q);
 P-P' % should be zero
 eig(P) % should be positive
 F = R\B'*P;
 F
 J
 x0
 x0'*P*x0

&ref(2014.09.25-1.jpg,left,noimg,whiteboard #1); ... sorry for the mistake in Uc ! The correct one is
\[ U_c := \left[\begin{array}{ccccc} B & AB & A^2 B & \cdots & A^{n-1} B \end{array}\right] \]
#ref(2014.09.25-2.jpg,left,noimg,whiteboard #2);
#ref(2014.09.25-3.jpg,left,noimg,whiteboard #3);
#ref(2014.09.25-4.jpg,left,noimg,whiteboard #4);

** &color(green){[lecture #4]}; 2014.10.2 Intro. to robust control theory (H infinity control theory) 1/3 [#d73502ce]

+ review &ref(map_v1.0_intro1.pdf);
++ advantage and disadvantage of the modern control theory
++ explicit consideration of plant uncertainty ---> robust control theory
//
+ Typical design problems of H infinity control theory
++ robust stabilization
++ &color(black,yellow){performance optimization};
++ robust performance problem (robust stability and performance optimization are simultaneously considered)
//
+ H infinity norm
-- definition
-- example
//
+ H infinity control problem
-- definition
//
+ performance optimization example : reference tracking problem
-- relation to the sensitivity function S(s) (S(s) -> 0 is desired but impossible)
-- given control system
&ref(ex1002_1.m);
-- controller design with H infinity control theory
&ref(ex1002_2.m);

 %-- 10/2/2014 12:57 PM --%
 ex1002_1
 ex1002_2
 s = tf('s')
 T1 = 1/(s+1)
 norm(T1,inf)
 T2 = s/(s+1)
 norm(T2,inf)
 bode(T1)
 T3 = 10/(s+2)
 bode(T3)
 ex1002_1
 ex1002_2
 K
 K_hinf
 ex1002_2
 eig(K_hinf.a)

#ref(2014.10.02-1.jpg,left,noimg,whiteboard #1);
#ref(2014.10.02-2.jpg,left,noimg,whiteboard #2);
#ref(2014.10.02-3.jpg,left,noimg,whiteboard #3);
#ref(2014.10.02-4.jpg,left,noimg,whiteboard #4);

** &color(green){[lecture #5]}; 2014.10.09 Intro. to Robust Control Theory (H infinity control theory) 2/3 [#pbd554d4]

- review &ref(map_v1.0_intro2.pdf);
- schedule of mini report and exam #1

+ Typical design problems
++ &color(black,yellow){robust stabilization};
++ performance optimization
++ robust performance problem (robust stability and performance optimization are simultaneously considered)
//
+ connection between [H infinity control problem] and [robust stabilization problem]
-- small gain theorem
-- normalized uncertainty \Delta
-- sketch proof ... Nyquist stability criterion
+ How to design robust stabilizing controller with H infinity control problem ?
-- practical example : unstable plant with perturbation
#ref(ex1009_1.m)
-- how to use uncertainty model (multiplicative uncertainty model)
#ref(ex1009_2.m)
-- how to set generalized plant G ?
#ref(ex1009_3.m)
-- simulation
#ref(mod1009.mdl)

 %-- 10/9/2014 1:01 PM --%
 ex1009_1
 ex1009_2
 ex1009_3
 mod1009
 c

#ref(2014.10.09-1.jpg,left,noimg,whiteboard #1);
#ref(2014.10.09-2.jpg,left,noimg,whiteboard #2);
#ref(2014.10.09-3.jpg,left,noimg,whiteboard #3);
#ref(2014.10.09-4.jpg,left,noimg,whiteboard #4);

** &color(green){[lecture #6]}; 2014.10.16 Intro. to robust control theory (H infinity control theory) (3/3) [#g6834f91]
+ review
-- robust stabilization ... (1) ||WT T||_inf < 1 (for multiplicative uncertainty)
-- performance optimization ... (2) ||WS S||_inf < gamma -> min
-- &color(black,yellow){mixed sensitivity problem}; ... simultaneous consideration of stability and performance
+ a sufficient condition for (1) and (2) ... (*) property of maximum singular value
+ definition of singular value
+ mini report #1
-- write by hand
-- submit at the beginning of next lecture on 23 Oct.
-- check if your answer is correct or not before submission by using Matlab
-- You will have a mini exam #1 related to this report on 30 Oct.
+ meaning of singular value ... singular value decomposition (SVD)
+ proof of (*)
+ example
#ref(ex1016.m);

 %-- 10/16/2014 1:00 PM --%
 M = [j, 0; -j, 1]
 M
 svd(M)
 sqrt((3+sqrt(5))/2)
 M'
 M'*M
 ex1016

#ref(2014.10.16-1.jpg,left,noimg,whiteboard #1);
#ref(2014.10.16-2.jpg,left,noimg,whiteboard #2);
#ref(2014.10.16-3.jpg,left,noimg,whiteboard #3);
#ref(2014.10.16-4.jpg,left,noimg,whiteboard #4);
#ref(2014.10.16-5.jpg,left,noimg,whiteboard #5);
#ref(2014.10.16-6.jpg,left,noimg,whiteboard #6);

** &color(green){[lecture #7]}; 2014.10.23 review of SVD, robust performance problem 1/3 (motivation of robust performance) [#d3dac11c]
- submission of mini report #1
- review of SVD : graphical image and rotation matrix for 2-by-2 real matrix case 
#ref(ex1023_1.m);
- motivation of robust performance : nominal performance to robust performance
#ref(ex1023_2.m);
#ref(ex1023_3.m);

 %-- 10/23/2014 12:58 PM --%
 ex1023_1
 A
 S
 V
 V'*V
 V'*V(:,1)
 ex1009_1
 ex1009_2
 ex1023_2
 ex1023_3

#ref(2014.10.23-1.jpg,left,noimg,whiteboard #1);
#ref(2014.10.23-2.jpg,left,noimg,whiteboard #2);
#ref(2014.10.23-3.jpg,left,noimg,whiteboard #3);

-Q1: Is H infinity control theory available for discrete-time system ?
-A1: The answer is yes. You can use the same tool hinfsyn in Matlab to design discrete-time controller. If the question is to ask how to implement digital (discrete-time) controller for given continuous-time plant, there are 3 ways to do this: (i) continuous-time H infinity control based design (continuous-time controller is designed, then it is discretized); (ii) discretized H infinity control design (continuous-time plant is discretized first, then discrete-time controller is designed); (iii) sampled-data H infinity control design (discrete-time controller is directly designed for continuous-time plant)     

** &color(green){[lecture #8]}; 2014.10.30 Robust performance problem (2/3) [#z50bb5a3]

+ return of mini report #1
+ review of the limitation of mixed sensitivity problem
+ diffinition of robust performance (R.P.) problem (cf. nominal performance problem on white board #6 in photo #4 of lecture #4) ... S is changed to S~
+ review of robust stability (R.S.) problem on white board #5 in photo #5 of lecture #3 ... robust stability against Delta <=> closed-loop system without Delta has less-than-or-equal-to-one H infinity norm (by small gain theorem) 
+ equivalent R.P. problem with structured uncertainty Delta_hat
+ a conservative problem to R.P. with 2-by-2 unstructured uncertainty Delta_tilde
-- example based on the one given in the last lecture
#ref(ex1030_1.m);
-- a check of the conservativeness
#ref(ex1030_2.m);
+ Delta_tilde is larger set than Delta_hat ... conservativeness
+ mini exam #1 
&ref(exam1.pdf);

 %-- 10/30/2014 1:10 PM --%
 ex1023_2
 ex1023_3
 gam_opt
 ex1030_1
 gam_opt

#ref(2014.10.30-1.jpg,left,noimg,whiteboard #1);
#ref(2014.10.30-2.jpg,left,noimg,whiteboard #2);
#ref(2014.10.30-3.jpg,left,noimg,whiteboard #3);
#ref(2014.10.30-4.jpg,left,noimg,whiteboard #4);

** &color(green){[lecture #9]}; 2014.11.13 Robust performance problem (3/3) [#y1331d96]

- return of mini exam #1 
- review
- inclusion relation of two uncertain set (structured and non-structured)
- scaled H infinity control problem
- effect of scaling matrix
- how to determine structure of scaling matrix
- example
#ref(ex1113_1.m);
#ref(ex1113_2.m);
- %%mini report #2%%

 %-- 11/13/2014 12:58 PM --%
 ex1023_2
 gam_opt
 ex1023_3
 ex1030_1
 gam_opt
 ex1113
 ex1113_1
 gam_opt
 ex1113_2

#ref(2014.11.13-1.jpg,left,noimg,whiteboard #1);
#ref(2014.11.13-2.jpg,left,noimg,whiteboard #2);
#ref(2014.11.13-3.jpg,left,noimg,whiteboard #3);
#ref(2014.11.13-4.jpg,left,noimg,whiteboard #4);

** &color(green){[lecture #10]}; 2014.11.20 Robust performance problem (1/3) (cont.) [#uecb44ef]

- review - effect of scaling
- mini report #2
-- write by hand
-- submit at the beginning of next lecture on 27 Nov.
-- check if your answer is correct or not before submission by using Matlab
-- You will have a mini exam #2 related to this report on 4 Dec.
- how to obtain generalized plant by hand
- %%example: H infinity controller design by hand%%

 %-- 11/20/2014 1:11 PM --%
 ex1023_2
 ex1023_3
 ex1030_1
 ex1030_2
 k
 ex1113_1
 ex1113_2
 k
 Delta_hat

#ref(2014.11.20-1.jpg,left,noimg,whiteboard #1);
#ref(2014.11.20-2.jpg,left,noimg,whiteboard #2);
#ref(2014.11.20-3.jpg,left,noimg,whiteboard #3);
#ref(2014.11.20-4.jpg,left,noimg,whiteboard #4);

** &color(green){[lecture #11]}; 2014.11.27 relation between H infinity control and modern control theory [#p5cdaaf2]

- mini report #2 submit
- outline
-- state-feedback H infinity control and LQR
-- output-feedback H infinity control and LQG
- simple example
- find controller by hand <---> ans. by hinfsyn
&ref(ex1127.m);
- relation to LQR
- H infinity norm = L2 induced norm
- how to construct the worst case input ?
&ref(mod1127.mdl);
- general case
&ref(J_hinf.pdf);

 %-- 11/27/2014 1:51 PM --%
 ex1127
 1/sqrt(2)
 ex1127
 2*(sqrt(2)-1)
 mod1127
 x0 = 0
 h = 1
 f = 1
 zz
 ww

#ref(2014.11.27-1.jpg,left,noimg,whiteboard #1);
#ref(2014.11.27-2.jpg,left,noimg,whiteboard #2);
#ref(2014.11.27-3.jpg,left,noimg,whiteboard #3);

-Q: \[ \dot x, z \] の導出過程（ホワイトボード◯2）がわからなかった
-A: \[ \left[ \begin{array}{c} w \\ u \end{array} \right], \left[ \begin{array}{c} z \\ y \end{array} \right] \] をそれぞれ入力、出力とする一般化プラント \[ G = \left[ \begin{array}{c|c} A & B \\ \hline C & D \end{array} \right] \] に対して、次式が成り立ちます。
\[ \dot x = A x + B \left[ \begin{array}{c} w \\ u \end{array} \right] \]
\[ \left[ \begin{array}{c} z \\ y \end{array} \right] = C x + D \left[ \begin{array}{c} w \\ u \end{array} \right] \]
ただし、x は G の状態で、
\[
A = a = -1, \quad 
B = \left[ \begin{array}{cc} 1 & b \end{array} \right] = \left[ \begin{array}{cc} 1 & 1 \end{array} \right], \quad 
C = \left[ \begin{array}{c} \sqrt{q} \\ 0 \\ 1 \end{array} \right] 
= \left[ \begin{array}{c} 1 \\ 0 \\ 1 \end{array} \right], \quad  
D = \left[ \begin{array}{cc} 0 & 0 \\ 0 & \sqrt{r} \\ 0 & 0 \end{array} \right]
= \left[ \begin{array}{cc} 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{array} \right]
\] 
です。状態空間表現の表記によります。これと、u = -f x より、導出されます。

** &color(green){[lecture #12]}; 2014.12.4 relation between H infinity control and modern control theory (cont.); %%Speed control of two inertia system with servo motor (1/4)%% [#c8305e92]
- return of mini report #2
- contents for the last lecture
- %%speed control of two inertia system with servo motor%%&br;
%%&ref(setup.pdf);%%
- %%frequency response experiment and physical model of speed control system%%&br;
%%&ref(ex1204_1.m);%%&br;
%%&ref(ex1204_2.dat);%%
- mini exam #2

 %-- 12/4/2014 1:28 PM --%
 ex1127
 mod1127
 x0 = 0
 h = 1
 f = 1
 ww
 zz
 h = 10
 ww
 zz
 h = 50
 zz
 h
 zz


#ref(2014.12.4-1.jpg,left,noimg,whiteboard #1);
#ref(2014.12.4-2.jpg,left,noimg,whiteboard #2);

** &color(green){[lecture #13]}; 2014.12.11 Robust control design for a practical system : Speed control of two inertia system with servo motor (1/3) [#q3906ee1]

- return of mini exam #2
- introduction of experimental setup
#ref(setup.pdf); 
#ref(photo1.jpg,left,noimg);
- objective of control system
++ to reduce the tracking error of the driving motor against disturbance torque
++ robust stabilization against plant variation due to aging degradation
- frequency response experiment and physical model of speed control system
#ref(ex1211_1.m);
#ref(ex1211_2.dat);
#ref(ex1211_3.dat);
#ref(ex1211_4.dat);
- %%determination of nominal plant%%&br;
%%&ref(ex1211_5.m);%%
- %%determination of weighting function%%&br;
%%&ref(ex1211_6.m);%%

 %-- 12/11/2014 1:24 PM --%
 ex1211_1
 frdata
 frdata(:,1)
 P1_jw
 P1_g
 ex1211_1

#ref(2014.12.11-1.jpg,left,noimg,whiteboard #1);

-Q: What is the inertia moment of the load disk ?
-A: It is about 0.0002 (kg m^2) (60mm in diameter, 16mm in inner diameter, 20mm in thick, made by SS400)

-Q: 周波数応答実験について、定常応答になるのにどのくらい待っているか？音が大きくなるのはゲインが高いから？周波数変化のきざみは？
-A: 待ち時間は3秒です。音の発生源はよくわかりませんし、騒音計などで計測したこともありませんが、大きな音が聞こえるのは共振周波数付近です。周波数変化の刻みは常用対数で0.01です。以下に掲載するプログラムソースの freqresp.h 中で指定しています。

** &color(green){[lecture #14]}; 2014.12.18 Robust control design for a practical system : Speed control of two inertia system with servo motor (2/3) [#z228b373]

- design example
//-- modeling based on frequency response experiment 
//- design example 1 : PI control
//-- control experiment
//#ref(cont_PI.dat,,,`cont.dat' file for P control);
//#ref(cont_P_order.dat,,,`cont_order.dat' file for P control);
//#ref(result_P.dat);
//#ref(result_openloop.dat);
//#ref(openloop.mp4);
//#ref(ex1.mp4);
//- design example 2 : H infinity control
-- m-files
#ref(freqresp.m);
#ref(nominal.m);
#ref(weight.m);
#ref(cont.m);
#ref(perf.m);
 >> freqresp
 >> nominal
 >> weight
 >> cont
 >> perf
-- control experiment ... see [[participant list2014]]
- report
+design your controller(s) so that the system performance is improved compared with the design example above
+Draw the following figures and explain the difference between two control systems &color(red){(your controller and the example above)};:
++bode diagram of controllers
++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: 9th(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 20  
--submit your &size(25){&color(red){cont.dat, cont_order.dat, and cont.mat};}; to kobayasi@nagaokaut.ac.jp &size(30){&color(red){not later than 26th(Fri) Dec};};

- program sources for frequency response experiment
#ref(freqresp.h)
#ref(freqresp_module.c)
#ref(freqresp_app.c)
-- format of datafile
--- 1st column ... frequency (Hz)
--- 2nd column ... gain from T_M to omega_M
--- 3rd column ... phase from T_M to omega_M
--- 4th column ... gain from T_M to omega_L
--- 5th column ... phase from T_M to omega_L
- program sources for control experiment
#ref(hinf.h)
#ref(hinf_module.c)
#ref(hinf_app.c)
-- format of result.dat file
--- 1st column: time (s)
--- 2nd column: omega_M (rad/s)
--- 3rd column: T_M (Nm)
--- 4th column: reference speed (rad/s)
--- 5th column: T_L (Nm)
- configuration of control experiment
-- reference signal is generated as described in hinf_module.c: 
 if((t > 1)&&(t < 4)){
   r = 20.0;
 }else{
   r = 10;
 }
-- disturbance torque is specified as described in hinf_module.c:
  if((t > 2)&&(t < 3)){
   d = -0.1;
 }else{
   d = 0;
 }
- 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 = (double)read_theta(0) / (double)Pn212 * 2 * M_PI;
 speed_rad = (theta_rad - theta_rad_before) / msg->sampling_period;
 theta_rad_before = theta_rad
where the sampling period is given as 0.25 ms.

[[participant list2014]]

 %-- 12/18/2014 1:01 PM --%
 freqresp
 nominal
 help fitfrd
 weight
 cont
 help c2d
 perf

#ref(2014.12.18-1.jpg,left,noimg,whiteboard #1);
#ref(2014.12.18-2.jpg,left,noimg,whiteboard #2);

//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is under construction (the contents below are from 2013) !!!};};

** &color(green){[lecture #15]}; 2014.12.25 Robust control design for a practical system : Speed control of two inertia system with servo motor (3/3) [#b2861acd]

-preparation of your own controller(s)

 %-- 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]]