授業/制御工学特論2019
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[[授業]]
*Advanced Automation [#xd2b10c7]
[[latest lecture>#j156dee6]]
** &color(green){[lecture #1]}; 2019.9.5 outline of the l...
- 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%
-- [[schedule2019]] (tentative)
-- map &ref(授業/制御工学特論2017/map_v1.1.pdf); for revi...
- review : stabilization of SISO unstable plant by classi...
-- transfer functions / differential equations
-- poles / eigenvalues
-- impulse response / initial value response
-- ...
%-- 2019/09/05 13:23 --%
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
k = 2
Tyr = feedback(P*k, 1)
step(Tyr)
k = 10
Tyr = feedback(P*k, 1)
step(Tyr)
#ref(2019.09.05-1.jpg,left,noimg,whiteboard #1);
#ref(2019.09.05-2.jpg,left,noimg,whiteboard #2);
#ref(2019.09.05-3.jpg,left,noimg,whiteboard #3);
#ref(2019.09.05-4.jpg,left,noimg,whiteboard #4);
-Q: MATLAB は使いこなせないと後になると大変ですか？
-Q: MATLAB は使いなれていないと授業はきびしいのか。
-A: 使いこなせなくても大丈夫です。多くの場合、mファイル（...
-Q: レポートおよびテストは日本語か英語か
-A: mini report、mini exam は、課題は英語で書かれています...
-Q: ブロック線図を左から右に描いて欲しい。
-A: 式との対応が良いので、右から左に描いています。たとえ...
** &color(green){[lecture #2]}; 2019.9.12 review of class...
+ introduction of Matlab and Simulink
&ref(授業/制御工学特論2015/text_fixed.pdf); Basic usage o...
-- interactive system (no compilation, no variable difini...
-- m file
//-- example: stabilization of inverted pendulum (sorry i...
//--- [[derivation of equation of motion>http://c.nagaoka...
//--- [[stabilization of 1-link pendulum>http://c.nagaoka...
//--- [[stabilization of 2-link pendulum>http://c.nagaoka...
//
//
+ system representation: Transfer Function(TF) / State-Sp...
//
-- 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(ex0912_1.m); &re...
--- 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)
%-- 2019/09/12 13:04 --%
a = 1
t = [1 2 3]
u = [4; 5; 6]
t'
who
b = u
who
ex0912_1
P
P.num
P.num{:}
P.den{:}
ex0912_2
#ref(2019.09.12-1.jpg,left,noimg,whiteboard #1);
#ref(2019.09.12-2.jpg,left,noimg,whiteboard #2);
#ref(2019.09.12-3.jpg,left,noimg,whiteboard #3);
#ref(2019.09.12-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #3]}; 2019.9.19 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- (semi)-positive definiteness
-- 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); ≒ &re...
%-- 2019/09/19 13:16 --%
ex0919_1
A
B
P
A
A'*P+P*A+Q-P*B*inv(R)*B'*P
P
Q
R
J
plot(t, J)
Jmin
#ref(2019.09.19-1.jpg,left,noimg,whiteboard #1);
#ref(2019.09.19-2.jpg,left,noimg,whiteboard #2);
#ref(2019.09.19-3.jpg,left,noimg,whiteboard #3);
#ref(2019.09.19-4.jpg,left,noimg,whiteboard #4);
-Q: --- 一つ、質問かどうか判断できないものがあり、質問の...
-Q: recommend text books for review ?
-A: the followings are only for example,
-- for the classical control theory: @book{BCKuo:Book,
author = "Benjamin C. Kuo",
title = "Automatic Control Systems",
edition = "7",
publisher = "Prentice Hall",
year = "1995"
}
-- for the modern control theory:
@book{Noton:Book,
author = "Maxwell Noton",
title = "Modern Control Engineering",
publisher = "Pergamon Press Inc.",
year = "1972"
}
** &color(green){[lecture #4]}; 2019.9.26 relation betwee...
- GOAL: to learn difference in concepts between LQR probl...
//- review of LQR problem and the simple example
+ a simple example relating LQR and H infinity control pr...
-- For given plant G
\[
G = \left[\begin{array}{c|c:c} a & 1 & b \\ \hline \sqrt{...
= \left\{ \begin{array}{l} \dot x = ax + bu + w\\ z = \le...
\]
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) = \de...
\| T_{zw} \|_\infty \mbox{($H_\infty$ control problem)}\e...
\end{eqnarray}
-- comparison of norms in (ii) (for a = -1, b = 1, q = 1,...
\[
\begin{array}{|c||c|c|}\hline
& \mbox{LQR}: f=-1+\sqrt{2} & \quad \quad H_\infty: f=1\q...
J=\|z\|_2^2 & & \\ \hline
\|T_{zw}\|_\infty & & \\ \hline
\end{array}
\]
+ an alternative description to LQR problem
++ J = (L2 norm of z)^2
++ impulse resp. with zero initial value = initial value ...
+ 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(ex0926_1.m);
%-- 2019/09/26 13:55 --%
s = tf('s')
G1 = 1/(s+1);
bode(G1);
norm(G1, inf)
help norm
G2 = 1/(s^2+0.1*s+1)
bode(G2)
ctrlpref
bode(G2)
grid on
norm(G2, inf)
format long e
norm(G2, inf)
#ref(2019.09.26-1.jpg,left,noimg,whiteboard #1);
#ref(2019.09.26-2.jpg,left,noimg,whiteboard #2);
#ref(2019.09.26-3.jpg,left,noimg,whiteboard #3);
#ref(2019.09.26-4.jpg,left,noimg,whiteboard #4);
#ref(2019.09.26-5.jpg,left,noimg,whiteboard #5);
** &color(green){[lecture #5]}; 2019.10.03 relation betwe...
+ complete the table in simple example
+ confirm the cost function J for both controllers by sim...
-- block diagram in the simulink model
-- how to approximate impulse disturbance with a step fun...
-- impulse disturbance resp. with zero initial condition ...
+ confirm the closed-loop H infinity norm for both contro...
-- H infinity norm = L2 induced norm
-- review: steady-state response; the worst-case disturba...
-- how to make the worst-case disturbance w(t)? w(t) for ...
+ general state-feedback case: &ref(授業/制御工学特論2015...
-- includes the simple example as a special case
-- LQR &ref(授業/制御工学特論2015/lqr.pdf); is included a...
%-- 2019/10/03 13:20 --%
ex0926_1
K
bode(K)
sqrt(2)/2
sqrt(2-sqrt(2))
mod1003
h
h = 0.01
x0
x0 = 0
f = -1+sqrt(2)
zz
zz(end)
h
h = 0.00001
zz(end)
f = 1
zz(end)
x0
x0 = 1
f
zz(end)
f = -1+sqrt(2)
zz(end)
h
h = 100
x0
x0 = 0
zz(end)
sqrt(zz(end)/ww(end))
h
h = 10000
sqrt(zz(end)/ww(end))
sqrt(2-sqrt(2))
#ref(2019.10.03-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.03-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.03-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #6]}; 2019.10.10 Mixed sensitiv...
+ outline: &ref(授業/制御工学特論2017/map_v1.1_mixedsens1...
-- sensitivity function S and complementary sensitivity f...
+ 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 in...
-- corresponding generalized plant G ?
-- introduction of weighting function for sensitivity fun...
+ design example &ref(ex1010_1.m); &ref(ex1010_2.m);
+ the small gain theorem
-- proof: Nyquist stability criterion
//+ from performance optimization to robust stabilization
%-- 2019/10/10 12:58 --%
ex1010_1
P
pole(P)
eig(P)
ex1010_2
K_hinf
K_hinf.a
eig(K_hinf.a)
ex1010_2
#ref(2019.10.10-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.10-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.10-3.jpg,left,noimg,whiteboard #3);
-Q: スモールゲインの定理と、極配置の実部が負となることの...
-A: ナイキストの安定判別と開ループ系の安定性の関係につい...
開ループ系が安定な場合、ナイキストの安定判別は簡単になり...
-Q: WSの決め方はプラントの感度に依存しますか？
-A: 感度関数も相補感度関数もプラント単独ではなくコントロ...
「プラントの感度」は「プラントのゲイン」の意味だと思って...
結論から言えば、プラントのゲインに依存します。
十分な目標値追従性能を達成したい周波数帯域でWSのゲインを...
** &color(green){[lecture #7]}; 2019.10.17 Mixed sensitiv...
+ outline: from point to set &ref(授業/制御工学特論2017/m...
+ review: the small gain theorem ... robust stability = H...
+ normalized uncertainty Delta
+ uncertainty model
+ how to determine P0 and WT
-- example: frequency response of plant with perturbation...
-- frequency response based procedure for P0 and WT &ref(...
+ robust stabilization problem and equivalent problem
-- design example and simulation &ref(ex1017_3.m); &ref(m...
%-- 2019/10/17 13:00 --%
ex1017_1
ex1017_2
ex1017_1
ex1017_2
ex1017_3
mod1017
c
c = 0.8
c = 2
#ref(2019.10.17-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.17-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.17-3.jpg,left,noimg,whiteboard #3);
#ref(2019.10.17-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #8]}; 2019.10.24 Mixed sensitiv...
//- schedule (no lecture will be given on Nov.31)
- review: &ref(授業/制御工学特論2017/map_v1.1_mixedsens2....
- outline:
++ how to design controllers considering both conditions ...
++ gap between NP(nominal performance) and RP(robust perf...
+ 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);
+ intro. to RP(problem of NP) &ref(ex1024_3.m);
%-- 2019/10/24 13:00 --%
ex1024_1
K
ex1024_1
ex1024_2
ex1024_3
#ref(2019.10.24-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.24-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.24-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #9]}; 2019.10.31 robust perform...
-- [[schedule2019]] mini report #1; exam #1; lec. on Nov....
+ review
-- mixed sensitivity problem : N.P. but not R.P.
-- robust performance problem (R.P.) c.f. the last whiteb...
-- the small gain theorem
+ an equivalent robust stability (R.S.) problem to R.P.
-- (i) introduction of a fictitious uncertainty Delta_p (...
-- (ii) for 2-by-2 uncertainty block Delta hat which incl...
+ definition of H infinity norm for general case (MIMO)
-- definition of singular values and the maximum singular...
M = [1, 1i/sqrt(2); 1, -1i/sqrt(2)]
M'
eig(M'*M)
svd(M)
-- mini report #1 &ref(report1.pdf); ... You will have a ...
+ 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 desig...
%-- 2019/10/31 14:07 --%
M = [1, 1i/sqrt(2); 1, -1i/sqrt(2)]
M'
eig(M'*M)
svd(M)
ex1031
ex1031_1
#ref(2019.10.31-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.31-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.31-3.jpg,left,noimg,whiteboard #3);
#ref(2019.10.31-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture --]}; 2019.11.7 canceled [#dd5e...
** &color(green){[lecture #10]}; 2019.11.14 Robust perfor...
+ return of mini report #1
//+ review
//-- robust performance but too conservative
// ex1108_1
//-- robust stability problem for Delta hat and its equiv...
//-- structured unertainty Delta hat and unstructured unc...
+ SVD: singular value decomposition
-- definition
-- meaning of the largest singular value (a property and ...
-- 2-norm of vectors (Euclidean norm)
-- SVD for 2-by-2 real matrix &ref(ex1114_1.m);
%-- 2019/11/14 13:17 --%
M = [1/sqrt(2), 1i; 1/sqrt(2), -1i]
[U, Sigma, V] = svd(M)
svd(M)
[U, Sigma, V] = svd(M)
U'*U
V'*V
U*Sigma*V'
U*Sigma*V'-M
ex1114_1
#ref(2019.11.14-1.jpg,left,noimg,whiteboard #1);
#ref(2019.11.14-2.jpg,left,noimg,whiteboard #2);
#ref(2019.11.14-3.jpg,left,noimg,whiteboard #3);
#ref(2019.11.14-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #11]}; 2019.11.21 Robust perfor...
+ review: R.S. prob. for Delta hat and Delta tilde
+ scaled H infinity control problem
+ relation between three problems
+ how to determine structure of scaling matrix
+ design example &ref(ex1121_1.m);
ex1031_1
gam2 = gam_opt
K2 = K_opt;
ex1121_1
gam_opt
+ %%mini report #2 &ref(report2.pdf);%%
+ mini exam #1 (10 min.)
%-- 2019/11/21 13:50 --%
ex1031_1
pwd
ex1031_1
gam2 = gam_opt
K2 = K_opt
gam2 = gam_opt
ex1121_1
gam_opt
d_opt
#ref(2019.11.21-1.jpg,left,noimg,whiteboard #1);
#ref(2019.11.21-2.jpg,left,noimg,whiteboard #2);
#ref(2019.11.21-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #12]}; 2019.11.28 Robust perfor...
+ return of mini exam #1
+ review of scaling &ref(ex1128_1.m);
+ mini report #2 &ref(report2.pdf);
+ introduction of a practical system: Speed control of tw...
-- experimental setup &br;
&ref(setup_fixed.pdf); &br;
&ref(photo.jpg,left,noimg);
-- objective of control system = disturbance attenuation ...
-- frequency response experiment and physical model of sp...
#ref(ex1128_2.m);
&ref(servo1.dat); &ref(servo2.dat); ... old data(the reso...
-- room 374 @ Dept. Mech. Bldg. 2
%-- 2019/11/28 12:59 --%
ex1128_1
gam2
gam3
format long e
norm(M3_d, 'inf')
ex1128_2
#ref(2019.11.28-1.jpg,left,noimg,whiteboard #1);
#ref(2019.11.28-2.jpg,left,noimg,whiteboard #2);
#ref(2019.11.28-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #13]}; 2019.12.5 Control system...
+ return of mini report #2; ... You will have a mini exa...
-- [[schedule2019]]
+ 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 consta...
-- frequency response data of G can be used; how to handl...
+ design example (modeling error for Gyu is only consider...
-- frequency response experiment data&br;
[[servo1.dat>/:~exp/seigyokougakutokuron_2019/exp/freqres...
[[servo2.dat>/:~exp/seigyokougakutokuron_2019/exp/freqres...
-- determination of plant model(nominal plant and additiv...
&ref(nominal.m);&br;
&ref(weight.m);
-- configuration of generalized plant and controller desi...
&ref(cont.m);
-- comparison of closed-loop gain characteristics with an...
&ref(compare.m);
-- result of control experiment&br;
[[result.dat>/:~exp/seigyokougakutokuron_2019/exp/design_...
&ref(perf.m);
//+ room 157 @ Dept. Mech. Bldg.2
+ final report and remote experimental system
++design your controller(s) so that the system performanc...
++Draw the following figures and explain the difference b...
+++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 unfor...
--&size(30){&color(red){due date: 6th(Mon) Jan 17:00};};
--submit your report(pdf or doc) by e-mail to kobayasi@na...
--You can use Japanese
--maximum controller order is 20
--submit your &size(25){&color(red){controller.dat, contr...
//--Your login password will be e-mailed on Dec 16.
--the system will be started until next lecture
--You can send up to 10 controllers
--&size(30){&color(black){[[control experimental results ...
--freqresp ... frequency response will be measured and up...
+ how to improve the performance ?
-- accuracy of the nominal(physical) model
-- weighting for robust stability
//+ detailed explanation of m-files in the previous lecture
+ specifications of the experimental system
++ program sources for frequency response experiment
--- [[freqresp.h>/:~exp/seigyokougakutokuron_2019/freqres...
--- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2019/...
--- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2019/fre...
--- format of servo1.dat (w is used instead of u for serv...
1st column ... frequency (Hz)
2nd column ... gain from u(Nm) to y(rad/s)
3rd column ... phase (deg) from u to y
4th column ... gain from u to z
5th column ... phase (deg) from u to z
++ program sources for control experiment
--- [[hinf.h>/:~exp/seigyokougakutokuron_2019/hinf.h]]
--- [[hinf_module.c>/:~exp/seigyokougakutokuron_2019/hinf...
--- [[hinf_app.c>/:~exp/seigyokougakutokuron_2019/hinf_ap...
--- format of result.dat
1st column: time (s)
2nd column: y (rad/s)
3rd column: z (rad/s)
4th column: u (Nm)
5th column: w (Nm)
++ configuration of control experiment
--- disturbance signal w is specified as described in hin...
w = 0; // disturbance torque for driven motor ...
if((t > 2)&&(t < 3)){
w = RATED_TORQ * -0.15;
}
if((t > 4)&&(t < 5)){
w = RATED_TORQ * -0.1 * sin(2*M_PI*5.0 * (t-4.0));
}
da_conv(torq_volt_conv_1(w), 1);
--- control signal u is limited as specified in hinf.h an...
#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 ...
++ calculation of rotational speed
--- The rotational speed is approximately calculated by u...
theta_rad[0] = (double)read_theta(0) / (double)Pn212 * 2...
theta_rad[1] = (double)read_theta(1) / (double)Pn212 * 2...
y = (theta_rad[0] - theta_rad_before[0]) / msg->sampling...
z = (theta_rad[1] - theta_rad_before[1]) / msg->sampling...
theta_rad_before[0] = theta_rad[0];
theta_rad_before[1] = theta_rad[1];
where the sampling period is given as 0.25 ms.
%-- 2019/12/05 13:03 --%
pwd
nominal
weight
cont
compare
ctrlpref
compare
perf
-Q: 実機パラメータのイナーシャはどうやって試験をして導出...
-A: 周波数応答実験の結果に物理モデルが近くなるように調整...
-Q: C_S, C_L はダンパなのか。C_S はどこの部分のダンパとい...
-A: C_S は軸の捩りに伴う減衰係数です（これが0の場合、二つ...
#ref(2019.12.05-1.jpg,left,noimg,whiteboard #1);
** &color(green){[lecture #14]}; 2019.12.12 Control syste...
- review & supplemental explanations
-- final report
--- control objective is to suppress speed fluctuation of...
--- unstable controller is admitted
--- no strict control objective is given ( there is a fre...
--- c2d() is used to discretize the resultant continuous-...
-- web based remote experiment system
--- now you can login after registration
--- room temperature is displayed and stored in temp.txt ...
//--- the system will be unavailable from %%21(Fri)%% &co...
- preparation of your own controller(s) by using the remo...
- mini exam #2 (14:00〜)
#ref(2019.12.12-1.jpg,left,noimg,whiteboard #1);
//#ref(nominal_ave.m);
//#ref(sub.m);
** &color(green){[lecture #15]}; 2019.12.19 Control syste...
- return of mini exam #2
- preparation of your own controller(s)
-- use [[design example #3>/:~exp/seigyokougakutokuron_20...
-- discuss relationship between the required figures (a),...
-- driving and driven torque are set to zero when driven-...
#define SPEED_MAX 60.0 // driving torque becomes zero wh...
and [[hinf_module.c>/:~exp/seigyokougakutokuron_2018/hinf...
int flag_speed_excess = 0; // 2018.12.19
if(flag_speed_excess == 1) w = 0; // 2018.12.20
da_conv(torq_volt_conv_1(w), 1);
if((fabs(z) > SPEED_MAX) || (fabs(y) > SPEED_MAX)) ...
if(flag_speed_excess == 1) u = 0; // 2018.12.19
#ifndef NO_CONTROL
da_conv(torq_volt_conv_0(u), 0);
#endif
//-- the system will be unavailable 7:30 -- 19:00 on 22(S...
- questionnaires
-- to university
-- for web-based experimental environment
%-- 2019/12/19 13:31 --%
nominal
weight
cont
compare
perf
nominal
weight
cont
compare
perf
weight
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
#ref(2019.12.19-1.jpg,left,noimg,whiteboard #1);
//- 2018.12.20 19:00
//&size(30){&color(red){due to unexpected alert occurred ...
//**related links [#g1a68a2b]
//-東ティモール工学部復興支援/support of rehabilitation f...
//--[[How to control objects>/:~kobayasi/easttimor/2009/i...
//--[[Prof. Kimura's page>http://sessyu.nagaokaut.ac.jp/~...
終了行:
[[授業]]
*Advanced Automation [#xd2b10c7]
[[latest lecture>#j156dee6]]
** &color(green){[lecture #1]}; 2019.9.5 outline of the l...
- 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%
-- [[schedule2019]] (tentative)
-- map &ref(授業/制御工学特論2017/map_v1.1.pdf); for revi...
- review : stabilization of SISO unstable plant by classi...
-- transfer functions / differential equations
-- poles / eigenvalues
-- impulse response / initial value response
-- ...
%-- 2019/09/05 13:23 --%
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
k = 2
Tyr = feedback(P*k, 1)
step(Tyr)
k = 10
Tyr = feedback(P*k, 1)
step(Tyr)
#ref(2019.09.05-1.jpg,left,noimg,whiteboard #1);
#ref(2019.09.05-2.jpg,left,noimg,whiteboard #2);
#ref(2019.09.05-3.jpg,left,noimg,whiteboard #3);
#ref(2019.09.05-4.jpg,left,noimg,whiteboard #4);
-Q: MATLAB は使いこなせないと後になると大変ですか？
-Q: MATLAB は使いなれていないと授業はきびしいのか。
-A: 使いこなせなくても大丈夫です。多くの場合、mファイル（...
-Q: レポートおよびテストは日本語か英語か
-A: mini report、mini exam は、課題は英語で書かれています...
-Q: ブロック線図を左から右に描いて欲しい。
-A: 式との対応が良いので、右から左に描いています。たとえ...
** &color(green){[lecture #2]}; 2019.9.12 review of class...
+ introduction of Matlab and Simulink
&ref(授業/制御工学特論2015/text_fixed.pdf); Basic usage o...
-- interactive system (no compilation, no variable difini...
-- m file
//-- example: stabilization of inverted pendulum (sorry i...
//--- [[derivation of equation of motion>http://c.nagaoka...
//--- [[stabilization of 1-link pendulum>http://c.nagaoka...
//--- [[stabilization of 2-link pendulum>http://c.nagaoka...
//
//
+ system representation: Transfer Function(TF) / State-Sp...
//
-- 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(ex0912_1.m); &re...
--- 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)
%-- 2019/09/12 13:04 --%
a = 1
t = [1 2 3]
u = [4; 5; 6]
t'
who
b = u
who
ex0912_1
P
P.num
P.num{:}
P.den{:}
ex0912_2
#ref(2019.09.12-1.jpg,left,noimg,whiteboard #1);
#ref(2019.09.12-2.jpg,left,noimg,whiteboard #2);
#ref(2019.09.12-3.jpg,left,noimg,whiteboard #3);
#ref(2019.09.12-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #3]}; 2019.9.19 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- (semi)-positive definiteness
-- 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); ≒ &re...
%-- 2019/09/19 13:16 --%
ex0919_1
A
B
P
A
A'*P+P*A+Q-P*B*inv(R)*B'*P
P
Q
R
J
plot(t, J)
Jmin
#ref(2019.09.19-1.jpg,left,noimg,whiteboard #1);
#ref(2019.09.19-2.jpg,left,noimg,whiteboard #2);
#ref(2019.09.19-3.jpg,left,noimg,whiteboard #3);
#ref(2019.09.19-4.jpg,left,noimg,whiteboard #4);
-Q: --- 一つ、質問かどうか判断できないものがあり、質問の...
-Q: recommend text books for review ?
-A: the followings are only for example,
-- for the classical control theory: @book{BCKuo:Book,
author = "Benjamin C. Kuo",
title = "Automatic Control Systems",
edition = "7",
publisher = "Prentice Hall",
year = "1995"
}
-- for the modern control theory:
@book{Noton:Book,
author = "Maxwell Noton",
title = "Modern Control Engineering",
publisher = "Pergamon Press Inc.",
year = "1972"
}
** &color(green){[lecture #4]}; 2019.9.26 relation betwee...
- GOAL: to learn difference in concepts between LQR probl...
//- review of LQR problem and the simple example
+ a simple example relating LQR and H infinity control pr...
-- For given plant G
\[
G = \left[\begin{array}{c|c:c} a & 1 & b \\ \hline \sqrt{...
= \left\{ \begin{array}{l} \dot x = ax + bu + w\\ z = \le...
\]
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) = \de...
\| T_{zw} \|_\infty \mbox{($H_\infty$ control problem)}\e...
\end{eqnarray}
-- comparison of norms in (ii) (for a = -1, b = 1, q = 1,...
\[
\begin{array}{|c||c|c|}\hline
& \mbox{LQR}: f=-1+\sqrt{2} & \quad \quad H_\infty: f=1\q...
J=\|z\|_2^2 & & \\ \hline
\|T_{zw}\|_\infty & & \\ \hline
\end{array}
\]
+ an alternative description to LQR problem
++ J = (L2 norm of z)^2
++ impulse resp. with zero initial value = initial value ...
+ 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(ex0926_1.m);
%-- 2019/09/26 13:55 --%
s = tf('s')
G1 = 1/(s+1);
bode(G1);
norm(G1, inf)
help norm
G2 = 1/(s^2+0.1*s+1)
bode(G2)
ctrlpref
bode(G2)
grid on
norm(G2, inf)
format long e
norm(G2, inf)
#ref(2019.09.26-1.jpg,left,noimg,whiteboard #1);
#ref(2019.09.26-2.jpg,left,noimg,whiteboard #2);
#ref(2019.09.26-3.jpg,left,noimg,whiteboard #3);
#ref(2019.09.26-4.jpg,left,noimg,whiteboard #4);
#ref(2019.09.26-5.jpg,left,noimg,whiteboard #5);
** &color(green){[lecture #5]}; 2019.10.03 relation betwe...
+ complete the table in simple example
+ confirm the cost function J for both controllers by sim...
-- block diagram in the simulink model
-- how to approximate impulse disturbance with a step fun...
-- impulse disturbance resp. with zero initial condition ...
+ confirm the closed-loop H infinity norm for both contro...
-- H infinity norm = L2 induced norm
-- review: steady-state response; the worst-case disturba...
-- how to make the worst-case disturbance w(t)? w(t) for ...
+ general state-feedback case: &ref(授業/制御工学特論2015...
-- includes the simple example as a special case
-- LQR &ref(授業/制御工学特論2015/lqr.pdf); is included a...
%-- 2019/10/03 13:20 --%
ex0926_1
K
bode(K)
sqrt(2)/2
sqrt(2-sqrt(2))
mod1003
h
h = 0.01
x0
x0 = 0
f = -1+sqrt(2)
zz
zz(end)
h
h = 0.00001
zz(end)
f = 1
zz(end)
x0
x0 = 1
f
zz(end)
f = -1+sqrt(2)
zz(end)
h
h = 100
x0
x0 = 0
zz(end)
sqrt(zz(end)/ww(end))
h
h = 10000
sqrt(zz(end)/ww(end))
sqrt(2-sqrt(2))
#ref(2019.10.03-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.03-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.03-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #6]}; 2019.10.10 Mixed sensitiv...
+ outline: &ref(授業/制御工学特論2017/map_v1.1_mixedsens1...
-- sensitivity function S and complementary sensitivity f...
+ 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 in...
-- corresponding generalized plant G ?
-- introduction of weighting function for sensitivity fun...
+ design example &ref(ex1010_1.m); &ref(ex1010_2.m);
+ the small gain theorem
-- proof: Nyquist stability criterion
//+ from performance optimization to robust stabilization
%-- 2019/10/10 12:58 --%
ex1010_1
P
pole(P)
eig(P)
ex1010_2
K_hinf
K_hinf.a
eig(K_hinf.a)
ex1010_2
#ref(2019.10.10-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.10-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.10-3.jpg,left,noimg,whiteboard #3);
-Q: スモールゲインの定理と、極配置の実部が負となることの...
-A: ナイキストの安定判別と開ループ系の安定性の関係につい...
開ループ系が安定な場合、ナイキストの安定判別は簡単になり...
-Q: WSの決め方はプラントの感度に依存しますか？
-A: 感度関数も相補感度関数もプラント単独ではなくコントロ...
「プラントの感度」は「プラントのゲイン」の意味だと思って...
結論から言えば、プラントのゲインに依存します。
十分な目標値追従性能を達成したい周波数帯域でWSのゲインを...
** &color(green){[lecture #7]}; 2019.10.17 Mixed sensitiv...
+ outline: from point to set &ref(授業/制御工学特論2017/m...
+ review: the small gain theorem ... robust stability = H...
+ normalized uncertainty Delta
+ uncertainty model
+ how to determine P0 and WT
-- example: frequency response of plant with perturbation...
-- frequency response based procedure for P0 and WT &ref(...
+ robust stabilization problem and equivalent problem
-- design example and simulation &ref(ex1017_3.m); &ref(m...
%-- 2019/10/17 13:00 --%
ex1017_1
ex1017_2
ex1017_1
ex1017_2
ex1017_3
mod1017
c
c = 0.8
c = 2
#ref(2019.10.17-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.17-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.17-3.jpg,left,noimg,whiteboard #3);
#ref(2019.10.17-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #8]}; 2019.10.24 Mixed sensitiv...
//- schedule (no lecture will be given on Nov.31)
- review: &ref(授業/制御工学特論2017/map_v1.1_mixedsens2....
- outline:
++ how to design controllers considering both conditions ...
++ gap between NP(nominal performance) and RP(robust perf...
+ 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);
+ intro. to RP(problem of NP) &ref(ex1024_3.m);
%-- 2019/10/24 13:00 --%
ex1024_1
K
ex1024_1
ex1024_2
ex1024_3
#ref(2019.10.24-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.24-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.24-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #9]}; 2019.10.31 robust perform...
-- [[schedule2019]] mini report #1; exam #1; lec. on Nov....
+ review
-- mixed sensitivity problem : N.P. but not R.P.
-- robust performance problem (R.P.) c.f. the last whiteb...
-- the small gain theorem
+ an equivalent robust stability (R.S.) problem to R.P.
-- (i) introduction of a fictitious uncertainty Delta_p (...
-- (ii) for 2-by-2 uncertainty block Delta hat which incl...
+ definition of H infinity norm for general case (MIMO)
-- definition of singular values and the maximum singular...
M = [1, 1i/sqrt(2); 1, -1i/sqrt(2)]
M'
eig(M'*M)
svd(M)
-- mini report #1 &ref(report1.pdf); ... You will have a ...
+ 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 desig...
%-- 2019/10/31 14:07 --%
M = [1, 1i/sqrt(2); 1, -1i/sqrt(2)]
M'
eig(M'*M)
svd(M)
ex1031
ex1031_1
#ref(2019.10.31-1.jpg,left,noimg,whiteboard #1);
#ref(2019.10.31-2.jpg,left,noimg,whiteboard #2);
#ref(2019.10.31-3.jpg,left,noimg,whiteboard #3);
#ref(2019.10.31-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture --]}; 2019.11.7 canceled [#dd5e...
** &color(green){[lecture #10]}; 2019.11.14 Robust perfor...
+ return of mini report #1
//+ review
//-- robust performance but too conservative
// ex1108_1
//-- robust stability problem for Delta hat and its equiv...
//-- structured unertainty Delta hat and unstructured unc...
+ SVD: singular value decomposition
-- definition
-- meaning of the largest singular value (a property and ...
-- 2-norm of vectors (Euclidean norm)
-- SVD for 2-by-2 real matrix &ref(ex1114_1.m);
%-- 2019/11/14 13:17 --%
M = [1/sqrt(2), 1i; 1/sqrt(2), -1i]
[U, Sigma, V] = svd(M)
svd(M)
[U, Sigma, V] = svd(M)
U'*U
V'*V
U*Sigma*V'
U*Sigma*V'-M
ex1114_1
#ref(2019.11.14-1.jpg,left,noimg,whiteboard #1);
#ref(2019.11.14-2.jpg,left,noimg,whiteboard #2);
#ref(2019.11.14-3.jpg,left,noimg,whiteboard #3);
#ref(2019.11.14-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #11]}; 2019.11.21 Robust perfor...
+ review: R.S. prob. for Delta hat and Delta tilde
+ scaled H infinity control problem
+ relation between three problems
+ how to determine structure of scaling matrix
+ design example &ref(ex1121_1.m);
ex1031_1
gam2 = gam_opt
K2 = K_opt;
ex1121_1
gam_opt
+ %%mini report #2 &ref(report2.pdf);%%
+ mini exam #1 (10 min.)
%-- 2019/11/21 13:50 --%
ex1031_1
pwd
ex1031_1
gam2 = gam_opt
K2 = K_opt
gam2 = gam_opt
ex1121_1
gam_opt
d_opt
#ref(2019.11.21-1.jpg,left,noimg,whiteboard #1);
#ref(2019.11.21-2.jpg,left,noimg,whiteboard #2);
#ref(2019.11.21-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #12]}; 2019.11.28 Robust perfor...
+ return of mini exam #1
+ review of scaling &ref(ex1128_1.m);
+ mini report #2 &ref(report2.pdf);
+ introduction of a practical system: Speed control of tw...
-- experimental setup &br;
&ref(setup_fixed.pdf); &br;
&ref(photo.jpg,left,noimg);
-- objective of control system = disturbance attenuation ...
-- frequency response experiment and physical model of sp...
#ref(ex1128_2.m);
&ref(servo1.dat); &ref(servo2.dat); ... old data(the reso...
-- room 374 @ Dept. Mech. Bldg. 2
%-- 2019/11/28 12:59 --%
ex1128_1
gam2
gam3
format long e
norm(M3_d, 'inf')
ex1128_2
#ref(2019.11.28-1.jpg,left,noimg,whiteboard #1);
#ref(2019.11.28-2.jpg,left,noimg,whiteboard #2);
#ref(2019.11.28-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #13]}; 2019.12.5 Control system...
+ return of mini report #2; ... You will have a mini exa...
-- [[schedule2019]]
+ 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 consta...
-- frequency response data of G can be used; how to handl...
+ design example (modeling error for Gyu is only consider...
-- frequency response experiment data&br;
[[servo1.dat>/:~exp/seigyokougakutokuron_2019/exp/freqres...
[[servo2.dat>/:~exp/seigyokougakutokuron_2019/exp/freqres...
-- determination of plant model(nominal plant and additiv...
&ref(nominal.m);&br;
&ref(weight.m);
-- configuration of generalized plant and controller desi...
&ref(cont.m);
-- comparison of closed-loop gain characteristics with an...
&ref(compare.m);
-- result of control experiment&br;
[[result.dat>/:~exp/seigyokougakutokuron_2019/exp/design_...
&ref(perf.m);
//+ room 157 @ Dept. Mech. Bldg.2
+ final report and remote experimental system
++design your controller(s) so that the system performanc...
++Draw the following figures and explain the difference b...
+++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 unfor...
--&size(30){&color(red){due date: 6th(Mon) Jan 17:00};};
--submit your report(pdf or doc) by e-mail to kobayasi@na...
--You can use Japanese
--maximum controller order is 20
--submit your &size(25){&color(red){controller.dat, contr...
//--Your login password will be e-mailed on Dec 16.
--the system will be started until next lecture
--You can send up to 10 controllers
--&size(30){&color(black){[[control experimental results ...
--freqresp ... frequency response will be measured and up...
+ how to improve the performance ?
-- accuracy of the nominal(physical) model
-- weighting for robust stability
//+ detailed explanation of m-files in the previous lecture
+ specifications of the experimental system
++ program sources for frequency response experiment
--- [[freqresp.h>/:~exp/seigyokougakutokuron_2019/freqres...
--- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2019/...
--- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2019/fre...
--- format of servo1.dat (w is used instead of u for serv...
1st column ... frequency (Hz)
2nd column ... gain from u(Nm) to y(rad/s)
3rd column ... phase (deg) from u to y
4th column ... gain from u to z
5th column ... phase (deg) from u to z
++ program sources for control experiment
--- [[hinf.h>/:~exp/seigyokougakutokuron_2019/hinf.h]]
--- [[hinf_module.c>/:~exp/seigyokougakutokuron_2019/hinf...
--- [[hinf_app.c>/:~exp/seigyokougakutokuron_2019/hinf_ap...
--- format of result.dat
1st column: time (s)
2nd column: y (rad/s)
3rd column: z (rad/s)
4th column: u (Nm)
5th column: w (Nm)
++ configuration of control experiment
--- disturbance signal w is specified as described in hin...
w = 0; // disturbance torque for driven motor ...
if((t > 2)&&(t < 3)){
w = RATED_TORQ * -0.15;
}
if((t > 4)&&(t < 5)){
w = RATED_TORQ * -0.1 * sin(2*M_PI*5.0 * (t-4.0));
}
da_conv(torq_volt_conv_1(w), 1);
--- control signal u is limited as specified in hinf.h an...
#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 ...
++ calculation of rotational speed
--- The rotational speed is approximately calculated by u...
theta_rad[0] = (double)read_theta(0) / (double)Pn212 * 2...
theta_rad[1] = (double)read_theta(1) / (double)Pn212 * 2...
y = (theta_rad[0] - theta_rad_before[0]) / msg->sampling...
z = (theta_rad[1] - theta_rad_before[1]) / msg->sampling...
theta_rad_before[0] = theta_rad[0];
theta_rad_before[1] = theta_rad[1];
where the sampling period is given as 0.25 ms.
%-- 2019/12/05 13:03 --%
pwd
nominal
weight
cont
compare
ctrlpref
compare
perf
-Q: 実機パラメータのイナーシャはどうやって試験をして導出...
-A: 周波数応答実験の結果に物理モデルが近くなるように調整...
-Q: C_S, C_L はダンパなのか。C_S はどこの部分のダンパとい...
-A: C_S は軸の捩りに伴う減衰係数です（これが0の場合、二つ...
#ref(2019.12.05-1.jpg,left,noimg,whiteboard #1);
** &color(green){[lecture #14]}; 2019.12.12 Control syste...
- review & supplemental explanations
-- final report
--- control objective is to suppress speed fluctuation of...
--- unstable controller is admitted
--- no strict control objective is given ( there is a fre...
--- c2d() is used to discretize the resultant continuous-...
-- web based remote experiment system
--- now you can login after registration
--- room temperature is displayed and stored in temp.txt ...
//--- the system will be unavailable from %%21(Fri)%% &co...
- preparation of your own controller(s) by using the remo...
- mini exam #2 (14:00〜)
#ref(2019.12.12-1.jpg,left,noimg,whiteboard #1);
//#ref(nominal_ave.m);
//#ref(sub.m);
** &color(green){[lecture #15]}; 2019.12.19 Control syste...
- return of mini exam #2
- preparation of your own controller(s)
-- use [[design example #3>/:~exp/seigyokougakutokuron_20...
-- discuss relationship between the required figures (a),...
-- driving and driven torque are set to zero when driven-...
#define SPEED_MAX 60.0 // driving torque becomes zero wh...
and [[hinf_module.c>/:~exp/seigyokougakutokuron_2018/hinf...
int flag_speed_excess = 0; // 2018.12.19
if(flag_speed_excess == 1) w = 0; // 2018.12.20
da_conv(torq_volt_conv_1(w), 1);
if((fabs(z) > SPEED_MAX) || (fabs(y) > SPEED_MAX)) ...
if(flag_speed_excess == 1) u = 0; // 2018.12.19
#ifndef NO_CONTROL
da_conv(torq_volt_conv_0(u), 0);
#endif
//-- the system will be unavailable 7:30 -- 19:00 on 22(S...
- questionnaires
-- to university
-- for web-based experimental environment
%-- 2019/12/19 13:31 --%
nominal
weight
cont
compare
perf
nominal
weight
cont
compare
perf
weight
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
#ref(2019.12.19-1.jpg,left,noimg,whiteboard #1);
//- 2018.12.20 19:00
//&size(30){&color(red){due to unexpected alert occurred ...
//**related links [#g1a68a2b]
//-東ティモール工学部復興支援/support of rehabilitation f...
//--[[How to control objects>/:~kobayasi/easttimor/2009/i...
//--[[Prof. Kimura's page>http://sessyu.nagaokaut.ac.jp/~...
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