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開始行:
[[授業]]
*Advanced Automation 2021 [#yfa8a823]
[[latest lecture>#m72f126d]]
** &color(green){[lecture #1]}; 2021.9.2 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%
-- [[schedule2021]] (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
-- ...
%-- 21/09/02 13:43 --%
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
k = 2
P
help step
Tyr = k/(s-1+k)
step(Tyr)
k = 10
Tyr = k/(s-1+k)
step(Tyr)
#ref(2021.09.02-1.jpg,left,noimg,whiteboard #1);
#ref(2021.09.02-2.jpg,left,noimg,whiteboard #2);
#ref(2021.09.02-3.jpg,left,noimg,whiteboard #3);
#ref(2021.09.02-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #2]}; 2021.9.9 review of classi...
//+ minute paper
+ 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 defini...
-- 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
-- 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(ex0909_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(ex0909_2.m); &ref(mod0909_2.mdl);
--- Gain Margin(GM); Phase Margin(PM)
%-- 21/09/09 13:05 --%
a = 1
a +2
a + 2
a +2
a
a + 2
ex0909_1
P
sqrt(k/m)
sqrt(k/m)/(2*pi)
ex0909_2
L
ex0909_2
ex0909_1
#ref(2021.09.09-1.jpg,left,noimg,whiteboard #1);
#ref(2021.09.09-2.jpg,left,noimg,whiteboard #2);
#ref(2021.09.09-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #3]}; 2021.9.23 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- positive (semi-)definite matrices
-- solution of LQR problem
-- example &ref(ex0923_1.m); &ref(mod0923_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...
Matlabのコマンド履歴の保存に失敗しました。すみません。
#ref(2021.09.23-1.jpg,left,noimg,whiteboard #1);
#ref(2021.09.23-2.jpg,left,noimg,whiteboard #2);
#ref(2021.09.23-3.jpg,left,noimg,whiteboard #3);
#ref(2021.09.23-4.jpg,left,noimg,whiteboard #4);
-Q: 最後の2枚目のホワイトの式と対応をとるとこがよくわから...
-A: □7のホワイトボードで、fに関する2次方程式とPに関する...
** &color(green){[lecture #4]}; 2021.9.30 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(ex0930_1.m);
%-- 2021/09/30 12:51 --%
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')
#ref(2021.09.30-1.jpg,left,noimg,whiteboard #1);
#ref(2021.09.30-2.jpg,left,noimg,whiteboard #2);
#ref(2021.09.30-3.jpg,left,noimg,whiteboard #3);
#ref(2021.09.30-4.jpg,left,noimg,whiteboard #4);
-Q: ホワイトボード前の照明が消えており、多少見ずらかった...
-A: ウェブカメラで白く反射しないよう消してました。次回、...
** &color(green){[lecture #5]}; 2021.10.07 relation betwe...
+ cont.
-- solve the problem by hand
-- solve the problem by tool(hinfsyn)
+ 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...
-- (unit) impulse disturbance resp. with zero initial con...
+ confirm the closed-loop H infinity norm for both contro...
-- H infinity norm = L2 induced norm
-- review: steady-state response for sinusoidal input sig...
-- the worst-case disturbance w(t) for the simple example...
+ 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...
%-- 2021/10/07 13:24 --%
ex0930_1
K
sqrt(2-sqrt(2))
mod1007
x0
x0 = 0
f = -1 + sqrt(2)
h
h = 0.01
zz
h
h = 0.001
zz
zz(end)
sqrt(2)-1
x0
x0 = 1
zz(end)
h = 10
x0
x0 = 0
f
zz
zz(end)/ww(end)
sqrt(zz(end)/ww(end))
h
h = 100
sqrt(zz(end)/ww(end))
f = 1
sqrt(zz(end)/ww(end))
#ref(2021.10.07-1.jpg,left,noimg,whiteboard #1);
#ref(2021.10.07-2.jpg,left,noimg,whiteboard #2);
#ref(2021.10.07-3.jpg,left,noimg,whiteboard #3);
-Q: 日本語での説明を増やしてほしいです。
-A: 今日は少なめでした。前回並に戻します。
-Q: Hインフィニティも万能じゃないことが意外だった
-A: good!
** &color(green){[lecture #6]}; 2021.10.14 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(ex1014_1.m); &ref(ex1014_2.m);
+ %%the small gain theorem%%
-- %%proof: Nyquist stability criterion%%
//+ from performance optimization to robust stabilization
%-- 2021/10/14 13:43 --%
ex1014_1
P
eig(P)
ex1014_2
K_hinf
eig(K)
K
eig(K_hinf)
#ref(2021.10.14-1.jpg,left,noimg,whiteboard #1);
#ref(2021.10.14-2.jpg,left,noimg,whiteboard #2);
-Q: ナイキスト線図について復習すべきだと感じた
-A: してください!
** &color(green){[lecture #7]}; 2021.10.21 Mixed sensitiv...
+ outline: from point to set &ref(授業/制御工学特論2017/m...
+ the small gain theorem ... robust stability = H infinit...
+ 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(ex1021_3.m); &ref(m...
%-- 2021/10/21 13:39 --%
ex1021_1
pwd
ls
cd MATLAB
ex1021_1
ex1021_2
#ref(2021.10.21-1.jpg,left,noimg,whiteboard #1);
#ref(2021.10.21-2.jpg,left,noimg,whiteboard #2);
#ref(2021.10.21-3.jpg,left,noimg,whiteboard #3);
#ref(2021.10.21-4.jpg,left,noimg,whiteboard #4);
-Q: 英語の説明が多くて難しかったです
-A: 基本的に、英語で説明した後に日本語でも説明しているの...
-Q: Pチルダの説明が難しかったので復習します。
-A: 復習して分からなかったらまた聞いてください。
** &color(green){[lecture #8]}; 2021.10.28 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(ex1028_1.m); minimize gamma by hand
+ gamma iteration by bisection method &ref(ex1028_2.m);
+ intro. to RP: weak point of mixed sensitivity problem(p...
%-- 2021/10/28 13:01 --%
ex1021_1
ex1021_2
ex1021_3
K
mod1021
c
c = 0.8
c = 2
ex1028_1
ex1028_2
ex1028_3
#ref(2021.10.28-1.jpg,left,noimg,whiteboard #1);
#ref(2021.10.28-2.jpg,left,noimg,whiteboard #2);
-Q: まだ一般化プラントへの変換がぱっとできないなと思いま...
-A: 時間をかけて考えてみてください。不明な点があればまた...
** &color(green){[lecture #9]}; 2021.11.4 robust performa...
-- [[schedule2021]]
+ review
-- mixed sensitivity problem : N.P. but not R.P.
//-- robust performance problem (R.P.) c.f. the last whit...
//-- the small gain theorem
+ robust performance problem (R.P.), but can not be solve...
+ 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 = [sqrt(2), -1i/sqrt(2); sqrt(2), 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(ex1104_1.m);
-- robust performance is achieved but large gap
-- non structured uncertainty is considered ... the desig...
%-- 2021/11/04 13:03 --%
ex1028_1
ex1028_2
ex1028_3
M = [sqrt(2), -1i/sqrt(2); sqrt(2), 1i/sqrt(2)]
M'
eig(M'*M)
svd(M)
help svd
ex1104_1
#ref(2021.11.04-1.jpg,left,noimg,whiteboard #1);
#ref(2021.11.04-2.jpg,left,noimg,whiteboard #2);
#ref(2021.11.04-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #10]}; 2021.11.11 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(ex1111_1.m);
%-- 2021/11/11 13:15 --%
M = [j, 0; j, 1]
[U, Sigma, V] = svd(M)
U*Sigma*V'
format long e
U*Sigma*V'
U
fomat short
help format
format
U
U'*U
U*U'
ex1111_1
#ref(2021.11.11-1.jpg,left,noimg,whiteboard #1);
#ref(2021.11.11-2.jpg,left,noimg,whiteboard #2);
#ref(2021.11.11-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #11]}; 2021.11.18 Robust perfor...
+ review : R.S. problems for structured and unstructured ...
+ scaled H infinity control problem
+ relation between three problems
+ how to determine structure of scaling matrix
+ design example &ref(ex1118_1.m);
ex1104_1
gam2 = gam_opt
K2 = K_opt;
ex1118_1
gam_opt
+ mini exam #1 (10 min.)
%-- 2021/11/18 12:49 --%
ex1104_1
ex1118_1
ex1104_1
gam2 = gam_opt
K2 = K_opt;
ex1118_1
gam_opt
gam2
format long
gam2
gam_opt
#ref(2021.11.18-1.jpg,left,noimg,whiteboard #1);
#ref(2021.11.18-2.jpg,left,noimg,whiteboard #2);
-Q: スケーリングが等価に扱えるのは便利だと思った
-A: 誤解を与えたかもしれません。「スケーリングが等価に扱...
** &color(green){[lecture #12]}; 2021.11.25 Robust perfor...
+ return of mini exam #1
+ review of scaling &ref(ex1125_1.m);
+ mini report #2 &ref(report2.pdf);
+ introduction of a practical system: active noise contro...
-- experimental setup
#ref(授業/制御工学特論2020/photo1.jpg,left,noimg);
#ref(授業/制御工学特論2020/photo2.jpg,left,noimg);
-- objective of control system: to drive control loudspea...
-- frequency response experiment
#ref(ex1125_2.m);
&ref(授業/制御工学特論2020/spk1.dat); &ref(授業/制御工学...
//-- %%room 157 @ Dept. Mech. Bldg. 2%%
%-- 2021/11/25 13:03 --%
ex1125_1
ex1125_2
346/(4*162)
346/(4*1.62)
#ref(2021.11.25-1.jpg,left,noimg,whiteboard #1);
#ref(2021.11.25-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #13]}; 2021.12.2 Control system...
+ return of mini report #2; ... You will have a mini exa...
-- [[schedule2021]]
+ 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 handl...
+ design example (modeling error for Gyu is only consider...
-- frequency response experiment data&br;
[[spk1.dat>/:~exp/seigyokougakutokuron_2021/exp/freqresp/...
[[spk2.dat>/:~exp/seigyokougakutokuron_2021/exp/freqresp/...
-- determination of plant model(nominal plant and additiv...
&ref(nominal.m);&br; &ref(n4sid_replaced.m); ... replacem...
&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 and evaluation&br;
[[result.dat>/:~exp/seigyokougakutokuron_2021/exp/design_...
&ref(perf.m);
+ 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: 5th(Wed) Jan 17:00};};
--submit your report(pdf file) by e-mail to kobayasi@naga...
--You can use Japanese
--maximum controller order is 35
--submit your &size(25){&color(red){controller.dat, contr...
--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 ?
-- 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_2021/freqres...
--- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2021/...
--- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2021/fre...
--- format of spk1.dat (u is used instead of w for spk2.d...
1st column ... frequency (Hz)
2nd column ... gain from w(V) to y(V) (signal's unit is ...
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_2021/hinf.h]]
--- [[hinf_module.c>/:~exp/seigyokougakutokuron_2021/hinf...
--- [[hinf_app.c>/:~exp/seigyokougakutokuron_2021/hinf_ap...
--- 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 hin...
//#define AMP 0.5 // amplitude for disturbance
#define AMP 1.0 // modified 2021.12.9
w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // unifo...
da_conv(V_OFFSET + w, 0); // D/A output to noise source
--- control signal u is limited to [-3, 3] as specified i...
#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...
%-- 2021/12/02 13:24 --%
nominal
G0
size(G0.a)
weight
cont
compare
perf
#ref(2021.12.02-1.jpg,left,noimg,whiteboard #1);
** &color(green){[lecture #14]}; 2021.12.9 Control system...
- 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 Technol...
//--- now you can login after registration
- supplemental explanations
-- room temperature is displayed and stored in temp.txt (...
//-- generating wav file [[filter.c>/:~exp/seigyokougakut...
-- c2d() is used to discretize the resultant continuous-t...
-- You can send up to 10 controllers (don't fall into tri...
-- no strict control objective is given ( there is a free...
- preparation of your own controller(s) by using the remo...
- mini exam #2 (14:10-14:20)
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
** &color(green){[lecture #15]}; 2021.12.16 Control syste...
- return of mini exam #2
- [[schedule2021]] no lecture will be given next week
//--- the system will be unavailable from %%21(Fri)%% &co...
- preparation of your own controller(s) by using the remo...
//**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 2021 [#yfa8a823]
[[latest lecture>#m72f126d]]
** &color(green){[lecture #1]}; 2021.9.2 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%
-- [[schedule2021]] (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
-- ...
%-- 21/09/02 13:43 --%
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
k = 2
P
help step
Tyr = k/(s-1+k)
step(Tyr)
k = 10
Tyr = k/(s-1+k)
step(Tyr)
#ref(2021.09.02-1.jpg,left,noimg,whiteboard #1);
#ref(2021.09.02-2.jpg,left,noimg,whiteboard #2);
#ref(2021.09.02-3.jpg,left,noimg,whiteboard #3);
#ref(2021.09.02-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #2]}; 2021.9.9 review of classi...
//+ minute paper
+ 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 defini...
-- 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
-- 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(ex0909_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(ex0909_2.m); &ref(mod0909_2.mdl);
--- Gain Margin(GM); Phase Margin(PM)
%-- 21/09/09 13:05 --%
a = 1
a +2
a + 2
a +2
a
a + 2
ex0909_1
P
sqrt(k/m)
sqrt(k/m)/(2*pi)
ex0909_2
L
ex0909_2
ex0909_1
#ref(2021.09.09-1.jpg,left,noimg,whiteboard #1);
#ref(2021.09.09-2.jpg,left,noimg,whiteboard #2);
#ref(2021.09.09-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #3]}; 2021.9.23 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- positive (semi-)definite matrices
-- solution of LQR problem
-- example &ref(ex0923_1.m); &ref(mod0923_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...
Matlabのコマンド履歴の保存に失敗しました。すみません。
#ref(2021.09.23-1.jpg,left,noimg,whiteboard #1);
#ref(2021.09.23-2.jpg,left,noimg,whiteboard #2);
#ref(2021.09.23-3.jpg,left,noimg,whiteboard #3);
#ref(2021.09.23-4.jpg,left,noimg,whiteboard #4);
-Q: 最後の2枚目のホワイトの式と対応をとるとこがよくわから...
-A: □7のホワイトボードで、fに関する2次方程式とPに関する...
** &color(green){[lecture #4]}; 2021.9.30 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(ex0930_1.m);
%-- 2021/09/30 12:51 --%
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')
#ref(2021.09.30-1.jpg,left,noimg,whiteboard #1);
#ref(2021.09.30-2.jpg,left,noimg,whiteboard #2);
#ref(2021.09.30-3.jpg,left,noimg,whiteboard #3);
#ref(2021.09.30-4.jpg,left,noimg,whiteboard #4);
-Q: ホワイトボード前の照明が消えており、多少見ずらかった...
-A: ウェブカメラで白く反射しないよう消してました。次回、...
** &color(green){[lecture #5]}; 2021.10.07 relation betwe...
+ cont.
-- solve the problem by hand
-- solve the problem by tool(hinfsyn)
+ 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...
-- (unit) impulse disturbance resp. with zero initial con...
+ confirm the closed-loop H infinity norm for both contro...
-- H infinity norm = L2 induced norm
-- review: steady-state response for sinusoidal input sig...
-- the worst-case disturbance w(t) for the simple example...
+ 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...
%-- 2021/10/07 13:24 --%
ex0930_1
K
sqrt(2-sqrt(2))
mod1007
x0
x0 = 0
f = -1 + sqrt(2)
h
h = 0.01
zz
h
h = 0.001
zz
zz(end)
sqrt(2)-1
x0
x0 = 1
zz(end)
h = 10
x0
x0 = 0
f
zz
zz(end)/ww(end)
sqrt(zz(end)/ww(end))
h
h = 100
sqrt(zz(end)/ww(end))
f = 1
sqrt(zz(end)/ww(end))
#ref(2021.10.07-1.jpg,left,noimg,whiteboard #1);
#ref(2021.10.07-2.jpg,left,noimg,whiteboard #2);
#ref(2021.10.07-3.jpg,left,noimg,whiteboard #3);
-Q: 日本語での説明を増やしてほしいです。
-A: 今日は少なめでした。前回並に戻します。
-Q: Hインフィニティも万能じゃないことが意外だった
-A: good!
** &color(green){[lecture #6]}; 2021.10.14 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(ex1014_1.m); &ref(ex1014_2.m);
+ %%the small gain theorem%%
-- %%proof: Nyquist stability criterion%%
//+ from performance optimization to robust stabilization
%-- 2021/10/14 13:43 --%
ex1014_1
P
eig(P)
ex1014_2
K_hinf
eig(K)
K
eig(K_hinf)
#ref(2021.10.14-1.jpg,left,noimg,whiteboard #1);
#ref(2021.10.14-2.jpg,left,noimg,whiteboard #2);
-Q: ナイキスト線図について復習すべきだと感じた
-A: してください!
** &color(green){[lecture #7]}; 2021.10.21 Mixed sensitiv...
+ outline: from point to set &ref(授業/制御工学特論2017/m...
+ the small gain theorem ... robust stability = H infinit...
+ 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(ex1021_3.m); &ref(m...
%-- 2021/10/21 13:39 --%
ex1021_1
pwd
ls
cd MATLAB
ex1021_1
ex1021_2
#ref(2021.10.21-1.jpg,left,noimg,whiteboard #1);
#ref(2021.10.21-2.jpg,left,noimg,whiteboard #2);
#ref(2021.10.21-3.jpg,left,noimg,whiteboard #3);
#ref(2021.10.21-4.jpg,left,noimg,whiteboard #4);
-Q: 英語の説明が多くて難しかったです
-A: 基本的に、英語で説明した後に日本語でも説明しているの...
-Q: Pチルダの説明が難しかったので復習します。
-A: 復習して分からなかったらまた聞いてください。
** &color(green){[lecture #8]}; 2021.10.28 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(ex1028_1.m); minimize gamma by hand
+ gamma iteration by bisection method &ref(ex1028_2.m);
+ intro. to RP: weak point of mixed sensitivity problem(p...
%-- 2021/10/28 13:01 --%
ex1021_1
ex1021_2
ex1021_3
K
mod1021
c
c = 0.8
c = 2
ex1028_1
ex1028_2
ex1028_3
#ref(2021.10.28-1.jpg,left,noimg,whiteboard #1);
#ref(2021.10.28-2.jpg,left,noimg,whiteboard #2);
-Q: まだ一般化プラントへの変換がぱっとできないなと思いま...
-A: 時間をかけて考えてみてください。不明な点があればまた...
** &color(green){[lecture #9]}; 2021.11.4 robust performa...
-- [[schedule2021]]
+ review
-- mixed sensitivity problem : N.P. but not R.P.
//-- robust performance problem (R.P.) c.f. the last whit...
//-- the small gain theorem
+ robust performance problem (R.P.), but can not be solve...
+ 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 = [sqrt(2), -1i/sqrt(2); sqrt(2), 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(ex1104_1.m);
-- robust performance is achieved but large gap
-- non structured uncertainty is considered ... the desig...
%-- 2021/11/04 13:03 --%
ex1028_1
ex1028_2
ex1028_3
M = [sqrt(2), -1i/sqrt(2); sqrt(2), 1i/sqrt(2)]
M'
eig(M'*M)
svd(M)
help svd
ex1104_1
#ref(2021.11.04-1.jpg,left,noimg,whiteboard #1);
#ref(2021.11.04-2.jpg,left,noimg,whiteboard #2);
#ref(2021.11.04-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #10]}; 2021.11.11 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(ex1111_1.m);
%-- 2021/11/11 13:15 --%
M = [j, 0; j, 1]
[U, Sigma, V] = svd(M)
U*Sigma*V'
format long e
U*Sigma*V'
U
fomat short
help format
format
U
U'*U
U*U'
ex1111_1
#ref(2021.11.11-1.jpg,left,noimg,whiteboard #1);
#ref(2021.11.11-2.jpg,left,noimg,whiteboard #2);
#ref(2021.11.11-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #11]}; 2021.11.18 Robust perfor...
+ review : R.S. problems for structured and unstructured ...
+ scaled H infinity control problem
+ relation between three problems
+ how to determine structure of scaling matrix
+ design example &ref(ex1118_1.m);
ex1104_1
gam2 = gam_opt
K2 = K_opt;
ex1118_1
gam_opt
+ mini exam #1 (10 min.)
%-- 2021/11/18 12:49 --%
ex1104_1
ex1118_1
ex1104_1
gam2 = gam_opt
K2 = K_opt;
ex1118_1
gam_opt
gam2
format long
gam2
gam_opt
#ref(2021.11.18-1.jpg,left,noimg,whiteboard #1);
#ref(2021.11.18-2.jpg,left,noimg,whiteboard #2);
-Q: スケーリングが等価に扱えるのは便利だと思った
-A: 誤解を与えたかもしれません。「スケーリングが等価に扱...
** &color(green){[lecture #12]}; 2021.11.25 Robust perfor...
+ return of mini exam #1
+ review of scaling &ref(ex1125_1.m);
+ mini report #2 &ref(report2.pdf);
+ introduction of a practical system: active noise contro...
-- experimental setup
#ref(授業/制御工学特論2020/photo1.jpg,left,noimg);
#ref(授業/制御工学特論2020/photo2.jpg,left,noimg);
-- objective of control system: to drive control loudspea...
-- frequency response experiment
#ref(ex1125_2.m);
&ref(授業/制御工学特論2020/spk1.dat); &ref(授業/制御工学...
//-- %%room 157 @ Dept. Mech. Bldg. 2%%
%-- 2021/11/25 13:03 --%
ex1125_1
ex1125_2
346/(4*162)
346/(4*1.62)
#ref(2021.11.25-1.jpg,left,noimg,whiteboard #1);
#ref(2021.11.25-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #13]}; 2021.12.2 Control system...
+ return of mini report #2; ... You will have a mini exa...
-- [[schedule2021]]
+ 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 handl...
+ design example (modeling error for Gyu is only consider...
-- frequency response experiment data&br;
[[spk1.dat>/:~exp/seigyokougakutokuron_2021/exp/freqresp/...
[[spk2.dat>/:~exp/seigyokougakutokuron_2021/exp/freqresp/...
-- determination of plant model(nominal plant and additiv...
&ref(nominal.m);&br; &ref(n4sid_replaced.m); ... replacem...
&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 and evaluation&br;
[[result.dat>/:~exp/seigyokougakutokuron_2021/exp/design_...
&ref(perf.m);
+ 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: 5th(Wed) Jan 17:00};};
--submit your report(pdf file) by e-mail to kobayasi@naga...
--You can use Japanese
--maximum controller order is 35
--submit your &size(25){&color(red){controller.dat, contr...
--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 ?
-- 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_2021/freqres...
--- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2021/...
--- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2021/fre...
--- format of spk1.dat (u is used instead of w for spk2.d...
1st column ... frequency (Hz)
2nd column ... gain from w(V) to y(V) (signal's unit is ...
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_2021/hinf.h]]
--- [[hinf_module.c>/:~exp/seigyokougakutokuron_2021/hinf...
--- [[hinf_app.c>/:~exp/seigyokougakutokuron_2021/hinf_ap...
--- 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 hin...
//#define AMP 0.5 // amplitude for disturbance
#define AMP 1.0 // modified 2021.12.9
w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // unifo...
da_conv(V_OFFSET + w, 0); // D/A output to noise source
--- control signal u is limited to [-3, 3] as specified i...
#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...
%-- 2021/12/02 13:24 --%
nominal
G0
size(G0.a)
weight
cont
compare
perf
#ref(2021.12.02-1.jpg,left,noimg,whiteboard #1);
** &color(green){[lecture #14]}; 2021.12.9 Control system...
- 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 Technol...
//--- now you can login after registration
- supplemental explanations
-- room temperature is displayed and stored in temp.txt (...
//-- generating wav file [[filter.c>/:~exp/seigyokougakut...
-- c2d() is used to discretize the resultant continuous-t...
-- You can send up to 10 controllers (don't fall into tri...
-- no strict control objective is given ( there is a free...
- preparation of your own controller(s) by using the remo...
- mini exam #2 (14:10-14:20)
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
** &color(green){[lecture #15]}; 2021.12.16 Control syste...
- return of mini exam #2
- [[schedule2021]] no lecture will be given next week
//--- the system will be unavailable from %%21(Fri)%% &co...
- preparation of your own controller(s) by using the remo...
//**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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