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[[授業]]
*Advanced Automation 2022 [#ua920811]
[[latest lecture>#lc121915]]
** &color(green){[lecture #1]}; 2022.9.8 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%
-- [[schedule2022]] (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
-- ...
%-- 2022/09/08 13:45 --%
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
t = 0:0.01:3
y
y = exp(t)
hold on
plot(t, y, 'r--')
help impulse
impulse(P, 3)
plot(t, y, 'r--')
impulse(P, 3)
hold on
plot(t, y, 'r--')
P
K = 2
help step
Tyr = K/(s-1+K)
step(Tyr)
K = 20
Tyr = K/(s-1+K)
step(Tyr)
#ref(2022.09.08-1.jpg,left,noimg,whiteboard #1);
#ref(2022.09.08-2.jpg,left,noimg,whiteboard #2);
#ref(2022.09.08-3.jpg,left,noimg,whiteboard #3);
#ref(2022.09.08-4.jpg,left,noimg,whiteboard #4);
-Q: 思っていた以上に忘れていることがあったため,有意義な...
-A: それは良かったです。
** &color(green){[lecture #2]}; 2022.9.15 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 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(ex0915_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(ex0915_2.m); &ref(mod0915_2.mdl);
--- Gain Margin(GM); Phase Margin(PM)
%-- 2022/09/15 13:02 --%
a = 3
b = 5
a + b
ex0915_1
sqrt(k/m)
sqrt(k/m)/(2*pi)
ex0915_1
ex0915_2
#ref(2022.09.15-1.jpg,left,noimg,whiteboard #1);
#ref(2022.09.15-2.jpg,left,noimg,whiteboard #2);
#ref(2022.09.15-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #3]}; 2022.9.22 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- positive (semi-)definite matrices
-- solution of LQR problem
-- example &ref(ex0922_1.m); &ref(mod0922_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...
%-- 2022/09/22 13:49 --%
ex0922_1
A
B
[B, A*B]
Uc
det(Uc)
J
J(end)
Jmin
F
F = [0 0]
ex0922_1
F
A'
A'*P + P*A + Q - P*B/R*B'*P
P
eig(P)
#ref(2022.09.22-1.jpg,left,noimg,whiteboard #1);
#ref(2022.09.22-2.jpg,left,noimg,whiteboard #2);
#ref(2022.09.22-3.jpg,left,noimg,whiteboard #3);
#ref(2022.09.22-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #4]}; 2022.9.29 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(ex0929_1.m);
%-- 2022/09/29 14:00 --%
s = tf('s');
G1 = 1/(s+1);
bode(G1);
norm(G1, 'inf')
G1d = s/(s+1);
bode(G1d);
norm(G1d, 'inf')
G2 = 1/(s^2 + 0.1*s + 1);
bode(G2);
norm(G2, 'inf')
grid on
#ref(2022.09.29-1.jpg,left,noimg,whiteboard #1);
#ref(2022.09.29-2.jpg,left,noimg,whiteboard #2);
#ref(2022.09.29-3.jpg,left,noimg,whiteboard #3);
#ref(2022.09.29-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #5]}; 2022.10.06 relation betwe...
+ cont.
-- solve the problem by hand
-- solve the problem by tool(hinfsyn) &ref(ex0929_1.m);
+ 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...
%-- 2022/10/06 13:14 --%
ex0929_1
clp
sqrt(2-sqrt(2))
format long e
sqrt(2-sqrt(2))
format long f
format f
sqrt(2-sqrt(2))
mod1006
h = 0.01
f = -1+sqrt(2)
x0 = 0
zz
zz(end)
f
f = 1
zz(end)
h
x0
x0 = 1
zz(end)
h = 100
x0 = 0
f
sqrt(zz(end)/ww(end))
f = -1+sqrt(2)
sqrt(zz(end)/ww(end))
#ref(2022.10.06-1.jpg,left,noimg,whiteboard #1);
#ref(2022.10.06-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #6]}; 2022.10.13 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(ex1013_1.m); &ref(ex1013_2.m);
+ the small gain theorem
-- proof: Nyquist stability criterion
//+ from performance optimization to robust stabilization
%-- 2022/10/13 13:54 --%
ex1013_1
P
eig(P)
ex1013_2
ex1013_1
ex1013_2
K_hinf
eig(K_hinf)
G
eig(K_hinf)
#ref(2022.10.13-1.jpg,left,noimg,whiteboard #1);
#ref(2022.10.13-2.jpg,left,noimg,whiteboard #2);
#ref(2022.10.13-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #7]}; 2022.10.20 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
+ simple example of plant set
-- given plant P tilde
--- frequency response of plant with perturbation &ref(ex...
-- how to determine P0 and WT ?
--- frequency response based procedure for P0 and WT &ref...
+ robust stabilization problem and equivalent problem
-- design example and simulation &ref(ex1020_3.m); &ref(m...
%-- 2022/10/20 13:31 --%
ex1020_1
ex1020_2
ctrlpref
ex1020_1
ex1020_2
#ref(2022.10.20-1.jpg,left,noimg,whiteboard #1);
#ref(2022.10.20-2.jpg,left,noimg,whiteboard #2);
#ref(2022.10.20-3.jpg,left,noimg,whiteboard #3);
#ref(2022.10.20-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #8]}; 2022.10.27 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(ex1027_1.m); minimize gamma by hand
+ gamma iteration by bisection method &ref(ex1027_2.m);
+ intro. to RP: weak point of mixed sensitivity problem(p...
%-- 2022/10/27 13:01 --%
ex1020_3
ex1020_1
ex1020_2
ex1020_3
pwd
mod1020
c
c = 0.8
c = 1.3
c = 2
ex1027_1
gam
ex1027_1
ex1027_2
gam
20*log10(gam)
ex1027_2
ex1027_3
ex1027_2
ex1027_3
#ref(2022.10.27-1.jpg,left,noimg,whiteboard #1);
#ref(2022.10.27-2.jpg,left,noimg,whiteboard #2);
#ref(2022.10.27-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #9]}; 2022.11.10 robust perform...
-- [[schedule2022]]
+ 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 = [1i, 0; 1i, 1]
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(ex1110_1.m);
-- robust performance is achieved but large gap
-- non structured uncertainty is considered ... the desig...
%-- 2022/11/10 14:16 --%
M = [1i, 0; 1i, 1]
M'
eig(M'*M)
svd(M)
sqrt((3+sqrt(2))/2)
sqrt((3+sqrt(5))/2)
ex1110_1
#ref(2022.11.10-1.jpg,left,noimg,whiteboard #1);
#ref(2022.11.10-2.jpg,left,noimg,whiteboard #2);
#ref(2022.11.10-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #10]}; 2022.11.17 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(ex1117_1.m);
%-- 2022/11/17 13:12 --%
M = [1, 1; 1i/sqrt(2), -1i/sqrt(2)]
svd(M)
[U, S, V] = svd(M)
U'
U'*U
format long e
U'*U
U*U'
format short
U*U'
V*V'
ex1117_1
rand(1)
help rand
ex1117_1
#ref(2022.11.17-1.jpg,left,noimg,whiteboard #1);
#ref(2022.11.17-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #11]}; 2022.11.24 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(ex1124_1.m);
ex1110_1
gam2 = gam_opt
ex1124_1
gam_opt
+ mini exam #1 (10 min.)
%-- 2022/11/24 13:04 --%
ex1110_1
gam_opt
gam2 = gam_opt
ex1124_1
gam_opt
gam2
ex1124_1
#ref(2022.11.24-1.jpg,left,noimg,whiteboard #1);
#ref(2022.11.24-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #12]}; 2022.12.1 Robust perform...
+ return of mini exam #1
+ review of scaling &ref(ex1201_1.m);
+ mini report #2 &ref(report2.pdf);
+ introduction of a practical system: Speed control of tw...
-- experimental setup &br;
&ref(授業/制御工学特論2019/setup_fixed.pdf); &br;
&ref(授業/制御工学特論2019/photo.jpg,left,noimg);
-- objective of control system = disturbance attenuation ...
-- frequency response experiment and physical model of sp...
#ref(ex1201_2.m);
&ref(servo1.dat); &ref(servo2.dat);
-- room 374 @ Dept. Mech. Bldg. 2
%-- 2022/12/01 13:12 --%
pwd
ex1201_1
ex1124_1
ex1110_1
ex1124_1
%-- 2022/12/01 13:22 --%
pwd
ex1110_1
ex1124_1
ex1201_1
ex1201_2
ctrlpref
ex1201_2
#ref(2022.12.01-1.jpg,left,noimg,whiteboard #1);
#ref(2022.12.01-2.jpg,left,noimg,whiteboard #2);
#ref(2022.12.01-3.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #13]}; 2022.12.8 Control system...
+ return of mini report #2; ... You will have a mini exa...
-- [[schedule2022]]
+ 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_2022/exp/freqres...
[[servo2.dat>/:~exp/seigyokougakutokuron_2022/exp/freqres...
-- determination of plant model(nominal plant and multipl...
&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 and evaluation&br;
[[result.dat>/:~exp/seigyokougakutokuron_2022/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: 4th(Wed) Jan 17:00};};
--submit your report(pdf file) by e-mail to kobayasi@naga...
--You can use Japanese
--maximum controller order is 20
--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 ?
-- 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_2022/freqres...
--- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2022/...
--- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2022/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_2022/hinf.h]]
--- [[hinf_module.c>/:~exp/seigyokougakutokuron_2022/hinf...
--- [[hinf_app.c>/:~exp/seigyokougakutokuron_2022/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.
%-- 2022/12/08 13:25 --%
nominal
pwd
cd ..
nominal
weight
cont
compare
perf
pwd
cd ..
perf
weight
#ref(2022.12.08-1.jpg,left,noimg,whiteboard #1);
** &color(green){[lecture #14]}; 2022.12.16 Control syste...
- 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
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
** &color(green){[lecture #15]}; 2022.12.22 Control syste...
- return of mini exam #2
- [[schedule2022]] 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 2022 [#ua920811]
[[latest lecture>#lc121915]]
** &color(green){[lecture #1]}; 2022.9.8 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%
-- [[schedule2022]] (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
-- ...
%-- 2022/09/08 13:45 --%
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
t = 0:0.01:3
y
y = exp(t)
hold on
plot(t, y, 'r--')
help impulse
impulse(P, 3)
plot(t, y, 'r--')
impulse(P, 3)
hold on
plot(t, y, 'r--')
P
K = 2
help step
Tyr = K/(s-1+K)
step(Tyr)
K = 20
Tyr = K/(s-1+K)
step(Tyr)
#ref(2022.09.08-1.jpg,left,noimg,whiteboard #1);
#ref(2022.09.08-2.jpg,left,noimg,whiteboard #2);
#ref(2022.09.08-3.jpg,left,noimg,whiteboard #3);
#ref(2022.09.08-4.jpg,left,noimg,whiteboard #4);
-Q: 思っていた以上に忘れていることがあったため,有意義な...
-A: それは良かったです。
** &color(green){[lecture #2]}; 2022.9.15 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 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(ex0915_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(ex0915_2.m); &ref(mod0915_2.mdl);
--- Gain Margin(GM); Phase Margin(PM)
%-- 2022/09/15 13:02 --%
a = 3
b = 5
a + b
ex0915_1
sqrt(k/m)
sqrt(k/m)/(2*pi)
ex0915_1
ex0915_2
#ref(2022.09.15-1.jpg,left,noimg,whiteboard #1);
#ref(2022.09.15-2.jpg,left,noimg,whiteboard #2);
#ref(2022.09.15-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #3]}; 2022.9.22 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- positive (semi-)definite matrices
-- solution of LQR problem
-- example &ref(ex0922_1.m); &ref(mod0922_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...
%-- 2022/09/22 13:49 --%
ex0922_1
A
B
[B, A*B]
Uc
det(Uc)
J
J(end)
Jmin
F
F = [0 0]
ex0922_1
F
A'
A'*P + P*A + Q - P*B/R*B'*P
P
eig(P)
#ref(2022.09.22-1.jpg,left,noimg,whiteboard #1);
#ref(2022.09.22-2.jpg,left,noimg,whiteboard #2);
#ref(2022.09.22-3.jpg,left,noimg,whiteboard #3);
#ref(2022.09.22-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #4]}; 2022.9.29 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(ex0929_1.m);
%-- 2022/09/29 14:00 --%
s = tf('s');
G1 = 1/(s+1);
bode(G1);
norm(G1, 'inf')
G1d = s/(s+1);
bode(G1d);
norm(G1d, 'inf')
G2 = 1/(s^2 + 0.1*s + 1);
bode(G2);
norm(G2, 'inf')
grid on
#ref(2022.09.29-1.jpg,left,noimg,whiteboard #1);
#ref(2022.09.29-2.jpg,left,noimg,whiteboard #2);
#ref(2022.09.29-3.jpg,left,noimg,whiteboard #3);
#ref(2022.09.29-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #5]}; 2022.10.06 relation betwe...
+ cont.
-- solve the problem by hand
-- solve the problem by tool(hinfsyn) &ref(ex0929_1.m);
+ 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...
%-- 2022/10/06 13:14 --%
ex0929_1
clp
sqrt(2-sqrt(2))
format long e
sqrt(2-sqrt(2))
format long f
format f
sqrt(2-sqrt(2))
mod1006
h = 0.01
f = -1+sqrt(2)
x0 = 0
zz
zz(end)
f
f = 1
zz(end)
h
x0
x0 = 1
zz(end)
h = 100
x0 = 0
f
sqrt(zz(end)/ww(end))
f = -1+sqrt(2)
sqrt(zz(end)/ww(end))
#ref(2022.10.06-1.jpg,left,noimg,whiteboard #1);
#ref(2022.10.06-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #6]}; 2022.10.13 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(ex1013_1.m); &ref(ex1013_2.m);
+ the small gain theorem
-- proof: Nyquist stability criterion
//+ from performance optimization to robust stabilization
%-- 2022/10/13 13:54 --%
ex1013_1
P
eig(P)
ex1013_2
ex1013_1
ex1013_2
K_hinf
eig(K_hinf)
G
eig(K_hinf)
#ref(2022.10.13-1.jpg,left,noimg,whiteboard #1);
#ref(2022.10.13-2.jpg,left,noimg,whiteboard #2);
#ref(2022.10.13-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #7]}; 2022.10.20 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
+ simple example of plant set
-- given plant P tilde
--- frequency response of plant with perturbation &ref(ex...
-- how to determine P0 and WT ?
--- frequency response based procedure for P0 and WT &ref...
+ robust stabilization problem and equivalent problem
-- design example and simulation &ref(ex1020_3.m); &ref(m...
%-- 2022/10/20 13:31 --%
ex1020_1
ex1020_2
ctrlpref
ex1020_1
ex1020_2
#ref(2022.10.20-1.jpg,left,noimg,whiteboard #1);
#ref(2022.10.20-2.jpg,left,noimg,whiteboard #2);
#ref(2022.10.20-3.jpg,left,noimg,whiteboard #3);
#ref(2022.10.20-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #8]}; 2022.10.27 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(ex1027_1.m); minimize gamma by hand
+ gamma iteration by bisection method &ref(ex1027_2.m);
+ intro. to RP: weak point of mixed sensitivity problem(p...
%-- 2022/10/27 13:01 --%
ex1020_3
ex1020_1
ex1020_2
ex1020_3
pwd
mod1020
c
c = 0.8
c = 1.3
c = 2
ex1027_1
gam
ex1027_1
ex1027_2
gam
20*log10(gam)
ex1027_2
ex1027_3
ex1027_2
ex1027_3
#ref(2022.10.27-1.jpg,left,noimg,whiteboard #1);
#ref(2022.10.27-2.jpg,left,noimg,whiteboard #2);
#ref(2022.10.27-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #9]}; 2022.11.10 robust perform...
-- [[schedule2022]]
+ 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 = [1i, 0; 1i, 1]
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(ex1110_1.m);
-- robust performance is achieved but large gap
-- non structured uncertainty is considered ... the desig...
%-- 2022/11/10 14:16 --%
M = [1i, 0; 1i, 1]
M'
eig(M'*M)
svd(M)
sqrt((3+sqrt(2))/2)
sqrt((3+sqrt(5))/2)
ex1110_1
#ref(2022.11.10-1.jpg,left,noimg,whiteboard #1);
#ref(2022.11.10-2.jpg,left,noimg,whiteboard #2);
#ref(2022.11.10-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #10]}; 2022.11.17 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(ex1117_1.m);
%-- 2022/11/17 13:12 --%
M = [1, 1; 1i/sqrt(2), -1i/sqrt(2)]
svd(M)
[U, S, V] = svd(M)
U'
U'*U
format long e
U'*U
U*U'
format short
U*U'
V*V'
ex1117_1
rand(1)
help rand
ex1117_1
#ref(2022.11.17-1.jpg,left,noimg,whiteboard #1);
#ref(2022.11.17-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #11]}; 2022.11.24 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(ex1124_1.m);
ex1110_1
gam2 = gam_opt
ex1124_1
gam_opt
+ mini exam #1 (10 min.)
%-- 2022/11/24 13:04 --%
ex1110_1
gam_opt
gam2 = gam_opt
ex1124_1
gam_opt
gam2
ex1124_1
#ref(2022.11.24-1.jpg,left,noimg,whiteboard #1);
#ref(2022.11.24-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #12]}; 2022.12.1 Robust perform...
+ return of mini exam #1
+ review of scaling &ref(ex1201_1.m);
+ mini report #2 &ref(report2.pdf);
+ introduction of a practical system: Speed control of tw...
-- experimental setup &br;
&ref(授業/制御工学特論2019/setup_fixed.pdf); &br;
&ref(授業/制御工学特論2019/photo.jpg,left,noimg);
-- objective of control system = disturbance attenuation ...
-- frequency response experiment and physical model of sp...
#ref(ex1201_2.m);
&ref(servo1.dat); &ref(servo2.dat);
-- room 374 @ Dept. Mech. Bldg. 2
%-- 2022/12/01 13:12 --%
pwd
ex1201_1
ex1124_1
ex1110_1
ex1124_1
%-- 2022/12/01 13:22 --%
pwd
ex1110_1
ex1124_1
ex1201_1
ex1201_2
ctrlpref
ex1201_2
#ref(2022.12.01-1.jpg,left,noimg,whiteboard #1);
#ref(2022.12.01-2.jpg,left,noimg,whiteboard #2);
#ref(2022.12.01-3.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #13]}; 2022.12.8 Control system...
+ return of mini report #2; ... You will have a mini exa...
-- [[schedule2022]]
+ 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_2022/exp/freqres...
[[servo2.dat>/:~exp/seigyokougakutokuron_2022/exp/freqres...
-- determination of plant model(nominal plant and multipl...
&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 and evaluation&br;
[[result.dat>/:~exp/seigyokougakutokuron_2022/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: 4th(Wed) Jan 17:00};};
--submit your report(pdf file) by e-mail to kobayasi@naga...
--You can use Japanese
--maximum controller order is 20
--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 ?
-- 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_2022/freqres...
--- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2022/...
--- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2022/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_2022/hinf.h]]
--- [[hinf_module.c>/:~exp/seigyokougakutokuron_2022/hinf...
--- [[hinf_app.c>/:~exp/seigyokougakutokuron_2022/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.
%-- 2022/12/08 13:25 --%
nominal
pwd
cd ..
nominal
weight
cont
compare
perf
pwd
cd ..
perf
weight
#ref(2022.12.08-1.jpg,left,noimg,whiteboard #1);
** &color(green){[lecture #14]}; 2022.12.16 Control syste...
- 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
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
** &color(green){[lecture #15]}; 2022.12.22 Control syste...
- return of mini exam #2
- [[schedule2022]] 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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