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[[授業]]
*Advanced Automation 2020 [#ja23d818]
//[[latest lecture>#j156dee6]]
** &color(green){[lecture #1]}; 2020.9.3 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%
-- [[schedule2020]] (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
-- ...
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)
k = 0.5
Tyr = feedback(P*k, 1)
step(Tyr)
#ref(2020.09.03-1.jpg,left,noimg,whiteboard #1);
#ref(2020.09.03-2.jpg,left,noimg,whiteboard #2);
#ref(2020.09.03-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #2]}; 2020.9.10 review of class...
+ 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(ex0910_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(ex0910_2.m); &ref(mod0910_2.mdl);
--- Gain Margin(GM); Phase Margin(PM)
%-- 20/09/10 12:48 --%
a = 1
b = 2
a + b
ex0910_1
P
ctrlpref
ex0910_1
P
P.num
P.num{:}
P.den{:}
ex0910_2
roots(P.den{:})
roots(L.den{:})
ex0910_2
K
ex0910_2
#ref(2020.09.10-1.jpg,left,noimg,whiteboard #1);
#ref(2020.09.10-2.jpg,left,noimg,whiteboard #2);
#ref(2020.09.10-3.jpg,left,noimg,whiteboard #3);
-Q: マトラボの使いかたが分かるオススメの本やサイトが知り...
-A: この授業では基本的な使い方ができれば十分のため、まず...
** &color(green){[lecture #3]}; 2020.9.17 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- positive (semi-)definite matrices
-- solution of LQR problem
-- example &ref(ex0917_1.m); &ref(mod0917_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...
%-- 20/09/17 12:42 --%
ex0917_1
A
B
Uc
det(Uc)
R
M
M = [2, 1; 1, 1]
eig(M)
ex0917_1
C
D
F
R
Q
P
A
eig(A)
eig(P)
R\B'
R\B'*P
F
J
J(end)
#ref(2020.09.17-1.jpg,left,noimg,whiteboard #1);
#ref(2020.09.17-2.jpg,left,noimg,whiteboard #2);
#ref(2020.09.17-3.jpg,left,noimg,whiteboard #3);
#ref(2020.09.17-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #4]}; 2020.9.24 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(ex0924_1.m);
%-- 20/09/24 12:36 --%
s = tf('s');
G1 = 1/(s+1);
bode(G1);
norm(G1, 'inf')
norm(G1)
norm(G1, inf)
bode(G2)
G2 = 1/(s^2 + 0.1*s + 1);
bode(G2);
G2
ctrlpref
bode(G2);
norm(G2, inf)
#ref(2020.09.24-1.jpg,left,noimg,whiteboard #1);
#ref(2020.09.24-2.jpg,left,noimg,whiteboard #2);
#ref(2020.09.24-3.jpg,left,noimg,whiteboard #3);
#ref(2020.09.24-4.jpg,left,noimg,whiteboard #4);
-Q: 日本語での講義が良いです。
-A: 反対意見が無かったため、次回から日本語で講義を行いま...
** &color(green){[lecture #5]}; 2020.10.08 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...
%-- 20/10/08 12:36 --%
ex0924_1
mod1008
f
f = -1+sqrt(2)
x0
x0 = 0
h
h = 0.01
f
zz
zz(end)
sqrt(2)-1
h = 0.001
zz(end)
f
f = 1
zz(end)
h
x0
x0 = 1
zz(end)
x0
h
h = 0.00001
zz(end)
x0 = 0
zz(end)
format long e
zz(end)
x0
f
h
h = 10
zz(end)
sqrt(zz(end)/ww(end))
format short
sqrt(zz(end)/ww(end))
h
h = 100
sqrt(zz(end)/ww(end))
f
f = -1+sqrt(2)
sqrt(zz(end)/ww(end))
#ref(2020.10.08-1.jpg,left,noimg,whiteboard #1);
#ref(2020.10.08-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #6]}; 2020.10.15 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(ex1015_1.m); &ref(ex1015_2.m);
+ the small gain theorem
-- proof: Nyquist stability criterion
//+ from performance optimization to robust stabilization
%-- 20/10/15 12:54 --%
ex1015_1
P
eig(P)
K
help step
ex1015_2
who
K_hinf
eig(K.a)
eig(K_hinf.a)
help hinfsyn
ex1015_2
P
#ref(2020.10.15-1.jpg,left,noimg,whiteboard #1);
#ref(2020.10.15-2.jpg,left,noimg,whiteboard #2);
#ref(2020.10.15-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #7]}; 2020.10.22 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(ex1022_3.m); &ref(m...
%-- 20/10/22 12:55 --%
ex1022_1
ex1022_2
ex1022_3
mod1022
c
c = 0.8
c = 2
#ref(2020.10.22-1.jpg,left,noimg,whiteboard #1);
#ref(2020.10.22-2.jpg,left,noimg,whiteboard #2);
#ref(2020.10.22-3.jpg,left,noimg,whiteboard #3);
#ref(2020.10.22-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #8]}; 2020.10.29 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(ex1029_1.m); minimize gamma by hand
+ gamma iteration by bisection method &ref(ex1029_2.m);
+ intro. to RP(problem of NP) &ref(ex1029_3.m);
%-- 20/10/29 13:00 --%
ex1029_1
ex1029_2
ex1029_3
#ref(2020.10.29-1.jpg,left,noimg,whiteboard #1);
#ref(2020.10.29-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #9]}; 2020.11.5 robust performa...
-- [[schedule2020]] due date for mini reports corrected
+ 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/sqrt(2), 1i; 1/sqrt(2), -1i]
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(ex1105_1.m);
-- robust performance is achieved but large gap
-- non structured uncertainty is considered ... the desig...
%-- 20/11/05 13:59 --%
M = [1/sqrt(2); 1i; 1/sqrt(2), -1i]
M'
eig(M'*M)
svd(M)
M = [1/sqrt(2), 1i; 1/sqrt(2), -1i]
M'
M'*M
eig(M'*M)
svd(M)
max(svd(M))
ex1105_1
#ref(2020.11.05-1.jpg,left,noimg,whiteboard #1);
#ref(2020.11.05-2.jpg,left,noimg,whiteboard #2);
#ref(2020.11.05-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #10]}; 2020.11.12 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(ex1112_1.m);
%-- 20/11/12 13:18 --%
M = [sqrt(2), -1i/sqrt(2); sqrt(2), 1i/sqrt(2)]
help svd
X
svd(M)
[U,Sigma,V] = svd(M);
Sigma
U
U'*U
V'*V
V*V'
ex1112_1
svd(M)
ex1112_1
help rand
ex1112_1
svd(M)
#ref(2020.11.12-1.jpg,left,noimg,whiteboard #1);
#ref(2020.11.12-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #11]}; 2020.11.19 Robust perfor...
+ review
-- H infinity norm (MIMO case)
-- R.S. problems for structured and unstructured uncertai...
+ scaled H infinity control problem
+ relation between three problems
+ how to determine structure of scaling matrix
+ design example &ref(ex1119_1.m);
ex1105_1
gam2 = gam_opt
K2 = K_opt;
ex1119_1
gam_opt
+ mini exam #1 (10 min.)
%-- 20/11/19 12:51 --%
ex1105_1
#ref(2020.11.19-1.jpg,left,noimg,whiteboard #1);
#ref(2020.11.19-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #12]}; 2020.11.26 Robust perfor...
+ return of mini exam #1
+ review of scaling &ref(ex1126_1.m);
+ mini report #2 &ref(report2.pdf);
+ introduction of a practical system: active noise contro...
-- experimental setup
#ref(photo1.jpg,left,noimg);
#ref(photo2.jpg,left,noimg);
-- objective of control system: to drive control loudspea...
-- frequency response experiment
#ref(ex1126_2.m);
&ref(spk1.dat); &ref(spk2.dat);
-- %%room 157 @ Dept. Mech. Bldg. 2%%
%-- 20/11/26 13:09 --%
ex1105_1
gam2 = gam_opt
K2 = K_opt;
ex1119_1
gam_opt
gam2
ex1126_1
gam2
gam3
close all
clear all
clf
ex1105_1; %
gam2 = gam_opt
K2 = K_opt;
ex1119_1
gam3 = gam_opt
K3 = K_opt;
Delta_tilde = [0, 1/sqrt(2); 0, -1/sqrt(2)]; % example o...
uncertainty
fprintf('***1st check for singular values of Delta_tilde...
svd(Delta_tilde) % less than or equal to 1
fprintf('***2nd check for closed-loop stability of M2(Gh...
Delta_tilde:');
M2 = lft(mdiag(1,1/gam2,1)*Ghat, K2, 1, 1);
clp2 = lft(Delta_tilde, M2, 2, 2);
real(eig(clp2.a)) % closed-loop stability regardless the...
uncertainty block
clp2
clp2.a
max(eig(clp2.a))
fprintf('***3rd check for closed-loop stability of M3(Gh...
Delta_tilde:');
M3 = lft(mdiag(1,1/gam3,1)*Ghat, K3, 1, 1);
clp3 = lft(Delta_tilde, M3, 2, 2);
real(eig(clp3.a)) % closed-loop instability by the non-s...
block
fprintf('***4th check for closed-loop H infinity norm of...
norm(M3, 'inf') % larger than 1
fprintf('***5th check for closed-loop H infinity norm of...
W = mdiag(d_opt,1);
M3_d = W\M3*W;
norm(M3_d, 'inf') % less than 1
d_opt
ex1126_2
ctrlpref
ex1126_2
346/(4*1.62)
#ref(2020.11.26-1.jpg,left,noimg,whiteboard #1);
#ref(2020.11.26-2.jpg,left,noimg,whiteboard #2);
#ref(2020.11.26-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #13]}; 2020.12.3 Control system...
+ return of mini report #2; ... You will have a mini exa...
-- [[schedule2020]]
+ 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_2020/exp/freqresp/...
[[spk2.dat>/:~exp/seigyokougakutokuron_2020/exp/freqresp/...
-- determination of plant model(nominal plant and additiv...
&ref(nominal.m);&br; &ref(subspace.m); ... replacement of...
&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_2020/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: 6th(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 %%5%% &color(red){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 %%FE-87%%&color(red){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_2020/freqres...
--- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2020/...
--- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2020/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_2020/hinf.h]]
--- [[hinf_module.c>/:~exp/seigyokougakutokuron_2020/hinf...
--- [[hinf_app.c>/:~exp/seigyokougakutokuron_2020/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
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 [-4, 4] 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...
%-- 20/12/03 12:59 --%
nominal
weight
cont
compare
plot(result(:,1),result(:,2))
load result.dat;
plot(result(:,1),result(:,2))
plot(result(:,1),result(:,4))
K_opt
size(K_opt.a)
plot(result(:,1),result(:,2))
plot(result(:,1),result(:,5))
#ref(2020.12.03-1.jpg,left,noimg,whiteboard #1);
** &color(green){[lecture #14]}; 2020.12.10 Control syste...
- web based remote experiment system
-- usage; how to upload controller's
-- your password were sent by e-mail
- mini exam #2
- preparation of your own controller(s) by using the remo...
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
** &color(green){[lecture #15]}; 2020.12.17 Control syste...
- return of mini exam #2
- [[schedule2020]] no lecture will be given next week
- review & supplemental explanations
-- final report
--- controller design procedure in the design example
--- generating wav file [[filter.c>/:~exp/seigyokougakuto...
--- &ref(perf.m); (sampling period is fixed to 0.2ms (don...
--- c2d() is used to discretize the resultant continuous-...
---You can send up to 10 controllers (don't fall into tri...
--- discuss relationship between the required figures (a)...
--- no strict control objective is given ( there is a fre...
-- web based remote experiment system
--- powered by prof. Takebe, National Institute of Techno...
//--- 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...
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
&color(black,red){&size(20){!!! the remaining page is und...
//
//**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 2020 [#ja23d818]
//[[latest lecture>#j156dee6]]
** &color(green){[lecture #1]}; 2020.9.3 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%
-- [[schedule2020]] (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
-- ...
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)
k = 0.5
Tyr = feedback(P*k, 1)
step(Tyr)
#ref(2020.09.03-1.jpg,left,noimg,whiteboard #1);
#ref(2020.09.03-2.jpg,left,noimg,whiteboard #2);
#ref(2020.09.03-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #2]}; 2020.9.10 review of class...
+ 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(ex0910_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(ex0910_2.m); &ref(mod0910_2.mdl);
--- Gain Margin(GM); Phase Margin(PM)
%-- 20/09/10 12:48 --%
a = 1
b = 2
a + b
ex0910_1
P
ctrlpref
ex0910_1
P
P.num
P.num{:}
P.den{:}
ex0910_2
roots(P.den{:})
roots(L.den{:})
ex0910_2
K
ex0910_2
#ref(2020.09.10-1.jpg,left,noimg,whiteboard #1);
#ref(2020.09.10-2.jpg,left,noimg,whiteboard #2);
#ref(2020.09.10-3.jpg,left,noimg,whiteboard #3);
-Q: マトラボの使いかたが分かるオススメの本やサイトが知り...
-A: この授業では基本的な使い方ができれば十分のため、まず...
** &color(green){[lecture #3]}; 2020.9.17 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- positive (semi-)definite matrices
-- solution of LQR problem
-- example &ref(ex0917_1.m); &ref(mod0917_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...
%-- 20/09/17 12:42 --%
ex0917_1
A
B
Uc
det(Uc)
R
M
M = [2, 1; 1, 1]
eig(M)
ex0917_1
C
D
F
R
Q
P
A
eig(A)
eig(P)
R\B'
R\B'*P
F
J
J(end)
#ref(2020.09.17-1.jpg,left,noimg,whiteboard #1);
#ref(2020.09.17-2.jpg,left,noimg,whiteboard #2);
#ref(2020.09.17-3.jpg,left,noimg,whiteboard #3);
#ref(2020.09.17-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #4]}; 2020.9.24 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(ex0924_1.m);
%-- 20/09/24 12:36 --%
s = tf('s');
G1 = 1/(s+1);
bode(G1);
norm(G1, 'inf')
norm(G1)
norm(G1, inf)
bode(G2)
G2 = 1/(s^2 + 0.1*s + 1);
bode(G2);
G2
ctrlpref
bode(G2);
norm(G2, inf)
#ref(2020.09.24-1.jpg,left,noimg,whiteboard #1);
#ref(2020.09.24-2.jpg,left,noimg,whiteboard #2);
#ref(2020.09.24-3.jpg,left,noimg,whiteboard #3);
#ref(2020.09.24-4.jpg,left,noimg,whiteboard #4);
-Q: 日本語での講義が良いです。
-A: 反対意見が無かったため、次回から日本語で講義を行いま...
** &color(green){[lecture #5]}; 2020.10.08 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...
%-- 20/10/08 12:36 --%
ex0924_1
mod1008
f
f = -1+sqrt(2)
x0
x0 = 0
h
h = 0.01
f
zz
zz(end)
sqrt(2)-1
h = 0.001
zz(end)
f
f = 1
zz(end)
h
x0
x0 = 1
zz(end)
x0
h
h = 0.00001
zz(end)
x0 = 0
zz(end)
format long e
zz(end)
x0
f
h
h = 10
zz(end)
sqrt(zz(end)/ww(end))
format short
sqrt(zz(end)/ww(end))
h
h = 100
sqrt(zz(end)/ww(end))
f
f = -1+sqrt(2)
sqrt(zz(end)/ww(end))
#ref(2020.10.08-1.jpg,left,noimg,whiteboard #1);
#ref(2020.10.08-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #6]}; 2020.10.15 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(ex1015_1.m); &ref(ex1015_2.m);
+ the small gain theorem
-- proof: Nyquist stability criterion
//+ from performance optimization to robust stabilization
%-- 20/10/15 12:54 --%
ex1015_1
P
eig(P)
K
help step
ex1015_2
who
K_hinf
eig(K.a)
eig(K_hinf.a)
help hinfsyn
ex1015_2
P
#ref(2020.10.15-1.jpg,left,noimg,whiteboard #1);
#ref(2020.10.15-2.jpg,left,noimg,whiteboard #2);
#ref(2020.10.15-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #7]}; 2020.10.22 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(ex1022_3.m); &ref(m...
%-- 20/10/22 12:55 --%
ex1022_1
ex1022_2
ex1022_3
mod1022
c
c = 0.8
c = 2
#ref(2020.10.22-1.jpg,left,noimg,whiteboard #1);
#ref(2020.10.22-2.jpg,left,noimg,whiteboard #2);
#ref(2020.10.22-3.jpg,left,noimg,whiteboard #3);
#ref(2020.10.22-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #8]}; 2020.10.29 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(ex1029_1.m); minimize gamma by hand
+ gamma iteration by bisection method &ref(ex1029_2.m);
+ intro. to RP(problem of NP) &ref(ex1029_3.m);
%-- 20/10/29 13:00 --%
ex1029_1
ex1029_2
ex1029_3
#ref(2020.10.29-1.jpg,left,noimg,whiteboard #1);
#ref(2020.10.29-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #9]}; 2020.11.5 robust performa...
-- [[schedule2020]] due date for mini reports corrected
+ 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/sqrt(2), 1i; 1/sqrt(2), -1i]
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(ex1105_1.m);
-- robust performance is achieved but large gap
-- non structured uncertainty is considered ... the desig...
%-- 20/11/05 13:59 --%
M = [1/sqrt(2); 1i; 1/sqrt(2), -1i]
M'
eig(M'*M)
svd(M)
M = [1/sqrt(2), 1i; 1/sqrt(2), -1i]
M'
M'*M
eig(M'*M)
svd(M)
max(svd(M))
ex1105_1
#ref(2020.11.05-1.jpg,left,noimg,whiteboard #1);
#ref(2020.11.05-2.jpg,left,noimg,whiteboard #2);
#ref(2020.11.05-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #10]}; 2020.11.12 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(ex1112_1.m);
%-- 20/11/12 13:18 --%
M = [sqrt(2), -1i/sqrt(2); sqrt(2), 1i/sqrt(2)]
help svd
X
svd(M)
[U,Sigma,V] = svd(M);
Sigma
U
U'*U
V'*V
V*V'
ex1112_1
svd(M)
ex1112_1
help rand
ex1112_1
svd(M)
#ref(2020.11.12-1.jpg,left,noimg,whiteboard #1);
#ref(2020.11.12-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #11]}; 2020.11.19 Robust perfor...
+ review
-- H infinity norm (MIMO case)
-- R.S. problems for structured and unstructured uncertai...
+ scaled H infinity control problem
+ relation between three problems
+ how to determine structure of scaling matrix
+ design example &ref(ex1119_1.m);
ex1105_1
gam2 = gam_opt
K2 = K_opt;
ex1119_1
gam_opt
+ mini exam #1 (10 min.)
%-- 20/11/19 12:51 --%
ex1105_1
#ref(2020.11.19-1.jpg,left,noimg,whiteboard #1);
#ref(2020.11.19-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #12]}; 2020.11.26 Robust perfor...
+ return of mini exam #1
+ review of scaling &ref(ex1126_1.m);
+ mini report #2 &ref(report2.pdf);
+ introduction of a practical system: active noise contro...
-- experimental setup
#ref(photo1.jpg,left,noimg);
#ref(photo2.jpg,left,noimg);
-- objective of control system: to drive control loudspea...
-- frequency response experiment
#ref(ex1126_2.m);
&ref(spk1.dat); &ref(spk2.dat);
-- %%room 157 @ Dept. Mech. Bldg. 2%%
%-- 20/11/26 13:09 --%
ex1105_1
gam2 = gam_opt
K2 = K_opt;
ex1119_1
gam_opt
gam2
ex1126_1
gam2
gam3
close all
clear all
clf
ex1105_1; %
gam2 = gam_opt
K2 = K_opt;
ex1119_1
gam3 = gam_opt
K3 = K_opt;
Delta_tilde = [0, 1/sqrt(2); 0, -1/sqrt(2)]; % example o...
uncertainty
fprintf('***1st check for singular values of Delta_tilde...
svd(Delta_tilde) % less than or equal to 1
fprintf('***2nd check for closed-loop stability of M2(Gh...
Delta_tilde:');
M2 = lft(mdiag(1,1/gam2,1)*Ghat, K2, 1, 1);
clp2 = lft(Delta_tilde, M2, 2, 2);
real(eig(clp2.a)) % closed-loop stability regardless the...
uncertainty block
clp2
clp2.a
max(eig(clp2.a))
fprintf('***3rd check for closed-loop stability of M3(Gh...
Delta_tilde:');
M3 = lft(mdiag(1,1/gam3,1)*Ghat, K3, 1, 1);
clp3 = lft(Delta_tilde, M3, 2, 2);
real(eig(clp3.a)) % closed-loop instability by the non-s...
block
fprintf('***4th check for closed-loop H infinity norm of...
norm(M3, 'inf') % larger than 1
fprintf('***5th check for closed-loop H infinity norm of...
W = mdiag(d_opt,1);
M3_d = W\M3*W;
norm(M3_d, 'inf') % less than 1
d_opt
ex1126_2
ctrlpref
ex1126_2
346/(4*1.62)
#ref(2020.11.26-1.jpg,left,noimg,whiteboard #1);
#ref(2020.11.26-2.jpg,left,noimg,whiteboard #2);
#ref(2020.11.26-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #13]}; 2020.12.3 Control system...
+ return of mini report #2; ... You will have a mini exa...
-- [[schedule2020]]
+ 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_2020/exp/freqresp/...
[[spk2.dat>/:~exp/seigyokougakutokuron_2020/exp/freqresp/...
-- determination of plant model(nominal plant and additiv...
&ref(nominal.m);&br; &ref(subspace.m); ... replacement of...
&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_2020/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: 6th(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 %%5%% &color(red){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 %%FE-87%%&color(red){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_2020/freqres...
--- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2020/...
--- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2020/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_2020/hinf.h]]
--- [[hinf_module.c>/:~exp/seigyokougakutokuron_2020/hinf...
--- [[hinf_app.c>/:~exp/seigyokougakutokuron_2020/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
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 [-4, 4] 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...
%-- 20/12/03 12:59 --%
nominal
weight
cont
compare
plot(result(:,1),result(:,2))
load result.dat;
plot(result(:,1),result(:,2))
plot(result(:,1),result(:,4))
K_opt
size(K_opt.a)
plot(result(:,1),result(:,2))
plot(result(:,1),result(:,5))
#ref(2020.12.03-1.jpg,left,noimg,whiteboard #1);
** &color(green){[lecture #14]}; 2020.12.10 Control syste...
- web based remote experiment system
-- usage; how to upload controller's
-- your password were sent by e-mail
- mini exam #2
- preparation of your own controller(s) by using the remo...
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
** &color(green){[lecture #15]}; 2020.12.17 Control syste...
- return of mini exam #2
- [[schedule2020]] no lecture will be given next week
- review & supplemental explanations
-- final report
--- controller design procedure in the design example
--- generating wav file [[filter.c>/:~exp/seigyokougakuto...
--- &ref(perf.m); (sampling period is fixed to 0.2ms (don...
--- c2d() is used to discretize the resultant continuous-...
---You can send up to 10 controllers (don't fall into tri...
--- discuss relationship between the required figures (a)...
--- no strict control objective is given ( there is a fre...
-- web based remote experiment system
--- powered by prof. Takebe, National Institute of Techno...
//--- 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...
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
&color(black,red){&size(20){!!! the remaining page is und...
//
//**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/~...
ページ名:
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