- 追加された行はこの色です。
- 削除された行はこの色です。
*Advanced Automation [#zc0d6c13]
** &color(green){[lecture #1]}; 2013.9.5 outline of the lecture, review of classical and modern control theory (1/3) [#we254b0e]
- outline of this lecture
-- syllabus
-- map
#ref(map_v1.0.pdf);
-- evaluation
--- mini report #1 ... 10%
--- mini exam #1 ... 10%
--- mini report #2 ... 10%
--- mini exam #2 ... 10%
--- final report ... 60%
- review
#ref(2013.09.05-1.jpg,left,noimg,whiteboard #1);
#ref(2013.09.05-2.jpg,left,noimg,whiteboard #2);
... missed (derivation of G(s) = 1/(ms^2+cs+k) by Laplace transformation from given equation of motion)
#ref(2013.09.05-3.jpg,left,noimg,whiteboard #3);
#ref(2013.09.05-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #2]}; 2013.9.12 CACSD introduction with review of classical and modern control theory (2/3) [#wb3e6bee]
+ introduction of Matlab and Simulink
--[[Basic usage of MATLAB and Simulink>/:~kobayasi/easttimor/2009/text/text_fixed.pdf]]
used for 情報処理演習及び考究II/Consideration and Practice of Information Processing II: Advanced Course of MATLAB
//
+ How to define open-loop system
//
++ &color(black,cyan){TF};
s = tf('s');
G1 = 1 / (s+1);
G2 = 1 / (s^2 + 0.1*s + 1);
++ &color(black,lightgreen){SSR};
A = [-0.3, -1; 1, 0];
B = [1; 0];
C = [0, 1];
D = 0;
G3 = ss(A, B, C, D);
-- Bode plot
bode(G1, 'b-', G2, 'g', G3, 'r--');
grid on;
//
+ open-loop stability can be checked by
//
++ &color(black,cyan){poles of TF};
roots(G2.den{:})
++ &color(black,lightgreen){eigenvalues of A-matrix in SSR};
eig(G3.a)
++ also by simulation
#ref(mod0912_1.mdl);
//
+ closed-loop stability
//
L = 1/(s^3+1.5*s^2+1.5*s+1); % example of open-loop system
roots(L.den{:}) % confirm the open-loop system is stable
//
++ graphical test by &color(black,yellow){Nyquist stability criterion}; and &color(black,yellow){Bode plot}; with &color(black,yellow){GM};(gain margin) and &color(black,yellow){PM};(phase margin)
nyquist(L)
//
bode(L)
++ numerical test by closed-loop system
clp_den = L.den{:} + L.num{:};
roots(clp_den)
++ simulation
#ref(mod0912_2.mdl);
a = 1
ver
t = [1 2 3]
pwd
ls
foo
pwd
bar
s = tf('s')
G1 = 1 / (s+1);
G2 = 1 / (s^2 + 0.1*s + 1);
G1
G2
A = [-0.3, -1; 1, 0];
B = [1; 0];
C = [0, 1];
D = 0;
G3 = ss(A, B, C, D);
G3
bode(G1, 'b-', G2, 'g', G3, 'r--');
grid on;
G2
G2.den
G2.den{:}
roots(G2.den{:})
G3.a
eig(G3.a)
mod0912_1
L = 1/(s^3+1.5*s^2+1.5*s+1); % example of open-loop system
roots(L.den{:}) % confirm the open-loop system is stable
nyquist(L)
nyquist(L*1.25)
nyquist(L)
clp_den = L.den{:} + L.num{:};
roots(clp_den)
ex0912_2
mod0912_2
L = 1.5*L
#ref(2013.09.12-1.jpg,left,noimg,whiteboard #1);
#ref(2013.09.12-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #3]}; 2013.9.19 CACSD introduction with review of classical and modern control theory (3/3) [#ubfee4e1]
+ LQR problem
+ ARE and quadratic equation
#ref(J.pdf);
+ (semi)-positive definiteness
+ example
#ref(mod0919.mdl);
mod0919
A = [1, 2, 3; 4, 5, 6; 7, 8, 9]
eig(A)
B = [1; 1; 1]
Uc = ctrb(A, B)
det(Uc)
B = [1; 0; 0]
Uc = ctrb(A, B)
det(Uc)
help are
P = are(A, B/R*B', Q)
Q = eye(3)
R = 1
P = are(A, B/R*B', Q)
P = are(A, B*inv(R)*B', Q)
P - P'
eig(P)
x(0)
x0 = [1; 1; 1]
x0'*P*x0
F = R\B'*P
C = eye(3)
D = [0; 0; 0]
J
#ref(2013.09.19-1.jpg,left,noimg,whiteboard #1);
#ref(2013.09.19-2.jpg,left,noimg,whiteboard #2);
#ref(2013.09.19-3.jpg,left,noimg,whiteboard #3);
#ref(2013.09.19-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #4]}; 2013.9.26 Intro. to robust control theory (H infinity control theory) 1/3 [#edb9111a]
+ Typical design problems
++ robust stabilization
++ &color(black,yellow){performance optimization};
++ robust performance problem (robust stability and performance optimization are simultaneously considered)
//
+ H infinity norm
-- definition
-- example
//
+ H infinity control problem
-- definition
-- application example : reference tracking problem
--- relation to the sensitivity function S(s) (S(s) -> 0 is desired but impossible)
--- given control system
#ref(ex0926_1.m)
--- controller design with H infinity control theory
#ref(ex0926_2.m)
s = tf('s')
G = 1/(s+1)
norm(G, 'inf')
G = s/(s+1)
norm(G, 'inf')
G = 1/(s^2+0.1*s+1)
bodemag(G)
norm(G, 'inf')
bodemag(G, 'b', ss(10.0125), 'r--')
G = 1/(s^2+0.5*s+1)
norm(G, 'inf')
bodemag(G, 'b', ss(2.0656), 'r--')
ex0926_1
ex0926_2
eig(K_hinf)
#ref(2013.09.26-1.jpg,left,noimg,whiteboard #1);
#ref(2013.09.26-2.jpg,left,noimg,whiteboard #2);
#ref(2013.09.26-3.jpg,left,noimg,whiteboard #3);
#ref(2013.09.26-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #5]}; %% 2013.10.3 Intro. to Robust Control Theory (H infinity control theory) 2/3 %% &color(red){&size(20){cancelled};}; [#r0a1de6e]
** &color(green){[lecture #5]}; 2013.10.10 Intro. to Robust Control Theory (H infinity control theory) 2/3 [#cf0799b9]
+ Typical design problems
++ &color(black,yellow){robust stabilization};
++ performance optimization
++ robust performance problem (robust stability and performance optimization are simultaneously considered)
//
+ connection between [H infinity control problem] and [robust stabilization problem]
-- small gain theorem
-- normalized uncertainty \Delta
-- sketch proof ... Nyquist stability criterion
+ How to design robust stabilizing controller with H infinity control problem ?
-- practical example : unstable plant with perturbation
#ref(ex1010_1.m)
-- how to use uncertainty model (multiplicative uncertainty model)
#ref(ex1010_2.m)
-- how to set generalized plant G ?
#ref(ex1010_3.m)
-- simulation
#ref(mod1010.mdl)
ex1010_1
ex1010_2
WT]
WT
P0
P0_jw
ex1010_3
mod1010
#ref(2013.10.10-1.jpg,left,noimg,whiteboard #1);
#ref(2013.10.10-2.jpg,left,noimg,whiteboard #2);
#ref(2013.10.10-3.jpg,left,noimg,whiteboard #3);
#ref(2013.10.10-4.jpg,left,noimg,whiteboard #4);
// ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
&color(black,red){&size(20){#################### the remaining page is under construction ###################};};
** &color(green){[lecture #6]}; 2013.10.17 Intro. to robust control theory (H infinity control theory) (3/3) [#kfa23202]
+ review
-- robust stabilization ... (1) ||WT T||_inf < 1 (for multiplicative uncertainty)
-- performance optimization ... (2) ||WS S||_inf < gamma -> min
-- &color(black,yellow){mixed sensitivity problem}; ... simultaneous consideration of stability and performance
+ a sufficient condition for (1) and (2) ... (*) property of maximum singular value
+ definition of singular value
+ mini report #1
+ meaning of singular value ... singular value decomposition (SVD)
+ proof of (*)
+ example
#ref(ex1017.m);
//#ref(mod1017.mdl);
%-- 10/11/2012 12:56 PM --%
ex1011_1
ex1011_2
ex1011_3
mod1011
ex1011_1
P
ex1011_2
1i
j
j = 2
1i = 2
ex1011_3
mod1011
c = 0.8
c = 1.2
c = 1.3
c = 2
A = [1, j; 0, 2]
A'
eig(A'*A)
sqrt(ans)
3+sqrt(5)
sqrt(3+sqrt(5))
sqrt(3-sqrt(5))
A
[U,S,V] = svd(A)
U'*U
U*U'
help svd
ex1017
#ref(2012.10.11-1.jpg,left,noimg,whiteboard #1);
#ref(2012.10.11-2.jpg,left,noimg,whiteboard #2);
#ref(2012.10.11-3.jpg,left,noimg,whiteboard #3);
#ref(2013.10.17-1.jpg,left,noimg,whiteboard #1);
#ref(2013.10.17-2.jpg,left,noimg,whiteboard #2);
#ref(2013.10.17-3.jpg,left,noimg,whiteboard #3);
... sorry for missing to take photo ... &color(red){&size(20){mini report #1: write by hand; submit at the beginning of the next lecture; You will have a mini exam #1 related to this report on 31st Oct.};};
#ref(2013.10.17-4.jpg,left,noimg,whiteboard #4);
#ref(2013.10.17-5.jpg,left,noimg,whiteboard #5);
#ref(2013.10.17-6.jpg,left,noimg,whiteboard #6);
// ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
&color(black,red){&size(20){#################### the remaining page is under construction ###################};};
** &color(green){[lecture #6]}; 2012.10.18 Introduction to Robust Control (cont.) [#kfa23202]
+ review
-- robust stabilization ... ||WT T||_inf < 1 (for multiplicative uncertainty)
-- performance optimization ... ||WS S||_inf < gamma -> min
-- mixed sensitivity problem gives &color(red){a sufficient condition}; such that both conditions hold ... nominal (not robust) performance problem
//
+ proof of small gain theorem ... Nyquist stability criterion
+ mixed sensitivity problem
-- the sufficient condition
-- exercise
#ref(ex1018_1.m);
#ref(ex1018_1_ans.m);
#ref(mod1018.mdl);
%-- 10/18/2012 12:56 PM --%
mod1018
ex1018_1
input_to_P0
WT
WSgam
systemnames
ex1018_1
G
ex1018_1
#ref(2012.10.18-1.jpg,left,noimg,whiteboard #1);
#ref(2012.10.18-2.jpg,left,noimg,whiteboard #2);
#ref(2012.10.18-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #7]}; 2012.10.25 singular value decomposition(SVD) [#v3874954]
- review of mixed sensitivity problem
- definition of SVD and maximum singular value
%-- 10/25/2012 1:27 PM --%
A = [1, 2, 3; 4, 5, 6]
[U, Sigma, V] = svd(A)
U*U'
U'*U
V'*V
[U, Sigma, V] = svd(A')
A = [1, 2, 3; 4, 5, 1i]
[U, Sigma, V] = svd(A)
#ref(2012.10.25-1.jpg,left,noimg,whiteboard #1);
#ref(2012.10.25-2.jpg,left,noimg,whiteboard #2);
#ref(2012.10.25-3.jpg,left,noimg,whiteboard #3);
#ref(2012.10.25-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #8]}; 2012.11.1 Robust performance problem [#e4893dbc]
- review of the proof
- nominal and robust performance (motivation of robust performance problem)
-- setting for P0 and WT
#ref(ex1101_1.m);
-- controller design with mixed sensitivity problem
#ref(ex1101_2.m);
-- robust performance is not satisfied
#ref(ex1101_3.m);
- %%how to tackle with the robust performance problem (a conservative design with H infinity control problem)%%
//#ref(ex1101_4.m);
- %%less conservative design with scaled H infinity control problem%%
//#ref(ex1101_5.m);
%-- 11/1/2012 1:37 PM --%
ex1101_1
ex1101_2
K
ex1101_2
ex1101_3
#ref(2012.11.01-1.jpg,left,noimg,whiteboard #1);
#ref(2012.11.01-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #9]}; 2012.11.8 Robust performance problem (cont.) [#w1002562]
- review: mixed sensitivity problem = robust stability + nominal performance(not robust performance)
- how to tackle with the robust performance problem
++ introduction of a fictitious uncertainty
++ the maximum singular value of block diagonal matrices is less than or equal to 1 ---> a conservative design with H infinity control problem
#ref(ex1101_4.m);
++ intruduction of scalling matrix ---> less conservative design with scaled H infinity control problem
#ref(ex1101_5.m);
%-- 11/8/2012 1:29 PM --%
ex1101_4
ex1101_1
ex1101_2
ex1101_3
ex1101_4
ex1101_5
#ref(2012.11.08-1.jpg,left,noimg,whiteboard #1);
#ref(2012.11.08-2.jpg,left,noimg,whiteboard #2);
#ref(2012.11.08-3.jpg,left,noimg,whiteboard #3);
#ref(2012.11.08-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #10]}; 2012.11.15 Robust performance problem (cont.) [#d51591df]
- review of scaled H infinity control problem
++ ex1101_4.m, ex1101_5.m ... What does the designed result mean ?
++ How was the class of scaling determined ?
++ effect of scaling on closed-loop H infinity norm ... motivation of scaling
%-- 11/15/2012 1:08 PM --%
ex1101_1
ex1101_2
ex1101_3
ex1101_4
ex1101_5
-13/20
10**-0.65
10^-0.65
ex1101_4
K
ex1101_4
ex1101_5
gam
K
d
gam
K
sigma(mdiag(1,1/gam)*lft(G,K))
sigma(mdiag(1/d,1/gam)*lft(G,K)*mdiag(d,1),'r')
sigma(mdiag(1,1/gam)*lft(G,K))
hold on
sigma(mdiag(1/d,1/gam)*lft(G,K)*mdiag(d,1),'r')
#ref(2012.11.15-1.jpg,left,noimg,whiteboard #1);
#ref(2012.11.15-2.jpg,left,noimg,whiteboard #2);
#ref(2012.11.15-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #11]}; 2012.11.22 will be given by Prof. Kimura [#b3bcf006]
** &color(green){[lecture #12]}; 2012.11.29 will be given by Prof. Kimura [#l3bcde04]
** &color(green){[lecture #13]}; 2012.12.6 Robust control design for a practical system (1/3) : Active vibration control of a pendulum using linear motor (loudspeaker) [#ucd0105e] &color(red){some mistakes fixed on 2012.12.13}; [#w0e25ce2]
- Experimental setup
#ref(photo1.jpg,left,noimg,photo1);
#ref(photo2.jpg,left,noimg,photo2);
#ref(photo3.jpg,left,noimg,photo3);
#ref(photo4.jpg,left,noimg,photo4);
#ref(photo5.jpg,left,noimg,photo5);
-- linear motor : FOSTEX &color(red){FW-};208N (corn, edge, and damper are partially removed to decrease damping)
-- PSD(Position Sensitive Detector) : Hamamatsu Photonics Corp. C3683-01
-- Potentio meter : Midori Precisions Model QP-2H
-- PC : Dell Dimension 2400 (Intel Celeron 2.4G, Linux 2.4.22, RTlinux-3.2pre3)
-- D/A : CONTEC DA12-4 (PCI) 12bit, 10us
-- A/D : CONTEC AD12-16 (PCI) 12bit, 10us
-- He-Ne Laser : NEC GLG5230
- Objective for control system design
++ to attenuate vibration due to pendulum oscillation
++ robust stability against modelling error due to non-linear dinamics etc.
- Modelling
-- frequency response experiment results:
#ref(frdata_amp005.dat);
#ref(frdata_amp010.dat);
#ref(frdata_amp020.dat);
&color(red){due to some change on experimental apparatus, please use following data as frequency response experimental results (2012.12.13)};
#ref(frdata_amp005_1st.dat);
#ref(frdata_amp010_1st.dat);
#ref(frdata_amp020_1st.dat);
#ref(frdata_amp005_2nd.dat);
#ref(frdata_amp005_3rd.dat);
#ref(frdata_amp005_4th.dat);
-- example of m-files &color(red){(the following files have been replaced by modified ones (2012.12.13)};
#ref(freqresp.m)
#ref(nominal.m)
#ref(weight.m);
>> freqresp
>> nominal
>> weight
- Controller design (robust performance problem)
-- example of m-file
#ref(cont.m)
>> cont
-- design example
#ref(cont.dat)
#ref(cont_order.dat)
#ref(cont.mat)
#ref(result.dat)
#ref(result_no.dat)
-- example of m-file to compare designed controllers
#ref(compare.m)
- report
+design your controller(s) so that the system performance is improved compared with the given example above
+Draw the following figures and explain the difference between two control systems &color(red){(your controller and the example above)};:
++bode diagram of controllers
++gain characteristic of closed-loop systems
++time response of control experiment
+Why is the performance of your system improved(or unfortunately decreased)?
--&size(30){&color(red){due date: 27th(Thu) Dec 17:00};};
--submit your report(pdf or doc) by e-mail to kobayasi@nagaokaut.ac.jp
--You can use Japanese
--maximum controller order is 20
--submit your cont.dat, cont_order.dat, and cont.mat to kobayasi@nagaokaut.ac.jp &size(30){&color(red){not later than %%21th(Fri)%% 25th(Tue) Dec};};
- program sources for frequency response experiment
#ref(freqresp.h)
#ref(freqresp_module.c)
#ref(freqresp_app.c)
-- format of frdata.dat file
--- 1st column: frequency (Hz)
--- 2nd column: gain
--- 3rd column: phase (deg)
--- 4th column: gain
--- 5th column: phase (deg)
- program sources for control experiment
#ref(hinf.h)
#ref(hinf_module.c)
#ref(hinf_app.c)
-- format of result.dat file
--- 1st column: time (s)
--- 2nd column: PSD output (V)
--- 3rd column: Potentio meter output (V)
--- 4th column: control input (V)
--- 5th column: input disturbance (V)
- configuration of control experiment
-- input disturbance is given as described in hinf_module.c:
if(t < 0.1){
w = DIST_AMP;
}else{
w = 0;
}
da_conv(V_OFFSET + w, 0); // D/A output for linear motor
freqresp
nominal
help n4sid
load result.dat
load result_no.dat
plot(result(:,1),result(:,3),'r',result_no(:,1),result_no(:,2),'g')
plot(result(:,1),result(:,3),'r',result_no(:,1),result_no(:,3),'g')
#ref(2012.12.06-1.jpg,left,noimg,whiteboard #1);
#ref(2012.12.06-2.jpg,left,noimg,whiteboard #2);
&color(red){... please use the potentio meter output as the measured output instead the PSD output (2012.12.13)};
#ref(2012.12.06-3.jpg,left,noimg,whiteboard #3);
#ref(2012.12.06-4.jpg,left,noimg,whiteboard #4);
** &color(green){[lecture #14]}; 2012.12.13 Robust control design for a practical system (2/3) [#ke2d6da6]
&size(20){&color(red){IMPOTANT};:due to some change on experimental apparatus, &color(red){please use the potentio meter output as the measured output}; instead of the PSD output. Moreover, some files have been re-uploaded due to this change. See previous links.};
-explanation of design example (cont. from the previous lecture)
-preparation of your own controller(s)
%-- 12/13/2012 12:59 PM --%
freqresp
pwd
freqresp
nominal
P0
weight
cont
compare
load result.dat
plot(result(:,1),result(:,3));
load result_no.dat
plot(result_no(:,1),result_no(:,3),'b',result(:,1),result(:,3),'r');
plot(result_no(:,1),result_no(:,2),'b',result(:,1),result(:,2),'r');
plot(result_no(:,1),result_no(:,4),'b',result(:,1),result(:,4),'r');
#ref(2012.12.13-1.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #15]}; 2012.12.20 Robust control design for a practical system (3/3) [#vb924552]
-preparation of your own controller(s)
#ref(load_frdata.m)
[[participant list2012]]
**related links [#g1a68a2b]
//-東ティモール工学部復興支援/support of rehabilitation for faculty of eng. National University of Timor-Leste
//--[[How to control objects>/:~kobayasi/easttimor/2009/index.html]] to design, to simulate and to experiment control system by using MATLAB/Simulink with an application of Inverted Pendulum
--[[Prof. Kimura's page>http://sessyu.nagaokaut.ac.jp/~kimuralab/index.php?%C0%A9%B8%E6%B9%A9%B3%D8%C6%C3%CF%C0]]