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[[授業]]
*Advanced Automation [#ye2d4ebb]
** &color(green){[lecture #1]}; 2015.9.3 outline of the l...
- outline of this lecture
-- syllabus
-- evaluation
--- mini report #1 ... 10%
--- mini exam #1 ... 10%
--- mini report #2 ... 10%
--- mini exam #2 ... 10%
--- final report ... 60%
-- [[schedule2015]] (tentative)
-- map
#ref(map_v1.0_review.pdf);
- review : stabilization of 1st-order unstable plant by c...
-- transfer function
-- differential equation
-- eigenvalue and pole
-- ...
%-- 9/3/2015 2:09 PM --%
s = tf('s')
Ptf = 1/(s+1)
pole(Ptf)
impulse(Ptf)
Pss = ss(Ptf)
initial(Pss, 1)
initial(Pss, 2)
#ref(2015.09.03-1.jpg,left,noimg,whiteboard #1);
#ref(2015.09.03-2.jpg,left,noimg,whiteboard #2);
#ref(2015.09.03-3.jpg,left,noimg,whiteboard #3);
#ref(2015.09.03-4.jpg,left,noimg,whiteboard #4);
&ref(ex0902.m);
** &color(green){[lecture #2]}; 2015.9.10 review of class...
+ introduction of Matlab and Simulink
&ref(text_fixed.pdf); Basic usage of MATLAB and Simulink ...
-- interactive system (no compilation, no variable difini...
-- m file
//
+ system representation: Transfer Function(TF) / State-Sp...
//
-- example: mass-spring-damper system
-- difinition of SSR
-- from SSR to TF
-- from TF to SSR: controllable canonical form
+ open-loop characteristic
-- open-loop stability: poles and eigenvalues
-- Bode plot and frequency response &ref(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)
%-- 9/10/2015 1:55 PM --%
ex0910_1
P
P.den
P.den{:}
P.num{:}
ex0910_1
ex0910_2
#ref(2015.09.10-1.jpg,left,noimg,whiteboard #1);
#ref(2015.09.10-2.jpg,left,noimg,whiteboard #2);
#ref(2015.09.10-3.jpg,left,noimg,whiteboard #3);
#ref(2015.09.10-4.jpg,left,noimg,whiteboard #4);
** &color(green){[]}; 2015.9.17 cancelled [#w7af2c97]
** &color(green){[]}; 2015.9.25 no lecture (lectures for ...
** &color(green){[lecture #3]}; 2015.10.1 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- (semi)-positive definiteness
+ solution of LQR problem
-- ARE and quadratic equation
-- closed loop stability ... Lyapunov criterion
-- Jmin
&ref(lqr.pdf); ≒ &ref(proof4.pdf); (from B3「動的システム...
+ example
&ref(mod1001.mdl);
A = [1, 2; 0, -1]; % unstable plant
B = [0; 1];
Uc = ctrb(A,B);
det(Uc) % should be nonzero
C = eye(2); % dummy
D = zeros(2,1); % dummy
F = [0, 0]; % without control
x0 = [1; 1]; % initial state
Q = eye(2);
R = 1;
P = are(A, B/R*B', Q);
eig(P) % should be positive
F = R\B'*P;
x0'*P*x0
%-- 10/1/2015 2:08 PM --%
mod1001
A = [1, 2; 0, -1]; % unstable plant
B = [0; 1];
Uc = ctrb(A,B);
A
B
Uc
det(Uc)
C = eye(2); % dummy
D = zeros(2,1); % dummy
F = [0, 0]; % without control
x0 = [1; 1]; % initial state
Q = eye(2);
R = 1;
F
P = are(A, B/R*B', Q);
P
eig(P)
F = R\B'*P;
F
J
x0
x0'*P*x0
A-B*F
eig(A-B*F)
#ref(2015.10.01-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.01-2.jpg,left,noimg,whiteboard #2);
#ref(2015.10.01-3.jpg,left,noimg,whiteboard #3);
... I'm sorry but all of equations are in the pdf file.
#ref(2015.10.01-4.jpg,left,noimg,whiteboard #4);
#ref(2015.10.01-5.jpg,left,noimg,whiteboard #5);
** &color(green){[lecture #4]}; 2015.10.8 relation betwee...
- GOAL: to learn difference in concepts between LQR probl...
- review of LQR problem and the simple example
+ an equivalent problem
+ a simple example of state-feedback H infinity control p...
+ definition of H infinity norm (SISO)
s = tf('s');
P1 = 1/(s+1);
bode(P1);
norm(P1, 'inf')
P2 = 1/(s^2 + 0.1*s + 1);
bode(P2);
norm(P2, 'inf')
+ definition of H infinity norm (SIMO)
+ solve the problem by hand
+ solve the problem by tool(hinfsyn)
&ref(ex1008.m);
%-- 10/8/2015 1:48 PM --%
s = tf('s');
P1 = 1/(s+1);
bode(P1);
norm(P1, 'inf')
P2 = 1/(s^2 + 0.1*s + 1);
bode(P2);
norm(P2, 'inf')
ex1008
#ref(2015.10.08-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.08-2.jpg,left,noimg,whiteboard #2);
#ref(2015.10.08-3.jpg,left,noimg,whiteboard #3);
#ref(2015.10.08-4.jpg,left,noimg,whiteboard #4);
#ref(2015.10.08-5.jpg,left,noimg,whiteboard #5);
-Q: 最後にfを求めてどうするのか分からなかった。
-A: 閉ループ系のH∞ノルムを最小化するfを求め、LQRの最適解...
-Q: |Tzw|∞ は感度関数になる?
-A: Tzw が感度関数になるか?という意味と思いますが、一般...
** &color(green){[lecture #5]}; 2015.10.15 relation betwe...
+ complete the table in simple example
+ behavior of hinfsyn in &ref(ex1008.m);
+ confirm the cost function J for both controllers by sim...
+ confirm the closed-loop H infinity norm for both contro...
-- review: steady-state response (see photo 8 @ lec. #2)
-- how to construct the worst-case disturbance w(t) which...
-- what is the worst-case disturbance in the simple examp...
+ general case: &ref(hinf.pdf); includes the simple examp...
-- LQR &ref(lqr.pdf); is included as a special case where...
%-- 10/15/2015 1:14 PM --%
ex1008
K
dcgain(K)
gopt
ex1008
mod1015
f
f = 1
x0 = 0
h = 0.1
zz
zz(end)
h = 1e-6
zz(end)
f = -1+sqrt(2)
h
zz(end)
x0 = 1
zz(end)
f
h
h = 10
zz(end)/ww(end)
x0
x0 = 0
zz(end)/ww(end)
sqrt(zz(end)/ww(end))
h
h = 100
sqrt(zz(end)/ww(end))
#ref(2015.10.15-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.15-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #6]}; 2015.10.22 Mixed sensitiv...
+ review &ref(map_v1.0_intro1.pdf); and outline
+ H infinity control problem (general form)
+ reference tracking problem
+ weighting function for sensitivity function
+ design example &ref(ex1022_1.m); &ref(ex1022_2.m);
+ the small gain theorem
-- proof: Nyquist stability criterion
+ from performance optimization to robust stabilization
%-- 10/22/2015 2:06 PM --%
ex1022_1
eig(P)
ex1022_2
#ref(2015.10.22-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.22-2.jpg,left,noimg,whiteboard #2);
#ref(2015.10.22-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #7]}; 2015.10.29 Mixed sensitiv...
+ review &ref(map_v1.0_intro2.pdf); and outline
+ an equivalent problem of robust stabilization for refer...
+ uncertainty model and normalized uncertainty Delta
+ robust stabilization problem and an equivalent problem
+ practical example of plant with perturbation &ref(ex102...
+ how to determine the model &ref(ex1029_2.m);
+ design example and simulation &ref(ex1029_3.m); &ref(mo...
%-- 10/29/2015 1:52 PM --%
ex1029_1
ex1029_2
ex1029_3
mod1029
c
c = 0.8
#ref(2015.10.29-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.29-2.jpg,left,noimg,whiteboard #2);
#ref(2015.10.29-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #8]}; 2015.11.5 Mixed sensitivi...
+ review : (1)robust stabilization and (2)performance opt...
+ mixed sensitivity problem : a sufficient condition for ...
-- proof by definition of H infinity norm
+ construction of the generalized plant
+ design example &ref(ex1105_1.m);
+ gamma iteration by bisection method &ref(ex1105_2.m);
+ a problem of the mixed sensitivity problem: nominal per...
+ introduction of robust performance problem
%-- 11/5/2015 1:37 PM --%
ex1105_1
ex1105_2
gam
ex1105_2
WT
ex1105_2
ex1105_3
ex1105_2
ex1105_3
#ref(2015.11.05-1.jpg,left,noimg,whiteboard #1);
#ref(2015.11.05-2.jpg,left,noimg,whiteboard #2);
#ref(2015.11.05-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #9]}; 2015.11.12 robust perform...
-- [[schedule2015]] mini report and exam
+ review: robust performance problem
+ an equivalent robust stability problem
+ definition of H infinity norm for general case (MIMO)
+ definition of (maximum) singular value
M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
+ mini report #1 &ref(report1.pdf);
-- write by hand
-- due date and place of submission -> see [[schedule2015]]
-- check if your answer is correct or not before submissi...
-- You will have a mini exam #1 related to this report
+ SVD: singular value decomposition
-- definition
[U,S,V] = svd(M)
M = [j, 0; -j, 1; 2, 3]
-- unitary matrix and 2 norm of vectors
-- a property of SVD: input-output interpretation
-- illustrative example: rotation matrix &ref(ex1112_1.m);
+ H infinity norm of Delta hat
%-- 11/12/2015 1:01 PM --%
M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
(3+sqrt(5))/2
sqrt((3+sqrt(5))/2)
help svd
[U,S,V] = svd(M)
U'*U
M = [j, 0; -j, 1; 2, 3]
[U,S,V] = svd(M)
V'*V
ex1112_1
#ref(2015.11.12-1.jpg,left,noimg,whiteboard #1);
#ref(2015.11.12-2.jpg,left,noimg,whiteboard #2);
#ref(2015.11.12-3.jpg,left,noimg,whiteboard #3);
#ref(2015.11.12-4.jpg,left,noimg,whiteboard #4);
-Q: Σでノルムが決まる?→固有値で
-A: 与えられた行列Mを、入力をベクトル a、出力をベクトル b...
\[ b = M a \]
とみなすと、出力の2ノルムの最大値は、入力の2ノルムの
\[ \bar \sigma(M) \] 倍となります(それを超えるような a ...
-Q: it was too fast
-A: This might be caused by my less explanation in Japane...
** &color(green){[lecture #10]}; 2015.11.19 Robust perfor...
+ return of mini report #1
+ review and outline: robust stability problem for Delta ...
+ signal vector's size is not restricted in H infinity co...
+ H infinity norm of Delta hat
+ design example: robust performance is achieved &ref(ex1...
+ non structured uncertainty is considered ... the design...
%-- 11/19/2015 1:23 PM --%
doc hinfsyn
ex1105_2
ex1105_3
gam_opt
ex1119_1
gam_opt
svd([1/sqrt(2), 0; 1/sqrt(2), 0])
#ref(2015.11.19-1.jpg,left,noimg,whiteboard #1);
#ref(2015.11.19-2.jpg,left,noimg,whiteboard #2);
-Q: Delta tilder is more tractability than Delta hat, Del...
-A: That is right. For problems
-- (P1) robust stabilization against Delta hat
-- (P2) robust stabilization against Delta tilde,&br;
P1 is more solvable than P2 because of the smaller uncert...
P2 is more tractable than P1 because of the ignorance of ...
** &color(green){[lecture #11]}; 2015.11.26 Robust perfor...
+ review
-- robust performance problem with Delta hat and conserva...
-- inclusion relation between two uncertain sets
+ introduction of the scaled H infinity control problem
+ how to determine structure of scaling matrix
//+ mini report #2 &ref(report2.pdf);
+ design example &color(red){moved to next lecture};
% less conservative design
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;
#ref(ex1126_1.m);
+ effect of scaling matrix &color(red){moved to next lect...
#ref(ex1126_2.m);
+ mini exam #1
#ref(2015.11.26-1.jpg,left,noimg,whiteboard #1);
#ref(2015.11.26-2.jpg,left,noimg,whiteboard #2);
-Q: 薄いマーカーが見辛い
-A: すみません。次回、新しいマーカーに交換してもらいます。
** &color(green){[lecture #12]}; 2015.12.3 Robust perform...
+ return of mini exam #1; schedule of mini report #2 and ...
+ review of the scaled H infinity control problem
+ comments on mu-synthesis prolem
+ design example &color(red){(moved from the previous lec...
% less conservative design
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;
#ref(ex1126_1.m);
+ effect of scaling matrix &color(red){(moved from the pr...
#ref(ex1126_2.m);
+ mini report #2 &ref(report2.pdf);
-- write by hand
-- due date and place of submission -> see [[schedule2015]]
-- check if your answer is correct or not before submissi...
-- You will have a mini exam #2 related to this report
+ controller design for practical system: active noise co...
-- introduction of experimental setup
#ref(exp_apparatus1.jpg,left,noimg);
#ref(exp_apparatus2.jpg,left,noimg);
-- objective of control system: to drive control loudspea...
-- frequency response experiment
#ref(ex1203_1.m);
#ref(spk1.dat);
#ref(spk2.dat);
%-- 12/3/2015 1:27 PM --%
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;
who
gam_opt0
ex1126_1
gam_opt
d_opt
ex1126_2
ex1203_1
#ref(2015.12.03-1.jpg,left,noimg,whiteboard #1);
#ref(2015.12.03-2.jpg,left,noimg,whiteboard #2);
#ref(2015.12.03-3.jpg,left,noimg,whiteboard #3);
-Q: Why
\[ \mbox{(i) } \hat G(0.0111) \mbox{ is stable, and} \]
\[ \mbox{(ii) } \hat G(0.0103) \mbox{ is unstable,} \]
hold ?
-A: In the first design (i), controller K_opt0 and minimi...
On the other hand, in the second design (ii), controller ...
If this answer is not sufficient for your question, pleas...
** &color(green){[lecture #13]}; 2015.12.10 Control syste...
+ return of mini report #2
+ review of the experimental apparatus and frequency resp...
+ design example
-- 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;
&ref(result.dat);&br; &ref(compare_result.m);
+ room 157 @ Dept. Mech. Bldg.2
ex1203_1
ctrlpref
ex1203_1
346/3.6
ex1203_1
nominal
weight
cont
nominal
compare
compare_result
#ref(2015.12.10-1.jpg,left,noimg,whiteboard #1);
#ref(2015.12.10-2.jpg,left,noimg,whiteboard #2);
-Q: Each of the files I wanted to clarify what they repre...
-A: A detailed explanation will be given in the next lect...
** &color(green){[lecture #14]}; 2015.12.17 Control syste...
+ final report
++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 and frequency spectrum (PSD) of control ...
++Why is the performance of your system improved(or unfor...
--&size(30){&color(red){due date: 8th(Fri) Jan 17:00};};
--submit your report(pdf or doc) by e-mail to kobayasi@na...
--You can use Japanese
--maximum controller order is 35
--submit your &size(25){&color(red){controller.dat, contr...
--You can send up to 5 controllers
--&size(25){&color(red){[[control experimental results wi...
--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: AURA SOUND NSW2-326-8A (2inch, 15W)
-- pressure sensors: NAGANO KEIKI KP15
-- 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_2015/freqresp...
-- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2015/f...
-- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2015/freq...
-- format of spk1.dat (u is used instead of w for spk2.da...
---1st column ... frequency (Hz)
---2nd column ... gain from w(V) to y(V) (signal's unit i...
---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_2015/hinf.h]]
-- [[hinf_module.c>/:~exp/seigyokougakutokuron_2015/hinf_...
-- [[hinf_app.c>/:~exp/seigyokougakutokuron_2015/hinf_app...
-- 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 hinf...
#define AMP 3.0 // amplitude for disturbance
#define DIST_INTERVAL 5 // interval step for updating w
count_dist++;
if(count_dist >= DIST_INTERVAL){
w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // uni...
count_dist = 0;
}
da_conv(V_OFFSET + w, 0); // D/A output to noise source
w is updated with 1ms period (sampling period 0.2ms times...
-- control signal u is limited to [-4, 4] as specified in...
#define U_MAX 4.00
if(u > U_MAX) u = U_MAX;
if(u < -U_MAX) u = -U_MAX;
u is set to 0 for t < 10(s). (controller is operated for ...
-- a high pass filter with cut-off frequency are used to ...
// HPF(1 rad/s) to cut DC in z and y
#define AF 9.9980001999866674e-01
#define BF 1.9998000133326669e-04
#define CF -1.0000000000000000e+00
#define DF 1.0000000000000000e+00
ad_conv(&yz); // A/D input
// HPFs
yf = CF*xf_y + DF*yz[0];
xf_y = AF*xf_y + BF*yz[0];
zf = CF*xf_z + DF*yz[1];
xf_z = AF*xf_z + BF*yz[1];
//
%%[[experiment directory>/:~exp/seigyokougakutokuron_2015...
//
+ mini exam #2
%-- 12/17/2015 1:36 PM --%
help bodemag
#ref(2015.12.17-1.jpg,left,noimg,whiteboard #1);
-Q: I wanted to make a print on the final report.
-A: Do you mean you want to submit your report in hard co...
//[[network camera>/:~exp/picture/image.jpg]]
//#ref(2014.12.18-1.jpg,left,noimg,whiteboard #1);
//#ref(2014.12.18-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #15]}; 2015.12.24 Control syste...
- preparation of your own controller(s)
-- submission procedure of controllers has been changed
-- wav file is available
- questionnaires
-- to university
-- for web-based experimental environment
%-- 12/24/2015 1:11 PM --%
compare
#ref(2015.12.24-1.jpg,left,noimg,whiteboard #1);
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
//**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 [#ye2d4ebb]
** &color(green){[lecture #1]}; 2015.9.3 outline of the l...
- outline of this lecture
-- syllabus
-- evaluation
--- mini report #1 ... 10%
--- mini exam #1 ... 10%
--- mini report #2 ... 10%
--- mini exam #2 ... 10%
--- final report ... 60%
-- [[schedule2015]] (tentative)
-- map
#ref(map_v1.0_review.pdf);
- review : stabilization of 1st-order unstable plant by c...
-- transfer function
-- differential equation
-- eigenvalue and pole
-- ...
%-- 9/3/2015 2:09 PM --%
s = tf('s')
Ptf = 1/(s+1)
pole(Ptf)
impulse(Ptf)
Pss = ss(Ptf)
initial(Pss, 1)
initial(Pss, 2)
#ref(2015.09.03-1.jpg,left,noimg,whiteboard #1);
#ref(2015.09.03-2.jpg,left,noimg,whiteboard #2);
#ref(2015.09.03-3.jpg,left,noimg,whiteboard #3);
#ref(2015.09.03-4.jpg,left,noimg,whiteboard #4);
&ref(ex0902.m);
** &color(green){[lecture #2]}; 2015.9.10 review of class...
+ introduction of Matlab and Simulink
&ref(text_fixed.pdf); Basic usage of MATLAB and Simulink ...
-- interactive system (no compilation, no variable difini...
-- m file
//
+ system representation: Transfer Function(TF) / State-Sp...
//
-- example: mass-spring-damper system
-- difinition of SSR
-- from SSR to TF
-- from TF to SSR: controllable canonical form
+ open-loop characteristic
-- open-loop stability: poles and eigenvalues
-- Bode plot and frequency response &ref(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)
%-- 9/10/2015 1:55 PM --%
ex0910_1
P
P.den
P.den{:}
P.num{:}
ex0910_1
ex0910_2
#ref(2015.09.10-1.jpg,left,noimg,whiteboard #1);
#ref(2015.09.10-2.jpg,left,noimg,whiteboard #2);
#ref(2015.09.10-3.jpg,left,noimg,whiteboard #3);
#ref(2015.09.10-4.jpg,left,noimg,whiteboard #4);
** &color(green){[]}; 2015.9.17 cancelled [#w7af2c97]
** &color(green){[]}; 2015.9.25 no lecture (lectures for ...
** &color(green){[lecture #3]}; 2015.10.1 review of class...
+ LQR problem
-- controllability
-- cost function J >= 0
-- (semi)-positive definiteness
+ solution of LQR problem
-- ARE and quadratic equation
-- closed loop stability ... Lyapunov criterion
-- Jmin
&ref(lqr.pdf); ≒ &ref(proof4.pdf); (from B3「動的システム...
+ example
&ref(mod1001.mdl);
A = [1, 2; 0, -1]; % unstable plant
B = [0; 1];
Uc = ctrb(A,B);
det(Uc) % should be nonzero
C = eye(2); % dummy
D = zeros(2,1); % dummy
F = [0, 0]; % without control
x0 = [1; 1]; % initial state
Q = eye(2);
R = 1;
P = are(A, B/R*B', Q);
eig(P) % should be positive
F = R\B'*P;
x0'*P*x0
%-- 10/1/2015 2:08 PM --%
mod1001
A = [1, 2; 0, -1]; % unstable plant
B = [0; 1];
Uc = ctrb(A,B);
A
B
Uc
det(Uc)
C = eye(2); % dummy
D = zeros(2,1); % dummy
F = [0, 0]; % without control
x0 = [1; 1]; % initial state
Q = eye(2);
R = 1;
F
P = are(A, B/R*B', Q);
P
eig(P)
F = R\B'*P;
F
J
x0
x0'*P*x0
A-B*F
eig(A-B*F)
#ref(2015.10.01-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.01-2.jpg,left,noimg,whiteboard #2);
#ref(2015.10.01-3.jpg,left,noimg,whiteboard #3);
... I'm sorry but all of equations are in the pdf file.
#ref(2015.10.01-4.jpg,left,noimg,whiteboard #4);
#ref(2015.10.01-5.jpg,left,noimg,whiteboard #5);
** &color(green){[lecture #4]}; 2015.10.8 relation betwee...
- GOAL: to learn difference in concepts between LQR probl...
- review of LQR problem and the simple example
+ an equivalent problem
+ a simple example of state-feedback H infinity control p...
+ definition of H infinity norm (SISO)
s = tf('s');
P1 = 1/(s+1);
bode(P1);
norm(P1, 'inf')
P2 = 1/(s^2 + 0.1*s + 1);
bode(P2);
norm(P2, 'inf')
+ definition of H infinity norm (SIMO)
+ solve the problem by hand
+ solve the problem by tool(hinfsyn)
&ref(ex1008.m);
%-- 10/8/2015 1:48 PM --%
s = tf('s');
P1 = 1/(s+1);
bode(P1);
norm(P1, 'inf')
P2 = 1/(s^2 + 0.1*s + 1);
bode(P2);
norm(P2, 'inf')
ex1008
#ref(2015.10.08-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.08-2.jpg,left,noimg,whiteboard #2);
#ref(2015.10.08-3.jpg,left,noimg,whiteboard #3);
#ref(2015.10.08-4.jpg,left,noimg,whiteboard #4);
#ref(2015.10.08-5.jpg,left,noimg,whiteboard #5);
-Q: 最後にfを求めてどうするのか分からなかった。
-A: 閉ループ系のH∞ノルムを最小化するfを求め、LQRの最適解...
-Q: |Tzw|∞ は感度関数になる?
-A: Tzw が感度関数になるか?という意味と思いますが、一般...
** &color(green){[lecture #5]}; 2015.10.15 relation betwe...
+ complete the table in simple example
+ behavior of hinfsyn in &ref(ex1008.m);
+ confirm the cost function J for both controllers by sim...
+ confirm the closed-loop H infinity norm for both contro...
-- review: steady-state response (see photo 8 @ lec. #2)
-- how to construct the worst-case disturbance w(t) which...
-- what is the worst-case disturbance in the simple examp...
+ general case: &ref(hinf.pdf); includes the simple examp...
-- LQR &ref(lqr.pdf); is included as a special case where...
%-- 10/15/2015 1:14 PM --%
ex1008
K
dcgain(K)
gopt
ex1008
mod1015
f
f = 1
x0 = 0
h = 0.1
zz
zz(end)
h = 1e-6
zz(end)
f = -1+sqrt(2)
h
zz(end)
x0 = 1
zz(end)
f
h
h = 10
zz(end)/ww(end)
x0
x0 = 0
zz(end)/ww(end)
sqrt(zz(end)/ww(end))
h
h = 100
sqrt(zz(end)/ww(end))
#ref(2015.10.15-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.15-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #6]}; 2015.10.22 Mixed sensitiv...
+ review &ref(map_v1.0_intro1.pdf); and outline
+ H infinity control problem (general form)
+ reference tracking problem
+ weighting function for sensitivity function
+ design example &ref(ex1022_1.m); &ref(ex1022_2.m);
+ the small gain theorem
-- proof: Nyquist stability criterion
+ from performance optimization to robust stabilization
%-- 10/22/2015 2:06 PM --%
ex1022_1
eig(P)
ex1022_2
#ref(2015.10.22-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.22-2.jpg,left,noimg,whiteboard #2);
#ref(2015.10.22-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #7]}; 2015.10.29 Mixed sensitiv...
+ review &ref(map_v1.0_intro2.pdf); and outline
+ an equivalent problem of robust stabilization for refer...
+ uncertainty model and normalized uncertainty Delta
+ robust stabilization problem and an equivalent problem
+ practical example of plant with perturbation &ref(ex102...
+ how to determine the model &ref(ex1029_2.m);
+ design example and simulation &ref(ex1029_3.m); &ref(mo...
%-- 10/29/2015 1:52 PM --%
ex1029_1
ex1029_2
ex1029_3
mod1029
c
c = 0.8
#ref(2015.10.29-1.jpg,left,noimg,whiteboard #1);
#ref(2015.10.29-2.jpg,left,noimg,whiteboard #2);
#ref(2015.10.29-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #8]}; 2015.11.5 Mixed sensitivi...
+ review : (1)robust stabilization and (2)performance opt...
+ mixed sensitivity problem : a sufficient condition for ...
-- proof by definition of H infinity norm
+ construction of the generalized plant
+ design example &ref(ex1105_1.m);
+ gamma iteration by bisection method &ref(ex1105_2.m);
+ a problem of the mixed sensitivity problem: nominal per...
+ introduction of robust performance problem
%-- 11/5/2015 1:37 PM --%
ex1105_1
ex1105_2
gam
ex1105_2
WT
ex1105_2
ex1105_3
ex1105_2
ex1105_3
#ref(2015.11.05-1.jpg,left,noimg,whiteboard #1);
#ref(2015.11.05-2.jpg,left,noimg,whiteboard #2);
#ref(2015.11.05-3.jpg,left,noimg,whiteboard #3);
** &color(green){[lecture #9]}; 2015.11.12 robust perform...
-- [[schedule2015]] mini report and exam
+ review: robust performance problem
+ an equivalent robust stability problem
+ definition of H infinity norm for general case (MIMO)
+ definition of (maximum) singular value
M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
+ mini report #1 &ref(report1.pdf);
-- write by hand
-- due date and place of submission -> see [[schedule2015]]
-- check if your answer is correct or not before submissi...
-- You will have a mini exam #1 related to this report
+ SVD: singular value decomposition
-- definition
[U,S,V] = svd(M)
M = [j, 0; -j, 1; 2, 3]
-- unitary matrix and 2 norm of vectors
-- a property of SVD: input-output interpretation
-- illustrative example: rotation matrix &ref(ex1112_1.m);
+ H infinity norm of Delta hat
%-- 11/12/2015 1:01 PM --%
M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
(3+sqrt(5))/2
sqrt((3+sqrt(5))/2)
help svd
[U,S,V] = svd(M)
U'*U
M = [j, 0; -j, 1; 2, 3]
[U,S,V] = svd(M)
V'*V
ex1112_1
#ref(2015.11.12-1.jpg,left,noimg,whiteboard #1);
#ref(2015.11.12-2.jpg,left,noimg,whiteboard #2);
#ref(2015.11.12-3.jpg,left,noimg,whiteboard #3);
#ref(2015.11.12-4.jpg,left,noimg,whiteboard #4);
-Q: Σでノルムが決まる?→固有値で
-A: 与えられた行列Mを、入力をベクトル a、出力をベクトル b...
\[ b = M a \]
とみなすと、出力の2ノルムの最大値は、入力の2ノルムの
\[ \bar \sigma(M) \] 倍となります(それを超えるような a ...
-Q: it was too fast
-A: This might be caused by my less explanation in Japane...
** &color(green){[lecture #10]}; 2015.11.19 Robust perfor...
+ return of mini report #1
+ review and outline: robust stability problem for Delta ...
+ signal vector's size is not restricted in H infinity co...
+ H infinity norm of Delta hat
+ design example: robust performance is achieved &ref(ex1...
+ non structured uncertainty is considered ... the design...
%-- 11/19/2015 1:23 PM --%
doc hinfsyn
ex1105_2
ex1105_3
gam_opt
ex1119_1
gam_opt
svd([1/sqrt(2), 0; 1/sqrt(2), 0])
#ref(2015.11.19-1.jpg,left,noimg,whiteboard #1);
#ref(2015.11.19-2.jpg,left,noimg,whiteboard #2);
-Q: Delta tilder is more tractability than Delta hat, Del...
-A: That is right. For problems
-- (P1) robust stabilization against Delta hat
-- (P2) robust stabilization against Delta tilde,&br;
P1 is more solvable than P2 because of the smaller uncert...
P2 is more tractable than P1 because of the ignorance of ...
** &color(green){[lecture #11]}; 2015.11.26 Robust perfor...
+ review
-- robust performance problem with Delta hat and conserva...
-- inclusion relation between two uncertain sets
+ introduction of the scaled H infinity control problem
+ how to determine structure of scaling matrix
//+ mini report #2 &ref(report2.pdf);
+ design example &color(red){moved to next lecture};
% less conservative design
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;
#ref(ex1126_1.m);
+ effect of scaling matrix &color(red){moved to next lect...
#ref(ex1126_2.m);
+ mini exam #1
#ref(2015.11.26-1.jpg,left,noimg,whiteboard #1);
#ref(2015.11.26-2.jpg,left,noimg,whiteboard #2);
-Q: 薄いマーカーが見辛い
-A: すみません。次回、新しいマーカーに交換してもらいます。
** &color(green){[lecture #12]}; 2015.12.3 Robust perform...
+ return of mini exam #1; schedule of mini report #2 and ...
+ review of the scaled H infinity control problem
+ comments on mu-synthesis prolem
+ design example &color(red){(moved from the previous lec...
% less conservative design
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;
#ref(ex1126_1.m);
+ effect of scaling matrix &color(red){(moved from the pr...
#ref(ex1126_2.m);
+ mini report #2 &ref(report2.pdf);
-- write by hand
-- due date and place of submission -> see [[schedule2015]]
-- check if your answer is correct or not before submissi...
-- You will have a mini exam #2 related to this report
+ controller design for practical system: active noise co...
-- introduction of experimental setup
#ref(exp_apparatus1.jpg,left,noimg);
#ref(exp_apparatus2.jpg,left,noimg);
-- objective of control system: to drive control loudspea...
-- frequency response experiment
#ref(ex1203_1.m);
#ref(spk1.dat);
#ref(spk2.dat);
%-- 12/3/2015 1:27 PM --%
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;
who
gam_opt0
ex1126_1
gam_opt
d_opt
ex1126_2
ex1203_1
#ref(2015.12.03-1.jpg,left,noimg,whiteboard #1);
#ref(2015.12.03-2.jpg,left,noimg,whiteboard #2);
#ref(2015.12.03-3.jpg,left,noimg,whiteboard #3);
-Q: Why
\[ \mbox{(i) } \hat G(0.0111) \mbox{ is stable, and} \]
\[ \mbox{(ii) } \hat G(0.0103) \mbox{ is unstable,} \]
hold ?
-A: In the first design (i), controller K_opt0 and minimi...
On the other hand, in the second design (ii), controller ...
If this answer is not sufficient for your question, pleas...
** &color(green){[lecture #13]}; 2015.12.10 Control syste...
+ return of mini report #2
+ review of the experimental apparatus and frequency resp...
+ design example
-- 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;
&ref(result.dat);&br; &ref(compare_result.m);
+ room 157 @ Dept. Mech. Bldg.2
ex1203_1
ctrlpref
ex1203_1
346/3.6
ex1203_1
nominal
weight
cont
nominal
compare
compare_result
#ref(2015.12.10-1.jpg,left,noimg,whiteboard #1);
#ref(2015.12.10-2.jpg,left,noimg,whiteboard #2);
-Q: Each of the files I wanted to clarify what they repre...
-A: A detailed explanation will be given in the next lect...
** &color(green){[lecture #14]}; 2015.12.17 Control syste...
+ final report
++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 and frequency spectrum (PSD) of control ...
++Why is the performance of your system improved(or unfor...
--&size(30){&color(red){due date: 8th(Fri) Jan 17:00};};
--submit your report(pdf or doc) by e-mail to kobayasi@na...
--You can use Japanese
--maximum controller order is 35
--submit your &size(25){&color(red){controller.dat, contr...
--You can send up to 5 controllers
--&size(25){&color(red){[[control experimental results wi...
--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: AURA SOUND NSW2-326-8A (2inch, 15W)
-- pressure sensors: NAGANO KEIKI KP15
-- 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_2015/freqresp...
-- [[freqresp_module.c>/:~exp/seigyokougakutokuron_2015/f...
-- [[freqresp_app.c>/:~exp/seigyokougakutokuron_2015/freq...
-- format of spk1.dat (u is used instead of w for spk2.da...
---1st column ... frequency (Hz)
---2nd column ... gain from w(V) to y(V) (signal's unit i...
---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_2015/hinf.h]]
-- [[hinf_module.c>/:~exp/seigyokougakutokuron_2015/hinf_...
-- [[hinf_app.c>/:~exp/seigyokougakutokuron_2015/hinf_app...
-- 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 hinf...
#define AMP 3.0 // amplitude for disturbance
#define DIST_INTERVAL 5 // interval step for updating w
count_dist++;
if(count_dist >= DIST_INTERVAL){
w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // uni...
count_dist = 0;
}
da_conv(V_OFFSET + w, 0); // D/A output to noise source
w is updated with 1ms period (sampling period 0.2ms times...
-- control signal u is limited to [-4, 4] as specified in...
#define U_MAX 4.00
if(u > U_MAX) u = U_MAX;
if(u < -U_MAX) u = -U_MAX;
u is set to 0 for t < 10(s). (controller is operated for ...
-- a high pass filter with cut-off frequency are used to ...
// HPF(1 rad/s) to cut DC in z and y
#define AF 9.9980001999866674e-01
#define BF 1.9998000133326669e-04
#define CF -1.0000000000000000e+00
#define DF 1.0000000000000000e+00
ad_conv(&yz); // A/D input
// HPFs
yf = CF*xf_y + DF*yz[0];
xf_y = AF*xf_y + BF*yz[0];
zf = CF*xf_z + DF*yz[1];
xf_z = AF*xf_z + BF*yz[1];
//
%%[[experiment directory>/:~exp/seigyokougakutokuron_2015...
//
+ mini exam #2
%-- 12/17/2015 1:36 PM --%
help bodemag
#ref(2015.12.17-1.jpg,left,noimg,whiteboard #1);
-Q: I wanted to make a print on the final report.
-A: Do you mean you want to submit your report in hard co...
//[[network camera>/:~exp/picture/image.jpg]]
//#ref(2014.12.18-1.jpg,left,noimg,whiteboard #1);
//#ref(2014.12.18-2.jpg,left,noimg,whiteboard #2);
** &color(green){[lecture #15]}; 2015.12.24 Control syste...
- preparation of your own controller(s)
-- submission procedure of controllers has been changed
-- wav file is available
- questionnaires
-- to university
-- for web-based experimental environment
%-- 12/24/2015 1:11 PM --%
compare
#ref(2015.12.24-1.jpg,left,noimg,whiteboard #1);
//■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
//&color(black,red){&size(20){!!! the remaining page is u...
//
//**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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