授業

Advanced Automation†

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[] 2014.9.4 cancelled†

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[lecture #1] 2014.9.11 outline of the lecture, review of classical and modern control theory (1/3)†

filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
filewhiteboard #4
filewhiteboard #5
filewhiteboard #6
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[lecture #2] 2014.9.18 CACSD introduction with review of classical and modern control theory (2/3)†

  1. introduction of Matlab and Simulink filetext_fixed.pdf Basic usage of MATLAB and Simulink used for 情報処理演習及び考究II/Consideration and Practice of Information Processing II: Advanced Course of MATLAB
  2. How to define open-loop system
    1. TF 
      s = tf('s');
T1 = 1 / (s+1);
T2 = 1 / (s^2 + 0.1*s + 1);
    2. SSR 
      A = [-0.3, -1; 1, 0];
B = [1; 0];
C = [0, 1];
D = 0;
S3 = ss(A, B, C, D);
    • Bode plot 
      bode(T1, 'b-', T2, 'g', S3, 'r--');
grid on;
  3. open-loop stability can be checked by
    1. poles of TF 
      roots(T2.den{:})
    2. eigenvalues of A-matrix in SSR 
      eig(S3.a)
    3. also by simulation
      filemod0918_1.mdl
  4. 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
    1. graphical test by Nyquist stability criterion and Bode plot with GM(gain margin) and PM(phase margin) 
      nyquist(L)
bode(L)
    2. numerical test by closed-loop system 
      clp_den = L.den{:} + L.num{:};
roots(clp_den)
    3. simulation
      filemod0918_2.mdl
%-- 9/18/2014 1:06 PM --%
t = [1 2 3]
u = [1;2;3]
V = [1 2 3; 4 5 6; 7 8 9]
t'
t'*t
who
k=0:0.1:10:
k=0:0.1:10;
y = sin(k);
whos
plot(x,y)
plot(k,y)
foo
print -djpeg sin.jpg
s = tf('s');
T1 = 1/(s+1)
T2 = 1/(s^2+0.1*s+1);
A = [-0.3, -1; 1, 0];
B = [1; 0];
C = [0, 1];
D = 0;
S3 = ss(A, B, C, D);
A
B
S3
bode(T1, 'b-', T2, 'g', S3, 'r--');
grid on;
T2
T3 = tf(S3);
T3
T2
T2.num
T2.num{:}
T2.den{:}
roots(T2.den{:})
S3
S3.a
eig(S3.a)
mod0918_1
bode(T1, 'b-', T2, 'g', S3, 'r--');
grid on;
roots(L.den{:})
L
L = 1/(s^3+1.5*s^2+1.5*s+1);
L
roots(L.den{:})
nyquist(L)
bode(L)
grid on
nyquist(L)
L
clp_den = L.den{:} + L.num{:};
clp_den
roots(clp_den)
mod0918_2
filewhiteboard #1
filewhiteboard #2
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[lecture #3] 2014.9.25 CACSD introduction with review of classical and modern control theory (3/3)†

  1. LQR problem
    • controllability
    • cost function J >= 0
    • (semi)-positive definiteness
  2. solution of LQR problem
    • ARE and quadratic equation
    • closed loop stability ... Lyapunov criterion
    • Jmin fileproof4.pdf (from B43「動的システムの解析と制御」)
  3. example filemod0925.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);
P-P' % should be zero
eig(P) % should be positive
F = R\B'*P;
%-- 9/25/2014 2:17 PM --%
A = [1, 2; 0, -1]; % unstable plant
B = [0; 1];
Uc = ctrb(A,B);
Uc
det(Uc)]
det(Uc)
C = eye(2); % dummy
D = zeros(2,1); % dummy
F = [0, 0]; % without control
x0 = [1; 1]; % initial state
mod0925
Q = eye(2);
R = 1;
P = are(A, B/R*B', Q);
P-P' % should be zero
eig(P) % should be positive
F = R\B'*P;
F
J
x0
x0'*P*x0

filewhiteboard #1 ... sorry for the mistake in Uc ! The correct one is \[ U_c := \left[\begin{array}{ccccc} B & AB & A^2 B & \cdots & A^{n-1} B \end{array}\right] \]

filewhiteboard #2
filewhiteboard #3
filewhiteboard #4
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[lecture #4] 2014.10.2 Intro. to robust control theory (H infinity control theory) 1/3†

  1. review filemap_v1.0_intro1.pdf
    1. advantage and disadvantage of the modern control theory
    2. explicit consideration of plant uncertainty ---> robust control theory
  2. Typical design problems of H infinity control theory
    1. robust stabilization
    2. performance optimization
    3. robust performance problem (robust stability and performance optimization are simultaneously considered)
  3. H infinity norm
    • definition
    • example
  4. H infinity control problem
    • definition
  5. performance optimization example : reference tracking problem
    • relation to the sensitivity function S(s) (S(s) -> 0 is desired but impossible)
    • given control system fileex1002_1.m
    • controller design with H infinity control theory fileex1002_2.m
%-- 10/2/2014 12:57 PM --%
ex1002_1
ex1002_2
s = tf('s')
T1 = 1/(s+1)
norm(T1,inf)
T2 = s/(s+1)
norm(T2,inf)
bode(T1)
T3 = 10/(s+2)
bode(T3)
ex1002_1
ex1002_2
K
K_hinf
ex1002_2
eig(K_hinf.a)
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
filewhiteboard #4
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[lecture #5] 2014.10.09 Intro. to Robust Control Theory (H infinity control theory) 2/3†

  1. Typical design problems
    1. robust stabilization
    2. performance optimization
    3. robust performance problem (robust stability and performance optimization are simultaneously considered)
  2. connection between [H infinity control problem] and [robust stabilization problem]
    • small gain theorem
    • normalized uncertainty \Delta
    • sketch proof ... Nyquist stability criterion
  3. How to design robust stabilizing controller with H infinity control problem ?
    • practical example : unstable plant with perturbation
      fileex1009_1.m
    • how to use uncertainty model (multiplicative uncertainty model)
      fileex1009_2.m
    • how to set generalized plant G ?
      fileex1009_3.m
    • simulation
      filemod1009.mdl
%-- 10/9/2014 1:01 PM --%
ex1009_1
ex1009_2
ex1009_3
mod1009
c
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
filewhiteboard #4
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[lecture #6] 2014.10.16 Intro. to robust control theory (H infinity control theory) (3/3)†

  1. review
    • robust stabilization ... (1) ||WT T||_inf < 1 (for multiplicative uncertainty)
    • performance optimization ... (2) ||WS S||_inf < gamma -> min
    • mixed sensitivity problem ... simultaneous consideration of stability and performance
  2. a sufficient condition for (1) and (2) ... (*) property of maximum singular value
  3. definition of singular value
  4. mini report #1
    • write by hand
    • submit at the beginning of next lecture on 23 Oct.
    • check if your answer is correct or not before submission by using Matlab
    • You will have a mini exam #1 related to this report on 30 Oct.
  5. meaning of singular value ... singular value decomposition (SVD)
  6. proof of (*)
  7. example
    fileex1016.m
%-- 10/16/2014 1:00 PM --%
M = [j, 0; -j, 1]
M
svd(M)
sqrt((3+sqrt(5))/2)
M'
M'*M
ex1016
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
filewhiteboard #4
filewhiteboard #5
filewhiteboard #6
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[lecture #7] 2014.10.23 review of SVD, robust performance problem 1/3 (motivation of robust performance)†

%-- 10/23/2014 12:58 PM --%
ex1023_1
A
S
V
V'*V
V'*V(:,1)
ex1009_1
ex1009_2
ex1023_2
ex1023_3
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
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[lecture #8] 2014.10.30 Robust performance problem (2/3)†

  1. return of mini report #1
  2. review of the limitation of mixed sensitivity problem
  3. diffinition of robust performance (R.P.) problem (cf. nominal performance problem on white board #6 in photo #4 of lecture #4) ... S is changed to S
  4. review of robust stability (R.S.) problem on white board #5 in photo #5 of lecture #3 ... robust stability against Delta <=> closed-loop system without Delta has less-than-or-equal-to-one H infinity norm (by small gain theorem)
  5. equivalent R.P. problem with structured uncertainty Delta_hat
  6. a conservative problem to R.P. with 2-by-2 unstructured uncertainty Delta_tilde
    • example based on the one given in the last lecture
      fileex1030_1.m
    • a check of the conservativeness
      fileex1030_2.m
  7. Delta_tilde is larger set than Delta_hat ... conservativeness
  8. mini exam #1 fileexam1.pdf
%-- 10/30/2014 1:10 PM --%
ex1023_2
ex1023_3
gam_opt
ex1030_1
gam_opt
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
filewhiteboard #4
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[lecture #9] 2014.11.13 Robust performance problem (3/3)†

%-- 11/13/2014 12:58 PM --%
ex1023_2
gam_opt
ex1023_3
ex1030_1
gam_opt
ex1113
ex1113_1
gam_opt
ex1113_2

!!! the remaining page is under construction (the contents below are from 2013) !!!

ex1024_2
ex1024_3
ex1024_4
ex1024_5
ex1031_1
gam
ex1024_3
gam
ex1031_2
help lft
ex1031_2
ex1031_1
ex1031_2

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[lecture #10] 2013.11.21 Robust performance problem (1/3) (cont.)†

ex1024_2
ex1024_3
gam
ex1024_4
ex1031_1
gam
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ex1114_1
gam

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[lecture #11] 2013.11.28 Robust stabilization of inverted pendulum†

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[lecture #12] 2013.12.5 Robust control design for a practical system : Active vibration control of a pendulum using linear motor (1/3)†

filephoto1

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[lecture #13] 2013.12.12 Robust control design for a practical system : Active vibration control of a pendulum using linear motor (2/3)†

  1. design your controller(s) so that the system performance is improved compared with the design example 3 (ex3) example 2 (ex2)
  2. Draw the following figures and explain the difference between two control systems (your controller and ex3 ex2):
    1. bode diagram of controllers
    2. gain characteristic of closed-loop systems
    3. time response of control experiment
  3. Why is the performance of your system improved(or unfortunately decreased)?
    • due date: 31th(Tue) 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 not later than 26th(Thu) Dec

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[lecture #14] 2013.12.19 Robust control design for a practical system : Active vibration control of a pendulum using linear motor (3/3)†

[IMPORTANT] Due to unavailability of n4sid in IPC which is used in cont_ex3.m, please compare your controller and example 2 (not 3) in your report. The explanation of the report has been modified due to this change. See above.

participant list2013

freqresp_fixed
frdata
check_pcont
weight_ex2
freqresp_fixed
weight_ex2
cont_ex2
nominal_ex3
weight_ex4
cont_ex4
compare

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[lecture #15] 2013.12.26 Robust control design for a practical system : Active vibration control of a pendulum using linear motor (cont.)†

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