Advanced Automation

[lecture #1] 2013.9.5 outline of the lecture, review of classical and modern control theory (1/3)

... missed (derivation of G(s) = 1/(ms^2+cs+k) by Laplace transformation from given equation of motion)

[lecture #2] 2013.9.12 CACSD introduction with review of classical and modern control theory (2/3)

  1. introduction of Matlab and Simulink
  2. How to define open-loop system
    1. TF
      s = tf('s');
      G1 = 1 / (s+1);
      G2 = 1 / (s^2 + 0.1*s + 1);
    2. 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;
  3. open-loop stability can be checked by
    1. poles of TF
      roots(G2.den{:})
    2. eigenvalues of A-matrix in SSR
      eig(G3.a)
    3. also by simulation
  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
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 

[lecture #3] 2013.9.19 CACSD introduction with review of classical and modern control theory (3/3)

  1. LQR problem
  2. ARE and quadratic equation
  3. (semi)-positive definiteness
  4. example
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

[lecture #4] 2013.9.26 Intro. to robust control theory (H infinity control theory) 1/3

  1. Typical design problems
    1. robust stabilization
    2. performance optimization
    3. robust performance problem (robust stability and performance optimization are simultaneously considered)
  2. H infinity norm
    • definition
    • example
  3. 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
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)

[lecture #5] 2013.10.3 Intro. to Robust Control Theory (H infinity control theory) 2/3 cancelled

[lecture #5] 2013.10.10 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
    • how to use uncertainty model (multiplicative uncertainty model)
    • how to set generalized plant G ?
    • simulation
ex1010_1
ex1010_2
WT]
WT
P0
P0_jw
ex1010_3
mod1010

[lecture #6] 2013.10.17 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
  5. meaning of singular value ... singular value decomposition (SVD)
  6. proof of (*)
  7. example
j
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

... sorry for missing to take photo ... 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.

[lecture #7] 2013.10.24 review of SVD, motivation of robust performance (robust performance problem 1/3), state space representation of generalized plant

ex1024_1
ex1024_2
ex1024_3
ex1024_4

[lecture #8] 2013.10.31 Robust performance problem (2/3)

  1. review of mini report #1
  2. review of the limitation of mixed sensitivity problem
  3. a solution of conservative design
    • example based on the one given in the last lecture
    • a check of the conservativeness
  4. mini exam #1
%-- 10/31/2013 1:02 PM --%
A = [j, 0; -j, 0]
A = [j, 0; -j, 1]
svd(A)
sqrt((3+sqrt(5))/2)
sqrt((3-sqrt(5))/2)
ex1024_2
ex1024_3
ex1024_4
ex1031_1

[lecture #9] 2013.11.14 Robust performance problem (3/3)

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

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

ex1024_2
ex1024_3
gam
ex1024_4
ex1031_1
gam
ex1031_2
ex1114_1
gam

[lecture #11] 2013.11.28 Robust stabilization of inverted pendulum

#################### the remaining page is under construction ###################

[lecture #13] 2012.12.6 Robust control design for a practical system (1/3) : Active vibration control of a pendulum using linear motor (loudspeaker) some mistakes fixed on 2012.12.13 [#w0e25ce2]

  1. design your controller(s) so that the system performance is improved compared with the given example above
  2. Draw the following figures and explain the difference between two control systems (your controller and the example above):
    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: 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 not later than 21th(Fri) 25th(Tue) Dec
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(): File not found: "2012.12.06-1.jpg" at page "授業/制御工学特論2013"

#ref(): File not found: "2012.12.06-2.jpg" at page "授業/制御工学特論2013"

... please use the potentio meter output as the measured output instead the PSD output (2012.12.13)

#ref(): File not found: "2012.12.06-3.jpg" at page "授業/制御工学特論2013"

#ref(): File not found: "2012.12.06-4.jpg" at page "授業/制御工学特論2013"

[lecture #14] 2012.12.13 Robust control design for a practical system (2/3)

IMPOTANT:due to some change on experimental apparatus, 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.

%-- 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(): File not found: "2012.12.13-1.jpg" at page "授業/制御工学特論2013"

[lecture #15] 2012.12.20 Robust control design for a practical system (3/3)

#ref(): File not found: "load_frdata.m" at page "授業/制御工学特論2013"

participant list2012

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