Advanced Automation†

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

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

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[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
      filemod0912_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
      filemod0912_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
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[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
    fileJ.pdf
  3. (semi)-positive definiteness
  4. example
    filemod0919.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
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[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
        fileex0926_1.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)
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[lecture #5] 2013.10.3 Intro. to Robust Control Theory (H infinity control theory) 2/3 cancelled†

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[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
      fileex1010_1.m
    • how to use uncertainty model (multiplicative uncertainty model)
      fileex1010_2.m
    • how to set generalized plant G ?
      fileex1010_3.m
    • simulation
      filemod1010.mdl
ex1010_1
ex1010_2
WT]
WT
P0
P0_jw
ex1010_3
mod1010
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[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
    fileex1017.m
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
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... 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.

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

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[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
      fileex1031_1.m
    • a check of the conservativeness
      fileex1031_2.m
  4. mini exam #1
    fileexam1.pdf
%-- 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
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[lecture #9] 2013.11.14 Robust performance problem (3/3)†

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ex1024_4
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ex1031_1
gam
ex1024_3
gam
ex1031_2
help lft
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ex1031_1
ex1031_2
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[lecture #10] 2013.11.21 Robust performance problem (1/3) (cont.)†

ex1024_2
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gam
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ex1031_1
gam
ex1031_2
ex1114_1
gam
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[lecture #11] 2013.11.28 Robust stabilization of inverted pendulum†

fileex1128_1.m
filemod1128_1.mdl
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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)†

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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)
  2. Draw the following figures and explain the difference between two control systems (your controller and ex3):
    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 #15] 2012.12.20 Robust control design for a practical system (3/3)†

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participant list2013

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