授業

Advanced Automation†

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[lecture #1] 2015.9.3 outline of the lecture, review of classical and modern control theory (1/3)†

!!! the remaining page is under construction (the contents below are from 2014) !!!

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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

      #ref(): File not found: "mod0918_1.mdl" at page "授業/制御工学特論2015"

  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

      #ref(): File not found: "mod0918_2.mdl" at page "授業/制御工学特論2015"

%-- 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

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[] 2015.9.17 cancelled†

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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 &ref(): File not found: "mod0925.mdl" at page "授業/制御工学特論2015"; 
    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

&ref(): File not found: "2014.09.25-1.jpg" at page "授業/制御工学特論2015"; ... sorry for the mistake in Uc ! The correct one is \[ U_c := \left[\begin{array}{ccccc} B & AB & A^2 B & \cdots & A^{n-1} B \end{array}\right] \]

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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 &ref(): File not found: "ex1002_1.m" at page "授業/制御工学特論2015";
    • controller design with H infinity control theory &ref(): File not found: "ex1002_2.m" at page "授業/制御工学特論2015";
%-- 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)

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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

      #ref(): File not found: "ex1009_1.m" at page "授業/制御工学特論2015"

    • how to use uncertainty model (multiplicative uncertainty model)

      #ref(): File not found: "ex1009_2.m" at page "授業/制御工学特論2015"

    • how to set generalized plant G ?

      #ref(): File not found: "ex1009_3.m" at page "授業/制御工学特論2015"

    • simulation

      #ref(): File not found: "mod1009.mdl" at page "授業/制御工学特論2015"

%-- 10/9/2014 1:01 PM --%
ex1009_1
ex1009_2
ex1009_3
mod1009
c

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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

    #ref(): File not found: "ex1016.m" at page "授業/制御工学特論2015"

%-- 10/16/2014 1:00 PM --%
M = [j, 0; -j, 1]
M
svd(M)
sqrt((3+sqrt(5))/2)
M'
M'*M
ex1016

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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

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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

      #ref(): File not found: "ex1030_1.m" at page "授業/制御工学特論2015"

    • a check of the conservativeness

      #ref(): File not found: "ex1030_2.m" at page "授業/制御工学特論2015"

  7. Delta_tilde is larger set than Delta_hat ... conservativeness
  8. mini exam #1 &ref(): File not found: "exam1.pdf" at page "授業/制御工学特論2015";
%-- 10/30/2014 1:10 PM --%
ex1023_2
ex1023_3
gam_opt
ex1030_1
gam_opt

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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

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[lecture #10] 2014.11.20 Robust performance problem (1/3) (cont.)†

%-- 11/20/2014 1:11 PM --%
ex1023_2
ex1023_3
ex1030_1
ex1030_2
k
ex1113_1
ex1113_2
k
Delta_hat

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[lecture #11] 2014.11.27 relation between H infinity control and modern control theory†

%-- 11/27/2014 1:51 PM --%
ex1127
1/sqrt(2)
ex1127
2*(sqrt(2)-1)
mod1127
x0 = 0
h = 1
f = 1
zz
ww

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[lecture #12] 2014.12.4 relation between H infinity control and modern control theory (cont.); Speed control of two inertia system with servo motor (1/4)†

%-- 12/4/2014 1:28 PM --%
ex1127
mod1127
x0 = 0
h = 1
f = 1
ww
zz
h = 10
ww
zz
h = 50
zz
h
zz

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[lecture #13] 2014.12.11 Robust control design for a practical system : Speed control of two inertia system with servo motor (1/3)†

%-- 12/11/2014 1:24 PM --%
ex1211_1
frdata
frdata(:,1)
P1_jw
P1_g
ex1211_1

#ref(): File not found: "2014.12.11-1.jpg" at page "授業/制御工学特論2015"

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[lecture #14] 2014.12.18 Robust control design for a practical system : Speed control of two inertia system with servo motor (2/3)†

  1. design your controller(s) so that the system performance is improved compared with the design 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 sensitivity function
    3. time response of control experiment
  3. Why is the performance of your system improved(or unfortunately deteriorated)?
    • due date: 9th(Fri) Jan 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(Fri) Dec

participant list2014

%-- 12/18/2014 1:01 PM --%
freqresp
nominal
help fitfrd
weight
cont
help c2d
perf

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[lecture #15] 2014.12.25 Robust control design for a practical system : Speed control of two inertia system with servo motor (3/3)†

%-- 12/25/2014 12:58 PM --%
load cont.mat
who
K_opt
who
Kd
who
Ghat
load result.dat
plot(result(:,1), result(:,2))
plot(result(:,1), result(:,3))
who
bode(K_opt)
bode(Kd)
Kd1 = Kd
K_opt1 = K_opt
load cont.mat
bode(K_opt1, 'b', K_opt, 'r')
bode(Kd1, 'b', Kd, 'r')
Kd_tmp = c2d(K_opt1, 0.000001);
bode(Kd1, 'b', Kd, 'r', Kd_tmp, 'm')
clear all
load cont.mat
who
bode(K_opt)
K_example = K_opt;
load cont.mat
bode(K_example, 'b', K_opt, 'r')

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