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)†

%-- 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)
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
filewhiteboard #4

fileex0902.m

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[lecture #2] 2015.9.10 review of classical and modern control theory (2/3) with introduction of Matlab/Simulink†

  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
    • interactive system (no compilation, no variable difinition)
    • m file
  2. system representation: Transfer Function(TF) / State-Space Representation (SSR)
    • example: mass-spring-damper system
    • difinition of SSR
    • from SSR to TF
    • from TF to SSR: controllable canonical form
  3. open-loop characteristic
    • open-loop stability: poles and eigenvalues
    • Bode plot and frequency response fileex0910_1.m filemod0910_1.mdl
      • cut off frequency; DC gain; -40dB/dec; variation of c
      • relation between P(jw) and steady-state response
  4. closed-loop stability
    • Nyquist stability criterion (for L(s):stable)
    • Nyquist plot fileex0910_2.m filemod0910_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
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
filewhiteboard #4
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[] 2015.9.17 cancelled†

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[] 2015.9.25 no lecture (lectures for Monday are given)†

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[lecture #3] 2015.10.1 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 filelqr.pdf ≒ fileproof4.pdf (from B3「動的システムの解析と制御」)
  3. example filemod1001.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)
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3

... I'm sorry but all of equations are in the pdf file.

filewhiteboard #4
filewhiteboard #5
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[lecture #4] 2015.10.8 relation between LQR and H infinity control problem (1/2)†

  1. an equivalent problem
  2. a simple example of state-feedback H infinity control problem
  3. 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')
  4. definition of H infinity norm (SIMO)
  5. solve the problem by hand
  6. solve the problem by tool(hinfsyn) fileex1008.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
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
filewhiteboard #4
filewhiteboard #5
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[lecture #5] 2015.10.15 relation between LQR and H infinity control problem (2/2)†

  1. complete the table in simple example
  2. behavior of hinfsyn in fileex1008.m
  3. confirm the cost function J for both controllers by simulation filemod1015.mdl
  4. confirm the closed-loop H infinity norm for both controllers by simulation (common mdl file is available)
    • review: steady-state response (see photo 8 @ lec. #2)
    • how to construct the worst-case disturbance w(t) which maximizes L2 norm of z(t) ?
    • what is the worst-case disturbance in the simple example ?
  5. general case: filehinf.pdf includes the simple example as a special case
    • LQR filelqr.pdf is included as a special case where gamma -> infinity, non-zero x(0), and B2 -> B
%-- 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))
filewhiteboard #1
filewhiteboard #2
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[lecture #6] 2015.10.22 Mixed sensitivity problem 1/3†

  1. review filemap_v1.0_intro1.pdf and outline
  2. H infinity control problem (general form)
  3. reference tracking problem
  4. weighting function for sensitivity function
  5. design example fileex1022_1.m fileex1022_2.m
  6. the small gain theorem
    • proof: Nyquist stability criterion
  7. from performance optimization to robust stabilization
%-- 10/22/2015 2:06 PM --%
ex1022_1
eig(P)
ex1022_2
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3
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[lecture #7] 2015.10.29 Mixed sensitivity problem 2/3†

  1. review filemap_v1.0_intro2.pdf and outline
  2. an equivalent problem of robust stabilization for reference tracking problem
  3. uncertainty model and normalized uncertainty Delta
  4. robust stabilization problem and an equivalent problem
  5. practical example of plant with perturbation fileex1029_1.m
  6. how to determine the model fileex1029_2.m
  7. design example and simulation fileex1029_3.m filemod1029.mdl
%-- 10/29/2015 1:52 PM --%
ex1029_1
ex1029_2
ex1029_3
mod1029
c
c = 0.8
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3

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

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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

#ref(): File not found: "2014.10.30-1.jpg" at page "授業/制御工学特論2015"

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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

#ref(): File not found: "2014.11.13-1.jpg" at page "授業/制御工学特論2015"

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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

#ref(): File not found: "2014.11.20-1.jpg" at page "授業/制御工学特論2015"

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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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