授業

Advanced Automation†

↑

[lecture #1] 2016.9.1 outline of the lecture, review of classical and modern control theory (1/3)†

%-- 2016/09/01 14:26 --%
s = tf('s')
k = 2
alpha = -1
Tyr = k/(s+alpha+k)
step(Tyr, 'b')
k = 100
Tyr2 = k/(s+alpha+k)
step(Tyr, 'b', Try2, 'r--')
step(Tyr, 'b', Tyr2, 'r--')
filewhiteboard #1
filewhiteboard #2
filewhiteboard #3

!!! the remaining page is under construction (the contents below are from 2015) !!!

↑

[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 &ref(): File not found: "text_fixed.pdf" at page "授業/制御工学特論2016"; 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 &ref(): File not found: "ex0910_1.m" at page "授業/制御工学特論2016"; &ref(): File not found: "mod0910_1.mdl" at page "授業/制御工学特論2016";
      • 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 &ref(): File not found: "ex0910_2.m" at page "授業/制御工学特論2016"; &ref(): File not found: "mod0910_2.mdl" at page "授業/制御工学特論2016";
      • 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

#ref(): File not found: "2015.09.10-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.09.10-2.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.09.10-3.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.09.10-4.jpg" at page "授業/制御工学特論2016"

↑

[] 2015.9.17 cancelled†

↑

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

↑

[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 &ref(): File not found: "lqr.pdf" at page "授業/制御工学特論2016"; ≒ &ref(): File not found: "proof4.pdf" at page "授業/制御工学特論2016"; (from B3「動的システムの解析と制御」)
  3. example &ref(): File not found: "mod1001.mdl" at page "授業/制御工学特論2016"; 
    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)

#ref(): File not found: "2015.10.01-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.01-2.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.01-3.jpg" at page "授業/制御工学特論2016"

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

#ref(): File not found: "2015.10.01-4.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.01-5.jpg" at page "授業/制御工学特論2016"

↑

[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) &ref(): File not found: "ex1008.m" at page "授業/制御工学特論2016";
%-- 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

#ref(): File not found: "2015.10.08-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.08-2.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.08-3.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.08-4.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.08-5.jpg" at page "授業/制御工学特論2016"

↑

[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 &ref(): File not found: "ex1008.m" at page "授業/制御工学特論2016";
  3. confirm the cost function J for both controllers by simulation &ref(): File not found: "mod1015.mdl" at page "授業/制御工学特論2016";
  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: &ref(): File not found: "hinf.pdf" at page "授業/制御工学特論2016"; includes the simple example as a special case
    • LQR &ref(): File not found: "lqr.pdf" at page "授業/制御工学特論2016"; 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))

#ref(): File not found: "2015.10.15-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.15-2.jpg" at page "授業/制御工学特論2016"

↑

[lecture #6] 2015.10.22 Mixed sensitivity problem 1/3†

  1. review &ref(): File not found: "map_v1.0_intro1.pdf" at page "授業/制御工学特論2016"; and outline
  2. H infinity control problem (general form)
  3. reference tracking problem
  4. weighting function for sensitivity function
  5. design example &ref(): File not found: "ex1022_1.m" at page "授業/制御工学特論2016"; &ref(): File not found: "ex1022_2.m" at page "授業/制御工学特論2016";
  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

#ref(): File not found: "2015.10.22-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.22-2.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.22-3.jpg" at page "授業/制御工学特論2016"

↑

[lecture #7] 2015.10.29 Mixed sensitivity problem 2/3†

  1. review &ref(): File not found: "map_v1.0_intro2.pdf" at page "授業/制御工学特論2016"; 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 &ref(): File not found: "ex1029_1.m" at page "授業/制御工学特論2016";
  6. how to determine the model &ref(): File not found: "ex1029_2.m" at page "授業/制御工学特論2016";
  7. design example and simulation &ref(): File not found: "ex1029_3.m" at page "授業/制御工学特論2016"; &ref(): File not found: "mod1029.mdl" at page "授業/制御工学特論2016";
%-- 10/29/2015 1:52 PM --%
ex1029_1
ex1029_2
ex1029_3
mod1029
c
c = 0.8

#ref(): File not found: "2015.10.29-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.29-2.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.10.29-3.jpg" at page "授業/制御工学特論2016"

↑

[lecture #8] 2015.11.5 Mixed sensitivity problem 3/3†

  1. review : (1)robust stabilization and (2)performance optimization
  2. mixed sensitivity problem : a sufficient condition for (1) and (2)
    • proof by definition of H infinity norm
  3. construction of the generalized plant
  4. design example &ref(): File not found: "ex1105_1.m" at page "授業/制御工学特論2016";
  5. gamma iteration by bisection method &ref(): File not found: "ex1105_2.m" at page "授業/制御工学特論2016";
  6. a problem of the mixed sensitivity problem: nominal performance and robust performance &ref(): File not found: "ex1105_3.m" at page "授業/制御工学特論2016";
  7. introduction of robust performance problem
%-- 11/5/2015 1:37 PM --%
ex1105_1
ex1105_2
gam
ex1105_2
WT
ex1105_2
ex1105_3
ex1105_2
ex1105_3

#ref(): File not found: "2015.11.05-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.11.05-2.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.11.05-3.jpg" at page "授業/制御工学特論2016"

↑

[lecture #9] 2015.11.12 robust performance problem 1/3†

  1. review: robust performance problem
  2. an equivalent robust stability problem
  3. definition of H infinity norm for general case (MIMO)
  4. definition of (maximum) singular value 
    M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
  5. mini report #1 filereport1.pdf
    • write by hand
    • due date and place of submission -> see schedule2015
    • check if your answer is correct or not before submission by using Matlab
    • You will have a mini exam #1 related to this report
  6. SVD: singular value decomposition
    • definition 
      [U,S,V] = svd(M)
M = [j, 0; -j, 1; 2, 3]
    • unitary matrix and 2 norm of vectors
    • a property of SVD: input-output interpretation
    • illustrative example: rotation matrix &ref(): File not found: "ex1112_1.m" at page "授業/制御工学特論2016";
  7. H infinity norm of Delta hat
%-- 11/12/2015 1:01 PM --%
M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
M = [j, 0; -j, 1]
M'
eig(M'*M)
svd(M)
(3+sqrt(5))/2
sqrt((3+sqrt(5))/2)
help svd
[U,S,V] = svd(M)
U'*U
M = [j, 0; -j, 1; 2, 3]
[U,S,V] = svd(M)
V'*V
ex1112_1

#ref(): File not found: "2015.11.12-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.11.12-2.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.11.12-3.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.11.12-4.jpg" at page "授業/制御工学特論2016"

↑

[lecture #10] 2015.11.19 Robust performance problem (2/3)†

  1. return of mini report #1
  2. review and outline: robust stability problem for Delta hat and its equivalent problem(?)
  3. signal vector's size is not restricted in H infinity control problem and small gain theorem
  4. H infinity norm of Delta hat
  5. design example: robust performance is achieved &ref(): File not found: "ex1119_1.m" at page "授業/制御工学特論2016";
  6. non structured uncertainty is considered ... the design problem is too conservative
%-- 11/19/2015 1:23 PM --%
doc hinfsyn
ex1105_2
ex1105_3
gam_opt
ex1119_1
gam_opt
svd([1/sqrt(2), 0; 1/sqrt(2), 0])

#ref(): File not found: "2015.11.19-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.11.19-2.jpg" at page "授業/制御工学特論2016"

↑

[lecture #11] 2015.11.26 Robust performance problem (3/3)†

  1. review
    • robust performance problem with Delta hat and conservative design problem with Delta tilde
    • inclusion relation between two uncertain sets
  2. introduction of the scaled H infinity control problem
  3. how to determine structure of scaling matrix
  4. design example moved to next lecture 
    % less conservative design 
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;

    #ref(): File not found: "ex1126_1.m" at page "授業/制御工学特論2016"

  5. effect of scaling matrix moved to next lecture

    #ref(): File not found: "ex1126_2.m" at page "授業/制御工学特論2016"

  6. mini exam #1

#ref(): File not found: "2015.11.26-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.11.26-2.jpg" at page "授業/制御工学特論2016"

↑

[lecture #12] 2015.12.3 Robust performance problem (3/3) (cont.), Control system design for practical system (1/3)†

  1. return of mini exam #1; schedule of mini report #2 and exam #2
  2. review of the scaled H infinity control problem
  3. comments on mu-synthesis prolem
  4. design example (moved from the previous lecture) 
    % less conservative design 
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;

    #ref(): File not found: "ex1126_1.m" at page "授業/制御工学特論2016"

  5. effect of scaling matrix (moved from the previous lecture)

    #ref(): File not found: "ex1126_2.m" at page "授業/制御工学特論2016"

  6. mini report #2 filereport2.pdf
    • write by hand
    • due date and place of submission -> see schedule2015
    • check if your answer is correct or not before submission by using Matlab
    • You will have a mini exam #2 related to this report
  7. controller design for practical system: active noise control in duct
    • introduction of experimental setup

      #ref(): File not found: "exp_apparatus1.jpg" at page "授業/制御工学特論2016"

      #ref(): File not found: "exp_apparatus2.jpg" at page "授業/制御工学特論2016"

    • objective of control system: to drive control loudspeaker by generating proper driving signal u using reference microphone output y such that the error microphone's output z is attenuated against the disturbance input w
    • frequency response experiment

      #ref(): File not found: "ex1203_1.m" at page "授業/制御工学特論2016"

      filespk1.dat
      filespk2.dat
%-- 12/3/2015 1:27 PM --%
ex1105_2
ex1105_3
ex1119_1
gam_opt0 = gam_opt;
K_opt0 = K_opt;
who
gam_opt0
ex1126_1
gam_opt
d_opt
ex1126_2
ex1203_1

#ref(): File not found: "2015.12.03-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.12.03-2.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.12.03-3.jpg" at page "授業/制御工学特論2016"

↑

[lecture #13] 2015.12.10 Control system design for practical system (2/3)†

  1. return of mini report #2
  2. review of the experimental apparatus and frequency response experiment
  3. design example
    • determination of plant model(nominal plant and additive uncertainty weight)
      filenominal.m
      filesubspace.m ... replacement of n4sid in System Identification Toolbox (not provided in IPC)
      fileweight.m
    • configuration of generalized plant and controller design by scaled H infinity control problem using one-dimensional search on the scaling d
      filecont.m
    • comparison of closed-loop gain characteristics with and without control
      filecompare.m
    • result of control experiment
      fileresult.dat
      filecompare_result.m
  4. room 157 @ Dept. Mech. Bldg.2
ex1203_1
ctrlpref
ex1203_1
346/3.6
ex1203_1
nominal
weight
cont
nominal
compare
compare_result

#ref(): File not found: "2015.12.10-1.jpg" at page "授業/制御工学特論2016"

#ref(): File not found: "2015.12.10-2.jpg" at page "授業/制御工学特論2016"

↑

[lecture #14] 2015.12.17 Control system design for practical system (3/3)†

  1. final report
    1. design your controller(s) so that the system performance is improved compared with the design example introduced in the previous lecture
    2. Draw the following figures and explain the difference between two control systems (your controller and the design example):
      1. bode diagram of controllers
      2. gain characteristic of closed-loop system from w to z
      3. time response and frequency spectrum (PSD) of control experiment
    3. Why is the performance of your system improved(or unfortunately deteriorated)?
  2. how to improve the performance ?
    • order of the nominal plant
    • weighting for robust stability
  3. detailed explanation of m-files in the previous lecture
  4. specifications of the experimental system
    1. experimental equipments
    • loudspeakers: AURA SOUND NSW2-326-8A (2inch, 15W)
    • pressure sensors: NAGANO KEIKI KP15
    • A/D, D/A converters: CONTEC AD12-16(PCI), DA12-4(PCI)
    • PC: Dell Dimension 1100
    • OS: Linux kernel 2.4.22 / Real Time Linux 3.2-pre3
    1. program sources for frequency response experiment
    • freqresp.h
    • freqresp_module.c
    • freqresp_app.c
    • format of spk1.dat (u is used instead of w for spk2.dat)
      • 1st column ... frequency (Hz)
      • 2nd column ... gain from w(V) to y(V) (signal's unit is voltage (V))
      • 3rd column ... phase from w to y
      • 4th column ... gain from w to z
      • 5th column ... phase from w to z
    1. program sources for control experiment
    • hinf.h
    • hinf_module.c
    • hinf_app.c
    • format of result.dat
      • 1st column: time (s)
      • 2nd column: z (V)
      • 3rd column: y (V)
      • 4th column: u (V)
      • 5th column: w (V)
    1. configuration of control experiment
    • disturbance signal w is specified as described in hinf.h and hinf_module.c: 
      #define AMP 3.0 // amplitude for disturbance
#define DIST_INTERVAL 5 // interval step for updating w

count_dist++;
if(count_dist >= DIST_INTERVAL){
  w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // uniform random number in [-AMP, AMP]
  count_dist = 0;
}

da_conv(V_OFFSET + w, 0); // D/A output to noise source
      w is updated with 1ms period (sampling period 0.2ms times DIST_INTERVAL 5)
    • control signal u is limited to [-4, 4] as specified in hinf.h and hinf_module.c: 
      #define U_MAX 4.00

if(u > U_MAX) u = U_MAX;
if(u < -U_MAX) u = -U_MAX;
      u is set to 0 for t < 10(s). (controller is operated for 10 <= t < 15.)
    • a high pass filter with cut-off frequency are used to cut DC components in z and y as described in hinf.h and hinf_module.c 
      // HPF(1 rad/s) to cut DC in z and y
#define AF 9.9980001999866674e-01  
#define BF 1.9998000133326669e-04
#define CF -1.0000000000000000e+00
#define DF 1.0000000000000000e+00

ad_conv(&yz); // A/D input

// HPFs
yf = CF*xf_y + DF*yz[0];
xf_y = AF*xf_y + BF*yz[0];
zf = CF*xf_z + DF*yz[1];
xf_z = AF*xf_z + BF*yz[1];
      experiment directory
  5. mini exam #2
%-- 12/17/2015 1:36 PM --%
help bodemag

#ref(): File not found: "2015.12.17-1.jpg" at page "授業/制御工学特論2016"

↑

[lecture #15] 2015.12.24 Control system design for practical system (cont.)†

%-- 12/24/2015 1:11 PM --%
compare

#ref(): File not found: "2015.12.24-1.jpg" at page "授業/制御工学特論2016"

トップ   新規 一覧 検索 最終更新   ヘルプ   最終更新のRSS