授業

Advanced Automation 2020

[lecture #1] 2020.9.3 outline of the lecture, review of classical and modern control theory (1/3)

  • review : stabilization of SISO unstable plant by classical and modern control theory
    • transfer functions / differential equations
    • poles / eigenvalues
    • impulse response / initial value response
    • ...
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
k = 2
Tyr = feedback(P*k, 1)
step(Tyr)
k = 10
Tyr = feedback(P*k, 1)
step(Tyr)
k = 0.5
Tyr = feedback(P*k, 1)
step(Tyr)

[lecture #2] 2020.9.10 review of classical and modern control theory (2/3) with introduction of Matlab/Simulink

  1. minute paper
  2. 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 definition)
    • m file
  3. system representation: Transfer Function(TF) / State-Space Representation (SSR)
    • example: mass-spring-damper system
    • definition of SSR
    • from SSR to TF
    • from TF to SSR: controllable canonical form
  4. 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
  5. closed-loop stability
    • Nyquist stability criterion (for L(s):stable)
    • Nyquist plot fileex0910_2.m filemod0910_2.mdl
      • Gain Margin(GM); Phase Margin(PM)
%-- 20/09/10 12:48 --%
a = 1
b = 2
a + b
ex0910_1
P
ctrlpref
ex0910_1
P
P.num
P.num{:}
P.den{:}
ex0910_2
roots(P.den{:})
roots(L.den{:})
ex0910_2
K
ex0910_2
  • Q: マトラボの使いかたが分かるオススメの本やサイトが知りたいです
  • A: この授業では基本的な使い方ができれば十分のため、まずは今日示したpdfファイルを見てもらえればと思います。

[lecture #3] 2020.9.17 review of classical and modern control theory (3/3)

  1. LQR problem
    • controllability
    • cost function J >= 0
    • positive (semi-)definite matrices
    • solution of LQR problem
    • example fileex0917_1.m filemod0917_1.mdl
  2. ARE and quadratic equation
    • scalar case (solve by hand)
    • matrix case filelqr.pdffileproof4.pdf (from B3「動的システムの解析と制御」)
%-- 20/09/17 12:42 --%
ex0917_1
A
B
Uc
det(Uc)
R
M
M = [2, 1; 1, 1]
eig(M)
ex0917_1
C
D
F
R
Q
P
A
eig(A)
eig(P)
R\B'
R\B'*P
F
J
J(end)

[lecture #4] 2020.9.24 relation between LQR and H infinity control problem (1/2)

  • GOAL: to learn difference in concepts between LQR problem and H infinity control problem
  1. a simple example relating LQR and H infinity control problems
    • For given plant G \[ G = \left[\begin{array}{c|c:c} a & 1 & b \\ \hline \sqrt{q} & 0 & 0 \\ 0 & 0 & \sqrt{r} \\ \hdashline 1 & 0 & 0 \end{array} \right] = \left\{ \begin{array}{l} \dot x = ax + bu + w\\ z = \left[ \begin{array}{c} \sqrt{q} x \\ \sqrt{r} u \end{array}\right] \\ x = x \end{array}\right. \] with zero initial state value x(0) = 0, find a state-feedback controller \[ u = -f x \] such that \begin{eqnarray} (i) &&\quad \mbox{closed loop is stable} \\ (ii) &&\quad \mbox{minimize} \left\{\begin{array}{l} \| z \|_2 \mbox{ for } w(t) = \delta(t) \quad \mbox{(LQR)} \\ \| T_{zw} \|_\infty \mbox{($H_\infty$ control problem)}\end{array}\right. \end{eqnarray}
    • comparison of norms in (ii) (for a = -1, b = 1, q = 1, r = 1) \[ \begin{array}{|c||c|c|}\hline & \mbox{LQR}: f=-1+\sqrt{2} & \quad \quad H_\infty: f=1\quad\quad \\ \hline\hline J=\|z\|_2^2 & & \\ \hline \|T_{zw}\|_\infty & & \\ \hline \end{array} \]
  2. an alternative description to LQR problem
    1. J = (L2 norm of z)^2
    2. impulse resp. with zero initial value = initial value resp. with zero disturbance
  3. definition of H infinity norm (SISO)
    s = tf('s');
    G1 = 1/(s+1);
    bode(G1);
    norm(G1, 'inf')
    G2 = 1/(s^2 + 0.1*s + 1);
    bode(G2);
    norm(G2, 'inf')
  4. definition of H infinity norm (SIMO)
  5. solve the problem by hand
  6. solve the problem by tool(hinfsyn) fileex0924_1.m
%-- 20/09/24 12:36 --%
s = tf('s');
G1 = 1/(s+1);
bode(G1);
norm(G1, 'inf')
norm(G1)
norm(G1, inf)
bode(G2)
G2 = 1/(s^2 + 0.1*s + 1);
bode(G2);
G2
ctrlpref
bode(G2);
norm(G2, inf)
  • Q: 日本語での講義が良いです。
  • A: 反対意見が無かったため、次回から日本語で講義を行います。

[lecture #5] 2020.10.08 relation between LQR and H infinity control problem (2/2)

  1. complete the table in simple example
  2. confirm the cost function J for both controllers by simulation filemod1008.mdl
    • block diagram in the simulink model
    • how to approximate impulse disturbance with a step function
    • impulse disturbance resp. with zero initial condition = initial condition resp. with zero disturbance
  3. confirm the closed-loop H infinity norm for both controllers by simulation
    • H infinity norm = L2 induced norm
    • review: steady-state response; the worst-case disturbance w(t) which maximizes L2 norm of z(t) ?
    • how to make the worst-case disturbance w(t)? w(t) for the simple example ?
  4. general state-feedback case: filehinf.pdf
    • includes the simple example as a special case
    • LQR filelqr.pdf is included as a special case in which gamma -> infinity, w(t) = 0, B2 -> B, and non-zero x(0) are considered
%-- 20/10/08 12:36 --%
ex0924_1
mod1008
f
f = -1+sqrt(2)
x0
x0 = 0
h
h = 0.01
f
zz
zz(end)
sqrt(2)-1
h = 0.001
zz(end)
f
f = 1
zz(end)
h
x0
x0 = 1
zz(end)
x0
h
h = 0.00001
zz(end)
x0 = 0
zz(end)
format long e
zz(end)
x0
f
h
h = 10
zz(end)
sqrt(zz(end)/ww(end))
format short
sqrt(zz(end)/ww(end))
h
h = 100
sqrt(zz(end)/ww(end))
f
f = -1+sqrt(2)
sqrt(zz(end)/ww(end))

[lecture #6] 2020.10.15 Mixed sensitivity problem 1/3

  1. outline: filemap_v1.1_mixedsens1.pdf
    • sensitivity function S and complementary sensitivity function T
  2. H infinity control problem (general case)
    • with generalized plant G
    • including the state-feedback case
  3. reference tracking problem
    • how to translate the condition (ii) into one with H infinity norm ?
    • corresponding generalized plant G ?
    • introduction of weighting function for sensitivity function in (ii)
  4. design example fileex1015_1.m fileex1015_2.m
  5. the small gain theorem
    • proof: Nyquist stability criterion
%-- 20/10/15 12:54 --%
ex1015_1
P
eig(P)
K
help step
ex1015_2
who
K_hinf
eig(K.a)
eig(K_hinf.a)
help hinfsyn
ex1015_2
P

[lecture #7] 2020.10.22 Mixed sensitivity problem 2/3

  1. outline: from point to set filemap_v1.1_mixedsens2.pdf
  2. review: the small gain theorem ... robust stability = H infinity norm condition
  3. normalized uncertainty Delta
  4. uncertainty model
  5. how to determine P0 and WT
    • example: frequency response of plant with perturbation fileex1022_1.m
    • frequency response based procedure for P0 and WT fileex1022_2.m
  6. robust stabilization problem and equivalent problem
%-- 20/10/22 12:55 --%
ex1022_1
ex1022_2
ex1022_3
mod1022
c
c = 0.8
c = 2

[lecture #8] 2020.10.29 Mixed sensitivity problem 3/3

  • review: filemap_v1.1_mixedsens2.pdf (1)robust stabilization and (2)performance optimization
  • outline:
    1. how to design controllers considering both conditions in (1) and (2)
    2. gap between NP(nominal performance) and RP(robust performance)
  1. mixed sensitivity problem => (1) and (2) : proof
  2. generalized plant for mixed senstivity problem
  3. design example fileex1029_1.m minimize gamma by hand
  4. gamma iteration by bisection method fileex1029_2.m
  5. intro. to RP(problem of NP) fileex1029_3.m
%-- 20/10/29 13:00 --%
ex1029_1
ex1029_2
ex1029_3

[lecture #9] 2020.11.5 robust performance problem 1/3

  1. review
    • mixed sensitivity problem : N.P. but not R.P.
    • robust performance problem (R.P.) c.f. the last whiteboard, but can not be solved by tool
    • the small gain theorem
  2. an equivalent robust stability (R.S.) problem to R.P.
    • (i) introduction of a fictitious uncertainty Delta_p (for performance)
    • (ii) for 2-by-2 uncertainty block Delta hat which includes Delta and Delta_p
  3. definition of H infinity norm for general case (MIMO)
    • definition of singular values and the maximum singular value
      M = [1/sqrt(2), 1i; 1/sqrt(2), -1i]
      M'
      eig(M'*M)
      svd(M)
    • mini report #1 filereport1.pdf ... You will have a mini exam #1 related to this report
  4. proof of ||Delta hat||_inf <= 1
  5. design example: fileex1105_1.m
    • robust performance is achieved but large gap
    • non structured uncertainty is considered ... the design problem is too conservative
%-- 20/11/05 13:59 --%
M = [1/sqrt(2); 1i; 1/sqrt(2), -1i]
M'
eig(M'*M)
svd(M)
M = [1/sqrt(2), 1i; 1/sqrt(2), -1i]
M'
M'*M
eig(M'*M)
svd(M)
max(svd(M))
ex1105_1

[lecture #10] 2020.11.12 Robust performance problem (2/3)

  1. return of mini report #1
  2. SVD: singular value decomposition
    • definition
    • meaning of the largest singular value (a property and proof)
    • 2-norm of vectors (Euclidean norm)
    • SVD for 2-by-2 real matrix fileex1112_1.m
%-- 20/11/12 13:18 --%
M = [sqrt(2), -1i/sqrt(2); sqrt(2), 1i/sqrt(2)]
help svd
X
svd(M)
[U,Sigma,V] = svd(M);
Sigma
U
U'*U
V'*V
V*V'
ex1112_1
svd(M)
ex1112_1
help rand
ex1112_1
svd(M)

[lecture #11] 2020.11.19 Robust performance problem (3/3)

  1. review
    • H infinity norm (MIMO case)
    • R.S. problems for structured and unstructured uncertainty
  2. scaled H infinity control problem
  3. relation between three problems
  4. how to determine structure of scaling matrix
  5. design example fileex1119_1.m
    ex1105_1
    gam2 = gam_opt
    K2 = K_opt;
    ex1119_1
    gam_opt
  6. mini exam #1 (10 min.)
%-- 20/11/19 12:51 --%
ex1105_1

[lecture #12] 2020.11.26 Robust performance problem (3/3) (cont.), Control system design for practical system (1/3)

  1. return of mini exam #1
  2. review of scaling fileex1126_1.m
  3. mini report #2 filereport2.pdf
  4. introduction of a practical system: active noise control in duct
    • experimental setup
    • 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 filespk1.dat filespk2.dat
    • room 157 @ Dept. Mech. Bldg. 2
%-- 20/11/26 13:09 --%
ex1105_1
gam2 = gam_opt
K2 = K_opt;
ex1119_1
gam_opt
gam2
ex1126_1
gam2
gam3
close all
clear all
clf
ex1105_1; %
gam2 = gam_opt
K2 = K_opt;
ex1119_1
gam3 = gam_opt
K3 = K_opt;
Delta_tilde = [0, 1/sqrt(2); 0, -1/sqrt(2)]; % example of non-structured 
uncertainty
fprintf('***1st check for singular values of Delta_tilde:');
svd(Delta_tilde) % less than or equal to 1
fprintf('***2nd check for closed-loop stability of M2(Ghat(gamma2) and K2) and 
Delta_tilde:');
M2 = lft(mdiag(1,1/gam2,1)*Ghat, K2, 1, 1);
clp2 = lft(Delta_tilde, M2, 2, 2);
real(eig(clp2.a)) % closed-loop stability regardless the structure of 
uncertainty block
clp2
clp2.a
max(eig(clp2.a))
fprintf('***3rd check for closed-loop stability of M3(Ghat(gamma3) and K3) and 
Delta_tilde:');
M3 = lft(mdiag(1,1/gam3,1)*Ghat, K3, 1, 1);
clp3 = lft(Delta_tilde, M3, 2, 2);
real(eig(clp3.a)) % closed-loop instability by the non-structured uncertainty 
block
fprintf('***4th check for closed-loop H infinity norm of M3:');
norm(M3, 'inf') % larger than 1
fprintf('***5th check for closed-loop H infinity norm of M3 with scaling:');
W = mdiag(d_opt,1);
M3_d = W\M3*W;
norm(M3_d, 'inf') % less than 1
d_opt
ex1126_2
ctrlpref
ex1126_2
346/(4*1.62)

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

[lecture #13] 2019.12.5 Control system design for practical system (2/3)

  1. return of mini report #2; ... You will have a mini exam #2 related to this report next week
  2. review of the experimental system
    • closed-loop system of 2-by-2 plant G and controller K
    • closed-loop gain is desired to be minimized for constant speed operation
    • frequency response data of G can be used; how to handle modeling error of G ?
  3. design example (modeling error for Gyu is only considered for simplicity)
    • frequency response experiment data
      servo1.dat
      servo2.dat
    • determination of plant model(nominal plant and additive uncertainty weight)
      &ref(): File not found: "nominal.m" at page "授業/制御工学特論2020";
      &ref(): File not found: "weight.m" at page "授業/制御工学特論2020";
    • configuration of generalized plant and controller design by scaled H infinity control problem using one-dimensional search on the scaling d
      &ref(): File not found: "cont.m" at page "授業/制御工学特論2020";
    • comparison of closed-loop gain characteristics with and without control
      &ref(): File not found: "compare.m" at page "授業/制御工学特論2020";
    • result of control experiment
      result.dat
      &ref(): File not found: "perf.m" at page "授業/制御工学特論2020";
  4. final report and remote experimental system
    1. design your controller(s) so that the system performance is improved compared with the design example
    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 of control experiment
    3. Why is the performance of your system improved(or unfortunately deteriorated)?
    • due date: 6th(Mon) 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 controller.dat, controller_order.dat, and controller.mat at this page:participant list2019(download is also possible) not later than 25th(Wed) Dec
    • the system will be started until next lecture
    • You can send up to 10 controllers
    • control experimental results will be uploaded here
    • freqresp ... frequency response will be measured and uploaded everyday
  5. how to improve the performance ?
    • accuracy of the nominal(physical) model
    • weighting for robust stability
  6. specifications of the experimental system
    1. program sources for frequency response experiment
      • freqresp.h
      • freqresp_module.c
      • freqresp_app.c
      • format of servo1.dat (w is used instead of u for servo2.dat)
        1st column ... frequency (Hz)
        2nd column ... gain from u(Nm) to y(rad/s)
        3rd column ... phase (deg) from u to y
        4th column ... gain from u to z
        5th column ... phase (deg) from u to z
    2. program sources for control experiment
      • hinf.h
      • hinf_module.c
      • hinf_app.c
      • format of result.dat
        1st column: time (s)
        2nd column: y (rad/s)
        3rd column: z (rad/s)
        4th column: u (Nm)
        5th column: w (Nm)
    3. configuration of control experiment
      • disturbance signal w is specified as described in hinf.h and hinf_module.c:
        w = 0; // disturbance torque for driven motor                       
        if((t > 2)&&(t < 3)){
          w = RATED_TORQ * -0.15;
        }
        if((t > 4)&&(t < 5)){
          w = RATED_TORQ * -0.1 * sin(2*M_PI*5.0 * (t-4.0));
        }
        da_conv(torq_volt_conv_1(w), 1);
      • control signal u is limited as specified in hinf.h and hinf_module.c:
        #define U_MAX (RATED_TORQ / 3.0)
        
        if(u > U_MAX) u = U_MAX;
        if(u < -U_MAX) u = -U_MAX;
        u is generated by PI control for t < 1(s). Your designed controller is started at t = 1(s).
    4. calculation of rotational speed
      • The rotational speed is approximately calculated by using difference for one sampling period in hinf_module.c and freqresp_module.c like:
        theta_rad[0] = (double)read_theta(0) / (double)Pn212 * 2.0 * M_PI;
        theta_rad[1] = (double)read_theta(1) / (double)Pn212 * 2.0 * M_PI;
        y = (theta_rad[0] - theta_rad_before[0]) / msg->sampling_period;
        z = (theta_rad[1] - theta_rad_before[1]) / msg->sampling_period;
        theta_rad_before[0] = theta_rad[0];
        theta_rad_before[1] = theta_rad[1];
        where the sampling period is given as 0.25 ms.
%-- 2019/12/05 13:03 --%
pwd
nominal
weight
cont
compare
ctrlpref
compare
perf
  • Q: 実機パラメータのイナーシャはどうやって試験をして導出しているか。
  • A: 周波数応答実験の結果に物理モデルが近くなるように調整しました。
  • Q: C_S, C_L はダンパなのか。C_S はどこの部分のダンパというかたちになるのか。
  • A: C_S は軸の捩りに伴う減衰係数です(これが0の場合、二つの慣性の捩り振動は減衰することなく永久に持続する)。C_L は従動側慣性の回転に伴う減衰係数です(これが0の場合、二慣性系全体は与えられた初速に応じて永久に回転し続ける)。

#ref(): File not found: "2019.12.05-1.jpg" at page "授業/制御工学特論2020"

[lecture #14] 2019.12.12 Control system design for practical system (3/3)

  • review & supplemental explanations
    • final report
      • control objective is to suppress speed fluctuation of the driven-side motor not the drive-side motor (a large vibration at the drive-side motor is allowed)
      • unstable controller is admitted
      • no strict control objective is given ( there is a freedom to define what is good performance; a frequency dependent weighting function can be introduced to evaluate the performance )
      • c2d() is used to discretize the resultant continuous-time controller
    • web based remote experiment system
      • now you can login after registration
      • room temperature is displayed and stored in temp.txt (Bosch Sensortec BME280 is used)
  • preparation of your own controller(s) by using the remote experiment system
  • mini exam #2 (14:00〜)

#ref(): File not found: "2019.12.12-1.jpg" at page "授業/制御工学特論2020"

[lecture #15] 2019.12.19 Control system design for practical system (cont.)

  • return of mini exam #2
  • preparation of your own controller(s)
    • use design example #3 for comparison with your design
    • discuss relationship between the required figures (a), (b), and (c)
    • driving and driven torque are set to zero when driven-motor speed z or driving-motor speed y exceeds a limit as in hinf.h:
      #define SPEED_MAX 60.0 // driving torque becomes zero when rotational speed is out of range from -60.0 rad/s to 60.0 rad/s in order to avoid shutting down the driving motor by the maximum speed excess alarm (2018.12.19)
      and hinf_module.c:
             int flag_speed_excess = 0; // 2018.12.19
      
      	    if(flag_speed_excess == 1) w = 0; // 2018.12.20
      	    da_conv(torq_volt_conv_1(w), 1);
      
      	    if((fabs(z) > SPEED_MAX) || (fabs(y) > SPEED_MAX)) flag_speed_excess = 1; // 2018.12.20 further modified after today's lecture
		if(flag_speed_excess == 1) u = 0; // 2018.12.19
                
#ifndef NO_CONTROL
                da_conv(torq_volt_conv_0(u), 0);
#endif
  • questionnaires
    • to university
    • for web-based experimental environment
%-- 2019/12/19 13:31 --%
nominal
weight
cont
compare
perf
nominal
weight
cont
compare
perf
weight

#ref(): File not found: "2019.12.19-1.jpg" at page "授業/制御工学特論2020"


添付ファイル: file2020.11.26-1.jpg 1件 [詳細] file2020.11.26-2.jpg 1件 [詳細] file2020.11.26-3.jpg 1件 [詳細] fileex1126_2.m 3件 [詳細] filespk1.dat 1件 [詳細] filespk2.dat 1件 [詳細] filephoto1.jpg 3件 [詳細] filephoto2.jpg 3件 [詳細] fileex1126_1.m 1件 [詳細] filereport2.pdf 5件 [詳細] file2020.11.19-1.jpg 7件 [詳細] file2020.11.19-2.jpg 4件 [詳細] fileex1119_1.m 1件 [詳細] file2020.11.12-1.jpg 13件 [詳細] file2020.11.12-2.jpg 8件 [詳細] fileex1112_1.m 5件 [詳細] file2020.11.05-1.jpg 22件 [詳細] file2020.11.05-2.jpg 21件 [詳細] file2020.11.05-3.jpg 29件 [詳細] fileex1105_1.m 9件 [詳細] filereport1.pdf 32件 [詳細] file2020.10.29-1.jpg 11件 [詳細] file2020.10.29-2.jpg 13件 [詳細] fileex1029_1.m 7件 [詳細] fileex1029_2.m 7件 [詳細] fileex1029_3.m 7件 [詳細] file2020.10.22-1.jpg 10件 [詳細] file2020.10.22-2.jpg 12件 [詳細] file2020.10.22-3.jpg 10件 [詳細] file2020.10.22-4.jpg 11件 [詳細] fileex1022_1.m 8件 [詳細] fileex1022_2.m 6件 [詳細] fileex1022_3.m 6件 [詳細] filemod1022.mdl 6件 [詳細] file2020.10.15-1.jpg 18件 [詳細] file2020.10.15-2.jpg 20件 [詳細] file2020.10.15-3.jpg 19件 [詳細] fileex1015_1.m 9件 [詳細] fileex1015_2.m 10件 [詳細] file2020.10.08-1.jpg 19件 [詳細] file2020.10.08-2.jpg 18件 [詳細] filemod1008.mdl 12件 [詳細] file2020.09.24-1.jpg 38件 [詳細] file2020.09.24-2.jpg 31件 [詳細] file2020.09.24-3.jpg 37件 [詳細] file2020.09.24-4.jpg 38件 [詳細] fileex0924_1.m 17件 [詳細] file2020.09.17-1.jpg 36件 [詳細] file2020.09.17-2.jpg 29件 [詳細] file2020.09.17-3.jpg 31件 [詳細] file2020.09.17-4.jpg 36件 [詳細] fileex0917_1.m 17件 [詳細] filemod0917_1.mdl 20件 [詳細] file2020.09.10-1.jpg 28件 [詳細] file2020.09.10-3.jpg 27件 [詳細] file2020.09.10-2.jpg 26件 [詳細] filemod0910_1.mdl 21件 [詳細] filemod0910_2.mdl 22件 [詳細] fileex0910_1.m 25件 [詳細] fileex0910_2.m 21件 [詳細] file2020.09.03-1.jpg 49件 [詳細] file2020.09.03-2.jpg 31件 [詳細] file2020.09.03-3.jpg 29件 [詳細]

トップ   編集 凍結 差分 バックアップ 添付 複製 名前変更 リロード   新規 一覧 単語検索 最終更新   ヘルプ   最終更新のRSS
Last-modified: 2020-11-26 (木) 15:21:04 (6h)