授業/応用数学2B_2012 の履歴(No.1)

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• 授業/応用数学2B_2012 へ行く。
  • 1 (2012-10-03 (水) 17:46:34)
  • 2 (2012-10-03 (水) 18:29:22)
  • 3 (2012-10-10 (水) 17:01:35)
  • 4 (2012-10-17 (水) 16:34:48)
  • 5 (2012-10-24 (水) 18:29:24)
  • 6 (2012-10-31 (水) 14:00:38)
  • 7 (2012-11-07 (水) 14:58:12)
  • 8 (2012-11-20 (火) 19:09:54)
  • 9 (2012-12-05 (水) 13:22:02)
  • 10 (2012-12-19 (水) 13:11:08)
  • 11 (2013-01-09 (水) 15:41:00)

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Advanced Automation†

[lecture #1] 2012.9.6 review of classical control theory†

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[lecture #2] 2012.9.20 given by Prof. Kimura†

[lecture #3] 2012.9.27 CACSD introduction with review of classical and modern control theory†

1. introduction of Matlab and Simulink
   • 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. relationship between TF and SSR

   #ref(): File not found: "ex0927_1.m" at page "授業/応用数学2B_2012"

3. open-loop stability can be checked by poles of TF and eigenvalues of A-matrix in SSR

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   #ref(): File not found: "mod0927_2.mdl" at page "授業/応用数学2B_2012"

4. closed-loop stability
   • graphical test by Nyquist stability criterion and Bode plot with GM(gain margin) and PM(phase margin)

     #ref(): File not found: "ex0927_3.m" at page "授業/応用数学2B_2012"

     #ref(): File not found: "mod0927_3.mdl" at page "授業/応用数学2B_2012"

   • numerical test by poles of closed-loop TF and eigenvalues of A-matrix in closed-loop SSR

     #ref(): File not found: "ex0927_4.m" at page "授業/応用数学2B_2012"

   • example: stabilization of unstable system

     #ref(): File not found: "ex0927_5.m" at page "授業/応用数学2B_2012"

     #ref(): File not found: "mod0927_5.mdl" at page "授業/応用数学2B_2012"

%-- 9/27/2012 1:07 PM --%
a = 1
pwd
G1 = 1/(s+1)
ex0927_1
G1 = 1/(s+1)
A
ex0927_2
G2_tf
G2_tf.den
G2_tf.den{:}
G2_tf.num{:}
eig(G2_tf)
G2_ss.a
eig(G2_ss.a)
G2_ss
G2_ss.a
G2_ss.b
ex0927_3
nyquist(1.5*G3_tf)
nyquist(-1.5*G3_tf)
bode(-1.5*G3_tf)
bode(1.5*G3_tf)
K
K = 1
ex0927_4
ex0927_5
A
eig(A)
K = 0
K=1
ex0927_5 

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! the followings are under construction !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[lecture #4] 2011.10.13 Intro. to robust control theory (H infinity control theory)†

• H infinity norm
  1. robust stabilization
  2. performance optimization

• Example of performance optimization problem : disturbance attenuation problem
  • controller design with trial and error

    #ref(): File not found: "ex1013_1.m" at page "授業/応用数学2B_2012"

  • controller design with H infinity control theory

    #ref(): File not found: "ex1013_2.m" at page "授業/応用数学2B_2012"

s = tf('s');
G = 1/(s+1);
norm(G, inf)
G = 1/(s^2+0.1*s+1);
norm(G, inf)
bode(G)
ex1013_1
ex1013_2

#ref(): File not found: "2011.10.13-1.jpg" at page "授業/応用数学2B_2012"

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[lecture #5] 2011.10.7 Introduction to Robust Control (cont.)†

• H infinity norm
  1. robust stabilization
  2. performance optimization

1. a class used to represent plant uncertainty and/or perturbation ... H infinity norm
2. small gain theorem
   • connects the closed-loop H infinity norm and robust stability condition
   • sketch proof ... Nyquist stability criterion
3. practical example of robust stabilization problem
   • unstable plant with perturbation

     #ref(): File not found: "ex1020_1.m" at page "授業/応用数学2B_2012"

   • multiplicative uncertainty model

     #ref(): File not found: "ex1020_2.m" at page "授業/応用数学2B_2012"

   • H infinity controller design

     #ref(): File not found: "ex1020_3.m" at page "授業/応用数学2B_2012"

     #ref(): File not found: "mod1020.mdl" at page "授業/応用数学2B_2012"

%-- 10/20/2011 1:41 PM --%
ex1020_1
ex1020_2
ex1020_3
mod1020
c
c = 0.8

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[lecture #6] 2011.10.27 norm (vector space, normed linear space), singular value, mixed sensitivity problem†

• H infinity norm ... {robust stabilization, performance improvement}
• norm

  #ref(): File not found: "norm.pdf" at page "授業/応用数学2B_2012"

  • size of {number, signal, system}
  • defined on vector space (linear space)
    • ---> normed linear space
    • ... optimization
• H infinity norm : scalar (SISO) ---> matrix (MIMO)
  • absolute value ---> maximum singular value

    #ref(): File not found: "norm2.pdf" at page "授業/応用数学2B_2012"

  • robust stabilization or performance optimization problem ---> mixed sensitivity problem
• reference
  • Feedback control theory / John C. Doyle, Bruce A. Francis and Allen R. Tannenbaum

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    • 2. Norms for signals and systems
    • 3. Basic concepts
    • 4. Uncertainty and robustness ... p.62 Eq.(4.10) H infinity norm condition in mixed sensitivity problem
  • Robust and optimal control / Kemin Zhou, John C. Doyle and Keith Glover

    #ref(): File not found: "robust_and_optimal_control.jpg" at page "授業/応用数学2B_2012"

    • 2. Linear algebra (2.7 Vector norms and matrix norms, 2.8 Singular value decomposition)
    • 4. Performance specifications (4.1 Normed spaces, 4.3 Hardy spaces H_2 and H_infinity)
    • 9. Model uncertainty and robustness
    • 16. H infinity control: simple case
    • 17. H infinity control: general case

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[lecture #8] 2011.11.10 state space representation of connected system, state space representation of generalized plant for various control problem, mixed sensitivity problem (by Prof. Kimura)†

see Prof. Kimura's page

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[lecture #9] 2011.12.1 robust control design example: Robust Control System Synthesis for Pneumatic Systems (given by Prof. Kimura)†

[lecture #10] 2011.12.8 Speed control of two inertia system with servo motor (1/3)†

• Experimental setup

  #ref(): File not found: "apparatus.ppt" at page "授業/応用数学2B_2012"

  #ref(): File not found: "setup.pdf" at page "授業/応用数学2B_2012"

  • Objective for control system design
  1. speed tracking
  2. robust against inertia-load variation
• Modelling
  • frequency response experiment results:

    #ref(): File not found: "frdata_small_offset0.dat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "frdata_small_offset5.dat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "frdata_small_offset10.dat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "frdata_large_offset0.dat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "frdata_large_offset5.dat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "frdata_large_offset10.dat" at page "授業/応用数学2B_2012"

  • example of m-file

    #ref(): File not found: "freqresp.m" at page "授業/応用数学2B_2012"

    #ref(): File not found: "nominal.m" at page "授業/応用数学2B_2012"

    #ref(): File not found: "weight.m" at page "授業/応用数学2B_2012"

    >> freqresp
    >> nominal
    >> weight

• Controller design (mixed sensitivity problem)
  • example of m-file

    #ref(): File not found: "cont.m" at page "授業/応用数学2B_2012"

    >> cont

  • design example

    #ref(): File not found: "cont.dat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "cont_order.dat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "cont.mat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "result_small.dat" at page "授業/応用数学2B_2012"

    #ref(): File not found: "result_large.dat" at page "授業/応用数学2B_2012"

• report

1. design your controller(s) so that the system performance is improved compared with the given 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 closed-loop systems
   3. time response of control experiment
3. Why is the performance of your system improved(or unfortunately decreased)?
   • due date: 28th(Wed) 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 23th(Fri) Dec

• program sources for frequency response experiment

  #ref(): File not found: "freqresp.h" at page "授業/応用数学2B_2012"

  #ref(): File not found: "freqresp_module.c" at page "授業/応用数学2B_2012"

  #ref(): File not found: "freqresp_app.c" at page "授業/応用数学2B_2012"

  • format of frdata.dat file
    • 1st column: frequency (Hz)
    • 2nd column: gain
    • 3rd column: phase (deg)
• program sources for control experiment

  #ref(): File not found: "hinf.h" at page "授業/応用数学2B_2012"

  #ref(): File not found: "hinf_module.c" at page "授業/応用数学2B_2012"

  #ref(): File not found: "hinf_app.c" at page "授業/応用数学2B_2012"

  • format of result.dat file
    • 1st column: time (s)
    • 2nd column: motor speed (rad/s)
    • 3rd column: motor torque (Nm)
    • 4th column: reference speed (rad/s)
• configuration of control experiment
  • reference signal is generated as described in hinf_module.c:

    if((t > 3)&&(t < 7)){
      r = 10.0;
    }else{
      r = 5;
    }

  • two inertia-load discs (small and large) are used

• Difficulties of our plant: As motor speed is approximately calculated by using difference for one sampling period in hinf_module.c like

  thetaM_rad = (double)read_theta(0) / (double)Pn212 * 2 * M_PI;
  speedM_rad = (thetaM_rad - thetaM_rad_before) / msg->sampling_period;
  thetaM_rad_before = thetaM_rad

  sampling period should not become too small. On the other hand, sampling period should be chosen as small as possible so that desiged continuous-time controller could be closely implemented by its descretized version. Therefore, we have a dilemma to control our plant. The sampling period 0.25 msec was chosen by traial and error so that noise in measured speed is not too large. The gain in high frequency range of continuous-time controller should be small enough for discretization.

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%-- 12/8/2011 1:22 PM --%
freqresp
nominal
weight
load result_small.dat
result_small
plot(result_small(:,1),result_small(:,2))
plot(result_small(:,1),result_small(:,2), 'b', result_small(:,1),result_small (:,4),'r')

[lecture #11] 2011.12.15 Speed control of two inertia system with servo motor (2/3)†

• explanation of design example (cont. from the previous lecture)
• preparation of your own controller(s)

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#ref(): File not found: "2011.12.15-2.jpg" at page "授業/応用数学2B_2012"

%-- 12/15/2011 1:06 PM --%
cont
help hinfsyn
K
Ktf = tf(K)
bode(K, 'r', Ktf, 'b')
legend('ss', 'tf')
help fitsys

[lecture #12] 2011.12.22 Speed control of two inertia system with servo motor (3/3)†

• preparation of your own controller(s)

participant list2011

supplementary lectures will be given by Prof. Kimura.†

• example for changing ws = 2 pi 0.1 -> 2 pi 0.01

  #ref(): The style ref(filename,pagename) is ambiguous and become obsolete. Please try ref(pagename/filename)

• example to compare time responces

  #ref(): File not found: "compare_result.m" at page "授業/応用数学2B_2012"

%-- 12/22/2011 1:20 PM --%
nominal
ctrlpref
nominal
weight
cont
ks
cont
compare_result

related links†

• Prof. Kimura's page