授業/制御工学特論2009

Advanced Automation†

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2009.9.10 CACSD introduction†

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
review of clasical control theory
1. transfer function
2. state-space representation
3. Bode diagram
4. characteristics of 2nd-order system
5. Nyquist stability criterion
6. gain margin and phase margin
7. characteristic polynomial
8. poles and impulse response

ex.m

mod1.mdl

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2009.9.17 Intro. to Robust Control†

robust stability
performance improvement
H infinity norm

2009.10.15 norm, vector space, normed linear space†

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2009.10.22 eigenvalue, eigenvector, singular value decomposition†

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2009.11.5 state space representation of connected system, state space representation of generalized plant for various control problem, mixed sensitivity problem†

ex61.m

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2009.11.19†

rub2.m

... generalized plant is described by hand

rub.m

... function sysic() is used to obtain generalized plant

simulink model

simurub.mdl

simulink model for noise problem

simurub_noise.mdl

execution example for Matlab

rub2
size(Ap)
size(Ap,1)
size(Ap,2)
M = zeros(3,2)
size(M,1)
size(M,2)
gopt
simulink
simout
plot(t, y)
plot(t, y, 'r', t, r, 'b')
figure(3);
bodemag(S, 'r', gopt/WS, 'r--', T, 'b', gopt/WT, 'b--');
ctrlpref
bodemag(S, 'r', gopt/WS, 'r--', T, 'b', gopt/WT, 'b--');
grid on
ctrlpref
figure(6);
plot(t, y, 'r', t, r, 'b')
grid on
figure(7);
bodemag(P0, 'r', K, 'b')
grid on
rub2
grid on
gopt
plot(t, y, 'r', t, r, 'b')
grid on
simulink
plot(t, y, 'r', t, r, 'b')
plot(t, y, 'r', t, n, 'b')
plot(t, y*10, 'r', t, n, 'b')
plot(t, y, 'r', t, n, 'b')
bodemag(P0, 'r', K, 'b')
grid on
bode(P0, 'r', K, 'b')
grid on
bode(P0, 'r', K, 'b', P0*K, 'm')
grid on
bode(P0, 'r', K, 'b', -P0*K, 'm')
grid on
size(K.a)
size(WS.a)
size(WT.a)
size(P0.a)

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2009.12.3†

rub3.m

rub4.m

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2009.12.10 Speed control of two inertia system with servo motor (1/3)†

Problem setup
Modelling (frequency response experiment)

setup.pdf

freqresp.m

frdata_amp2_offset10.dat

frdata_amp3_offset10.dat

frdata_amp4_offset10.dat

frdata_amp3_offset5.dat

frdata_amp3_offset0.dat

nominal.m

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2009.12.17 Speed control of two inertia system with servo motor (2/3)†

Controller design

frequency response experiments are re-executed so that all data files have same frequency range. The format of data files are also changed. Use the following links instead those given in the previous lecture!:
freqresp_fixed.m

frdata_amp2_offset10_fixed.dat

frdata_amp3_offset10_fixed.dat

frdata_amp4_offset10_fixed.dat

frdata_amp3_offset5_fixed.dat

frdata_amp3_offset0_fixed.dat

frdata_amp10_offset20_fixed.dat

You can design controllers based on mixed sensitivity problem by executing the following m-files in the order:
nominal_fixed.m

weight.m

cont.m

example of results
cont.dat

cont_order.dat

result.dat

report

design your controller so that the system performance is improved compared with the given example
Draw the following figures and explain the difference between two control systems:
1. bode diagram of controllers
2. gain characteristic of closed-loop systems
3. time response of control experiment
Why is the performance of your system improved(or unfortunately decreased)?
- due date: ~~28th(Mon) Dec 17:00~~ --> 8th(Fri) Jan 2010
- submit your report by e-mail to kobayasi@nagaokaut.ac.jp(pdf or doc files is OK)
- You can use Japanese
- maximum controller order is 20
- submit your cont.dat and cont_order.dat to kobayasi@nagaokaut.ac.jp ~~until 24th Dec~~ --> 8th(Fri) Jan 2010

2009.12.22 announcement to design controller

Please use LMI based H infinity control method instead of Riccati based method (default). Specifically, please use hinfsyn function in cont.m as follows:

[K, clp, gopt] = hinfsyn(G, 1, 1, 'DISPLAY', 'on', 'METHOD', 'lmi')

There were some numerical problems in the execution of cont.m last week: the gain characteristics of sensitivity(and complementary sensitivity) function was not bounded by the weighting function in the resultant figure as following:

You can also check if the design result is OK or not as following:

>> cont
Test bounds:      0.0000 <  gamma  <=      0.9051 

  gamma    hamx_eig  xinf_eig  hamy_eig   yinf_eig   nrho_xy   p/f
    0.905   4.1e-03 -2.2e-08   1.5e-03   -1.9e-20    0.0674    p 
    0.453   4.1e-03# ********   1.5e-03   -3.8e-05# ********    f 
    0.679   4.1e-03# ********   1.5e-03   -1.5e-04# ********    f 
    0.792   4.1e-03 -2.2e-08   1.5e-03  -3.9e-04#   0.0031    f 
    0.849   4.1e-03 -2.2e-08   1.5e-03  -9.0e-04#   0.0064    f 
    0.877   4.1e-03 -2.2e-08   1.5e-03  -2.1e-03#   0.0138    f 
    0.891   4.1e-03 -2.2e-08   1.5e-03  -5.3e-03#   0.0343    f 
    0.898   4.1e-03 -2.2e-08   1.5e-03  -2.2e-02#   0.1384    f   

Gamma value achieved:     0.9051
>> norm(clp, 'inf')

ans = 

    1.2593

In the above result, the closed-loop H infinity norm were 0.9051 and 1.2593 by hinfsyn() and norm() functions, respectively. This inconsistency implies that the design result might be incorrect since these two values of gamma should be the same.

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2009.12.24 Speed control of two inertia system with servo motor (3/3)†

Control experiment

2009.12.25†

program sources for frequency response experiment
freqresp.h

freqresp_module.c

freqresp_app.c
program sources for control experiment
hinf.h

hinf_module.c

hinf_app.c
format of result.dat file
- 1st column: time [sec]
- 2nd column: motor speed [rad/sec] (measured output)
- 3rd column: load speed [rad/sec]
- 4th column: motor torque [Nm] (control input)
- 5th column: reference speed [rad/sec]

configuration of control experiment

reference signal is generated as described in hinf_module.c:
```
if((t > 5)&&(t < 15)){
  r = 15.0;
}else{
  r = 5;
}
```

30% of rated torque is loaded from 10 to 12 sec as specifying in hinf_module.c:

if((t > 10.)&&(t < 12.)){
  daconv(0, bit16_conv(torq_volt_conv(RATED_TORQ*-0.3)));
}else{
  daconv(0, bit16_conv(torq_volt_conv(0)));
}

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2009.12.28†

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(1) / (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.

There are some ways to tackle the problem:

Use loose weight WT(s) (over estimate of modeling error in high frequency range as explained in Dec. 24.)
Loop shaping design procedure
sampled-data H infinity control design
Evaluate of control input u with a high pass filter
...

The final option will be explained below:

Design example with evaluation of control input u

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