### 2009.9.10 CACSD introduction †

• introduction of Matlab and Simulink
• Basic usage of MATLAB and Simulink used for О№ЪѓНшЭ§БщНЌЕкЄгЙЭЕцII/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

### 2009.9.17 Intro. to Robust Control †

• robust stability
• performance improvement
• H infinity norm

### 2009.11.19 †

... generalized plant is described by hand

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

execution example for Matlab

rub2
size(Ap)
size(Ap,1)
size(Ap,2)
M = zeros(3,2)
size(M,1)
size(M,2)
gopt
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
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)

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

• Problem setup
• Modelling (frequency response experiment)

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

• Controller design
• report
1. design your controller so that the system performance is improved compared with the given example
2. 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
3. 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.

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

• Control experiment

participant list

### 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;
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: