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

1. introduction of Matlab and Simulink
2. relationship between TF and SSR
3. open-loop stability can be checked by poles of TF and eigenvalues of A-matrix in SSR
4. closed-loop stability
• graphical test by Nyquist stability criterion and Bode plot with GM(gain margin) and PM(phase margin)
• numerical test by poles of closed-loop TF and eigenvalues of A-matrix in closed-loop SSR
• example: stabilization of unstable system
%-- 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 

### [lecture #4] 2012.10.4 Intro. to robust control theory (H infinity control theory) †

1. Typical design problems
1. robust stabilization
2. performance optimization
3. robust performance problem (robust stability and performance optimization are simultaneously considered)
2. H infinity norm
• definition
• example
3. H infinity control problem
• definition
• application example : reference tracking problem
• relation to the sensitivity function S(s) (S(s) -> 0 is desired but impossible)
• given control system please change the line K = 1' to K = ss(1)'
• controller design with H infinity control theory
%-- 10/4/2012 1:11 PM --%
s = tf('s')
G = 1/(s+1)
norm(G, 'inf')
help norm
ex1004_1
ex1004_2
K
K_hinf
size(K_hinf.a)

### [lecture #5] 2012.10.11 Introduction to Robust Control (cont.) †

1. Typical design problems
1. robust stabilization ... plant is given as class
2. performance optimization; ... plant has no variation
3. robust performance problem (robust stability and performance optimization are simultaneously considered)
2. connection between H infinity control problem and robust stabilization
• normalized uncertainty \Delta
• small gain theorem
• sketch proof ... Nyquist stability criterion
3. How to design robust stabilizing controller with H infinity control problem ?
• connection between \Delta and plant ?
• generalized plant G ?
• practical example of robust stabilization problem
%-- 10/11/2012 12:56 PM --%
ex1011_1
ex1011_2
ex1011_3
mod1011
ex1011_1
P
ex1011_2
1i
j
j = 2
1i = 2
ex1011_3
mod1011
c = 0.8
c = 1.2
c = 1.3
c = 2

### [lecture #6] 2012.10.18 Introduction to Robust Control (cont.) †

1. review
• robust stabilization ... ||WT T||_inf < 1 (for multiplicative uncertainty)
• performance optimization ... ||WS S||_inf < gamma -> min
• mixed sensitivity problem gives a sufficient condition such that both conditions hold ... nominal (not robust) performance problem
2. proof of small gain theorem ... Nyquist stability criterion
3. mixed sensitivity problem
%-- 10/18/2012 12:56 PM --%
mod1018
ex1018_1
input_to_P0
WT
WSgam
systemnames
ex1018_1
G
ex1018_1

### [lecture #7] 2012.10.25 singular value decomposition(SVD) †

• review of mixed sensitivity problem
• definition of SVD and maximum singular value
%-- 10/25/2012 1:27 PM --%
A = [1, 2, 3; 4, 5, 6]
[U, Sigma, V] = svd(A)
U*U'
U'*U
V'*V
[U, Sigma, V] = svd(A')
A = [1, 2, 3; 4, 5, 1i]
[U, Sigma, V] = svd(A)

### [lecture #8] 2012.11.1 Robust performance problem †

• review of the proof
• nominal and robust performance (motivation of robust performance problem)
• how to tackle with the robust performance problem (a conservative design with H infinity control problem)
• less conservative design with scaled H infinity control problem
%-- 11/1/2012 1:37 PM --%
ex1101_1
ex1101_2
K
ex1101_2
ex1101_3

### [lecture #9] 2012.11.8 Robust performance problem (cont.) †

• review: mixed sensitivity problem = robust stability + nominal performance(not robust performance)
• how to tackle with the robust performance problem
1. introduction of a fictitious uncertainty
2. the maximum singular value of block diagonal matrices is less than or equal to 1 ---> a conservative design with H infinity control problem
3. intruduction of scalling matrix ---> less conservative design with scaled H infinity control problem
%-- 11/8/2012 1:29 PM --%
ex1101_4
ex1101_1
ex1101_2
ex1101_3
ex1101_4
ex1101_5

### [lecture #10] 2012.11.15 Robust performance problem (cont.) †

• review of scaled H infinity control problem
1. ex1101_4.m, ex1101_5.m ... What does the designed result mean ?
2. How was the class of scaling determined ?
3. effect of scaling on closed-loop H infinity norm ... motivation of scaling
%-- 11/15/2012 1:08 PM --%
ex1101_1
ex1101_2
ex1101_3
ex1101_4
ex1101_5
-13/20
10**-0.65
10^-0.65
ex1101_4
K
ex1101_4
ex1101_5
gam
K
d
gam
K
sigma(mdiag(1,1/gam)*lft(G,K))
sigma(mdiag(1/d,1/gam)*lft(G,K)*mdiag(d,1),'r')
sigma(mdiag(1,1/gam)*lft(G,K))
hold on
sigma(mdiag(1/d,1/gam)*lft(G,K)*mdiag(d,1),'r')

### [lecture #13] 2012.12.6 Robust control design for a practical system (1/3) : Active vibration control of a pendulum using linear motor (loudspeaker) some mistakes fixed on 2012.12.13 [#k776c145] †

• linear motor : FOSTEX FW-208N (corn, edge, and damper are partially removed to decrease damping)
• PSD(Position Sensitive Detector) : Hamamatsu Photonics Corp. C3683-01
• Potentio meter : Midori Precisions Model QP-2H
• PC : Dell Dimension 2400 (Intel Celeron 2.4G, Linux 2.4.22, RTlinux-3.2pre3)
• D/A : CONTEC DA12-4 (PCI) 12bit, 10us
• A/D : CONTEC AD12-16 (PCI) 12bit, 10us
• He-Ne Laser : NEC GLG5230
• Objective for control system design
1. to attenuate vibration due to pendulum oscillation
2. robust stability against modelling error due to non-linear dinamics etc.
• example of m-files (the following files have been replaced by modified ones (2012.12.13)
>> freqresp
>> nominal
>> weight
• example of m-file to compare designed controllers
• 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: 27th(Thu) 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 21th(Fri) 25th(Tue) Dec
• program sources for frequency response experiment
• format of frdata.dat file
• 1st column: frequency (Hz)
• 2nd column: gain
• 3rd column: phase (deg)
• 4th column: gain
• 5th column: phase (deg)
• program sources for control experiment
• format of result.dat file
• 1st column: time (s)
• 2nd column: PSD output (V)
• 3rd column: Potentio meter output (V)
• 4th column: control input (V)
• 5th column: input disturbance (V)
• configuration of control experiment
• input disturbance is given as described in hinf_module.c:
if(t < 0.1){
w = DIST_AMP;
}else{
w = 0;
}
da_conv(V_OFFSET + w, 0); // D/A output for linear motor
freqresp
nominal
help n4sid
plot(result(:,1),result(:,3),'r',result_no(:,1),result_no(:,2),'g')
plot(result(:,1),result(:,3),'r',result_no(:,1),result_no(:,3),'g')

... please use the potentio meter output as the measured output instead the PSD output (2012.12.13)

### [lecture #14] 2012.12.13 Robust control design for a practical system (2/3) †

IMPOTANT:due to some change on experimental apparatus, please use the potentio meter output as the measured output instead of the PSD output. Moreover, some files have been re-uploaded due to this change. See previous links.

• explanation of design example (cont. from the previous lecture)
• preparation of your own controller(s)
%-- 12/13/2012 12:59 PM --%
freqresp
pwd
freqresp
nominal
P0
weight
cont
compare
plot(result(:,1),result(:,3));
plot(result_no(:,1),result_no(:,3),'b',result(:,1),result(:,3),'r');
plot(result_no(:,1),result_no(:,2),'b',result(:,1),result(:,2),'r');
plot(result_no(:,1),result_no(:,4),'b',result(:,1),result(:,4),'r');

### [lecture #15] 2012.12.20 Robust control design for a practical system (3/3) †

• preparation of your own controller(s)

participant list2012