### [lecture #1] 2016.9.1 outline of the lecture, review of classical and modern control theory (1/3) †

• outline of this lecture
• syllabus
• evaluation
• mini report #1 ... 10%
• mini exam #1 ... 10%
• mini report #2 ... 10%
• mini exam #2 ... 10%
• final report ... 60%
• schedule2016 (tentative)
• map
• review : stabilization of SISO 1st-order unstable plant by classical and modern control theory
• transfer functions / differential equations
• poles / eigenvalues
• impulse response / initial condition response
• ...
%-- 2016/09/01 14:26 --%
s = tf('s')
k = 2
alpha = -1
Tyr = k/(s+alpha+k)
step(Tyr, 'b')
k = 100
Tyr2 = k/(s+alpha+k)
step(Tyr, 'b', Try2, 'r--')
step(Tyr, 'b', Tyr2, 'r--')

### [lecture #2] 2016.9.8 review of classical and modern control theory (2/3) with introduction of Matlab/Simulink †

1. introduction of Matlab and Simulink text_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 difinition)
• m file
2. system representation: Transfer Function(TF) / State-Space Representation (SSR)
• example: mass-spring-damper system
• difinition of SSR
• from SSR to TF
• from TF to SSR: controllable canonical form
3. open-loop characteristic
• open-loop stability: poles and eigenvalues
• Bode plot and frequency response ex0908_1.m mod0908_1.mdl
• cut off frequency; DC gain; -40dB/dec; variation of c
• relation between P(jw) and steady-state response
4. closed-loop stability
• Nyquist stability criterion (for L(s):stable)
• Nyquist plot ex0908_2.m mod0908_2.mdl
• Gain Margin(GM); Phase Margin(PM)
%-- 2016/09/08 13:08 --%
a = 1
a
pwd
ls
ex0908_1
sqrt(m/k)/(2*pi)
sqrt(k/m)/(2*pi)

### [lecture #3] 2016.9.15 review of classical and modern control theory (3/3) †

1. LQR problem
• controllability
• cost function J >= 0
• (semi)-positive definiteness
2. solution of LQR problem
• closed loop stability ... Lyapunov criterion
• Jmin lqr.pdfproof4.pdf (from B3「動的システムの解析と制御」)
3. example mod0915.mdl
A = [1, 2; 0, -1]; % unstable plant
B = [0; 1];
Uc = ctrb(A,B);
det(Uc) % should be nonzero
C = eye(2); % dummy
D = zeros(2,1); % dummy
F = [0, 0]; % without control
x0 = [1; 1]; % initial state
Q = eye(2);
R = 1;
P = are(A, B/R*B', Q);
eig(P) % should be positive
F = R\B'*P;
x0'*P*x0
%-- 2016/09/15 14:18 --%
mod0915
A = [1, 2; 0, -1]; % unstable plant
B = [0; 1];
Uc = ctrb(A,B);
det(Uc) % should be nonzero
C = eye(2); % dummy
D = zeros(2,1); % dummy
F = [0, 0]; % without control
x0 = [1; 1]; % initial state
Q = eye(2);
R = 1;
P = are(A, B/R*B', Q);
eig(P) % should be positive
F = R\B'*P;
x0'*P*x0
x0
J
x0'*P*x0
plot(t, J)
• Q: 各行列の意味がよくわからなかった。
• A: Matlab で入力した行列のことと仮定して回答します（そうでない場合はまた聞いてください）。
A = [1, 2; 0, -1]; % unstable plant
B = [0; 1];

は、制御対象の例として与えました。何でも良いですが、ここでは不安定な方が分かり易いかと思い、不安定な制御対象としました。

Uc = ctrb(A,B);
det(Uc) % should be nonzero

は、可制御性を確認している部分です。(A,B)の可制御性行列を Uc として作り、det で行列式を表示しています。

C = eye(2); % dummy
D = zeros(2,1); % dummy

おそらくこの部分を全く説明しなかったので混乱していると思います（すみません）が、 この C と D を y = Cx + Du に代入すると、y = x となります。つまり、出力に状態をそのまま取り出すための設定です。これで、Simulink の制御対象のブロックの出力が、状態ベクトル x となります。

F = [0, 0]; % without control

フィードバック制御無しの設定です。ここまで入力して一度、シミュレーションスタートするべきでした（不安定な、信号が発散する様子が確認できます）。が、時間なくスキップしたと思います。すみません。

x0 = [1; 1]; % initial state

Q = eye(2);
R = 1;

P = are(A, B/R*B', Q);

リカッチ方程式の正定解Pを計算します。この結果を実際にリカッチ方程式に代入すると0となるところも見せるべきでした。やってみてください。

eig(P) % should be positive

F = R\B'*P;

x0'*P*x0

### [lecture #4] 2016.9.29 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 condition x(0) = 0, find a state-feedback controller $u = -f x$ such that \begin{eqnarray} (i) &&\quad \mbox{closed loop is stable} \\ (ii) &&\quad \left\{\begin{array}{l} \| z \|_2 \rightarrow \mbox{min for } w(t) = \delta(t) \quad \mbox{(LQR)} \\ \| T_{zw} \|_\infty \rightarrow \mbox{min} \quad \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 and L2 norm of z
2. impulse disturbance resp. with zero initial condition and initial condition resp. with zero disturbance
3. definition of H infinity norm (SISO)
s = tf('s');
P1 = 1/(s+1);
bode(P1);
norm(P1, 'inf')
P2 = 1/(s^2 + 0.1*s + 1);
bode(P2);
norm(P2, 'inf')
4. definition of H infinity norm (SIMO)
5. solve the problem by hand
6. solve the problem by tool(hinfsyn) ex0929.m
%-- 2016/09/29 13:46 --%
s = tf('s');
P1 = 1/(s+1);
bode(P1);
norm(P1, 'inf')
P2 = 1/(s^2 + 0.1*s + 1);
bode(P2);
norm(P2, 'inf')
format long e
norm(P2, 'inf')
grid on
ex0929
• Q: H∞ノルムが何を表しているのか
• A: 簡単に言うと安定なシステムの最大ゲインです。

### [lecture #5] 2016.10.6 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 mod1006.mdl
• block diagram in the simulink model
• how to approximate impulse disturbance
• 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 (common mdl file is available)
• review: steady-state response (see whiteboard#4 @ lec. #2)
• H infinity norm = L2 induced norm
• how to construct the worst-case disturbance w(t) which maximizes L2 norm of z(t) ?
• what is the worst-case disturbance in the simple example ?
4. general case: hinf.pdf includes the simple example as a special case
• LQR lqr.pdf is included as a special case in which gamma -> infinity, w(t) = 0, B2 -> B, and non-zero x(0) are considered
%-- 2016/10/06 13:09 --%
1/sqrt(2)
sqrt(2+sqrt(2))
sqrt(2-sqrt(2))
mod1006
ex0929
A
B
C
D
h
h = 0.01
x0 = 0
f
f = -1+sqrt(2)
zz
zz(end)
h = 0.0001
zz(end)
x0
x0 = 1
zz(end)
f
f = 1
x0
x0 = 0
zz(end)
x0
h
h = 10
zz(end)/ww(end)
sqrt(zz(end)/ww(end))
f
h = 100
sqrt(zz(end)/ww(end))
f
f = -1+sqrt(2)
sqrt(zz(end)/ww(end))

whiteboard #1 ... sorry for the missing picture in which A, B1, B2, C1, and D12 were additionally written in red color.

• Q: Some things can not be described clearly.
• A: Can you ask me in detail ?

### [lecture #6] 2016.10.13 Mixed sensitivity problem 1/3 †

1. review: LQR and H infinity
2. outline: map_v1.0_intro1.pdf ; schedule2016
3. H infinity control problem (general case)
4. reference tracking problem
5. weighting function for sensitivity function
6. design example ex1013_1.m ex1013_2.m
7. the small gain theorem
• proof: Nyquist stability criterion
8. from performance optimization to robust stabilization
%-- 2016/10/13 13:58 --%
ex1013_1
P
pole(P)
ex1013_2
K
K_hinf
eig(K_hinf.a)

### [lecture #7] 2016.10.20 Mixed sensitivity problem 2/3 †

1. review map_v1.0_intro2.pdf and outline
2. an equivalent problem of robust stabilization for reference tracking problem
3. uncertainty model and normalized uncertainty Delta
4. how to determine P0 and WT
• example: frequency response of plant with perturbation ex1020_1.m
• frequency response based procedure for P0 and WT ex1020_2.m
5. robust stabilization problem and equivalent problem
%-- 2016/10/20 13:36 --%
ex1020_1
ex1020_2
ex1020_3
mod1020
c
• Q: Δは、何を表しているのか忘れてしまったので、もう一度説明お願いします。
• A: 安定で、H∞ノルムが1以下の1入出力系です（最大ゲインが1以下の安定な伝達関数、でも良いです）。

### [lecture #8] 2016.10.27 Mixed sensitivity problem 3/3 †

• review: map_v1.0_intro2.pdf (1)robust stabilization and (2)performance optimization
• outline:
1. how to design controllers considering both conditions in (1) and (2)
2. gap between nominal performance and robust performance
1. mixed sensitivity problem ---> (1) and (2) : proof
2. generalized plant for mixed senstivity problem
3. design example ex1027_1.m minimize gamma by hand
4. gamma iteration by bisection method ex1027_2.m
5. nominal performance and robust performance ex1027_3.m
6. introduction of robust performance problem
%-- 2016/10/27 13:46 --%
ex1027_1
K
ex1027_2
ex1027_3

... sorry for the mistake in the left-upper inequality, please correct it as pointed out by a student as below

• Q: $\left[\begin{array}{cc} \overline{m_1(j\omega)} & \overline{m_2(j\omega)} \end{array} \right] \left[\begin{array}{c} m_1(j\omega) \\ m_2(j\omega) \end{array} \right] = |m_1(j\omega)|^2 + |m_2(j\omega)|^2$ とならないでしょうか？
• A: ごめんなさい、その通りです。できれば授業中に訂正してもらえると助かります。

### [lecture #9] 2016.11.17 robust performance problem 1/3 †

1. review
• mixed sensitivity problem
ex1027_1
ex1027_2
ex1027_3
• robust performance problem c.f. the last whiteboard
2. an equivalent robust stability problem
3. definition of H infinity norm for general case (MIMO)
• definition of singular values and the maximum singular value
M = [1/sqrt(2), 1/sqrt(2); j, -j]
M'
eig(M'*M)
svd(M)
• mini report #1 report1.pdf ... You will have a mini exam #1 related to this report
4. proof of ||Delta hat||_inf <= 1
5. design example: ex1117_1.m
• robust performance is achieved but large gap
• non structured uncertainty is considered ... the design problem is too conservative
%-- 2016/11/17 13:05 --%
ex1027_1
ex1027_2
ex1027_3
M = [1/sqrt(2), 1/sqrt(2); j, -j]
M'
eig(M'*M)
svd(M)
ex1117_1
• Q: 特異値は何を意味しているのか？
• Q: 特異点（値：小林註）は何を表していますか？（最大特異点（値）$\bar \sigma$ も）
• A: 次回、特異値分解(singular value decomposition)と共に説明します。

### [lecture #10] 2016.11.24 Robust performance problem (2/3) †

1. return of mini report #1
2. review
• robust performance but too conservative
ex1027_2
ex1117_1
• structured unertainty Delta hat and unstructured uncertainty Delta tilde
• robust stability problem for Delta hat and its equivalent problem(?) with Delta tilde
3. SVD: singular value decomposition
• definition
• meaning off the largest singular value
• 2-norm of vectors
• SVD for 2-by-2 real matrix ex1124_1.m
%-- 2016/11/24 13:04 --%
ex1027_2
ex1117_1
M = [1/sqrt(2), 0; 1/sqrt(2), 0]
svd(M)
M
[U, S, V] = svd(M)
[U, S, V] = svd(M);
U
V
U*S*V'
U*S*V' - M
S
U'*'
U'*U
V'*V
help norm
ex1124_1
• Q: ホワイトボード◯6の a=v1 の v1 が V* の成分だった理由が分かりませんでした。
• A: 説明が悪かったと思いますが、m×m行列 V の第一列を列ベクトル v1 と決めた、というだけです。この結果、V* の第一行は行ベクトル v1* となります。わからなければまた聞いてください。

### [lecture #11] 2016.12.1 Robust performance problem (3/3) †

1. review: conservative design with Delta tilde
2. scaled H infinity control problem
3. how to determine structure of scaling matrix
4. design example ex1201_1.m
ex1027_2
ex1117_1
gam_opt0 = gam_opt
K_opt0 = K_opt;
ex1201_1
gam_opt
5. mini exam #1
%-- 2016/12/01 13:59 --%
ex1027_2
ex1117_1
gam_opt
gam_opt0 = gam_opt
K_opt0 = K_opt;
ex1201_1
gam_opt
gam_opt0
d_opt

### [lecture #12] 2016.12.8 Robust performance problem (3/3) (cont.), Control system design for practical system (1/3) †

1. return of mini exam #1;
2. review of the scaled H infinity control problem
3. effect of scaling ex1208_1.m
ex1027_2
ex1117_1
gam_opt0 = gam_opt
K_opt0 = K_opt;
ex1201_1
gam_opt
ex1208_1
4. mini report #2 report2.pdf please use modified file report2_fixed.pdf ... You will have a mini exam #2 related to this report
5. 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
%-- 2016/12/08 13:25 --%
ex1027_2
ex1117_1
gam_opt0 = gam_opt
K_opt0 = K_opt;
ex1201_1
gam_opt
ex1208_1
ex1208_1
format long
ex1208_1
M = [0, 0.5; sqrt(2), 0]
W = mdiag(1/sqrt(3), 0)
W = mdiag(1/sqrt(3), 1)
svd(M)
Mhat = inv(W)*M*W
Mhat = W\M*W
svd(Mhat)
ex1208_2

### [lecture #13] 2016.12.15 Control system design for practical system (2/3) †

1. return of mini report #2
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
• how to handle modeling error of G ?
3. design example
• determination of plant model(nominal plant and additive uncertainty weight)
nominal.m
subspace.m ... replacement of n4sid in System Identification Toolbox (not provided in IPC)
weight.m
• configuration of generalized plant and controller design by scaled H infinity control problem using one-dimensional search on the scaling d
cont.m
• comparison of closed-loop gain characteristics with and without control
compare.m
• result of control experiment
result.dat
compare_result.m
4. room 157 @ Dept. Mech. Bldg.2
5. 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 and frequency spectrum (PSD) of control experiment
3. Why is the performance of your system improved(or unfortunately deteriorated)?
• due date: 6th(Fri) 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 35
• You can send up to 5 controllers
• control experimental results will be uploaded here
• freqresp ... frequency response will be measured and uploaded everyday
6. how to improve the performance ?
• order of the nominal plant
• weighting for robust stability
7. specifications of the experimental system
1. experimental equipments
• loudspeakers: AURA SOUND NSW2-326-8A (2inch, 15W)
• pressure sensors: NAGANO KEIKI KP15
• A/D, D/A converters: CONTEC AD12-16(PCI), DA12-4(PCI)
• PC: Dell Dimension 1100
• OS: Linux kernel 2.4.22 / Real Time Linux 3.2-pre3
1. program sources for frequency response experiment
• freqresp.h
• freqresp_module.c
• freqresp_app.c
• format of spk1.dat (u is used instead of w for spk2.dat)
• 1st column ... frequency (Hz)
• 2nd column ... gain from w(V) to y(V) (signal's unit is voltage (V))
• 3rd column ... phase from w to y
• 4th column ... gain from w to z
• 5th column ... phase from w to z
1. program sources for control experiment
• hinf.h
• hinf_module.c
• hinf_app.c
• format of result.dat
• 1st column: time (s)
• 2nd column: z (V)
• 3rd column: y (V)
• 4th column: u (V)
• 5th column: w (V)
1. configuration of control experiment
• disturbance signal w is specified as described in hinf.h and hinf_module.c:
#define AMP 3.0 // amplitude for disturbance
#define DIST_INTERVAL 5 // interval step for updating w

count_dist++;
if(count_dist >= DIST_INTERVAL){
w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // uniform random number in [-AMP, AMP]
count_dist = 0;
}

da_conv(V_OFFSET + w, 0); // D/A output to noise source
w is updated with 1ms period (sampling period 0.2ms times DIST_INTERVAL 5)
• control signal u is limited to [-4, 4] as specified in hinf.h and hinf_module.c:
#define U_MAX 4.00

if(u > U_MAX) u = U_MAX;
if(u < -U_MAX) u = -U_MAX;
u is set to 0 for t < 10(s). (controller is operated for 10 <= t < 15.)
• a high pass filter with cut-off frequency are used to cut DC components in z and y as described in hinf.h and hinf_module.c
// HPF(1 rad/s) to cut DC in z and y
#define AF 9.9980001999866674e-01
#define BF 1.9998000133326669e-04
#define CF -1.0000000000000000e+00
#define DF 1.0000000000000000e+00

// HPFs
yf = CF*xf_y + DF*yz[0];
xf_y = AF*xf_y + BF*yz[0];
zf = CF*xf_z + DF*yz[1];
xf_z = AF*xf_z + BF*yz[1];
%-- 2016/12/15 13:18 --%
ls
nominal
ls data
nominal
ctrlpre
ctrlpref
nominal
345/3.6
who
G0
size(G0.a)
weight
help n4sid
eig(A)
max(real(eig(A)))
weight
cont
compare
compare_result

### [lecture #14] 2016.12.20 Control system design for practical system (3/3) †

• preparation of your own controller(s) by using the remote experiment system but no explanation will be given (Kobayashi will not be in Nagaoka. I'm sorry)

### [lecture #15] 2016.12.22 Control system design for practical system (cont.) †

• preparation of your own controller(s)
• mini exam #2
• questionnaires
• to university
• for web-based experimental environment
%-- 2016/12/22 13:09 --%
weight
compare