授業/制御工学特論2017 の履歴(No.14) - Kobayashi Laboratory Homepage

Advanced Automation†

[lecture #1] 2017.9.7 outline of the lecture, review of classical and modern control theory (1/3)†

outline of this lecture
- syllabus([https://vos-lc-web01.nagaokaut.ac.jp/])
- evaluation
  - mini report #1 ... 10%
  - mini exam #1 ... 10%
  - mini report #2 ... 10%
  - mini exam #2 ... 10%
  - final report ... 60%
- schedule2017 (tentative)
- map map_v1.1.pdf for review map_v1.1_review.pdf

review : stabilization of SISO unstable plant by classical and modern control theory
- transfer functions / differential equations
- poles / eigenvalues
- impulse response / initial value response
- ...

%-- 2017/09/07 13:29 --%
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
Tyr = feedback(P*k, 1)
k = 2
Tyr = feedback(P*k, 1)
P
Tyr = feedback(P*k, 1)
step(Tyr)
k = 0.5
Tyr = feedback(P*k, 1)
step(Tyr)
k = 10
Tyr = feedback(P*k, 1)
step(Tyr)

Q: I don't know how to use the MATLAB.
Q: マトラボの使い方が分かりません。
A: A brief explanation of MATLAB and Simulink will be given in the next lecture by using simple examples.
A: 次週、MATLAB（とSimulink）の使い方について説明します。

Q: 個人的には倒立振子のようなモデルから制御系の説明が最初にあると制御工学のイメージがつきやすい
A: 確かに、不安定＝倒れる、という分かり易さは魅力なのですが、1次系で表すことができません。今回は数式の取扱いの容易さから1次系を選びました。倒立振子については、次回、別の演習で用いていたシミュレーションモデルを紹介したいと思います。

Q: when you conduct please speak slowly.
A: Thank you for the sugestion. I will improve my speaking.

↑

[lecture #2] 2017.9.14 review of classical and modern control theory (2/3) with introduction of Matlab/Simulink†

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
- example: stabilization of inverted pendulum (sorry in Japanese)

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
open-loop characteristic
- open-loop stability: poles and eigenvalues
- Bode plot and frequency response ex0914_1.m mod0914_1.mdl
  - cut off frequency; DC gain; -40dB/dec; variation of c
  - relation between P(jw) and steady-state response
closed-loop stability
- Nyquist stability criterion (for L(s):stable)
- Nyquist plot ex0914_2.m mod0914_2.mdl
  - Gain Margin(GM); Phase Margin(PM)

%-- 2017/09/14 13:05 --%
a = 1
who
a + 2
demo
lookfor demo
demo toolbox
demo toolbox]
demo toolbox
t = [1, 2, 3]
t = [1 2 3]
u = [1; 2; 3]
t
t'
penddemo
help penddemo
penddemo
ls
ex0914_1
who
P

Q: ○7について、 \[ \frac{b_1 s + b_0}{s^2 + a_1 s + a_0} + d \] とした時、対応するSSRは、 \[ \left[\begin{array}{cc|c} 0 & 1 & 0 \\ -a_0 & -a_1 & 1 \\ \hline b_0 & b_1 & 0 \end{array}\right] \] となるという理解で良いでしょうか。（ホワイトボードの説明では、 \[ \left[\begin{array}{cc|c} 0 & 1 & 0 \\ -a_1 & -a_2 & 1 \\ \hline b_1 & b_2 & 0 \end{array}\right] \] に見える為）
A: その理解で正しいです。赤枠で囲んだ際に添え字が一つずれていることに気付いていませんでした。申し訳ありません。

Q: I didn't understand the usefulness of SSR.
A: LQR problem is formulated and solved by using SSR, which will be explain in the next lecture.

Q: TF から SSR への変換の過程がよくわからなかった。
A: 与えられた TF に対応する SSR は、状態ベクトルの取り方によって一意に決まらない（無限に存在する）ため、代表的な SSR の型を利用します。今日扱ったのは controllable canonical form （可制御正準形）で、伝達関数の分母・分子多項式及び直達項をA,C,D行列の特定の場所に当てはめることで、TFからSSRの変換を行ったことになります。この双対の可観測正準形も同様です。他に、今日は紹介しませんでしたが、共振モード毎のブロック対角構造がA行列に表れる型などもあり、目的に応じて使い分けられます。

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[lecture #3] 2017.9.21 review of classical and modern control theory (3/3)†

LQR problem
- controllability
- cost function J >= 0
- (semi)-positive definiteness
solution of LQR problem
- ARE and quadratic equation
- closed loop stability ... Lyapunov criterion
- Jmin lqr.pdf ≒ proof4.pdf (from B3「動的システムの解析と制御」)
example ex0921_1.m mod0921_1.mdl

%-- 2017/09/21 13:17 --%
ex0921_1
B
A
A*B
P
eig(P)
F
J
x0
x0'*P*x0

whiteboard #1

whiteboard #2

whiteboard #3

Q: MATLABの使い方が所々わからない。
A: 今日のように時間の余ったときにでも聞いてください。

Q: QとRの値は何で決まるのですか？
A: 設計者が決めます。大雑把に言うと、Qを大きく（Rを小さく）すると、状態xの収束性が上がるが大きな制御入力が必要となり、Qを小さく（Rを大きく）すると、状態xの収束性が犠牲になるが小さな制御入力で済む、というトレードオフがあるので、それを考慮して決めます。

Q: f = b/r P と出来る理由が分からなくなってしまった
Q: ホワイトボード○5のfとPの関係がどこから出てきたのか分からなかった
A: LQR問題の解において、状態フィードバックゲインFは代数リカッチ方程式の解Pを用いて与えられます。この関係式（スカラの例題の場合は f = b/r p）を使うと、（リカッチ方程式とは無関係に導出した）f に関する二次方程式が、Pに関する代数リカッチ方程式と同一であることが示されます。わからなければまた聞いてください。

Q: In m-file, line 13 F = R\B'"P what is this (Yen mark) operator? the same process this? inv(R)\B'

A: Yen mark means `\'. X\Y and Y/X respectively mean \[ X^{-1} Y, \quad Y X^{-1}. \]

Q: LQR：L→ 非線形は扱えない？その影響は偏差？時間？パラメータ変動になる？
A: 先週紹介した状態空間実現は、状態ベクトルに関する一階の線形微分方程式です。よって、非線形特性を直接表現することはできず、平衡点回りで線形化するのが典型的な対処法です。一方、安定となるよう設計された制御系は通常、余裕を持ち、若干の非線形特性は許容されます。この意味で大きな安定余裕を持つように制御系を設計すれば、間接的に非線形特性を考慮できる場合もあります。

Q: If it is possible, I want to practice more during the lecture, may I get more additional example materials to try on Matlab, for the each topics.
A: I think this is because a lack of detailed explanation on examples: The mass-spring-damper system given in the today's lecture was unstable one choosing the damping coefficient as a negative value. Please try to change those values; I completely forgot to explain physical meaning of Q and R and how to choose them. As in the Q&A above (sorry in Japanese), matrices Q and R are given by designer considering trade-off between performance and cost e.g. energy consumption. Please try with other values of Q and R, to confirm the trade-off; Therefore, I would like to recommend you to change numerical values in examples for better understandings. If this answer is not sufficient for you, please ask again.

Q: 少しスピードを落としてほしい
A: 日本語の説明を少なくしたせいと思います。やはり同じ説明を日本語でもするようにします。

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[lecture #4] 2017.9.28 relation between LQR and H infinity control problem (1/2)†

GOAL: to learn difference in concepts between LQR problem and H infinity control problem

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 state value x(0) = 0, find a state-feedback controller \[ u = -f x \] such that \begin{eqnarray} (i) &&\quad \mbox{closed loop is stable} \\ (ii) &&\quad \mbox{minimize} \left\{\begin{array}{l} \| z \|_2 \mbox{ for } w(t) = \delta(t) \quad \mbox{(LQR, $H_2$ control)} \\ \| T_{zw} \|_\infty \mbox{($H_\infty$ control problem)}\end{array}\right. \end{eqnarray}
- comparison of norms in (ii) (for a = -1, b = 1, q = 1, r = 1) \[ \begin{array}{|c||c|c|}\hline & \mbox{LQR}: f=-1+\sqrt{2} & \quad \quad H_\infty: f=1\quad\quad \\ \hline\hline J=\|z\|_2^2 & & \\ \hline \|T_{zw}\|_\infty & & \\ \hline \end{array} \]
an alternative description to LQR problem
1. J = (L2 norm of z)^2
2. impulse resp. with zero initial value = initial value resp. with zero disturbance

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')

definition of H infinity norm (SIMO)
solve the problem by hand
solve the problem by tool(hinfsyn) ex0928.m

%-- 2017/09/28 13:40 --%
s = tf('s');
P1 = 1/(s+1);
bode(P1);
norm(P1, 'inf')
P1
norm(P1, 'inf')
P2 = 1/(s^2 + 0.1*s + 1);
bode(P2)
grid on
norm(P3, 'inf')
norm(P2, 'inf')
format long e
norm(P2, 'inf')
ex0928
ex0929
ex0928
f
K
dcgain(K)
gopt
1/sqrt(29
1/sqrt(2)
gopt

Q: H∞制御がまず、どのような設計で使われているなどの具体例があれば嬉しいと思いました
A: 話が抽象的過ぎてピンとこないという指摘ですね。ただ、この制御対象ならばこの制御理論を使うべき、というルールは特に無く、設計者がどうしたいか、どんな制御系を良しとするかによって制御理論は選んで使われ（無ければ新たに開発され）ます。その意味で、H∞制御が使われる典型的な場面は、制御対象の平均的なモデルとその不確かさ（バラツキ具合、変動）がおおよそ分かっている場合に、それに対する安定性（と性能も）を保証するコントローラ（ロバストコントローラ）の設計です。古い例では小型自動着陸実験機ALFLEXが挙げられます（大野他、ALFLEXのロバスト飛行制御則設計、計測自動制御学会論文集, 34(12):1905--1912（1998））。開発期間が限られており多くの試行錯誤が許されなかったことなどからH∞制御が採用されています。もう一つ、これも古い例ですが、レインボーブリッジや明石海峡橋など、建設中の主塔が強風で振動するのを抑制するためにH∞制御が採用されています（B.F.Spencer Jr. and Michael K.Sain: Controlling Buildings: A New Frontier in Feedback, IEEE Control Systems, 17(6):19{35, 1997）。他にも自動車のサスペンションの制御など、数多くの応用事例があります。調べてみてください。

Q: 具体的にどのようなシステムを制御する時にH∞、状態FBを使い分けるのか？
A: 説明が悪かったと思いますが、H∞制御と状態フィードバックは無関係です。つまり、全状態が利用できる状況が状態フィードバック、状態は直接利用できず観測出力を利用する場合が出力フィードバックです。どちらもH∞制御との組み合わせがあります。今日は、手計算では状態フィードバックコントローラをH∞制御で求め、hinfsynでは出力フィードバックH∞コントローラを求めました。出力フィードバックと状態フィードバックをどう使い分けるかという質問に対しては、コストはかかるがセンサが十分設置できるなら制御対象の全状態を利用して性能の良い制御系となるのが状態フィードバックで、多少性能は犠牲になっても制御できれば良いならコストが比較的安い出力フィードバック、ということは言えます。ただ、無限次元系など、センサをいくら設置しても状態フィードバックは不可能な場合もありますが。

Q: 授業ではふれませんでしたが sup と max の違いを教えていただけないでしょうか？
A: sup = 最小上界、または上限、max = 最大値、です。1入出力システム P1, P2 \[ P_1(s) = \frac{1}{s+1}, \quad P_2(s) = \frac{s}{s+1} \] を考えると、P1 については ω=0 のゲインがH∞ノルムを与えるので（今日言ったように）sup でも max でも同じですが、P2 については、ω→∞ の極限がH∞ノルム（1）を与えるので、max はとれません。ω=∞ という点がないためです。例えば、十分大きな正実数（10000000とか）をaとすると、 \[ \max_{\omega\in[0,a]} |P_2(j\omega)| = \max_{\omega\in[0,a]} \left|\frac{j\omega}{j\omega+1}\right| = \max_{\omega\in[0,a]} \frac{\omega}{\sqrt{\omega^2 + 1}} = \frac{a}{\sqrt{a^2 + 1}} < 1 \] となって、maxをとる範囲（0からa）を広げればいくらでも真のH∞ノルム 1 に近い値が得られますが、決して1に一致させることはできません。supなら、できます。

Q: Explanation of H∞ problem was not detailed enough
A: The difference to LQR is (ii) the target to be minimized: In H infinity control problem, closed-loop H infinity norm is minimized instead of the cost function in LQR. The difference in their concept will be explained in the next lecture. If this answer is not sufficient for you, please ask again.

Q: What purpose to derive "Tzw" ?
A: Because our task was to find optimal controller in terms of closed-loop H infinity norm by hand, the f-dependent closed-loop system, Tzw, was first derived. (Then, differentiate with respect to f ...)

Q: 波形があると分かりやすい
A: 今日の講義では波形が出ていなくてわかりにくかった、という指摘かと思って回答します：次回、H∞制御とLQRの違いを時間応答で説明します。それぞれで最適な場合として扱われる時間応答です。1回で終わらせられると内容的には区切りが良く分かり易いと思うのですが、分量的に無理のため、現状のような二部構成としています。

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[lecture #5] 2017.10.5 relation between LQR and H infinity control problem (2/2)†

complete the table in simple example
confirm the cost function J for both controllers by simulation mod1005.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
confirm the closed-loop H infinity norm for both controllers by simulation
- H infinity norm = L2 induced norm
- review: steady-state response; the worst-case disturbance w(t) which maximizes L2 norm of z(t) ?
- what is the worst-case disturbance in the simple example ?
general state-feedback 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

!!! the remaining page is under construction (the contents below are from 2016) !!!

%-- 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))

&ref(): File not found: "2016.10.06-1.jpg" at page "授業/制御工学特論2017"; ... sorry for the missing picture in which A, B1, B2, C1, and D12 were additionally written in red color.

#ref(): File not found: "2016.10.06-2.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.10.06-3.jpg" at page "授業/制御工学特論2017"

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†

review: LQR and H infinity
outline: map_v1.0_intro1.pdf ; schedule2016
H infinity control problem (general case)
reference tracking problem
weighting function for sensitivity function
design example &ref(): File not found: "ex1013_1.m" at page "授業/制御工学特論2017"; &ref(): File not found: "ex1013_2.m" at page "授業/制御工学特論2017";
the small gain theorem
- proof: Nyquist stability criterion
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)

#ref(): File not found: "2016.10.13-1.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.10.13-2.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.10.13-3.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.10.13-4.jpg" at page "授業/制御工学特論2017"

↑

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

review map_v1.0_intro2.pdf and outline
an equivalent problem of robust stabilization for reference tracking problem
uncertainty model and normalized uncertainty Delta
how to determine P0 and WT
- example: frequency response of plant with perturbation &ref(): File not found: "ex1020_1.m" at page "授業/制御工学特論2017";
- frequency response based procedure for P0 and WT &ref(): File not found: "ex1020_2.m" at page "授業/制御工学特論2017";
robust stabilization problem and equivalent problem
- design example and simulation &ref(): File not found: "ex1020_3.m" at page "授業/制御工学特論2017"; &ref(): File not found: "mod1020.mdl" at page "授業/制御工学特論2017";

%-- 2016/10/20 13:36 --%
ex1020_1
ex1020_2
ex1020_3
mod1020
c

#ref(): File not found: "2016.10.20-1.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.10.20-2.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.10.20-3.jpg" at page "授業/制御工学特論2017"

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）。

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[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

mixed sensitivity problem ---> (1) and (2) : proof
generalized plant for mixed senstivity problem
design example &ref(): File not found: "ex1027_1.m" at page "授業/制御工学特論2017"; minimize gamma by hand
gamma iteration by bisection method &ref(): File not found: "ex1027_2.m" at page "授業/制御工学特論2017";
nominal performance and robust performance &ref(): File not found: "ex1027_3.m" at page "授業/制御工学特論2017";
introduction of robust performance problem

%-- 2016/10/27 13:46 --%
ex1027_1
K
ex1027_2
ex1027_3

#ref(): File not found: "2016.10.27-1.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.10.27-2.jpg" at page "授業/制御工学特論2017"

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

#ref(): File not found: "2016.10.27-3.jpg" at page "授業/制御工学特論2017"

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。

↑

[] 2016.11.10 canceled†

↑

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

schedule2016 (modified) mini report and exam

review
- mixed sensitivity problem
```
ex1027_1
ex1027_2
ex1027_3
```
- robust performance problem c.f. the last whiteboard
an equivalent robust stability problem
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
proof of ||Delta hat||_inf <= 1
design example: &ref(): File not found: "ex1117_1.m" at page "授業/制御工学特論2017";
- 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

#ref(): File not found: "2016.11.17-1.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.11.17-2.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.11.17-3.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.11.17-4.jpg" at page "授業/制御工学特論2017"

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。

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

return of mini report #1
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
SVD: singular value decomposition
- definition
- meaning off the largest singular value
- 2-norm of vectors
- SVD for 2-by-2 real matrix &ref(): File not found: "ex1124_1.m" at page "授業/制御工学特論2017";

%-- 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

#ref(): File not found: "2016.11.24-1.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.11.24-2.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.11.24-3.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.11.24-4.jpg" at page "授業/制御工学特論2017"

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)†

review: conservative design with Delta tilde
scaled H infinity control problem
how to determine structure of scaling matrix
design example &ref(): File not found: "ex1201_1.m" at page "授業/制御工学特論2017";
```
ex1027_2
ex1117_1
gam_opt0 = gam_opt
K_opt0 = K_opt;
ex1201_1
gam_opt
```
mini exam #1

#ref(): File not found: "2016.12.01-1.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.12.01-2.jpg" at page "授業/制御工学特論2017"

%-- 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)†

return of mini exam #1;
review of the scaled H infinity control problem
effect of scaling &ref(): File not found: "ex1208_1.m" at page "授業/制御工学特論2017";
```
ex1027_2
ex1117_1
gam_opt0 = gam_opt
K_opt0 = K_opt;
ex1201_1
gam_opt
ex1208_1
```
mini report #2 ~~report2.pdf~~ please use modified file &ref(): File not found: "report2_fixed.pdf" at page "授業/制御工学特論2017"; ... You will have a mini exam #2 related to this report
introduction of a practical system: active noise control in duct
- experimental setup
  exp_apparatus1.jpg
  
  exp_apparatus2.jpg
- 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
  #ref(): File not found: "ex1208_2.m" at page "授業/制御工学特論2017"
  
  spk1.dat
  
  spk2.dat

%-- 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

#ref(): File not found: "2016.12.08-1.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.12.08-2.jpg" at page "授業/制御工学特論2017"

#ref(): File not found: "2016.12.08-3.jpg" at page "授業/制御工学特論2017"

↑

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

return of mini report #2
- mini exam #2
- schedule2016 Dec.20(added)
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 ?
design example
- determination of plant model(nominal plant and additive uncertainty weight)
  &ref(): File not found: "nominal.m" at page "授業/制御工学特論2017";
  &ref(): File not found: "subspace.m" at page "授業/制御工学特論2017"; ... replacement of n4sid in System Identification Toolbox (not provided in IPC)
  &ref(): File not found: "weight.m" at page "授業/制御工学特論2017";
- configuration of generalized plant and controller design by scaled H infinity control problem using one-dimensional search on the scaling d
  &ref(): File not found: "cont.m" at page "授業/制御工学特論2017";
- comparison of closed-loop gain characteristics with and without control
  &ref(): File not found: "compare.m" at page "授業/制御工学特論2017";
- result of control experiment
  &ref(): File not found: "result.dat" at page "授業/制御工学特論2017";
  &ref(): File not found: "compare_result.m" at page "授業/制御工学特論2017";
room 157 @ Dept. Mech. Bldg.2
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
- submit your controller.dat, controller_order.dat, and controller.mat at this page:participant list2016(download is also possible) not later than 28th(Wed) Dec
- Your login password will be e-mailed on Dec 16.
- You can send up to 5 controllers
- control experimental results will be uploaded here
- freqresp ... frequency response will be measured and uploaded everyday
how to improve the performance ?
- order of the nominal plant
- weighting for robust stability
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

ad_conv(&yz); // A/D input

// 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

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↑

[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
load result.dat
pwd
load data/result.dat;
plot(result(:,1),result(:,4),'-');
compare

#ref(): File not found: "2016.12.22-1.jpg" at page "授業/制御工学特論2017"

Q:ミニ試験2の返却は行いますか？
A:部屋まで来てもらえれば週明けに返します。