授業/制御工学特論2024

授業

Advanced Automation 2024†

latest lecture

[lecture #1] 2024.9.5 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%
  • schedule2024 (tentative)
  • map 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
  • ...

s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
K = 2
Tyr = K/(s-1+K)
step(Tyr)

• minute paper https://cera-e1.nagaokaut.ac.jp/ilias/

%-- 2024/09/05 13:40 --%
s = tf('s')
P = 1/(s-1)
pole(P)
impulse(P)
K = 2
Tyr = K/(s-1+K)
step(Tyr)

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[lecture #2] 2024.9.12 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 definition)
   • m file
2. system representation: Transfer Function(TF) / State-Space Representation (SSR)
   • example: mass-spring-damper system
   • definition 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 ex0912_1.m mod0912_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 ex0912_2.m mod0912_2.mdl
     • Gain Margin(GM); Phase Margin(PM)

%-- 2024/09/12 13:07 --%
a = 2
a + 1
t=[1 2 3]
t + 2
pwd
ex0912_1
sqrt(k/m)
sqrt(k/m)/(2*pi)

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[lecture #3] 2024.9.19 review of classical and modern control theory (3/3)†

1. LQR problem
   • controllability
   • cost function J >= 0
   • positive (semi-)definite matrices
   • solution of LQR problem
   • example ex0919_1.m mod0919_1.mdl
2. ARE and quadratic equation
   • scalar case (solve by hand)
   • matrix case lqr.pdf ≒ proof4.pdf (from B3「動的システムの解析と制御」)

%-- 2024/09/19 13:53 --%
ex0919_1
Uc
[B A*B]
ex0919_1
F

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[lecture #4] 2024.9.26 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 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)} \\ \| 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) \[ \begin{array}{|c||c|c|}\hline & \mbox{LQR}: f=-1+\sqrt{2} & \quad \quad H_\infty: f=1\quad\quad \\ \hline\hline J=\|z\|_2^2 & & \\ \hline \|T_{zw}\|_\infty & & \\ \hline \end{array} \]
2. 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
3. definition of H infinity norm (SISO)

   s = tf('s');
   G1 = 1/(s+1);
   bode(G1);
   norm(G1, 'inf')
   G2 = s/(s+1);
   bode(G2);
   norm(G2, 'inf')
   G3 = 1/(s^2 + 0.1*s + 1);
   bode(G3);
   norm(G3, 'inf')

4. definition of H infinity norm (SIMO)

   G4 = [1/(s+2); -1/(s+2)];
   norm(G4, 'inf') 

5. solve the problem by hand
6. solve the problem by tool(hinfsyn) ex0926_1.m

%-- 2024/09/26 14:04 --%
s = tf('s');
G1 = 1/(s+1);
bode(G1);
norm(G1, 'inf')
G2 = s/(s+1);
bode(G2);
norm(G2, 'inf')
bode(G1);
norm(G1, 'inf')
bode(G2);
norm(G2, 'inf')
G3 = 1/(s^2 + 0.1*s + 1);
bode(G3);
norm(G3, 'inf')
ctrlpref
bode(G3);
grid on
G4 = [1/(s+2); -1/(s+2)];
norm(G4, 'inf')
1/sqrt(2)

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[lecture #5] 2024.10.03 relation between LQR and H infinity control problem (2/2)†

1. cont.
   • solve the problem by hand
   • solve the problem by tool(hinfsyn) ex0926_1.m
2. complete the table in simple example
3. confirm the cost function J for both controllers by simulation mod1003.mdl
   • block diagram in the simulink model
   • how to approximate impulse disturbance with a step function
   • (unit) impulse disturbance resp. with zero initial condition = (unit) initial condition resp. with zero disturbance
4. confirm the closed-loop H infinity norm for both controllers by simulation
   • H infinity norm = L2 induced norm
   • review: steady-state response for sinusoidal input signal; how to choose the frequency such that the output amplitude is maximized ?
   • the worst-case disturbance w(t) for the simple example ?
5. 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

%-- 2024/10/03 13:17 --%
ex0926_1
sqrt(2-sqrt(2))
mod1003
h = 0.01
x0 = 0
f = 1
plot(t, x)
zz
plot(t, zz)
zz(end)
x0
zz(end)
x0 = 1
zz(end)
f = -1+sqrt(2)
zz(end)
h = 100
x0 = 0
f
sqrt(zz(end)/ww(end))
f = 1
sqrt(zz(end)/ww(end))

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[lecture #6] 2024.10.10 Mixed sensitivity problem 1/3†

1. outline: map_v1.1_mixedsens1.pdf
   • sensitivity function S and complementary sensitivity function T
2. H infinity control problem (general case)
   • with generalized plant G
   • including the state-feedback case
3. reference tracking problem
   • how to translate the condition (ii) into one with H infinity norm ?
   • corresponding generalized plant G ?
   • introduction of weighting function for sensitivity function in (ii)
4. design example ex1010_1.m ex1010_2.m
5. the small gain theorem
   • proof: Nyquist stability criterion

%-- 2024/10/10 13:47 --%
ex1010_1
pole(P)
ex1010_2
K_hinf
K_hinf.a
eig(K_hinf.a)
ex1010_2

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[lecture #7] 2024.10.17 Mixed sensitivity problem 2/3†

1. outline: from point to set map_v1.1_mixedsens2.pdf
2. the small gain theorem ... robust stability = H infinity norm condition
3. normalized uncertainty Delta
4. uncertainty model
5. simple example of plant set
   • given plant P tilde
     • frequency response of plant with perturbation ex1017_1.m
   • how to determine P0 and WT ?
     • frequency response based procedure for P0 and WT ex1017_2.m
6. robust stabilization problem and equivalent problem
   • design example and simulation ex1017_3.m mod1017.mdl

%-- 2024/10/17 13:36 --%
ex1017_1
ex1017_2
ex1017_1
ex1017_2
ex1017_1
ex1017_2
ex1017_3
mod1017
c
c = 0.8
c = 2
c = 1.5

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[lecture #8] 2024.10.24 Mixed sensitivity problem 3/3†

• review: map_v1.1_mixedsens2.pdf (1)performance optimization and (2)robust stabilization
• outline:
  1. how to design controllers considering both conditions in (1) and (2)
  2. gap between NP(nominal performance) and RP(robust performance)

1. mixed sensitivity problem => (1) and (2) : proof
2. generalized plant for mixed senstivity problem
3. design example ex1024_1.m minimize gamma by hand
4. gamma iteration by bisection method ex1024_2.m tradeoff between model robust stability and performance
5. intro. to RP: weak point of mixed sensitivity problem(problem of NP) ex1024_3.m

%-- 2024/10/24 13:41 --%
ex1024_1
K
ex1024_1
ex1024_2
gam
ex1024_2
gam
ex1024_2
gam
help hinfsyn
ex1024_3
ex1024_2
ex1024_3

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[lecture #9] 2024.10.31 robust performance problem 1/3†

• schedule2024

1. review
   • mixed sensitivity problem : N.P. but not R.P.
2. robust performance problem (R.P.), but can not be solved by tool
3. an equivalent robust stability (R.S.) problem to R.P.
   • (i) introduction of a fictitious uncertainty Delta_p (for performance)
   • (ii) for 2-by-2 uncertainty block Delta hat which includes Delta and Delta_p
4. 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); 1i, -1i]
     M'
     eig(M'*M)
     svd(M)

   • mini report #1 report1.pdf ... You will have a mini exam #1 related to this report
5. proof of ||Delta hat||_inf <= 1
6. design example: ex1031_1.m
   • robust performance is achieved but large gap
   • non structured uncertainty is considered ... the design problem is too conservative

%-- 2024/10/31 14:05 --%
M = [1/sqrt(2), 1/sqrt(2); 1i, -1i]
M'
eig(M'*M)
svd(M)
ex1031_1

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[lecture #10] 2024.11.7 Robust performance problem (2/3)†

1. return of mini report #1
2. SVD: singular value decomposition
   • definition
   • meaning of the largest singular value (a property and proof)
   • 2 norm of vectors (Euclidean norm)
   • SVD for 2-by-2 real matrix ex1107_1.m

%-- 2024/11/07 13:24 --%
M = [sqrt(2), sqrt(2); -1i/sqrt(2), 1i/sqrt(2)]
1i
j
i
[U, Sigma, V] = svd(M)
U'*U
format long e
U'*U
V*V'
ex1107_1

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[lecture #11] 2024.11.14 Robust performance problem (3/3)†

1. review : R.S. problems for structured and unstructured uncertainty
2. scaled H infinity control problem
3. relation between three problems
4. how to determine structure of scaling matrix
5. design example ex1114_1.m dgam.jpg

   ex1031_1
   gam2 = gam_opt
   ex1114_1
   gam_opt

6. mini exam #1 (10 min.)

%-- 2024/11/14 13:22 --%
ex1031_1
gam_opt
format long e
gam_opt
gam2 = gam_opt
ex1114_1
gam_opt
gam2

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[lecture #12] 2024.11.21 Robust performance problem (3/3) (cont.), Control system design for practical system (1/3)†

1. return of mini exam #1
2. review of scaling ex1121_1.m
3. mini report #2 report2.pdf
4. introduction of a practical system: active noise control in duct
   • experimental setup
     photo1.jpg photo2.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 ex1121_2.m spk1.dat spk2.dat
   • room 157 @ Dept. Mech. Bldg. 2

%-- 2024/11/21 13:03 --%
ex1121_1
gam2
gam3
ex1121_2
ctrlpref
ex1121_2
c0 = 346
L = 1.55
c0/(2*L)

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[lecture #13] 2024.11.28 [CANCELLED]†

Due to a hardware problem in our experimental environment, today's lecture is cancelled. I'm sorry for the inconvenience.

Please check the modified schedule at schedule2024

[lecture #13] 2024.12.5 Control system design for practical system (2/3)†

1. return of mini report #2; ... You will have a mini exam #2 related to this report next week
   • schedule2024
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
   • frequency response data of G can be used; how to handle modeling error of G ?
3. design example (modeling error for Gyu is only considered for simplicity)
   • frequency response experiment data
     spk1.dat
     spk2.dat
     
   • determination of plant model(nominal plant and additive uncertainty weight)
     nominal.m
     n4sid_replaced.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 and evaluation
     result.dat
     perf.m
4. 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 of control experiment
   3. Why is the performance of your system improved(or unfortunately deteriorated)?
   • due date: 6th(Mon) Jan 17:00
   • submit your report(pdf file) 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 list2024(download is also possible) not later than 24th(Tue) Dec
   • the system will be started until next lecture
   • You can send up to 10 controllers
   • control experimental results will be uploaded here
   • freqresp ... frequency response will be measured and uploaded everyday
5. how to improve the performance ?
   • order of the nominal plant
   • weighting for robust stability
6. specifications of the experimental system
   1. experimental equipments
      • loudspeakers: FOSTEX FE87E(10W)
      • 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
   2. 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

   3. 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)

   4. configuration of control experiment
      • disturbance signal w is specified as described in hinf.h and hinf_module.c:

        #define AMP 0.5 // amplitude for disturbance
        
        w = AMP * (2. * rand() / (RAND_MAX + 1.) - 1.); // uniform random number in [-AMP, AMP]
        
        da_conv(V_OFFSET + w, 0); // D/A output to noise source

      • control signal u is limited to [-3, 3] as specified in hinf.h and hinf_module.c:

        #define U_MAX 3.00
        
        if(u > U_MAX) u = U_MAX;
        if(u < -U_MAX) u = -U_MAX;

        u is set to 0 for t < 5(s). (controller is operated for 5 <= t < 10)

%-- 2024/12/05 13:31 --%
nominal
pwd
nominal
ls data
ls data/spk1.dat
save data/nominal.mat G0 G_g G0_g w
nominal
weight
cont
compare

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[lecture #14] 2024.12.12 Control system design for practical system (3/3)†

• review
  • experimental system
    • block diagram photo3.jpg
    • objective: z → 0 ... virtual open end (`absorption' was not correct)
  • frequency response experiment
    • AMP in freqresp.h has been decreased to avoid a saturation problem

      //#define AMP_CH0 0.5
      #define AMP_CH0 0.4
      //#define AMP_CH1 0.5
      #define AMP_CH1 0.4

  • design example

    (download and save spk1.dat and spk2.dat in data/)
    >> nominal
    >> weight
    >> cont
    >> compare
    (upload the controller, carry out control experiment, and download the result.dat)
    >> perf

• web based remote experiment system
  • usage; how to upload controller's
  • powered by prof. Takebe, National Institute of Technology, Nagaoka College
• supplemental explanations
  • room temperature is displayed and stored in temp.txt (Bosch Sensortec BME280 is used)
  • c2d() is used to discretize the resultant continuous-time controller in cont.m
  • You can send up to 10 controllers (don't fall into trial and error; think always about the reason)
  • no strict control objective is given ( there is a freedom to define what is good performance )
• preparation of your own controller(s) by using the remote experiment system
• mini exam #2

%-- 2024/12/12 12:58 --%
nominal
weight
cont
compare
perf

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[lecture #15] 2024.12.19 Control system design for practical system (cont.)†

• return of mini exam #2
• preparation of your own controller(s) by using the remote experiment system

%-- 2024/12/19 14:18 --%
pwd
cd ..
ls
perf
result
plot(t, result(:,2))
plot(result(:,1), result(:,2))
size(result)
plot(result(49000:,1), result(49000:,2))
plot(result(49000:end,1), result(49000:end,2))
plot(result(49000:end,1), result(49000:end,2), '.-')
plot(result(49000:end,1), result(49000:end,2), '-.')
plot(result(49000:end,1), result(49000:end,2), 'bo-')

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