Chapter 5¶
In [1]:
from control.matlab import *
import matplotlib.pyplot as plt
import numpy as np
plt.rcParams['font.family'] ='sans-serif' #使用するフォント
plt.rcParams['xtick.direction'] = 'in' #x軸の目盛線が内向き('in')か外向き('out')か双方向か('inout')
plt.rcParams['ytick.direction'] = 'in' #y軸の目盛線が内向き('in')か外向き('out')か双方向か('inout')
plt.rcParams['xtick.major.width'] = 1.0 #x軸主目盛り線の線幅
plt.rcParams['ytick.major.width'] = 1.0 #y軸主目盛り線の線幅
plt.rcParams['font.size'] = 10 #フォントの大きさ
plt.rcParams['axes.linewidth'] = 1.0 # 軸の線幅edge linewidth。囲みの太さ
plt.rcParams['mathtext.default'] = 'regular'
plt.rcParams['axes.xmargin'] = '0' #'.05'
plt.rcParams['axes.ymargin'] = '0.05'
plt.rcParams['savefig.facecolor'] = 'None'
plt.rcParams['savefig.edgecolor'] = 'None'
In [2]:
def linestyle_generator():
linestyle = ['-', '--', '-.', ':']
lineID = 0
while True:
yield linestyle[lineID]
lineID = (lineID + 1) % len(linestyle)
In [3]:
def plot_set(fig_ax, *args):
fig_ax.set_xlabel(args[0])
fig_ax.set_ylabel(args[1])
fig_ax.grid(ls=':')
if len(args)==3:
fig_ax.legend(loc=args[2])
In [4]:
def bodeplot_set(fig_ax, *args):
fig_ax[0].grid(which="both", ls=':')
fig_ax[0].set_ylabel('Gain [dB]')
fig_ax[1].grid(which="both", ls=':')
fig_ax[1].set_xlabel('$\omega$ [rad/s]')
fig_ax[1].set_ylabel('Phase [deg]')
if len(args) > 0:
fig_ax[1].legend(loc=args[0])
if len(args) > 1:
fig_ax[0].legend(loc=args[1])
垂直駆動アームの角度追従制御¶
In [5]:
g = 9.81 # 重力加速度[m/s^2]
l = 0.2 # アームの長さ[m]
M = 0.5 # アームの質量[kg]
mu = 1.5e-2 # 粘性摩擦係数[kg*m^2/s]
J = 1.0e-2 # 慣性モーメント[kg*m^2]
P = tf( [0,1], [J, mu, M*g*l] )
ref = 30 # 目標角度 [deg]
P制御¶
In [6]:
LS = linestyle_generator()
fig, ax = plt.subplots(figsize=(3, 2.3))
kp = (0.5, 1, 2)
for i in range(3):
K = tf([0, kp[i]], [0, 1])
Gyr = feedback(P*K, 1)
y,t = step(Gyr, np.arange(0, 2, 0.01))
pltargs = {'ls': next(LS), 'label': '$k_P$='+str(kp[i])}
ax.plot(t, y*ref, **pltargs)
ax.axhline(ref, color="k", linewidth=0.5)
plot_set(ax, 't', 'y', 'best')
ax.set_xlim(0, 2)
ax.set_ylim(0, 50)
# fig.savefig("pcont.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
Out[6]:
In [7]:
LS = linestyle_generator()
fig, ax = plt.subplots(2, 1, figsize=(4, 3.5))
for i in range(len(kp)):
K = tf([0, kp[i]], [0, 1])
Gyr = feedback(P*K, 1)
gain, phase, w = bode(Gyr, logspace(-1,2), Plot=False)
pltargs = {'ls': next(LS), 'label': '$k_P$='+str(kp[i])}
ax[0].semilogx(w, 20*np.log10(gain), **pltargs)
ax[1].semilogx(w, phase*180/np.pi, **pltargs)
bodeplot_set(ax, 'lower left')
ax[1].set_ylim(-190,10)
ax[1].set_yticks([-180,-90,0])
fig.tight_layout()
# fig.savefig("pcont_bode.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
PD制御¶
In [8]:
kp = 2
kd = (0, 0.1, 0.2)
LS = linestyle_generator()
fig, ax = plt.subplots(figsize=(3, 2.3))
for i in range(3):
K = tf([kd[i], kp], [0, 1])
Gyr = feedback(P*K, 1)
y,t = step(Gyr,np.arange(0, 2, 0.01))
pltargs = {'ls': next(LS), 'label': '$k_D$='+str(kd[i])}
ax.plot(t, y*ref, **pltargs)
ax.axhline(ref, color="k", linewidth=0.5)
plot_set(ax, 't', 'y', 'best')
ax.set_xlim(0, 2)
ax.set_ylim(0, 50)
#fig.savefig("pdcont.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
Out[8]:
In [9]:
LS = linestyle_generator()
fig, ax = plt.subplots(2, 1, figsize=(4, 3.5))
for i in range(3):
K = tf([kd[i], kp], [0,1])
Gyr = feedback(P*K, 1)
gain, phase, w = bode(Gyr, logspace(-1,2), dB=True, Plot=False)
pltargs = {'ls': next(LS), 'label': '$k_D$='+str(kd[i])}
ax[0].semilogx(w, 20*np.log10(gain), **pltargs)
ax[1].semilogx(w, phase*180/np.pi, **pltargs)
bodeplot_set(ax, 'lower left')
ax[1].set_ylim(-190,10)
ax[1].set_yticks([-180,-90,0])
fig.tight_layout()
#fig.savefig("pdcont_bode.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
PID制御¶
In [10]:
kp = 2
kd = 0.1
ki = (0, 5, 10)
LS = linestyle_generator()
fig, ax = plt.subplots(figsize=(3, 2.3))
for i in range(3):
K = tf([kd, kp, ki[i]], [1, 0])
Gyr = feedback(P*K, 1)
y, t = step(Gyr, np.arange(0, 2, 0.01))
pltargs = {'ls': next(LS), 'label': '$k_I$='+str(ki[i])}
ax.plot(t, y*ref, **pltargs)
ax.axhline(ref, color="k", linewidth=0.5)
plot_set(ax, 't', 'y', 'upper left')
ax.set_xlim(0, 2)
ax.set_ylim(0,50)
# fig.savefig("pidcont.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
Out[10]:
In [11]:
LS = linestyle_generator()
fig, ax = plt.subplots(2, 1, figsize=(4, 3.5))
for i in range(3):
K = tf([kd, kp, ki[i]], [1, 0])
Gyr = feedback(P*K,1)
gain, phase, w = bode(Gyr, logspace(-1,2), Plot=False)
pltargs = {'ls': next(LS), 'label': '$k_I$='+str(ki[i])}
ax[0].semilogx(w, 20*np.log10(gain), **pltargs)
ax[1].semilogx(w, phase*180/np.pi, **pltargs)
bodeplot_set(ax, 'best')
ax[1].set_ylim(-190,10)
ax[1].set_yticks([-180,-90,0])
fig.tight_layout()
#fig.savefig("pidcont_bode.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
練習問題（外乱抑制）¶
In [12]:
LS = linestyle_generator()
fig, ax = plt.subplots(figsize=(3, 2.3))
for i in range(3):
K = tf([kd, kp, ki[i]], [1, 0])
Gyd = feedback(P, K)
y, t = step(Gyd, np.arange(0, 2, 0.01))
pltargs = {'ls': next(LS), 'label': '$k_I$='+str(ki[i])}
ax.plot(t, y, **pltargs)
plot_set(ax, 't', 'y', 'center right')
ax.set_xlim(0, 2)
ax.set_ylim(-0.05, 0.5)
# fig.savefig("pidcont_dis.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
Out[12]:
In [13]:
LS = linestyle_generator()
fig, ax = plt.subplots(2, 1, figsize=(4, 3.5))
for i in range(3):
K = tf([kd, kp, ki[i]], [1,0])
Gyd = feedback(P, K)
gain, phase, w = bode(Gyd, logspace(-1,2), Plot=False)
pltargs = {'ls': next(LS), 'label': '$k_I$='+str(ki[i])}
ax[0].semilogx(w, 20*np.log10(gain), **pltargs)
ax[1].semilogx(w, phase*180/np.pi, **pltargs)
bodeplot_set(ax, 'best')
ax[1].set_ylim(-190,100)
ax[1].set_yticks([-180,-90, 0, 90])
fig.tight_layout()
# fig.savefig("pidcont_dis_bode.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
2自由度制御¶
In [14]:
kp = 2
ki = 10
kd = 0.1
K1 = tf([kd, kp, ki], [1, 0])
K2 = tf([0, ki], [kd, kp, ki])
K3 = tf([kp, ki], [kd, kp, ki])
PI-D制御¶
In [15]:
Gyz = feedback(P*K1, 1)
Td = np.arange(0, 2, 0.01)
r = 1*(Td>0)
z, t, _ = lsim(K3, r, Td, 0)
fig, ax = plt.subplots(1, 2, figsize=(6, 2.3))
y, _, _ = lsim(Gyz, r, Td, 0)
ax[0].plot(t, r*ref)
ax[1].plot(t, y*ref, ls='--', label='PID')
y, _, _ = lsim(Gyz, z, Td, 0)
ax[0].plot(t, z*ref)
ax[1].plot(t, y*ref, label='PI-D')
plot_set(ax[0], 't', 'r')
ax[0].set_xlim(0, 2)
ax[0].set_ylim(0,50)
ax[1].axhline(ref, color="k", linewidth=0.5)
plot_set(ax[1], 't', 'y', 'best')
ax[1].set_xlim(0, 2)
ax[1].set_ylim(0,50)
fig.tight_layout()
# fig.savefig("2deg1.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
制御入力の計算
PID制御では，$G_{ur}$ がインプロパーになるので，擬似微分を用いて計算する
In [16]:
tau = 0.0000001 # ローパスフィルタ
Klp = tf([kd, 0], [tau, 1]) # 擬似微分器
Ktau = tf([kp, ki], [1, 0]) + Klp
Gyz = feedback(P*Ktau, 1)
Guz = Ktau/(1+P*Ktau)
Td = np.arange(0, 2, 0.01)
r = 1*(Td>0)
z, t, _ = lsim(K3, r, Td, 0)
fig, ax = plt.subplots(1, 2, figsize=(6, 2.3))
u, _, _ = lsim(Guz, r, Td, 0)
ax[0].plot(t, u, ls='--', label='PID')
u, _, _ = lsim(Guz, z, Td, 0)
ax[0].plot(t, u, label='PI-D')
y, _, _ = lsim(Gyz, r, Td, 0)
ax[1].plot(t, y*ref, ls='--', label='PID')
y, _, _ = lsim(Gyz, z, Td, 0)
ax[1].plot(t, y*ref, label='PI-D')
ax[0].set_xlim(0, 0.5)
ax[1].axhline(ref, color="k", linewidth=0.5)
plot_set(ax[0], 't', 'u', 'best')
plot_set(ax[1], 't', 'y', 'best')
ax[1].set_xlim(0, 2)
ax[1].set_ylim(0,50)
fig.tight_layout()
# fig.savefig("2deg1_u.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
I-PD制御¶
In [17]:
Gyz = feedback(P*K1, 1)
Td = np.arange(0, 2, 0.01)
r = 1*(Td>0)
z, t, _ = lsim(K2, r, Td, 0)
fig, ax = plt.subplots(1, 2, figsize=(6, 2.3))
y, _, _ = lsim(Gyz, r, Td, 0)
ax[0].plot(t, r*ref)
ax[1].plot(t, y*ref, ls='--', label='PID')
y, _, _ = lsim(Gyz, z, Td, 0)
ax[0].plot(t, z*ref)
ax[1].plot(t, y*ref, label='I-PD')
plot_set(ax[0], 't', 'r')
ax[0].set_xlim(0, 2)
ax[0].set_ylim(0,50)
ax[1].axhline(ref, linewidth=0.5)
plot_set(ax[1], 't', 'y', 'best')
ax[1].set_xlim(0, 2)
ax[1].set_ylim(0,50)
fig.tight_layout()
# fig.savefig("2deg2.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
In [18]:
tau = 0.0000001 # ローパスフィルタ
Klp = tf([kd, 0], [tau, 1]) # 擬似微分器
Ktau = tf([kp, ki], [1, 0]) + Klp
Gyz = feedback(P*Ktau, 1)
Guz = Ktau/(1+P*Ktau)
Td = np.arange(0, 2, 0.01)
r = 1*(Td>0)
z, t, _ = lsim(K2, r, Td, 0)
fig, ax = plt.subplots(1, 2, figsize=(6, 2.3))
u, _, _ = lsim(Guz, r, Td, 0)
ax[0].plot(t, u, ls='--', label='PID')
u, _, _ = lsim(Guz, z, Td, 0)
ax[0].plot(t, u, label='I-PD')
y, _, _ = lsim(Gyz, r, Td, 0)
ax[1].plot(t, y*ref, ls='--', label='PID')
y, _, _ = lsim(Gyz, z, Td, 0)
ax[1].plot(t, y*ref, label='I-PD')
ax[0].set_xlim(0, 0.5)
ax[1].axhline(ref, color="k", linewidth=0.5)
plot_set(ax[0], 't', 'u', 'best')
plot_set(ax[1], 't', 'y', 'best')
ax[1].set_xlim(0, 2)
ax[1].set_ylim(0,50)
fig.tight_layout()
# fig.savefig("2deg2_u.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
限界感度法¶
無駄時間要素¶
In [19]:
# num_delay, den_delay = pade( 0.005, 5)
num_delay, den_delay = pade( 0.005, 1)
Pdelay = P * tf(num_delay, den_delay)
Pdelay
print(pole(Pdelay))
In [20]:
kvect = np.arange(0, 5, 0.001)
# rlist, klist = rlocus(Pdelay)
rlist, klist = rlocus(Pdelay, kvect, Plot=False)
fig, ax = plt.subplots(figsize=(3,3))
ax.plot(rlist.real, rlist.imag)
ax.set_xlim(-3, 1)
ax.grid(ls=':')
In [21]:
rlist, klist = rlocus(P, kvect, Plot=False)
fig, ax = plt.subplots(figsize=(3,3))
ax.plot(rlist.real, rlist.imag)
ax.set_xlim(-3, 1)
ax.grid(ls=':')
チューニング¶
In [22]:
fig, ax = plt.subplots(figsize=(3, 2.3))
kp0 = 2.9
K = tf([0, kp0], [0, 1])
Gyr = feedback(Pdelay*K, 1)
y,t = step(Gyr, np.arange(0, 2, 0.01))
ax.plot(t, y*ref, color='k')
ax.axhline(ref, color='k', linewidth=0.5)
ax.set_xlim(0, 2)
ax.set_ylim(0, 50)
plot_set(ax, 't', 'y')
# fig.savefig("tune_zn.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
In [23]:
kp = [0, 0]
ki = [0, 0]
kd = [0, 0]
Rule = ['', '']
T0 = 0.3
# Classic ZN
Rule[0] = 'Classic'
kp[0] = 0.6 * kp0
ki[0] = kp[0] / (0.5 * T0)
kd[0] = kp[0] * (0.125 * T0)
# No overshoot
Rule[1] = 'No Overshoot'
kp[1] = 0.2 * kp0
ki[1] = kp[1] / (0.5 * T0)
kd[1] = kp[1] * (0.33 * T0)
LS = linestyle_generator()
fig, ax = plt.subplots(figsize=(3, 2.3))
for i in range(2):
K = tf([kd[i], kp[i], ki[i]], [1, 0])
Gyr = feedback(Pdelay*K, 1)
y, t = step(Gyr, np.arange(0, 2, 0.01))
ax.plot(t, y*ref, ls=next(LS), label=Rule[i])
print(Rule[i])
print('kP=', kp[i])
print('kI=', ki[i])
print('kD=', kd[i])
print('------------------')
ax.axhline(ref, color="k", linewidth=0.5)
ax.set_xlim(0, 2)
ax.set_ylim(0, 50)
plot_set(ax, 't', 'y', 'best')
# fig.savefig("tune_zn_result.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
モデルマッチング¶
In [24]:
import sympy as sp
s = sp.Symbol('s')
kp, kd, ki = sp.symbols('k_p k_d k_i')
Mgl, mu, J = sp.symbols('Mgl mu J')
sp.init_printing()
G = (kp*s+ki)/(J*s**3 +(mu+kd)*s**2 + (Mgl + kp)*s + ki)
sp.series(1/G, s, 0, 4)
Out[24]:
In [25]:
import sympy as sp
z, wn = sp.symbols('zeta omega_n')
kp, kd, ki = sp.symbols('k_p k_d k_i')
Mgl,mu,J = sp.symbols('Mgl mu J')
sp.init_printing()
f1 = Mgl/ki-2*z/wn
f2 = (mu+kd)/ki-Mgl*kp/(ki**2)-1/(wn**2)
f3 = J/ki-kp*(mu+kd)/(ki**2)+Mgl*kp**2/(ki**3)
sp.solve([f1, f2, f3],[kp, kd, ki])
Out[25]:
In [26]:
g = 9.81 # 重力加速度[m/s^2]
l = 0.2 # アームの長さ[m]
M = 0.5 # アームの質量[kg]
mu = 1.5e-2 # 粘性摩擦係数
J = 1.0e-2 # 慣性モーメント
P = tf( [0,1], [J, mu, M*g*l] )
ref = 30 # 目標角度 [deg]
In [27]:
omega_n = 15
zeta = (1, 1/np.sqrt(2))
Label = ('Binomial coeff.', 'Butterworth')
LS = linestyle_generator()
fig, ax = plt.subplots(figsize=(3, 2.3))
for i in range(2):
Msys = tf([0,omega_n**2], [1,2*zeta[i]*omega_n,omega_n**2])
y, t = step(Msys, np.arange(0, 2, 0.01))
ax.plot(t, y*ref, ls=next(LS), label=Label[i])
ax.set_xlim(0, 2)
ax.set_ylim(0, 50)
plot_set(ax, 't', 'y', 'best')
# fig.savefig("ref_model_2nd.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
In [28]:
omega_n = 15
zeta = 0.707
Msys = tf([0,omega_n**2], [1,2*zeta*omega_n,omega_n**2])
kp = omega_n**2*J
ki = omega_n*M*g*l/(2*zeta)
kd = 2*zeta*omega_n*J + M*g*l/(2*zeta*omega_n) - mu
print('kP=', kp)
print('kI=', ki)
print('kD=', kd)
Gyr = tf([kp,ki], [J, mu+kd, M*g*l+kp, ki])
yM, tM = step(Msys, np.arange(0, 2, 0.01))
y, t = step(Gyr, np.arange(0, 2, 0.01))
fig, ax = plt.subplots(figsize=(3, 2.3))
ax.plot(tM, yM*ref, label='M', ls = '-.')
ax.plot(t, y*ref, label='Gyr')
ax.set_xlim(0, 2)
ax.set_ylim(0, 50)
plot_set(ax, 't', 'y', 'best')
# fig.savefig("model_match.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
In [29]:
alpha1 = (3, 2, 2.15)
alpha2 = (3, 2, 1,75)
omega_n = 15
Label = ('Binomial coeff.', 'Butterworth', 'ITAE')
LS = linestyle_generator()
fig, ax = plt.subplots(figsize=(3, 2.3))
for i in range(3):
M = tf([0, omega_n**3], [1, alpha2[i]*omega_n, alpha1[i]*omega_n**2, omega_n**3])
y,t = step(M, np.arange(0, 2, 0.01))
ax.plot(t, y*ref, ls=next(LS), label=Label[i])
ax.set_xlim(0, 2)
ax.set_ylim(0, 50)
plot_set(ax, 't', 'y', 'best')
# fig.savefig("ref_model_3rd.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
状態フィードバック¶
極配置¶
In [30]:
from control.matlab import *
import scipy
A = '0 1; -4 5'
B = '0; 1'
C = '1 0 ; 0 1'
D = '0; 0'
P = ss(A, B, C, D)
print(P)
In [31]:
np.linalg.eigvals(P.A)
Out[31]:
In [32]:
Pole = [-1, -1]
F = -acker(P.A, P.B, Pole)
F
Out[32]:
In [33]:
scipy.linalg.eigvals(P.A+P.B*F)
Out[33]:
In [34]:
np.linalg.eigvals(P.A+P.B*F)
Out[34]:
In [35]:
Acl = P.A + P.B*F
Pfb = ss(Acl, P.B, P.C, P.D)
Td = np.arange(0, 5, 0.01)
X0 = [-0.3, 0.4]
x, t = initial(Pfb, Td, X0) #ゼロ入力応答
fig, ax = plt.subplots(figsize=(3, 2.3))
ax.plot(t, x[:,0], label = '$x_1$')
ax.plot(t, x[:,1], ls = '-.', label = '$x_2$')
plot_set(ax, 't', 'x', 'best')
# fig.savefig("sf_pole.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
最適レギュレータ¶
In [36]:
Q = [ [100, 0], [0, 1]]
R = 1
F, X, E = lqr(P.A, P.B, Q, R)
F = -F
print('--- フィードバックゲイン ---')
print(F)
print(-(1/R)*P.B.T*X)
print('--- 閉ループ極 ---')
print(E)
print(np.linalg.eigvals(P.A+P.B*F))
In [37]:
FF = - (1/R)*(P.B.T)*X
FF
Out[37]:
In [38]:
X, E, F = care(P.A, P.B, Q, R)
F = -F
print('--- フィードバックゲイン ---')
print(F)
print('--- 閉ループ極 ---')
print(E)
In [39]:
Acl = P.A + P.B*F
Pfb = ss(Acl, P.B, P.C, P.D)
tdata = np.arange(0, 5, 0.01)
xini, tini = initial(Pfb, tdata, [-0.3, 0.4]) #ゼロ入力応答
fig, ax = plt.subplots(figsize=(3, 2.3))
ax.plot(tini, xini[:,0], label = '$x_1$')
ax.plot(tini, xini[:,1], ls = '-.', label = '$x_2$')
ax.set_xlabel('t')
ax.set_ylabel('x')
ax.grid(ls=':')
ax.legend()
# fig.savefig("sf_lqr.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
Out[39]:
円条件（最適レギュレータのロバスト性）¶
In [40]:
A = '0 1; -4 5'
B = '0; 1'
C = '1 0 ; 0 1'
D = '0; 0'
P = ss(A, B, C, D)
L = ss(P.A, P.B, -F, 0)
print(L)
In [41]:
import matplotlib.patches as patches
fig, ax = plt.subplots(figsize=(3, 3))
x, y, w = nyquist(L, logspace(-2,3,1000), Plot=False)
ax.plot(x, y)
ax.plot(x, -y, ls='--')
ax.scatter(-1, 0, color='k')
ax.grid(ls=':')
ax.set_xlim(-2.5, 2.5)
ax.set_ylim(-2.5, 2.5)
c = patches.Circle(xy=(-1, 0), radius=1, fill=False, ec='k')
ax.add_patch(c)
fig.tight_layout()
開ループ系のナイキスト軌跡が (-1, 0j) を中心とする単位円の中に入らない．
これにより，位相余裕が 60 [deg] 以上であることが保証される．
ハミルトン行列¶
In [42]:
H1 = np.c_[P.A, -P.B*(1/R)*P.B.T]
H2 = np.c_[ Q, P.A.T]
H = np.r_[H1, -H2]
eigH = np.linalg.eigvals(H)
print(eigH)
print('--- ハミルトン行列の安定固有値 ---')
eigH_stable = [ i for i in eigH if i < 0]
print(eigH_stable)
F = -acker(P.A, P.B, eigH_stable)
print('--- フィードバックゲイン ---')
print(F)
In [43]:
H1 = np.hstack((P.A, -P.B*(1/R)*P.B.T))
H2 = np.hstack(( Q, P.A.T))
H = np.vstack((H1, -H2))
eigH = np.linalg.eigvals(H)
print(eigH)
print('--- ハミルトン行列の安定固有値 ---')
eigH_stable = [ i for i in eigH if i < 0]
print(eigH_stable)
F = -acker(P.A, P.B, eigH_stable)
print('--- フィードバックゲイン ---')
print(F)
In [44]:
scipy.linalg.eigvals(H)
Out[44]:
In [45]:
E
Out[45]:
積分サーボ系¶
In [46]:
A = '0 1; -4 5'
B = '0; 1'
C = '1 0 ; 0 1'
D = '0; 0'
P = ss(A, B, C, D)
print(P)
In [47]:
Pole = [-1, -1]
F = -acker(P.A, P.B, Pole)
F
Out[47]:
In [48]:
Acl = P.A + P.B*F
Pfb = ss(Acl, P.B, P.C, P.D)
Td = np.arange(0, 8, 0.01)
Ud = 0.2 * (Td>0)
x, t, _ = lsim(Pfb, Ud, Td, [-0.3, 0.4])
fig, ax = plt.subplots(figsize=(3, 2.3))
ax.plot(t, x[:,0], label = '$x_1$')
ax.plot(t, x[:,1], ls = '-.', label = '$x_2$')
plot_set(ax, 't', 'x', 'best')
# fig.savefig("sf_dis.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
In [49]:
A = '0 1; -4 5'
B = '0; 1'
C = '1 0'
D = '0'
P = ss(A, B, C, D)
print(P)
In [50]:
Abar = np.r_[ np.c_[P.A, np.zeros((2,1))], -np.c_[ P.C, 0 ] ]
Bbar = np.c_[ P.B.T, 0 ].T
Cbar = np.c_[ P.C, 0 ]
In [51]:
Pole = [-1, -1, -5]
F = -acker(Abar, Bbar, Pole)
F
Out[51]:
In [52]:
# Acl = P.A + P.B*F[0,0:2]
Acl = Abar + Bbar*F
Pfb = ss(Acl, Bbar, np.eye(3), np.zeros((3,1)))
Td = np.arange(0, 8, 0.01)
Ud = 0.2 * (Td>0)
x, t, _ = lsim(Pfb, Ud, Td, [-0.3, 0.4, 0])
fig, ax = plt.subplots(figsize=(3, 2.3))
ax.plot(t, x[:,0], label = '$x_1$')
ax.plot(t, x[:,1], ls = '-.',label = '$x_2$')
# ax.plot(t, Ud, c='k')
plot_set(ax, 't', 'x', 'best')
# fig.savefig("servo.pdf", transparent=True, bbox_inches="tight", pad_inches=0.0)
可制御性，可観測性¶
In [53]:
A = '0 1; -4 5'
B = '0; 1'
C = '1 0'
D = '0'
P = ss(A, B, C, D)
print(P)
In [54]:
Uc = ctrb(P.A, P.B)
print('Uc=\n',Uc)
print('det(Uc)=', np.linalg.det(Uc))
print('rank(Uc)=', np.linalg.matrix_rank(Uc))
In [55]:
Uo = obsv(P.A, P.C)
print('Uo=\n', Uo)
print('det(Uo)=', np.linalg.det(Uo))
print('rank(Uo)=', np.linalg.matrix_rank(Uo))