def G(s):
G = 0.01**(-5*s)/((s + 1.72e-4)*(4.32*s + 1))*np.matrix([[-34.54*(s + 0.0572), 1.913], [-30.22*s, -9.188*(s + 6.95e-4)]])
return G
I1 = np.asmatrix(np.identity(2))
I2 = np.matrix([[0, 1], [1, 0]])
plt.figure('Figure 3.8')
plt.subplot(1, 2, 1)
plt.title('(a) Magnitude of RGA elements')
rga_plot(G, -5, 1, [None, None, 0, 1], plot_type='all')
plt.text(3e-4, 0.8, '|$\lambda$$_1$$_2$| = |$\lambda$$_2$$_1$|', fontsize=15)
plt.text(3e-4, 0.2, '|$\lambda$$_1$$_1$| = |$\lambda$$_2$$_2$|', fontsize=15)
plt.subplot(1, 2, 2)
plt.title('(b) RGA numbers')
rga_nm_plot(G, [I1, I2], ['Diagonal pairing','Off-diagonal pairing'], -5, 1, plot_type='all')
plt.show()
# The section below demonstrates more utilsplot functions
plt.figure('RGA per element')
rga_plot(G, -5, 1, plot_type='elements')
plt.figure('RGA per output')
rga_plot(G, -5, 1, plot_type='outputs')
plt.figure('RGA per input')
rga_plot(G, -5, 1, plot_type='inputs')
plt.show()
# a) Controllability analysis
G_poles = G.poles()
G_rhp_poles = utils.RHPonly(G_poles)
# G has stable poles, -10 with multiplicity of 2 and -1
G_zeros = G.zeros()
G_rhp_zeros = utils.RHPonly(G_zeros)
# G has a LHP zero @ -10 and a RHP zero @ 0.1
# Zero at 0.1 limits bandwidth to wB* < 0.05
pairing1 = np.asmatrix(np.eye(2))
pairing2 = np.asmatrix(pairing1[[1, 0]])
utilsplot.rga_nm_plot(G, pairing_list=[pairing1, pairing2],
pairing_names=['Diagonal pairing', 'Off diagonal pairing'],
w_start=-3, w_end=3, points=1000, plot_type='all')
# RGA Number of Off-diagonal pairing exhibits the lowest RGA Number for w < wB*
# b) Disturbance rejection and input saturation
w = np.logspace(-3, 3, 1000)
dim = G(0).shape[0]
Sv_G = np.zeros((len(w), dim))
Sv_G_min = np.zeros((len(w), 1))
Sv_G_max = np.zeros((len(w), 1))
wB_index = 0
for i in range(len(w)):
_, Sv_G[i, :], _ = np.linalg.svd(G(1j*w[i]))
Sv_G_min[i] = np.min(Sv_G[i, :])
G_poles = G.poles()
G_rhp_poles = utils.RHPonly(G_poles)
# G has stable poles, -10 with multiplicity of 2 and -1
G_zeros = G.zeros()
G_rhp_zeros = utils.RHPonly(G_zeros)
# G has a LHP zero @ -10 and a RHP zero @ 0.1
# Zero at 0.1 limits bandwidth to wB* < 0.05
pairing1 = np.asmatrix(np.eye(2))
pairing2 = np.asmatrix(pairing1[[1, 0]])
utilsplot.rga_nm_plot(
G,
pairing_list=[pairing1, pairing2],
pairing_names=['Diagonal pairing', 'Off diagonal pairing'],
w_start=-3,
w_end=3,
points=1000,
plot_type='all')
# RGA Number of Off-diagonal pairing exhibits the lowest RGA Number for w < wB*
# b) Disturbance rejection and input saturation
w = np.logspace(-3, 3, 1000)
dim = G(0).shape[0]
Sv_G = np.zeros((len(w), dim))
Sv_G_min = np.zeros((len(w), 1))
Sv_G_max = np.zeros((len(w), 1))
wB_index = 0
for i in range(len(w)):
G_poles = G.poles()
G_rhp_poles = G_poles[np.real(G_poles) >= 0]
# G has stable poles, -10 with multiplicity of 2 and -1
G_zeros = G.zeros()
G_rhp_zeros = G_zeros[np.real(G_zeros) >= 0]
# G has a LHP zero @ -10 and a RHP zero @ 0.1
# Zero at 0.1 limits bandwidth to wB* < 0.05
pairing1 = np.asmatrix(np.eye(2))
pairing2 = np.asmatrix(pairing1[[1, 0]])
utilsplot.rga_nm_plot(
G,
pairing_list=[pairing1, pairing2],
pairing_names=["Diagonal pairing", "Off diagonal pairing"],
w_start=-3,
w_end=3,
points=1000,
plot_type="all",
)
# RGA Number of Off-diagonal pairing exhibits the lowest RGA Number for w < wB*
# b) Disturbance rejection and input saturation
w = np.logspace(-3, 3, 1000)
dim = G(0).shape[0]
Sv_G = np.zeros((len(w), dim))
Sv_G_min = np.zeros((len(w), 1))
Sv_G_max = np.zeros((len(w), 1))
wB_index = 0