A second order system has a rise time of \(5~s\) and a overshoot of \(20 \%\). What is the settling time for this system?
Consider a ship of length \(150~m\) and design speed \(18~knots\) with Nomoto parameters \(T'=3\) and \(K'=-1.2\). A control engineer has designed a heading autopilot with the following dimensional gains:
Is the closed loop system stable? Use Routh-Hurwitz criteria to establish stability.
Consider a ship of length \(120~m\) and design speed \(20~knots\) with Nomoto parameters \(T'=2\) and \(K'=-1\). You are tasked with designing an underdamped PD heading autopilot for this ship that must satisfy the following performance metrics:
What should be the values of the proportional and derivtive gains that achieve the limiting values of the performance metrics?
Consider a ship of length \(100~m\) and design speed \(10~knots\) with Nomoto parameters \(T'=3\) and \(K'=-1\). For a PD heading autopilot with dimensional gains \(K_p = -1.0\) and \(K_d = -0.3\), determine the bandwidth of the closed loop system \(\Psi(s) / \Psi_d(s)\) and compare it with \(1 / T\). What can you infer?
A second order system has a rise time of \(5~s\) and a overshoot of \(20 \%\). What is the settling time for this system?
import numpy as np
import matplotlib.pyplot as plt
def solve_problem_01():
Tr2 = 5
Mp = 0.2
zeta = 0.6 * (1 - Mp)
wn = np.pi / (2 * Tr2)
Ts = 4 / (zeta * wn)
print(f"Settling time: {Ts:.2f} s")
solve_problem_01()Settling time: 26.53 sConsider a ship of length \(150~m\) and design speed \(18~knots\) with Nomoto parameters \(T'=3\) and \(K'=-1.2\). A control engineer has designed a heading autopilot with the following dimensional gains:
Is the closed loop system stable? Use Routh-Hurwitz criteria to establish stability.
import numpy as np
import matplotlib.pyplot as plt
def solve_problem_02():
L = 150
U = 18 * 0.5144
Tprime = 3
Kprime = -1.3
Kp = -1.0
Kd = -0.5
Ki = -0.1
T = Tprime * L / U
K = Kprime * U / L
a3 = T
a2 = 1 + K * Kd
a1 = K * Kp
a0 = K * Ki
b1 = (a2*a1 - a3*a0) / a2
b2 = 0
c1 = a0
c2 = 0
d1 = 0
print(f"Routh Array")
print(f"{a3:.3f} \t{a1:.3f}\n{a2:.3f} \t{a0:.3f}\n{b1:.3f} \t{b2:.3f}\n{c1:.3f} \t{c2:.3f}\n{d1:.3f}")
roots = np.roots(np.array([a3, a2, a1, a0]))
print("\nActual roots")
print(f"{roots[0]:.4f}\n{roots[1]:.4f}\n{roots[2]:.4f}")
solve_problem_02()Routh Array
48.600 0.080
1.040 0.008
-0.295 0.000
0.008 0.000
0.000
Actual roots
0.0151+0.0546j
0.0151-0.0546j
-0.0515+0.0000jConsider a ship of length \(120~m\) and design speed \(20~knots\) with Nomoto parameters \(T'=2\) and \(K'=-1\). You are tasked with designing an underdamped PD heading autopilot for this ship that must satisfy the following performance metrics:
What should be the values of the proportional and derivtive gains that achieve the limiting values of the performance metrics?
import numpy as np
import matplotlib.pyplot as plt
def solve_problem_03():
Tprime = 2
Kprime = -1
L = 120
U = 20 * 0.5144
T = Tprime * L / U
K = Kprime * U / L
Kp = T * np.pi**2 / (400 * K)
Kd = (2 * T / 5 - 1) / K
print(f"Kp: {Kp:.2f}, Kd:{Kd:.2f}")
solve_problem_03()Kp: -6.71, Kd:-97.18Consider a ship of length \(100~m\) and design speed \(10~knots\) with Nomoto parameters \(T'=3\) and \(K'=-1\). For a PD heading autopilot with dimensional gains \(K_p = -1.0\) and \(K_d = -0.3\), determine the bandwidth of the closed loop system \(\Psi(s) / \Psi_d(s)\) and compare it with \(1 / T\). What can you infer?
def solve_problem_04():
Tprime = 3
Kprime = -1
L = 100
U = 10 * 0.5144
T = Tprime * L / U
K = Kprime * U / L
Kp = -1.0
Kd = -0.3
omg = 10**(np.linspace(-4,1,501))
TF1 = K / (T * (1j*omg)**2 + (1j*omg))
TF2 = K*Kp / (T * (1j*omg)**2 + (1 + K*Kd) * (1j*omg) + K*Kp)
wn = np.sqrt(K*Kp/T)
zeta = (1 + K*Kd)/(2*np.sqrt(T*K*Kp))
w_bw2 = wn * np.sqrt(1 - 2* zeta**2 + np.sqrt(4*zeta**4 - 4*zeta**2 + 2))
print(f"wn:{wn:.4f}, w_bw:{w_bw2:.4f}")
plt.loglog(omg, np.abs(TF1),label="open loop TF")
plt.loglog(omg, np.abs(TF2),label="closed loop TF")
yl = plt.xlim()
plt.loglog(np.array([w_bw2, w_bw2]), yl, label="closed loop bandwidth")
plt.loglog(np.array([1/T, 1/T]), yl, label="open loop bandwidth")
plt.legend()
plt.show()
solve_problem_04()wn:0.0297, w_bw:0.0433