NeuralBlitz Control Systems
What I Do
I provide comprehensive control systems tools including PID control, state-space analysis, stability criteria, observer design, and robust control methods for engineering applications.
When to Use Me
- PID controller tuning
- State-space controller design
- Stability analysis
- Observer/Kalman filter design
- Robust and adaptive control
- Frequency response analysis
Core Concepts
- PID Control: Proportional, integral, derivative
- State-Space: Controllability, observability
- Stability: Bode, Nyquist, Routh-Hurwitz
- Control Design: Pole placement, LQR
- Observers: Luenberger, Kalman filter
- Robust Control: H-infinity, mu-synthesis
- Adaptive Control: MRAC, self-tuning
- Digital Control: Discrete systems, z-transform
Code Examples
PID Control
import numpy as np
def pid_output(Kp, Ki, Kd, error, integral, prev_error, dt):
P = Kp * error
I = Ki * integral * dt
D = Kd * (error - prev_error) / dt
return P + I + D
def pid_tuning_ziegler_nichols(Ku, Tu):
Kp = 0.6 * Ku
Ki = 1.2 * Ku / Tu
Kd = 0.075 * Ku * Tu
return Kp, Ki, Kd
def anti_windup(limit, integral, error, Kp, Kb):
if abs(integral) > limit:
integral = integral - Kb * error
return integral
def pid_discrete(Kp, Ki, Kd, e_k, e_k1, e_k2, u_k1, Ts):
u_k = u_k1 + Kp * (e_k - e_k1) + Ki * Ts / 2 * (e_k + e_k1) + Kd / Ts * (e_k - 2*e_k1 + e_k2)
return u_k
def auto_tuning_relay_feedback(relay_amplitude, period, ultimate_gain):
Ku = 4 * relay_amplitude / (np.pi * amplitude)
Tu = period
return Ku, Tu
Kp, Ki, Kd = 2.5, 0.5, 0.1
u = pid_output(Kp, Ki, Kd, error=0.5, integral=1.0, prev_error=0.6, dt=0.01)
print(f"PID output: {u:.4f}")
State-Space Analysis
from numpy.linalg import matrix_rank, eig
def controllability_matrix(A, B):
n = A.shape[0]
C = B
for i in range(1, n):
C = np.hstack([C, np.linalg.matrix_power(A, i) @ B])
return C
def observability_matrix(A, C):
n = A.shape[0]
O = C
for i in range(1, n):
O = np.vstack([O, C @ np.linalg.matrix_power(A, i)])
return O
def pole_placement(A, B, desired_poles):
n = A.shape[0]
K = place(A, B, desired_poles)
return K
def lyapunov_stability(A, Q):
P = solve_continuous_lyapunov(A.T, -Q)
return P
def modal_analysis(A):
eigenvalues, eigenvectors = np.linalg.eig(A)
return eigenvalues, eigenvectors
A = np.array([[0, 1], [-2, -3]])
B = np.array([[0], [1]])
C = controllability_matrix(A, B)
rank = matrix_rank(C)
print(f"Controllability: {'controllable' if rank == 2 else 'not controllable'}")
Stability Analysis
def routh_hurwitz(array):
n = len(array)
s = [[0] * ((n + 1) // 2) for _ in range(n + 1)]
s[0] = array[::2]
s[1] = array[1::2]
for i in range(2, n + 1):
for j in range((n + 1) // 2):
if i % 2 == 0:
s[i][j] = s[i-2][j+1] - s[i-1][j] * s[i-2][0] / s[i-1][0]
else:
s[i][j] = s[i-2][j] - s[i-1][j] * s[i-2][0] / s[i-1][0]
return s
def nyquist_stability(G, s_range):
return G(s_range)
def gain_margin(phase_cross, gain_at_phase_cross):
return 1 / gain_at_phase_cross
def phase_margin(gain_cross, phase_at_gain_cross):
return 180 + phase_at_gain_cross
def bode_plot_magnitude(G, omega):
return 20 * np.log10(np.abs(G(1j * omega)))
def bode_plot_phase(G, omega):
return np.angle(G(1j * omega), deg=True)
omega = np.logspace(-2, 2, 100)
GM, PM = 15, 45
print(f"Gain margin: {GM:.1f} dB, Phase margin: {PM:.1f}°")
Observer Design
def luenberger_observer(A, C, desired_poles):
L = place(A.T, C.T, desired_poles).T
return L
def kalman_gain(A, C, Q, R):
P = solve_continuous_are(A.T, C.T, Q, R)
return P @ C.T @ np.linalg.inv(R)
def reduced_order_observer(A, C, L):
pass
def disturbance_observer(K_d, G_p):
return K_d / (1 + K_d * G_p)
def sensor_fusion_kalman(GPS_variance, IMU_variance):
return GPS_variance / (GPS_variance + IMU_variance)
def notch_filter_design(f0, Q, fs):
w0 = 2 * np.pi * f0 / fs
alpha = np.sin(w0) / (2 * Q)
b0 = 1
b1 = -2 * np.cos(w0)
b2 = 1
a0 = 1 + alpha
a1 = -2 * np.cos(w0)
a2 = 1 - alpha
return [b0/a0, b1/a0, b2/a0], [1, a1/a0, a2/a0]
L = luenberger_observer(A, C, [-10, -12])
print(f"Observer gain: {L}")
Discrete Control
def bilinear_transform(s, Ts):
return (2/Ts) * (1 - s) / (1 + s)
def forward_euler(s, Ts):
return (z - 1) / Ts
def discrete_poles(continuous_poles, Ts):
return np.exp(continuous_poles * Ts)
def c2d_continuous_discrete(A, B, Ts):
n = A.shape[0]
M = np.eye(n)
N = np.zeros((n, n))
for k in range(1, 20):
M = M @ A / k + np.eye(n)
N = N + M
Ad = np.linalg.matrix_power(A, 19) @ Ts
Bd = N @ B * Ts
return Ad, Bd
def sample_and_hold(Gc, Ts):
return Gc * (1 - np.exp(-s * Ts)) / s
s_poles = [-10, -20]
z_poles = discrete_poles(s_poles, 0.01)
print(f"Discrete poles: {z_poles}")
Best Practices
- Model Accuracy: Validate models with experimental data
- Tuning: Systematic tuning procedures
- Constraints: Account for actuator limits
- Robustness: Test under uncertainty
- Implementation: Consider discretization effects
Common Patterns
# LQR controller
def lqr(A, B, Q, R):
P = solve_continuous_are(A, B, Q, R)
return np.linalg.inv(R) @ B.T @ P
Core Competencies
- PID control and tuning
- State-space methods
- Stability analysis
- Observer design
- Digital control