ffsshhttiikk What I do
- Design and analyze control systems using various methods
- Implement PID and advanced control algorithms
- Perform stability and robustness analysis
- Design state observers and estimators
- Develop MIMO control strategies
- Tune controller parameters for optimal performance
- Analyze system frequency response
- Implement adaptive and nonlinear control
When to use me
When designing feedback control systems, analyzing stability, tuning controllers, or implementing advanced control strategies for dynamic systems.
Core Concepts
- Transfer function and state-space representation
- PID control and tuning methods
- Stability analysis (Routh-Hurwitz, Nyquist, Bode)
- Controllability and observability
- State feedback and LQR control
- Observer design (Luenberger, Kalman)
- Frequency response analysis
- Robust control (H-infinity, mu-synthesis)
- Adaptive control
- Nonlinear control systems
Code Examples
Transfer Function Analysis
import numpy as np
from dataclasses import dataclass
from typing import List, Tuple
import control
import matplotlib.pyplot as plt
@dataclass
class TransferFunction:
num: List[float]
den: List[float]
def eval(self, s: complex) -> complex:
"""Evaluate transfer function at s."""
num_val = sum(self.num[i] * s**(len(self.num) - 1 - i)
for i in range(len(self.num)))
den_val = sum(self.den[i] * s**(len(self.den) - 1 - i)
for i in range(len(self.den)))
return num_val / den_val if den_val != 0 else complex(float('inf'))
def step_response(tf: TransferFunction, t: np.ndarray) -> np.ndarray:
"""Calculate step response."""
sys = control.TransferFunction(tf.num, tf.den)
t_out, y = control.step(sys, t)
return y
def impulse_response(tf: TransferFunction, t: np.ndarray) -> np.ndarray:
"""Calculate impulse response."""
sys = control.TransferFunction(tf.num, tf.den)
t_out, y = control.impulse(sys, t)
return y
def bode_plot(tf: TransferFunction, w: np.ndarray) -> Tuple[np.ndarray, np.ndarray]:
"""Calculate Bode plot data."""
sys = control.TransferFunction(tf.num, tf.den)
w_out, mag, phase = control.bode_plot(sys, w, plot=False)
return mag, phase
def root_locus(tf: TransferFunction, k: np.ndarray) -> Tuple[np.ndarray, np.ndarray]:
"""Calculate root locus."""
sys = control.TransferFunction(tf.num, tf.den)
k_out, poles = control.root_locus(sys, k, plot=False)
return k_out, poles
def routh_hurwitz(coeffs: List[float]) -> Tuple[List[List[float]], str]:
"""Construct Routh-Hurwitz array and check stability."""
n = len(coeffs) - 1
array = []
row1 = coeffs[::2] if len(coeffs) % 2 else coeffs[::2] + [0]
row2 = coeffs[1::2] if len(coeffs) % 2 else coeffs[1::2]
array.append(row1)
array.append(row2)
for i in range(2, n + 1):
row = []
for j in range(len(row1) - 1):
a = array[i-1][0]
row.append((array[i-2][j+1] * row1[0] - array[i-2][0] * array[i-1][j+1]) / a)
array.append(row)
if i == n:
break
# Check stability
sign_changes = 0
for i in range(len(array[0])):
if i == 0:
continue
if array[0][i-1] * array[0][i] < 0:
sign_changes += 1
stable = all(x > 0 for x in array[0])
return array, "stable" if stable else f"{sign_changes} sign changes"
# Example: Second-order system analysis
sys_tf = TransferFunction([100], [1, 10, 100])
t = np.linspace(0, 2, 1000)
y = step_response(sys_tf, t)
print(f"Peak time: {t[np.argmax(y)]:.3f} s")
print(f"Overshoot: {(max(y) - 1) * 100:.1f}%")
print(f"Settling time: {t[np.argmax(y > 0.98)]:.3f} s")
PID Controller Design
@dataclass
class PIDGains:
Kp: float
Ki: float
Kd: float
Tf: float = 0.0 # Filter time constant
def pid_transfer_function(gains: PIDGains) -> Tuple[List[float], List[float]]:
"""Get PID transfer function coefficients."""
if gains.Tf > 0:
num = [gains.Kp + gains.Ki * gains.Tf + gains.Kd,
gains.Ki + gains.Kd / gains.Tf,
gains.Kd * gains.Ki / gains.Tf]
den = [gains.Tf, 1, 0]
else:
num = [gains.Kp + gains.Kd, gains.Ki]
den = [1, 0]
return num, den
def ziegler_nichols_tuning(
Ku: float, # ultimate gain
Tu: float # ultimate period
) -> PIDGains:
"""Ziegler-Nichols tuning method."""
return PIDGains(
Kp=0.6 * Ku,
Ki=1.2 * Ku / Tu,
Kd=0.075 * Ku * Tu
)
def cohen_coon_tuning(
K: float,
L: float,
T: float
) -> PIDGains:
"""Cohen-Coon tuning method."""
return PIDGains(
Kp=(1.35 * T / (K * L)) * (1 + 0.2 * (L / T)),
Ki=1.35 / L,
Kd=0.37 * T
)
def imc_tuning(
K: float,
tau: float,
lambda_c: float # closed-loop time constant
) -> PIDGains:
"""IMC (Internal Model Control) tuning."""
Kc = tau / (K * (lambda_c + tau))
tau_I = tau
tau_D = lambda_c * tau / (tau + lambda_c)
return PIDGains(Kp=Kc, Ki=Kc/tau_I, Kd=Kc*tau_D)
def antiwindup_pid(
Kp: float,
Ki: float,
Kd: float,
u_max: float,
u_min: float,
Kb: float = 1.0
) -> Tuple[float, float, float]:
"""Anti-windup gain adjustment."""
return Kp, Ki, Kd
# Example: PID tuning
gains = ziegler_nichols_tuning(Ku=2.5, Tu=1.2)
print(f"Ziegler-Nichols: Kp={gains.Kp:.2f}, Ki={gains.Ki:.2f}, Kd={gains.Kd:.3f}")
imc_gains = imc_tuning(K=2.0, tau=0.5, lambda_c=0.3)
print(f"IMC: Kp={imc_gains.Kp:.2f}, Ki={imc_gains.Ki:.2f}, Kd={imc_gains.Kd:.3f}")
State-Space Control
@dataclass
class StateSpaceModel:
A: np.ndarray
B: np.ndarray
C: np.ndarray
D: np.ndarray
def controllability_matrix(sys: StateSpaceModel) -> np.ndarray:
"""Check controllability."""
n = sys.A.shape[0]
Cc = sys.B.copy()
for i in range(1, n):
Cc = np.hstack([Cc, np.linalg.matrix_power(sys.A, i) @ sys.B])
return Cc
def observability_matrix(sys: StateSpaceModel) -> np.ndarray:
"""Check observability."""
n = sys.A.shape[0]
Co = sys.C.copy()
for i in range(1, n):
Co = np.vstack([Co, sys.C @ np.linalg.matrix_power(sys.A, i)])
return Co
def lyapunov_equation(A: np.ndarray, Q: np.ndarray) -> np.ndarray:
"""Solve Lyapunov equation A'P + PA = -Q."""
n = A.shape[0]
P = np.zeros((n, n))
# Use SciPy's lyapunov_solve for numerical solution
from scipy.linalg import lyapunov_solve
return lyapunov_solve(A.T, -Q)
def lqr_design(
A: np.ndarray,
B: np.ndarray,
Q: np.ndarray,
R: float
) -> Tuple[np.ndarray, np.ndarray]:
"""Design LQR state feedback controller."""
from scipy.linalg import solve
n = A.shape[0]
P = solve_continuous_lyapunov(A.T @ Q, -Q @ B)
K = np.linalg.inv(R) @ B.T @ P
return K, P
def place_poles(A: np.ndarray, B: np.ndarray, desired_poles: np.ndarray) -> np.ndarray:
"""Pole placement controller design."""
return control.place(A, B, desired_poles)
def kalman_filter(
A: np.ndarray,
C: np.ndarray,
Q: np.ndarray,
R: np.ndarray
) -> Tuple[np.ndarray, np.ndarray]:
"""Design Kalman filter."""
from scipy.linalg import solve_discrete_lyapunov
P = solve_discrete_lyapunov(A.T, C.T @ R @ C + Q)
L = P @ C.T @ np.linalg.inv(C @ P @ C.T + R)
return L, P
# Example: State-space system
A = np.array([[0, 1], [-100, -10]])
B = np.array([[0], [1]])
C = np.array([[1, 0]])
D = np.array([[0]])
sys_ss = StateSpaceModel(A, B, C, D)
Cc = controllability_matrix(sys_ss)
rank = np.linalg.matrix_rank(Cc)
print(f"Controllability matrix rank: {rank} (system is {'controllable' if rank == 2 else 'not controllable'})")
Frequency Response Analysis
def gain_margin_phase_margin(
sys: control.TransferFunction
) -> Tuple[float, float]:
"""Calculate gain and phase margins."""
w, mag, phase = control.bode_plot(sys, plot=False)
# Find phase crossover frequency (where phase = -180)
phase_deg = np.rad2deg(phase)
idx_phase = np.where(phase_deg[:-1] * phase_deg[1:] < 0)[0]
if len(idx_phase) > 0:
w_pc = w[idx_phase[0]]
gm = 1 / np.abs(mag[idx_phase[0]])
else:
w_pc, gm = 0, float('inf')
# Find gain crossover frequency (where gain = 1 or 0 dB)
mag_db = 20 * np.log10(mag)
idx_gain = np.where(mag_db[:-1] * mag_db[1:] < 0)[0]
if len(idx_gain) > 0:
w_gc = w[idx_gain[0]]
pm = 180 + phase_deg[idx_gain[0]]
else:
w_gc, pm = 0, float('inf')
return gm, pm
def nyquist_stability(sys: control.TransferFunction) -> Tuple[int, bool]:
"""Analyze Nyquist plot for stability."""
from control import nyquist_plot
s = 1j * np.logspace(-3, 3, 1000)
sys_val = np.array([sys(complex(s_i)) for s_i in s])
N = -np.sum(np.diff(np.angle(sys_val)) < -np.pi) - 1
P = 0 # Number of unstable open-loop poles
Z = N + P
return N, Z == 0
def Nichols_analysis():
pass
Adaptive and Robust Control
def model_reference_adaptive_control(
ref_model: np.ndarray,
plant: np.ndarray,
gamma: float = 100
) -> Tuple[np.ndarray, np.ndarray]:
"""MRAC parameter adaptation."""
pass
def sliding_mode_control(
x: np.ndarray,
A: np.ndarray,
B: np.ndarray,
K: np.ndarray,
lambda_smc: float,
eta: float
) -> np.ndarray:
"""Sliding mode control with reaching law."""
s = K @ x
sat_s = np.sign(s)
return np.linalg.inv(B) @ (A @ x + lambda_smc * sat_s + eta * sat_s)
def h_infinity_design(
G: control.TransferFunction,
W1: control.TransferFunction,
W2: control.TransferFunction,
gamma: float = 1.0
) -> control.TransferFunction:
"""H-infinity controller synthesis."""
pass
# Example: Sliding mode control
A_cl = np.array([[0, 1], [-2, -3]])
B = np.array([[0], [1]])
K = np.array([[-3, -2]]) # Desired closed-loop poles
lambda_smc = 10
eta = 2.0
x = np.array([[0.5], [0.2]])
u = sliding_mode_control(x, A_cl, B, K, lambda_smc, eta)
print(f"SMC input: {u[0,0]:.3f}")
Best Practices
- Always verify stability before deploying controllers
- Use anti-windup techniques for integral control
- Consider measurement noise in derivative terms
- Validate models with experimental data
- Use multiple tuning methods and compare results
- Consider robustness to parameter variations
- Implement safety limits and saturation handling
- Use proper signal conditioning (filtering, scaling)
- Document controller design and tuning procedures
- Test under realistic conditions before deployment