363 lines
13 KiB
Python
363 lines
13 KiB
Python
"""
|
||
Magnetic Levitation Jacobian Linearizer
|
||
|
||
Computes the local linear (Jacobian) approximation of the degree-6 polynomial
|
||
force/torque model at any operating point. The result is an LTI gain matrix
|
||
that relates small perturbations in (currL, currR, roll, gap_height) to
|
||
perturbations in (Force, Torque):
|
||
|
||
[ΔF ] [∂F/∂currL ∂F/∂currR ∂F/∂roll ∂F/∂gap] [ΔcurrL ]
|
||
[ΔTau] ≈ J [∂T/∂currL ∂T/∂currR ∂T/∂roll ∂T/∂gap] [ΔcurrR ]
|
||
[Δroll ]
|
||
[Δgap ]
|
||
|
||
Since the polynomial is analytic, all derivatives are computed exactly
|
||
(symbolic differentiation of the power-product terms), NOT by finite
|
||
differences.
|
||
|
||
The chain rule is applied automatically to convert the internal invGap
|
||
(= 1/gap_height) variable back to physical gap_height [mm].
|
||
|
||
Usage:
|
||
lin = MaglevLinearizer("maglev_model.pkl")
|
||
plant = lin.linearize(currL=-15, currR=-15, roll=0.0, gap_height=10.0)
|
||
print(plant)
|
||
print(plant.dF_dcurrL) # single gain
|
||
print(plant.control_jacobian) # 2×2 matrix mapping ΔcurrL/R → ΔF/ΔT
|
||
f, t = plant.predict(delta_currL=0.5) # quick what-if
|
||
"""
|
||
|
||
import numpy as np
|
||
import joblib
|
||
import os
|
||
|
||
|
||
class LinearizedPlant:
|
||
"""
|
||
Holds the Jacobian linearization of the force/torque model at one
|
||
operating point.
|
||
|
||
Attributes
|
||
----------
|
||
operating_point : dict
|
||
The (currL, currR, roll, gap_height) where linearization was computed.
|
||
f0 : float
|
||
Force [N] at the operating point.
|
||
tau0 : float
|
||
Torque [mN·m] at the operating point.
|
||
jacobian : ndarray, shape (2, 4)
|
||
Full Jacobian:
|
||
Row 0 = Force derivatives, Row 1 = Torque derivatives.
|
||
Columns = [currL [A], currR [A], rollDeg [deg], gap_height [mm]]
|
||
"""
|
||
|
||
INPUT_LABELS = ['currL [A]', 'currR [A]', 'rollDeg [deg]', 'gap_height [mm]']
|
||
|
||
def __init__(self, operating_point, f0, tau0, jacobian):
|
||
self.operating_point = operating_point
|
||
self.f0 = f0
|
||
self.tau0 = tau0
|
||
self.jacobian = jacobian
|
||
|
||
# ---- Individual gain accessors ----
|
||
@property
|
||
def dF_dcurrL(self):
|
||
"""∂Force/∂currL [N/A] at operating point."""
|
||
return self.jacobian[0, 0]
|
||
|
||
@property
|
||
def dF_dcurrR(self):
|
||
"""∂Force/∂currR [N/A] at operating point."""
|
||
return self.jacobian[0, 1]
|
||
|
||
@property
|
||
def dF_droll(self):
|
||
"""∂Force/∂roll [N/deg] at operating point."""
|
||
return self.jacobian[0, 2]
|
||
|
||
@property
|
||
def dF_dgap(self):
|
||
"""∂Force/∂gap_height [N/mm] at operating point.
|
||
Typically positive (unstable): force increases as gap shrinks.
|
||
"""
|
||
return self.jacobian[0, 3]
|
||
|
||
@property
|
||
def dT_dcurrL(self):
|
||
"""∂Torque/∂currL [mN·m/A] at operating point."""
|
||
return self.jacobian[1, 0]
|
||
|
||
@property
|
||
def dT_dcurrR(self):
|
||
"""∂Torque/∂currR [mN·m/A] at operating point."""
|
||
return self.jacobian[1, 1]
|
||
|
||
@property
|
||
def dT_droll(self):
|
||
"""∂Torque/∂roll [mN·m/deg] at operating point."""
|
||
return self.jacobian[1, 2]
|
||
|
||
@property
|
||
def dT_dgap(self):
|
||
"""∂Torque/∂gap_height [mN·m/mm] at operating point."""
|
||
return self.jacobian[1, 3]
|
||
|
||
@property
|
||
def control_jacobian(self):
|
||
"""2×2 sub-matrix mapping control inputs [ΔcurrL, ΔcurrR] → [ΔF, ΔT].
|
||
|
||
This is the "B" portion of the linearized plant that the PID
|
||
controller acts through.
|
||
"""
|
||
return self.jacobian[:, :2]
|
||
|
||
@property
|
||
def state_jacobian(self):
|
||
"""2×2 sub-matrix mapping state perturbations [Δroll, Δgap] → [ΔF, ΔT].
|
||
|
||
Contains the magnetic stiffness (∂F/∂gap) and roll coupling.
|
||
This feeds into the "A" matrix of the full mechanical state-space.
|
||
"""
|
||
return self.jacobian[:, 2:]
|
||
|
||
def predict(self, delta_currL=0.0, delta_currR=0.0,
|
||
delta_roll=0.0, delta_gap=0.0):
|
||
"""
|
||
Predict force and torque using the linear approximation.
|
||
|
||
Returns (force, torque) including the nominal operating-point values.
|
||
"""
|
||
delta = np.array([delta_currL, delta_currR, delta_roll, delta_gap])
|
||
perturbation = self.jacobian @ delta
|
||
return self.f0 + perturbation[0], self.tau0 + perturbation[1]
|
||
|
||
def __repr__(self):
|
||
op = self.operating_point
|
||
lines = [
|
||
f"LinearizedPlant @ currL={op['currL']:.1f}A, "
|
||
f"currR={op['currR']:.1f}A, "
|
||
f"roll={op['roll']:.2f}°, "
|
||
f"gap={op['gap_height']:.2f}mm",
|
||
f" F₀ = {self.f0:+.4f} N τ₀ = {self.tau0:+.4f} mN·m",
|
||
f" ∂F/∂currL = {self.dF_dcurrL:+.4f} N/A "
|
||
f"∂T/∂currL = {self.dT_dcurrL:+.4f} mN·m/A",
|
||
f" ∂F/∂currR = {self.dF_dcurrR:+.4f} N/A "
|
||
f"∂T/∂currR = {self.dT_dcurrR:+.4f} mN·m/A",
|
||
f" ∂F/∂roll = {self.dF_droll:+.4f} N/deg "
|
||
f"∂T/∂roll = {self.dT_droll:+.4f} mN·m/deg",
|
||
f" ∂F/∂gap = {self.dF_dgap:+.4f} N/mm "
|
||
f"∂T/∂gap = {self.dT_dgap:+.4f} mN·m/mm",
|
||
]
|
||
return '\n'.join(lines)
|
||
|
||
|
||
class MaglevLinearizer:
|
||
"""
|
||
Jacobian linearizer for the polynomial maglev force/torque model.
|
||
|
||
Loads the same .pkl model as MaglevPredictor, but instead of just
|
||
evaluating the polynomial, computes exact analytical partial derivatives
|
||
at any operating point.
|
||
"""
|
||
|
||
def __init__(self, model_path='maglev_model.pkl'):
|
||
if not os.path.exists(model_path):
|
||
raise FileNotFoundError(
|
||
f"Model file '{model_path}' not found. "
|
||
"Train and save the model from Function Fitting.ipynb first."
|
||
)
|
||
|
||
data = joblib.load(model_path)
|
||
poly_transformer = data['poly_features']
|
||
linear_model = data['model']
|
||
|
||
# powers_: (n_terms, n_inputs) — exponent matrix from sklearn
|
||
# Transpose to (n_inputs, n_terms) for broadcasting with x[:, None]
|
||
self.powers = poly_transformer.powers_.T.astype(np.float64)
|
||
|
||
self.force_coef = linear_model.coef_[0] # (n_terms,)
|
||
self.torque_coef = linear_model.coef_[1] # (n_terms,)
|
||
self.force_intercept = linear_model.intercept_[0]
|
||
self.torque_intercept = linear_model.intercept_[1]
|
||
|
||
self.degree = data['degree']
|
||
self.n_terms = self.powers.shape[1]
|
||
|
||
def _to_internal(self, currL, currR, roll, gap_height):
|
||
"""Convert physical inputs to the polynomial's internal variables."""
|
||
invGap = 1.0 / max(gap_height, 1e-6)
|
||
return np.array([currL, currR, roll, invGap], dtype=np.float64)
|
||
|
||
def evaluate(self, currL, currR, roll, gap_height):
|
||
"""
|
||
Evaluate the full (nonlinear) polynomial at a single point.
|
||
|
||
Returns
|
||
-------
|
||
force : float [N]
|
||
torque : float [mN·m]
|
||
"""
|
||
x = self._to_internal(currL, currR, roll, gap_height)
|
||
poly_features = np.prod(x[:, None] ** self.powers, axis=0)
|
||
force = np.dot(self.force_coef, poly_features) + self.force_intercept
|
||
torque = np.dot(self.torque_coef, poly_features) + self.torque_intercept
|
||
return float(force), float(torque)
|
||
|
||
def _jacobian_internal(self, x):
|
||
"""
|
||
Compute the 2×4 Jacobian w.r.t. the internal polynomial variables
|
||
(currL, currR, rollDeg, invGap).
|
||
|
||
For each variable x_k, the partial derivative of a polynomial term
|
||
c · x₁^a₁ · x₂^a₂ · x₃^a₃ · x₄^a₄
|
||
is:
|
||
c · a_k · x_k^(a_k - 1) · ∏_{j≠k} x_j^a_j
|
||
|
||
This is computed vectorised over all terms simultaneously.
|
||
"""
|
||
jac = np.zeros((2, 4))
|
||
for k in range(4):
|
||
# a_k for every term — this becomes the multiplicative scale
|
||
scale = self.powers[k, :] # (n_terms,)
|
||
|
||
# Reduce the k-th exponent by 1 (floored at 0; the scale
|
||
# factor of 0 for constant-in-x_k terms zeros those out)
|
||
deriv_powers = self.powers.copy()
|
||
deriv_powers[k, :] = np.maximum(deriv_powers[k, :] - 1.0, 0.0)
|
||
|
||
# Evaluate the derivative polynomial
|
||
poly_terms = np.prod(x[:, None] ** deriv_powers, axis=0)
|
||
deriv_features = scale * poly_terms # (n_terms,)
|
||
|
||
jac[0, k] = np.dot(self.force_coef, deriv_features)
|
||
jac[1, k] = np.dot(self.torque_coef, deriv_features)
|
||
|
||
return jac
|
||
|
||
def linearize(self, currL, currR, roll, gap_height):
|
||
"""
|
||
Compute the Jacobian linearization at the given operating point.
|
||
|
||
Parameters
|
||
----------
|
||
currL : float Left coil current [A]
|
||
currR : float Right coil current [A]
|
||
roll : float Roll angle [deg]
|
||
gap_height : float Air gap [mm]
|
||
|
||
Returns
|
||
-------
|
||
LinearizedPlant
|
||
Contains the operating-point values (F₀, τ₀) and the 2×4
|
||
Jacobian with columns [currL, currR, roll, gap_height].
|
||
"""
|
||
x = self._to_internal(currL, currR, roll, gap_height)
|
||
f0, tau0 = self.evaluate(currL, currR, roll, gap_height)
|
||
|
||
# Jacobian in internal coordinates (w.r.t. invGap in column 3)
|
||
jac_internal = self._jacobian_internal(x)
|
||
|
||
# Chain rule: ∂f/∂gap = ∂f/∂invGap · d(invGap)/d(gap)
|
||
# = ∂f/∂invGap · (−1 / gap²)
|
||
jac = jac_internal.copy()
|
||
jac[:, 3] *= -1.0 / (gap_height ** 2)
|
||
|
||
return LinearizedPlant(
|
||
operating_point={
|
||
'currL': currL, 'currR': currR,
|
||
'roll': roll, 'gap_height': gap_height,
|
||
},
|
||
f0=f0,
|
||
tau0=tau0,
|
||
jacobian=jac,
|
||
)
|
||
|
||
def gain_schedule(self, gap_heights, currL, currR, roll=0.0):
|
||
"""
|
||
Precompute linearizations across a range of gap heights at fixed
|
||
current and roll. Useful for visualising how plant gains vary
|
||
and for designing a gain-scheduled PID.
|
||
|
||
Parameters
|
||
----------
|
||
gap_heights : array-like of float [mm]
|
||
currL, currR : float [A]
|
||
roll : float [deg], default 0
|
||
|
||
Returns
|
||
-------
|
||
list of LinearizedPlant, one per gap height
|
||
"""
|
||
return [
|
||
self.linearize(currL, currR, roll, g)
|
||
for g in gap_heights
|
||
]
|
||
|
||
|
||
# ==========================================================================
|
||
# Demo / quick validation
|
||
# ==========================================================================
|
||
if __name__ == '__main__':
|
||
import sys
|
||
|
||
model_path = os.path.join(os.path.dirname(__file__), 'maglev_model.pkl')
|
||
lin = MaglevLinearizer(model_path)
|
||
|
||
# --- Single-point linearization ---
|
||
print("=" * 70)
|
||
print("SINGLE-POINT LINEARIZATION")
|
||
print("=" * 70)
|
||
plant = lin.linearize(currL=-15, currR=-15, roll=0.0, gap_height=10.0)
|
||
print(plant)
|
||
print()
|
||
|
||
# Quick sanity check: compare linear prediction vs full polynomial
|
||
dc = 0.5 # small current perturbation
|
||
f_lin, t_lin = plant.predict(delta_currL=dc)
|
||
f_act, t_act = lin.evaluate(-15 + dc, -15, 0.0, 10.0)
|
||
print(f"Linearised vs Actual (ΔcurrL = {dc:+.1f} A):")
|
||
print(f" Force: {f_lin:.4f} vs {f_act:.4f} (err {abs(f_lin-f_act):.6f} N)")
|
||
print(f" Torque: {t_lin:.4f} vs {t_act:.4f} (err {abs(t_lin-t_act):.6f} mN·m)")
|
||
print()
|
||
|
||
# --- Gain schedule across gap heights ---
|
||
print("=" * 70)
|
||
print("GAIN SCHEDULE (currL = currR = -15 A, roll = 0°)")
|
||
print("=" * 70)
|
||
gaps = [6, 8, 10, 12, 15, 20, 25]
|
||
plants = lin.gain_schedule(gaps, currL=-15, currR=-15, roll=0.0)
|
||
|
||
header = f"{'Gap [mm]':>10} {'F₀ [N]':>10} {'∂F/∂iL':>10} {'∂F/∂iR':>10} {'∂F/∂gap':>10} {'∂T/∂iL':>12} {'∂T/∂iR':>12}"
|
||
print(header)
|
||
print("-" * len(header))
|
||
for p in plants:
|
||
g = p.operating_point['gap_height']
|
||
print(
|
||
f"{g:10.1f} {p.f0:10.3f} {p.dF_dcurrL:10.4f} "
|
||
f"{p.dF_dcurrR:10.4f} {p.dF_dgap:10.4f} "
|
||
f"{p.dT_dcurrL:12.4f} {p.dT_dcurrR:12.4f}"
|
||
)
|
||
print()
|
||
|
||
# --- Note on PID usage ---
|
||
print("=" * 70)
|
||
print("NOTES FOR PID DESIGN")
|
||
print("=" * 70)
|
||
print("""
|
||
At each operating point, the linearized electromagnetic plant is:
|
||
|
||
[ΔF ] [∂F/∂iL ∂F/∂iR] [ΔiL] [∂F/∂roll ∂F/∂gap] [Δroll]
|
||
[ΔTau] = [∂T/∂iL ∂T/∂iR] [ΔiR] + [∂T/∂roll ∂T/∂gap] [Δgap ]
|
||
^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^
|
||
control_jacobian state_jacobian
|
||
|
||
The full mechanical dynamics (linearized) are:
|
||
m · Δg̈ap = ΔF - m·g (vertical — note ∂F/∂gap > 0 means unstable)
|
||
Iz · Δroll̈ = ΔTau (roll)
|
||
|
||
So the PID loop sees:
|
||
control_jacobian → the gain from current commands to force/torque
|
||
state_jacobian → the coupling from state perturbations (acts like
|
||
a destabilising spring for gap, restoring for roll)
|
||
""")
|