""" 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) """)