Feedback Linearization

Feedback Linearization (gnc-autonomy/control/feedback-linearization)

Use when the task is input-output feedback linearization of a nonlinear plant with a known, exact model: the affine single-input plant xdot = f(x) + g(x) u, y = h(x) is linearized exactly by canceling its nonlinear terms through the control channel, so the chosen output obeys the linear relation y^(r) = v under an outer tracking loop, and the design is accepted only after the unobservable internal dynamics are checked stable through their zero dynamics. This leaf implements the Lie-derivative relative-degree construction (Slotine and Li; Isidori) in pure Python, stdlib only. It pairs with gnc-autonomy/control/state-space-analysis for the linear-model counterpart and with gnc-autonomy/control/sliding-mode-control, the switching-law sibling of the same nonlinear vein, for a plant with matched uncertainty instead of an exact model.

Domain quick reference

• Plant: xdot1 = -A11 x1 + x2, xdot2 = CUBIC x1^3 + QUAD x2^2 + x1 u, the affine model xdot = f(x) + g(x) u with g(x) = (0, x1), a control effectiveness that vanishes at x1 = 0. The model is exact and given, never identified or adapted.
• Lie derivatives: L_f^k h(x) is the k-th derivative of the output h along the drift field f, computed by analytic chain rule (no finite differences); L_g L_f^k h(x) = <d(L_f^k h)(x), g(x)> is the relative-degree probe along the control field.
• Relative degree r: the first r with L_g L_f^(r-1) h nonzero at x, preceded by L_g L_f^k h = 0 for k = 0 .. r - 2. No finite relative degree exists when every probe up to the system order N vanishes.
• Decoupling scalar: a(x) = L_g L_f^(r-1) h, the quantity that must be inverted to form the control. It is x1 for both registered outputs of this plant, invertible on x1 != 0.
• Linearizing control: u = (v - L_f^r h) / a(x) cancels the nonlinear terms exactly, leaving y^(r) = v on the chosen channel.
• Outer tracking loop: v = -K (y - y_ref) for relative degree 1, and v = -K^2 (y - y_ref) - 2 K ydot for relative degree 2, placing the closed-loop pole at s = -K (a double pole at r = 2). K is one fixed given constant, never tuned, scheduled, or adapted online.
• Zero dynamics: the internal dynamics left over when the output and its derivatives are held at zero. For this plant's relative-degree-1 design the internal dynamics reduce to x1dot = -A11 x1; the design is accepted only when that eigenvalue is strictly negative.
• Units are SI-neutral state variables; the module runs a closed-loop RK4 simulation with the control law recomputed fresh at every integration stage, so its residual against the exact closed form is RK4 truncation error only.
• ARP4754A frames the development assurance context for the airborne or space function that hosts the linearizing loop; the relations above are standard nonlinear control methodology, summary-only.

Workflow

1. Fix the plant and the operating state x, and register the output channel (x1 or x2) whose relative degree you need.
2. Compute the Lie derivatives L_f^k h along the drift field with Lf_power_h, and probe the control channel with Lg_Lf_power_h.
3. Establish the relative degree r with relative_degree: the defining property L_g L_f^k h = 0 for k < r - 1 with a nonzero probe at r - 1; a ValueError signals no finite relative degree exists at that state.
4. Read the decoupling scalar a(x) with decoupling_scalar and invert it, either directly with linearizing_control_from or at the state level with linearizing_control, to form the linearizing control u = (v - L_f^r h) / a that cancels the nonlinear terms.
5. Set the outer linear command v with outer_command at the assigned closed-loop pole rate K.
6. Run the closed-loop RK4 simulation with closed_loop_sim and compare the output against the exact closed form (closed_form_r1 or closed_form_r2) with series_max_abs_error, and confirm the pole placement with measured_decay_rate.
7. Check the internal dynamics of the relative-degree-1 design against the zero-dynamics stability verdict with zero_dynamics_analysis before accepting the design.
8. Confirm the deterministic checks with the contract test scripts/test_feedback_linearization.py.

Worked example

Plant: xdot1 = -x1 + x2, xdot2 = x1^3 + x2^2 + x1 u (A11 = CUBIC = QUAD = 1.0). Worked initial state x0 = (0.5, 0.0), unit-step reference y_ref = 1.0, assigned closed-loop rate K = 2.0.

• Relative degree at x0: output x2 gives r = 1 with probe table [(0, 0.5)] (L_g h2 = x1 = 0.5, nonzero at r - 1 = 0); output x1 gives r = 2 with table [(0, 0.0), (1, 0.5)] (L_g h1 = 0 for k = 0 < r - 1 = 1, L_g L_f h1 = x1 = 0.5 nonzero at r - 1). The decoupling scalar a(x0) = 0.5 for both outputs, and a(1.0, 0.75) = 1.0.
• Lie terms and controls: L_f h2 = f2 = 0.125 and L_f^2 h1 = x1 - x2 + x1^3 + x2^2 = 0.625. Outer commands v = 2.0 (r = 1) and v = 4.0 (r = 2) give linearizing controls u = 3.75 (r = 1) and u = 6.75 (r = 2). The cancellation residual f2 + a u - v is 0.000e+00 at x0 and at (1.0, 0.75), so the nonlinear terms leave the channel exactly.
• r = 1 closed loop: the simulated output matches the assigned linear dynamics 1 - exp(-2 t) with max abs error 4.929e-14 over 10001 RK4 samples (truncation only); y(0.5) = 0.632120558829, y(1) = 0.864664716763, y(2) = 0.981684361111, y(5) = 0.999954600070; settling to the 1e-3 band at 3.454 s; measured decay rate between t = 0.5 and t = 1.0 is 2.000000, the pole witness for s = -K.
• Internal state of the r = 1 design: x1 matches the closed form 1 - 1.5 exp(-t) + exp(-2 t) with max abs error 4.552e-14, dipping to 0.437500056838 at t = 0.288 s (theory ln(4/3) = 0.287682 s) before rising to x1(10) = 0.999931902167, approaching y_ref while staying bounded.
• r = 2 companion loop: the fully linearized output matches the double-pole closed form 1 - (0.5 + 1.5 t) exp(-2 t) with max abs error 1.275e-13; the coefficient B = -1.5 is nonzero, the surviving t exp(-2 t) signature of the repeated pole (s + 2)^2.
• Zero-dynamics verdict: eigenvalue -1.0, verdict "asymptotically stable"; the constrained samples x1 = 0.5 exp(-t) at t = 1, 2, 5 s are 0.183939721, 0.067667642, 0.003368973, so the design is accepted.

Verification

• Confirm relative_degree('x2', (0.5, 0.0)) returns (1, [(0, 0.5)]) and relative_degree('x1', (0.5, 0.0)) returns (2, [(0, 0.0), (1, 0.5)]).
• Confirm the cancellation identity f2 + a u - v = 0 within 1e-12 at the worked state and at (1.0, 0.75).
• Confirm the r = 1 closed loop matches 1 - exp(-2 t) within 1e-9 and the measured decay rate matches K = 2.0 within 1e-3.
• Confirm the r = 2 closed loop matches the double-pole closed form within 1e-9 and carries a nonzero double-root coefficient B.
• Confirm the zero-dynamics verdict is "asymptotically stable" with eigenvalue -1.0, and that relative_degree raises ValueError at x1 = 0 (the decoupling scalar vanishes) and for the constant output.
• Run the contract test offline: python3 scripts/test_feedback_linearization.py (29 tests, deterministic).

Related leaves

• gnc-autonomy/control/state-space-analysis: linear-model eigenvalue and controllability analysis, the fixed-linear counterpart when the plant needs no nonlinear cancellation.
• gnc-autonomy/control/sliding-mode-control: the switching-law sibling of the same nonlinear vein, sizing a discontinuous control against a matched-uncertainty bound instead of canceling a known model exactly.
• gnc-autonomy/control/adaptive-control: model-reference adaptation of gains for an unknown plant coefficient, the case where no exact model is available to cancel.
• gnc-autonomy/control/gain-scheduling: interpolating existing linear gains across the envelope, the alternative when K should vary with the operating point instead of staying fixed.

Pitfalls

• Holding the control fixed across an integration step: the linearizing control must be recomputed at every RK4 stage from that stage's state, not sampled once per step and held; a held control introduces first-order-in-dt error instead of the RK4 truncation residual near 1e-13 that the worked example reports.
• Evaluating the relative-degree probe at a singular state: the decoupling scalar a(x) = x1 vanishes at x1 = 0, so no finite relative degree exists there and the linearizing control is undefined; check x1 != 0 before inverting.
• Reading the internal state against the reference instead of its own closed form: x1 trails y_ref by the 1.5 exp(-t) mode of the internal dynamics (a 6.81e-5 gap at t = 10 s), so assert against 1 - 1.5 exp(-t) + exp(-2 t), not against y_ref directly.
• Treating K as tunable online: K is one fixed given closed-loop rate in this leaf; scheduling or adapting it belongs to gnc-autonomy/control/gain-scheduling or the adaptive-control siblings.
• Skipping the zero-dynamics check: canceling the output dynamics exactly says nothing about the unobservable internal state; the relative-degree-1 design is accepted only after its zero-dynamics eigenvalue is confirmed strictly negative.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_feedback_linearization.py

The test covers the Lie-derivative and relative-degree identities at the worked state and at a probe state, the decoupling-scalar inversion and the cancellation identity, the linearizing control values for both the relative-degree-1 and relative-degree-2 designs, the RK4 closed-loop simulation against both exact closed forms with the measured pole-placement witness, the internal-dynamics closed form and its minimum, the zero-dynamics stability verdict and its constrained samples, the relative-degree rejection cases (a singular state, the constant output, an unknown output), the decoupling-inversion guard, and ValueError rejection of non-positive closed-loop rate, sample time, simulation time, reference, and non-physical state inputs.

Compliance

• Standards referenced, not reproduced: ARP4754A frames development assurance for the airborne or space functions that host a feedback-linearizing control loop; the Lie-derivative relative-degree construction above is standard nonlinear control methodology (Slotine and Li; Isidori), summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.