# Feedback Linearization > Use when you must apply feedback linearization to a nonlinear plant with a known exact model: compute the Lie derivatives of the output along the drift and control vector fields to establish the relative degree, invert the decoupling scalar at the operating state, form the linearizing control that cancels the nonlinear terms so the output channel obeys the linear relation y^(r) = v, apply the outer linear tracking loop with the assigned closed-loop pole placement, and check the internal dynamics via their zero dynamics before accepting the design. Produces the relative-degree verdict, the linearizing control with the decoupling scalar, the exactly linearized closed-loop response against the assigned closed form, and the zero-dynamics stability verdict that gates the design. Trigger: feedback linearization, input output linearization, lie derivative, relative degree, decoupling matrix, linearizing control, zero dynamics, internal dynamics stability. - Skill: `ashfordeou/feedback-linearization` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/feedback-linearization` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/feedback-linearization/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: AI & ML - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/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) = 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.