# Deadbeat Control > Use when you must design a deadbeat-control law for a discrete-time plant given by its pulse transfer function: verify admissibility for direct deadbeat synthesis (every plant pole and zero strictly inside the unit circle), solve the deadbeat design equation that places every closed loop pole at the origin of the z plane, form the finite-settling-time controller difference equation from the plant polynomials, and simulate the closed loop to confirm the output reaches and holds the reference in the minimum number of sample periods with the minimum-settling-time control sequence. Produces the admissibility verdict, the controller coefficients, the settling sample and settling time, the step response and control effort histories, the steady-state tracking check and the control effort summary that gate a deadbeat digital control assessment. Trigger: deadbeat-control, finite-settling-time, pole-placement-at-origin, minimum-settling-time, z-domain-deadbeat. - Skill: `ashfordeou/deadbeat-control` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/deadbeat-control` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/deadbeat-control/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/deadbeat-control --- # Deadbeat Control (gnc-autonomy/control/deadbeat-control) Use when the task is a direct deadbeat synthesis for a discrete-time plant: the plant is already given as a pulse transfer function G(z) = B(z)/A(z) (no discretization is performed here), and the controller must place every closed loop pole at the origin of the z plane so the step response settles in a finite number of sample periods. This leaf implements the deadbeat response design of Franklin, Powell and Workman (Digital Control of Dynamic Systems, ch. 4-5) and Ogata (Discrete-Time Control Systems, ch. 6) in pure Python, stdlib only, deterministic and offline. It pairs with digital-control-design for the sampled-data toolbox that produces a discrete plant from a continuous one, and with pid-control-design for the continuous-plant analogue of direct pole placement. ## Domain quick reference - Plant: G(z) = B(z)/A(z), coefficient lists in descending powers of z; A is normalized monic (scaled by A[0]); the plant must be strictly proper (deg B < deg A) and first or second order (1 <= deg A <= 2). - Admissibility gate: deadbeat synthesis cancels the plant modes exactly, so every canceled mode must be strictly stable: every root of A and every root of B must have modulus strictly below 1.0. A pole or zero on or outside the unit circle makes the design inadmissible. Roots come from closed forms only (degree 1 and 2); degree 3 and higher is out of scope. - Deadbeat controller: unity feedback D(z) = N_c(z)/D_c(z) = A(z)/(B(z) *(z^d - 1)) with relative degree d = deg A - deg B, the plant's pure sample delay. The closed-loop characteristic polynomial is A*D_c + B*N_c = A*B*z^d, so after the strictly-stable cancellations the observable closed loop is T(z) = z^-d: a pure d-sample delay, every closed-loop pole at the origin. - Controller recursion on the tracking error: u(k) = sum_i num_e[i]* e(k - i) + sum_j den_u[j]*u(k - 1 - j), with num_e[i] = A[i]/mu0 and mu0 the leading coefficient of B; den_u comes from the coefficients of B(z)*(1 - z^-d) shifted by the relative degree d. - Plant recursion for the closed-loop simulation (zero initial state): y(k) = -sum_i A[i]*y(k - i) + sum_j B[j]*u(k - d - j), advanced from past samples only. - Deadbeat response: the step response reaches the reference exactly at sample d and holds it (settling time d*T_s); tracking error is exactly r for 0 <= k < d and exactly zero for k >= d. - Steady-state tracking: the control that holds y = r is u* = A(1)/B(1) (the inverse of the plant DC gain). For a plant with no numerator zeros the control reaches u* exactly at sample d; a plant with an interior zero keeps the output deadbeat while the control converges geometrically to u* at the plant-zero rate. - Sample time T_s is reporting-only: it converts sample counts to a time axis (settling time = d*T_s) and is never a design rule; the sample-rate selection rule belongs to digital-control-design. ## Workflow 1. Fix the plant: the pulse transfer function G(z) = B(z)/A(z) as descending-power coefficient lists, and confirm it is strictly proper and first or second order. 2. Verify admissibility with admissibility_check: every pole and zero modulus must be strictly below 1.0, using the closed-form roots from roots_moduli_desc. 3. Solve the deadbeat design equation with deadbeat_design: place every closed loop pole at the origin of the z plane and get the controller coefficients num_e, den_u, the z-domain controller d_num_desc and d_den_desc, and the pole-placement identity char_poly_desc. 4. Form the finite-settling-time controller difference equation from num_e and den_u (the causal recursion on the tracking error). 5. Simulate the closed loop with simulate under the pinned per-sample ordering to get the step response, tracking error and control sequence histories. 6. Read the settling sample and settling time from settling_report and confirm the output has settled (settled True). 7. Check the steady-state tracking with steady_control (u* = A(1)/B(1)) and summarize the control sequence with control_effort. 8. Confirm the pure-delay identity with closed_loop_impulse: the closed-loop impulse response is a d-sample delay. 9. Confirm the deterministic checks with the contract test scripts/test_deadbeat_control.py. ## Worked example Scenario A (first order, no zeros): G(z) = 0.5/(z - 0.5), A = [1.0, -0.5], B = [0.5], T_s = 0.1 s, unit step r = 1. - admissibility_check: admissible True, pole_moduli [0.5], zero_moduli []. - deadbeat_design: n = 1, m = 0, d = 1; num_e = [2.0, -1.0], den_u = [1.0], mu0 = 0.5; controller recursion u(k) = 2.0*e(k) - 1.0*e(k - 1) + 1.0*u(k - 1); u_star = 1.0. - Step response reaches y = 1.0 exactly at sample 1 (settling time 0.1 s) and holds it; error is 1.0 at sample 0 and 0.0 from sample 1 on; control effort u = [2.0, 1.0, 1.0, ...], peak 2.0, u* = 1.0 from sample 1 on. Scenario B (second order, no zeros): G(z) = 0.2/(z^2 - 1.1*z + 0.24), A = [1.0, -1.1, 0.24], B = [0.2], T_s = 0.02 s, unit step r = 1. - admissibility_check: pole_moduli [0.8, 0.3] (real anchors within 1e-9 of these values), admissible True. - deadbeat_design: n = 2, m = 0, d = 2; num_e = [5.0, -5.5, 1.2], den_u = [0.0, 1.0], mu0 = 0.2; char_poly_desc = [0.2, -0.22, 0.048, 0.0, 0.0]; u_star within 1e-9 of 0.7, plant_dc_gain within 1e-6 relative of 10/7. - Step response reaches y = 1.0 at sample 2 (settling time 0.04 s) and holds it to float precision; error area sum(e) = d = 2 within 1e-6 relative; control u = [5.0, -0.5, 0.7, 0.7, ...], peak 5.0, u* = 0.7 from sample 2 on; the pole-placement identity A*D_c + B*N_c equals char_poly_desc within 1e-12; the closed-loop impulse response is a pure 2-sample delay. Scenario C (interior zero, relative-degree generality): G(z) = (0.1*z + 0.05)/(z^2 - 1.1*z + 0.24), A = [1.0, -1.1, 0.24], B = [0.1, 0.05], T_s = 0.02 s, unit step r = 1. - admissibility_check: pole_moduli [0.8, 0.3], zero_moduli [0.5], all admissible. - deadbeat_design: n = 2, m = 1, d = n - m = 1: the interior zero shortens the minimum settling time; num_e = [10.0, -11.0, 2.4], den_u = [0.5, 0.5], mu0 = 0.1; u_star within 1e-9 of 14/15. - Step response reaches y = 1.0 at sample 1 (settling time 0.02 s) and holds it; the OUTPUT is deadbeat from sample 1 while the CONTROL converges geometrically to u* at the plant-zero rate (control effort u(0) = 10.0, u(1) = -6.0, u(39) within 1e-6 relative of 14/15). A complex-pole guard plant G(z) = 0.1/(z^2 - 1.6*z + 0.65) has pole pair 0.8 +/- j*0.1, pole_moduli sqrt(0.65) approx 0.8062257748298549 (both entries), admissible True: the closed-form quadratic branch covers complex poles. ## Verification - Confirm admissibility_check returns the exact dict keys {'admissible', 'pole_moduli', 'zero_moduli', 'reason'} and matches the pole and zero moduli of scenarios A, B, C and the complex-pole guard plant. - Confirm deadbeat_design on scenarios A, B and C matches the worked num_e, den_u, mu0 and u_star values within tolerance, and that the pole-placement identity A*D_c + B*N_c equals char_poly_desc = A*B*z^d within 1e-12. - Confirm simulate reproduces the step response, error and control histories of the worked examples within tolerance, and that closed_loop_impulse gives a pure d-sample delay. - Confirm deadbeat_design raises ValueError for an unstable plant, a pole or zero on the unit circle, a non-minimum-phase zero, a degree-3 plant, a non-strictly-proper plant, an empty plant list and a zero leading A coefficient. - Confirm simulate raises ValueError for a non-positive sample time, an unknown reference string, too few steps, and an inadmissible plant. - Run the contract test offline: python3 scripts/test_deadbeat_control.py (deterministic, no imports beyond math). ## Related leaves - gnc-autonomy/control/digital-control-design: the sampled-data toolbox that discretizes a continuous plant with a zero-order hold and emulates a continuous compensator, the source of a discrete plant this leaf can then consume. - gnc-autonomy/control/pid-control-design: continuous s-domain pole placement for a first or second order plant, the continuous analogue of the direct pole-placement idea used here in the z-domain. - gnc-autonomy/control/state-space-analysis: eigenvalue and controllability context for the state-space view of a discrete plant. ## Pitfalls - Feeding a continuous plant directly: this leaf never discretizes; the plant coefficients A and B must already be the pulse transfer function coefficients of a discrete-time plant (use digital-control-design first if the plant starts continuous). - Skipping the admissibility gate: a plant with a pole or zero outside or on the unit circle cannot be canceled safely; deadbeat_design raises ValueError rather than returning a marginal design, since an uncanceled unstable mode would appear unbounded in the closed loop. - Assuming the settling sample always equals the plant order: with an interior plant zero (scenario C) the settling sample is d = n - m, strictly less than n; only a plant with no numerator zeros settles at exactly n samples. - Reading the control sequence as deadbeat too: only the OUTPUT is guaranteed to reach and hold the reference in d samples; when the plant has a zero, the control sequence itself converges geometrically to u* at the plant-zero rate rather than jumping to it. - Treating T_s as part of the design: sample time only converts sample counts to a time axis (settling time = d*T_s); it never changes the controller coefficients and it is not a sample-rate selection rule. ## Behavior contract (gate 3) Run the deterministic contract test (stdlib unittest, offline): python3 scripts/test_deadbeat_control.py The test covers the admissibility truth table for scenarios A, B, C and the complex-pole guard plant, the deadbeat design coefficients and the pole-placement identity, the closed-loop step, error and control histories of all three worked scenarios within tolerance, the steady-state tracking identity, the pure-delay impulse identity, determinism of repeated simulate runs, and ValueError rejection of unstable, boundary, non-minimum-phase, degree-3 and non-strictly-proper plants and of invalid simulate arguments. ## Compliance - Standards referenced, not reproduced: ARP4754A is a proprietary SAE standard (name plus paraphrase only, per standards-map.yaml); the deadbeat design relations above are standard engineering methodology from Franklin, Powell and Workman and from Ogata, summary-only. - compliance: STANDARDS-REF, gated: false.