DeadbeatControl: non-unity DC gain causes static tracking error for Steps > 1

[gabrielsantosphilips]

Summary

controllers::DeadbeatControl with Steps > 1 never reaches its reference when used in receding-horizon mode (the mode ComputeControl implies). The closed loop settles to a fraction of the commanded value. Steps == 1 is correct and unaffected.

Introduced in #271.

Root cause

ComputeGains assigns the reference gain directly from the pseudo-inverse block:

auto gammaPinv = solvers::SolveSystem<T, StateSize, ReachSize>(gamma * gamma.Transpose(), gamma).Transpose();

gainRef = gammaPinv.template GetBlock<InputSize, StateSize>(0, 0);
gainState = gainRef * AN;

This is the minimum-norm solution for the whole input sequence [u[k] … u[k+N-1]], and it is correct if that entire sequence is applied open-loop — the state then reaches the reference in exactly N steps.

But ComputeControl returns only the first element and is called again every sample, so the effective closed loop is:

x[k+1] = (A - B·gainState)·x[k] + B·gainRef·r

Taking only the first element of a min-norm sequence and recomputing does not preserve the terminal condition. For a scalar plant the closed-loop pole is Ad/(Ad²+1) — stable, geometric, not deadbeat — and the DC gain is:

$$\frac{A_d}{A_d^2 - A_d + 1} < 1$$

For Steps == 1 the same algebra gives a pole at the origin and DC gain exactly 1, which is why the defect is invisible there.

Reproduction

Scalar plant A = [Ad], B = [Bd], DeadbeatControl<float, 1, 1, 2>, simulated closed-loop to steady state with r = 1.0:

Ad	Steps	Steady state	Error
0.9048	1	1.000000	0.00%
0.9048	2	0.990091	0.99%
0.8465	2	0.972912	2.71%
0.6065	2	0.796653	20.33%

The error grows as Ad decreases, i.e. as the plant gets faster relative to the sample rate.

Why the existing tests miss it

All three tests in TestDeadbeatControl.cpp assert open-loop N-step convergence from a fixed reference, or zero control when already at the reference. None checks closed-loop steady state, and none sweeps Ad. A regression test should simulate to steady state and assert x → r.

Suggested fix

Solve for a reference gain that makes the closed-loop DC gain unity (the standard N̄ precompensator) instead of taking the pseudo-inverse block:

B·gainRef = I - A + B·gainState

For the scalar case this reduces to gainRef = (1 - Ad + Bd·gainState) / Bd. Steps == 1 already satisfies it, so the change is backward compatible.

Caveat worth deciding on

This is not universally solvable. B·gainRef has rank ≤ InputSize, so exact unity DC gain on every state requires B to have full row rank. It works for StateSize == InputSize, but an under-actuated plant with Steps > 1 cannot achieve it exactly.

So there are two defensible resolutions:

1. Apply the N̄ correction where B has full row rank, and document the limitation otherwise.
2. Declare that DeadbeatControl is intended for open-loop application of the full N-step sequence, and add an API that exposes the whole sequence — making the current receding-horizon use of ComputeControl the misuse.

Option 2 is arguably more faithful to what the min-norm solution actually computes, but it conflicts with ComputeControl being the StateFeedbackController interface method, which strongly implies per-sample feedback use.

Context

Found while adopting DeadbeatControl for the PMSM current loop in the downstream e-foc project (per-axis scalar RL plant, Ad = e^(-Rs·Ts/Ls), 20 kHz). We are currently shipping Steps = 1 only; the two-step variant is intended for low-inductance motors, which is exactly the small-Ad regime where the error is largest.