Add alpha_beta and alpha_beta_gamma trackers

[baggepinnen]

Adds alpha_beta and alpha_beta_gamma: fixed-gain trackers of the position and rate (and, third order, the acceleration) of a target, returned as discrete-time StateSpace systems whose input is the measured position and whose outputs are the estimates.

Both are observer_filter of an integrator chain with K = [α, β/T_s] and K = [α, β/T_s, γ/T_s^2], so the state is the a-posteriori estimate and the systems are strictly proper. The rate output is a filtered derivative of the input, available without a separate differentiator.

julia> sys = alpha_beta(0.5, 0.1);          # α alone is a complete tuning

julia> size(sys)
(2, 1)

Why the third-order one

An α-β tracker predicts with a locally constant rate, so on a signal that is genuinely accelerating its rate estimate lags by an amount proportional to the acceleration, and no choice of alpha and beta removes that bias. Predicting with the acceleration removes it. That pays when the measurement is oversampled relative to the signal, where a longer effective memory costs little lag — differentiating a quantized encoder to estimate a joint velocity is the case these were written for.

Parameterization

alpha alone is a complete tuning. beta defaults to Kalata's steady-state relation β = 2(2-α) - 4\sqrt{1-α} and gamma to β^2/(2α), which make the filters the steady-state Kalman filters for constant-velocity and constant-acceleration targets. At α = 0.5 these are β ≈ 0.1716 and γ ≈ 0.0294.

The extended help also gives the critically damped alternative, because the defaults leave a complex pole pair for every alpha — the error always decays with an oscillatory mode. To place the error poles on the real axis at a common radius s:

second order	α = 1 - s^2, β = (1-s)^2
third order	α = 1 - s^3, β = \tfrac{3}{2}(1-s)^2(1+s), γ = (1-s)^3

with the trap stated: setting only alpha leaves the other gains at their Kalata defaults and puts the poles straight back off the axis.

These relations are stated for the gain convention used here, in which the acceleration correction is γ/T_s^2 alongside the rate correction β/T_s. References differ by factors of two depending on whether γ or 2γ is written in that position, so the docstring says which one it means, and the tests check the relations rather than leaving them as prose. I derived rather than quoted them for exactly this reason — my first draft of the second-order relation had a spurious factor of two and the test caught it.

One-sample bookkeeping

Documented in a !!! note, since it is easy to trip over: the state is x̂(k|k), so with the usual x(k+1) = Ax(k) + Bu(k) update the estimate that absorbed y(k) appears at output index k+1 of a simulation. This is the same convention observer_filter already has.

Tests

48 assertions in test_synthesis.jl: shapes, Ts, strict properness and that B == K; the default gains; agreement with the documented recurrence under lsim (transcribed as a Julia loop, compared tick by tick); ramp tracking; equality with observer_filter of the corresponding integrator chain, which is the structural check that these really are that filter; the critical-damping recipes at five radii and three sample rates; that the defaults are not critically damped; and argument validation.

No new dependencies. lib/ControlSystemsBase/test/test_synthesis.jl passes 141 pre-existing plus the 48 new.

Happy to convert this to an issue for discussion first if you would rather — CONTRIBUTING suggests that, and I went straight to a PR because the change is self-contained and additive.

🤖 Generated with Claude Code

https://claude.ai/code/session_01PZNSHjGe7PQ7MdJg3itmCM

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 91.74%. Comparing base (4784e94) to head (554fc26).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1069      +/-   ##
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+ Coverage   87.94%   91.74%   +3.80%     
==========================================
  Files          42       42              
  Lines        5649     5730      +81     
==========================================
+ Hits         4968     5257     +289     
+ Misses        681      473     -208     

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