Do not report a gain margin at zero frequency for systems with a zero in the origin

[baggepinnen]

margin prepends the frequency -0.0 to the frequency grid (_add_zero). This detects crossings of the negative real axis at zero frequency, as in #1045. For a system with more zeros than poles in the origin, the frequency response at ω = 0 is zero, and its computed value is determined by rounding error. When this value is a negative real number, its phase is ±180°. _findCrossings then reports a phase crossover at ω = 0, and sisomargin returns a gain margin 1/|G(0)| of the order 1e15–1e17.

s = tf("s")
G = balreal(ss(s/(s+1)^2))[1]
margin(G; allMargins=true) # master: wgm = [-0.0], gm = [6.5e15]; this PR: wgm = [], gm = [Inf]

This PR removes such crossings. A crossover at ω = 0 is discarded when the system has more zeros than poles in the origin (z = 1 in discrete time). The decision uses integrator_excess, i.e., the same classification as the phase adjustment in sisomargin. For a DelayLtiSystem, every delay equals the identity at ω = 0, so the classification is applied to the delay-free system lft(sys.P.P, I).

System	G(0) as computed	master: wgm, gm	this PR
balreal(ss(s/(s+1)^2))[1]	-1.5e-16	[-0.0], [6.5e15]	none
balreal(ss(s^2/((s+1)*(s^2 + 0.2s + 4))))[1]	-3.0e-17	[-0.0], [8.3e16]	none
c2d(ss(s/(s+1)^2), 0.1)	-7.1e-16	[-0.0, 31.4], [1.3e15, 20.1]	[31.4], [20.1]
balreal(ss(s/(s+1)^2))[1] * delay(0.1)	-1.1e-16	[-0.0, 16.9, 78.8], [9.0e15, 17.0, 78.8]	[16.9, 78.8], [17.0, 78.8]
-(s+1)/((s+2)*(s+3)) (genuine crossing)	-0.17	[-0.0, 1.0], [6.0, 5.0]	unchanged

The tests use zpk([1e-17], [-1, -1], 1) and zpk([1 + 1e-15], [0.5, 0.5], 1, 0.1) with and without delays. For these systems the computed G(0) is negative independently of the eigenvalue solver. With the check disabled, 8 of the 9 new assertions fail; the ninth checks the remaining gain margin. The test of #1045 and the new tests with a negative DC gain verify that genuine crossings at ω = 0 are retained.

This problem was found while reviewing #1076, which changes the same function in other lines.

Tests

Linux, Julia 1.13.0, with this branch of ControlSystemsBase developed into each environment:

Package	Result
ControlSystemsBase	21352 passed, 6 broken
DyadControlSystems	3483 passed, 2 failed, 1 errored, 20 broken; identical with master (ControlSystemsBase 1.23.0)

DyadControlSystems is the only one of RobustAndOptimalControl, ControlSystemsMTK, ModelPredictiveControl and DyadControlSystems that calls margin. Its failures are an allocation test, a convergence tolerance of the linear MPC, and the MethodError of bodeplot for a ControlSystemIdentification.FRD described in #1077.

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 91.70%. Comparing base (f1a56a9) to head (42bfff5).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1078      +/-   ##
==========================================
+ Coverage   91.69%   91.70%   +0.01%     
==========================================
  Files          42       42              
  Lines        5789     5796       +7     
==========================================
+ Hits         5308     5315       +7     
  Misses        481      481              

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