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Fix matched c2d NaN for a pole or zero at the origin
TransferFunction.sample(Ts, method='matched') (and c2d) returned an all-NaN numerator for any system with a pole or zero at s = 0 - integrators, PI/PID controllers, differentiators and other type-1/2 systems. The DC-gain match divides the continuous DC gain (inf for an origin pole, 0 for an origin zero) by a zgain that carries the same vanishing 1 - z factor, so the gain is 0/0 or inf/inf. Match the gain from the non-origin factors only and restore the origin scaling analytically through the z - 1 ~ s*Ts limit, which recovers the textbook Ts/(z - 1) for 1/s, Ts**2/(z - 1)**2 for 1/s**2, and so on. Finite-DC systems are byte-for-byte unchanged. Completes the gain fix from #950/#951.
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Lines changed: 63 additions & 13 deletions

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‎control/tests/discrete_test.py‎

Lines changed: 33 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -561,3 +561,36 @@ def test_c2d_matched(num, den, dt, method):
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for czero in sys_ct.zeros():
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zzero = zzeros[(np.abs(zzeros - cmath.exp(czero * dt))).argmin()]
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assert cmath.isclose(cmath.exp(czero * dt), zzero)
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@pytest.mark.parametrize("num, den", [
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([1.], [1., 0.]), # integrator (pole at s = 0)
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([1.], [1., 0., 0.]), # double integrator
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([2., 5.], [1., 0.]), # PI controller
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([1.], [1., 1., 0.]), # type 1 plant, 1/(s(s+1))
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([1., 0.], [1., 2., 5.]), # differentiator (zero at s = 0)
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([2., 0., 0.], [1., 3., 3., 1.]), # double zero at s = 0
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([1., 0.], [1., 0., 0.]), # zero and pole at s = 0
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])
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@pytest.mark.parametrize("dt", [0.1, 0.5])
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def test_c2d_matched_origin(num, den, dt):
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# A pole or zero at s = 0 (integrators, PI/PID, type-1/2 plants) used to
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# give an all-NaN numerator: the DC-gain match divides by the vanishing
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# 1 - z factor of the origin pole/zero (#950, #951).
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sys_ct = ct.tf(num, den)
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sys_dt = ct.sample_system(sys_ct, dt, method='matched')
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assert np.all(np.isfinite(sys_dt.num[0][0]))
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assert np.all(np.isfinite(sys_dt.den[0][0]))
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# the gain is matched just off the origin, so |G_d(e^jwT)| -> |G_c(jw)|
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w = 1e-3 / dt
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assert np.isclose(abs(sys_ct(1j * w)),
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abs(sys_dt(cmath.exp(1j * w * dt))), rtol=1e-4)
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@pytest.mark.parametrize("dt", [0.1, 0.5, 2])
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@pytest.mark.parametrize("k", [1, 2, 3])
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def test_c2d_matched_integrator(k, dt):
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# matched discretization of 1/s**k is the textbook Ts**k / (z - 1)**k
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sys_dt = ct.tf([1.], [1.] + [0.] * k).sample(dt, method='matched')
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np.testing.assert_allclose(sys_dt.num[0][0], [dt**k])
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np.testing.assert_allclose(sys_dt.den[0][0], np.poly([1.] * k))

‎control/xferfcn.py‎

Lines changed: 30 additions & 13 deletions
Original file line numberDiff line numberDiff line change
@@ -1334,23 +1334,40 @@ def _c2d_matched(sysC, Ts, **kwargs):
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raise ControlMIMONotImplemented("Not implemented for MIMO systems")
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# Pole-zero match method of continuous to discrete time conversion
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szeros, spoles, _ = tf2zpk(sysC.num[0][0], sysC.den[0][0])
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szeros, spoles, sgain = tf2zpk(sysC.num[0][0], sysC.den[0][0])
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zzeros = [0] * len(szeros)
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zpoles = [0] * len(spoles)
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pregainnum = [0] * len(szeros)
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pregainden = [0] * len(spoles)
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# The gain is matched at the origin (z = 1). A pole or zero at s = 0 maps
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# to z = 1, so its 1 - z factor vanishes and matching the DC gain there is
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# 0/0 or inf/inf -> a NaN numerator for integrators, PI/PID and other
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# type-1/type-2 systems. Keep the origin factors out of the gain product
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# and restore their scaling through the z - 1 ~ s*Ts limit, which recovers
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# Ts/(z - 1) for 1/s, Ts**2/(z - 1)**2 for 1/s**2, etc. (completes #951).
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origin_zeros = origin_poles = 0
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numgain, dengain = sgain, 1.0
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pregainnum = pregainden = 1.0
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for idx, s in enumerate(szeros):
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sTs = s * Ts
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z = exp(sTs)
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zzeros[idx] = z
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pregainnum[idx] = 1 - z
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zzeros[idx] = exp(s * Ts)
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if s == 0:
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origin_zeros += 1
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else:
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numgain *= -s
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pregainnum *= 1 - zzeros[idx]
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for idx, s in enumerate(spoles):
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sTs = s * Ts
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z = exp(sTs)
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zpoles[idx] = z
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pregainden[idx] = 1 - z
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zgain = np.multiply.reduce(pregainnum) / np.multiply.reduce(pregainden)
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gain = sysC.dcgain() / zgain.real
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zpoles[idx] = exp(s * Ts)
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if s == 0:
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origin_poles += 1
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else:
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dengain *= -s
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pregainden *= 1 - zpoles[idx]
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zgain = pregainnum / pregainden
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if origin_zeros or origin_poles:
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# DC gain of the system with the origin factors divided out, rescaled
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# by Ts**(origin poles - origin zeros) from the z - 1 ~ s*Ts limit
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gain = (numgain / dengain).real \
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* Ts**(origin_poles - origin_zeros) / zgain.real
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else:
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gain = sysC.dcgain() / zgain.real
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sysDnum, sysDden = zpk2tf(zzeros, zpoles, gain)
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return TransferFunction(sysDnum, sysDden, Ts, **kwargs)
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