python-control · murrayrm · Aug 11, 2026 · Jul 21, 2026
diff --git a/control/tests/discrete_test.py b/control/tests/discrete_test.py
@@ -561,3 +561,36 @@ def test_c2d_matched(num, den, dt, method):
         for czero in sys_ct.zeros():
             zzero = zzeros[(np.abs(zzeros - cmath.exp(czero * dt))).argmin()]
             assert cmath.isclose(cmath.exp(czero * dt), zzero)
+
+
+@pytest.mark.parametrize("num, den", [
+    ([1.], [1., 0.]),                   # integrator (pole at s = 0)
+    ([1.], [1., 0., 0.]),               # double integrator
+    ([2., 5.], [1., 0.]),               # PI controller
+    ([1.], [1., 1., 0.]),               # type 1 plant, 1/(s(s+1))
+    ([1., 0.], [1., 2., 5.]),           # differentiator (zero at s = 0)
+    ([2., 0., 0.], [1., 3., 3., 1.]),   # double zero at s = 0
+    ([1., 0.], [1., 0., 0.]),           # zero and pole at s = 0
+])
+@pytest.mark.parametrize("dt", [0.1, 0.5])
+def test_c2d_matched_origin(num, den, dt):
+    # A pole or zero at s = 0 (integrators, PI/PID, type-1/2 plants) used to
+    # give an all-NaN numerator: the DC-gain match divides by the vanishing
+    # 1 - z factor of the origin pole/zero (#950, #951).
+    sys_ct = ct.tf(num, den)
+    sys_dt = ct.sample_system(sys_ct, dt, method='matched')
+    assert np.all(np.isfinite(sys_dt.num[0][0]))
+    assert np.all(np.isfinite(sys_dt.den[0][0]))
+    # the gain is matched just off the origin, so |G_d(e^jwT)| -> |G_c(jw)|
+    w = 1e-3 / dt
+    assert np.isclose(abs(sys_ct(1j * w)),
+                      abs(sys_dt(cmath.exp(1j * w * dt))), rtol=1e-4)
+
+
+@pytest.mark.parametrize("dt", [0.1, 0.5, 2])
+@pytest.mark.parametrize("k", [1, 2, 3])
+def test_c2d_matched_integrator(k, dt):
+    # matched discretization of 1/s**k is the textbook Ts**k / (z - 1)**k
+    sys_dt = ct.tf([1.], [1.] + [0.] * k).sample(dt, method='matched')
+    np.testing.assert_allclose(sys_dt.num[0][0], [dt**k])
+    np.testing.assert_allclose(sys_dt.den[0][0], np.poly([1.] * k))
diff --git a/control/xferfcn.py b/control/xferfcn.py
@@ -1334,23 +1334,40 @@ def _c2d_matched(sysC, Ts, **kwargs):
         raise ControlMIMONotImplemented("Not implemented for MIMO systems")
 
     # Pole-zero match method of continuous to discrete time conversion
-    szeros, spoles, _ = tf2zpk(sysC.num[0][0], sysC.den[0][0])
+    szeros, spoles, sgain = tf2zpk(sysC.num[0][0], sysC.den[0][0])
     zzeros = [0] * len(szeros)
     zpoles = [0] * len(spoles)
-    pregainnum = [0] * len(szeros)
-    pregainden = [0] * len(spoles)
+    # The gain is matched at the origin (z = 1).  A pole or zero at s = 0 maps
+    # to z = 1, so its 1 - z factor vanishes and matching the DC gain there is
+    # 0/0 or inf/inf -> a NaN numerator for integrators, PI/PID and other
+    # type-1/type-2 systems.  Keep the origin factors out of the gain product
+    # and restore their scaling through the z - 1 ~ s*Ts limit, which recovers
+    # Ts/(z - 1) for 1/s, Ts**2/(z - 1)**2 for 1/s**2, etc. (completes #951).
+    origin_zeros = origin_poles = 0
+    numgain, dengain = sgain, 1.0
+    pregainnum = pregainden = 1.0
     for idx, s in enumerate(szeros):
-        sTs = s * Ts
-        z = exp(sTs)
-        zzeros[idx] = z
-        pregainnum[idx] = 1 - z
+        zzeros[idx] = exp(s * Ts)
+        if s == 0:
+            origin_zeros += 1
+        else:
+            numgain *= -s
+            pregainnum *= 1 - zzeros[idx]
     for idx, s in enumerate(spoles):
-        sTs = s * Ts
-        z = exp(sTs)
-        zpoles[idx] = z
-        pregainden[idx] = 1 - z
-    zgain = np.multiply.reduce(pregainnum) / np.multiply.reduce(pregainden)
-    gain = sysC.dcgain() / zgain.real
+        zpoles[idx] = exp(s * Ts)
+        if s == 0:
+            origin_poles += 1
+        else:
+            dengain *= -s
+            pregainden *= 1 - zpoles[idx]
+    zgain = pregainnum / pregainden
+    if origin_zeros or origin_poles:
+        # DC gain of the system with the origin factors divided out, rescaled
+        # by Ts**(origin poles - origin zeros) from the z - 1 ~ s*Ts limit
+        gain = (numgain / dengain).real \
+            * Ts**(origin_poles - origin_zeros) / zgain.real
+    else:
+        gain = sysC.dcgain() / zgain.real
     sysDnum, sysDden = zpk2tf(zzeros, zpoles, gain)
     return TransferFunction(sysDnum, sysDden, Ts, **kwargs)