diff --git a/control/statesp.py b/control/statesp.py
index bff14d241..8f7861434 100644
--- a/control/statesp.py
+++ b/control/statesp.py
@@ -54,8 +54,7 @@
import math
import numpy as np
from numpy import all, angle, any, array, asarray, concatenate, cos, delete, \
- dot, empty, exp, eye, matrix, ones, poly, poly1d, roots, shape, sin, \
- zeros, squeeze
+ dot, empty, exp, eye, isinf, matrix, ones, pad, shape, sin, zeros, squeeze
from numpy.random import rand, randn
from numpy.linalg import solve, eigvals, matrix_rank
from numpy.linalg.linalg import LinAlgError
@@ -516,17 +515,40 @@ def pole(self):
def zero(self):
"""Compute the zeros of a state space system."""
- if self.inputs > 1 or self.outputs > 1:
- raise NotImplementedError("StateSpace.zeros is currently \
-implemented only for SISO systems.")
+ if not self.states:
+ return np.array([])
- den = poly1d(poly(self.A))
- # Compute the numerator based on zeros
- #! TODO: This is currently limited to SISO systems
- num = poly1d(poly(self.A - dot(self.B, self.C)) + ((self.D[0, 0] - 1) *
- den))
-
- return roots(num)
+ # Use AB08ND from Slycot if it's available, otherwise use
+ # scipy.lingalg.eigvals().
+ try:
+ from slycot import ab08nd
+
+ out = ab08nd(self.A.shape[0], self.B.shape[1], self.C.shape[0],
+ self.A, self.B, self.C, self.D)
+ nu = out[0]
+ return sp.linalg.eigvals(out[8][0:nu,0:nu], out[9][0:nu,0:nu])
+ except ImportError: # Slycot unavailable. Fall back to scipy.
+ if self.C.shape[0] != self.D.shape[1]:
+ raise NotImplementedError("StateSpace.zero only supports "
+ "systems with the same number of "
+ "inputs as outputs.")
+
+ # This implements the QZ algorithm for finding transmission zeros
+ # from
+ # https://dspace.mit.edu/bitstream/handle/1721.1/841/P-0802-06587335.pdf.
+ # The QZ algorithm solves the generalized eigenvalue problem: given
+ # `L = [A, B; C, D]` and `M = [I_nxn 0]`, find all finite λ for
+ # which there exist nontrivial solutions of the equation `Lz - λMz`.
+ #
+ # The generalized eigenvalue problem is only solvable if its
+ # arguments are square matrices.
+ L = concatenate((concatenate((self.A, self.B), axis=1),
+ concatenate((self.C, self.D), axis=1)), axis=0)
+ M = pad(eye(self.A.shape[0]), ((0, self.C.shape[0]),
+ (0, self.B.shape[1])), "constant")
+ return np.array([x for x in sp.linalg.eigvals(L, M,
+ overwrite_a=True)
+ if not isinf(x)])
# Feedback around a state space system
def feedback(self, other=1, sign=-1):
diff --git a/control/tests/statesp_test.py b/control/tests/statesp_test.py
index 5f2d7d244..1677f6afe 100644
--- a/control/tests/statesp_test.py
+++ b/control/tests/statesp_test.py
@@ -43,14 +43,35 @@ def testPole(self):
np.testing.assert_array_almost_equal(p, true_p)
def testZero(self):
- """Evaluate the zeros of a SISO system."""
+ """Evaluate the zeros of a MIMO system."""
+
+ z = np.sort(self.sys1.zero())
+ true_z = np.sort([44.41465, -0.490252, -5.924398])
+
+ np.testing.assert_array_almost_equal(z, true_z)
+
+ A = np.array([[1, 0, 0, 0, 0, 0],
+ [0, 1, 0, 0, 0, 0],
+ [0, 0, 3, 0, 0, 0],
+ [0, 0, 0,-4, 0, 0],
+ [0, 0, 0, 0,-1, 0],
+ [0, 0, 0, 0, 0, 3]])
+ B = np.array([[0,-1],
+ [-1,0],
+ [1,-1],
+ [0, 0],
+ [0, 1],
+ [-1,-1]])
+ C = np.array([[1, 0, 0, 1, 0, 0],
+ [0, 1, 0, 1, 0, 1],
+ [0, 0, 1, 0, 0, 1]])
+ D = np.zeros((3,2))
+ sys = StateSpace(A, B, C, D)
- sys = StateSpace(self.sys1.A, [[3.], [-2.], [4.]], [[-1., 3., 2.]], [[-4.]])
- z = sys.zero()
+ z = np.sort(sys.zero())
+ true_z = np.sort([2., -1.])
- np.testing.assert_array_almost_equal(z, [4.26864638637134,
- -3.75932319318567 + 1.10087776649554j,
- -3.75932319318567 - 1.10087776649554j])
+ np.testing.assert_array_almost_equal(z, true_z)
def testAdd(self):
"""Add two MIMO systems."""
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