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ENH: add linform to compute linear system L-infinity norm #729

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Apr 27, 2022
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ENH: add linform to compute linear system L-infinity norm #729

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66 changes: 65 additions & 1 deletion control/statesp.py
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Original file line number Diff line number Diff line change
Expand Up @@ -55,6 +55,7 @@
from numpy.linalg import solve, eigvals, matrix_rank
from numpy.linalg.linalg import LinAlgError
import scipy as sp
import scipy.linalg
from scipy.signal import cont2discrete
from scipy.signal import StateSpace as signalStateSpace
from warnings import warn
Expand All @@ -63,7 +64,12 @@
from . import config
from copy import deepcopy

__all__ = ['StateSpace', 'tf2ss', 'ssdata']
try:
from slycot import ab13dd
except ImportError:
ab13dd = None

__all__ = ['StateSpace', 'tf2ss', 'ssdata', 'linfnorm']

# Define module default parameter values
_statesp_defaults = {
Expand Down Expand Up @@ -1888,3 +1894,61 @@ def ssdata(sys):
"""
ss = _convert_to_statespace(sys)
return ss.A, ss.B, ss.C, ss.D


def linfnorm(sys, tol=1e-10):
"""L-infinity norm of a linear system

Parameters
----------
sys : LTI (StateSpace or TransferFunction)
system to evalute L-infinity norm of
tol : real scalar
tolerance on norm estimate

Returns
-------
gpeak : non-negative scalar
L-infinity norm
fpeak : non-negative scalar
Frequency, in rad/s, at which gpeak occurs

For stable systems, the L-infinity and H-infinity norms are equal;
for unstable systems, the H-infinity norm is infinite, while the
L-infinity norm is finite if the system has no poles on the
imaginary axis.

See also
--------
slycot.ab13dd : the Slycot routine linfnorm that does the calculation
"""

if ab13dd is None:
raise ControlSlycot("Can't find slycot module 'ab13dd'")

a, b, c, d = ssdata(_convert_to_statespace(sys))
e = np.eye(a.shape[0])

n = a.shape[0]
m = b.shape[1]
p = c.shape[0]

if n == 0:
# ab13dd doesn't accept empty A, B, C, D;
# static gain case is easy enough to compute
gpeak = scipy.linalg.svdvals(d)[0]
# max svd is constant with freq; arbitrarily choose 0 as peak
fpeak = 0
return gpeak, fpeak
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Note that this will return a value even if sys.dt == None, contrary to the documentation.


dico = 'C' if sys.isctime() else 'D'
jobe = 'I'
equil = 'S'
jobd = 'Z' if all(0 == d.flat) else 'D'

gpeak, fpeak = ab13dd(dico, jobe, equil, jobd, n, m, p, a, e, b, c, d, tol)

if dico=='D':
fpeak /= sys.dt

return gpeak, fpeak
53 changes: 52 additions & 1 deletion control/tests/statesp_test.py
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Original file line number Diff line number Diff line change
Expand Up @@ -19,13 +19,15 @@
from control.dtime import sample_system
from control.lti import evalfr
from control.statesp import StateSpace, _convert_to_statespace, tf2ss, \
_statesp_defaults, _rss_generate
_statesp_defaults, _rss_generate, linfnorm
from control.iosys import ss, rss, drss
from control.tests.conftest import ismatarrayout, slycotonly
from control.xferfcn import TransferFunction, ss2tf


from .conftest import editsdefaults


class TestStateSpace:
"""Tests for the StateSpace class."""

Expand Down Expand Up @@ -1097,3 +1099,52 @@ def test_latex_repr_testsize(editsdefaults):

gstatic = ss([], [], [], 1)
assert gstatic._repr_latex_() is None


class TestLinfnorm:
# these are simple tests; we assume ab13dd is correct
# python-control specific behaviour is:
# - checking for continuous- and discrete-time
# - scaling fpeak for discrete-time
# - handling static gains

# the underdamped gpeak and fpeak are found from
# gpeak = 1/(2*zeta*(1-zeta**2)**0.5)
# fpeak = wn*(1-2*zeta**2)**0.5
@pytest.fixture(params=[
('static', ct.tf, ([1.23],[1]), 1.23, 0),
('underdamped', ct.tf, ([100],[1, 2*0.5*10, 100]), 1.1547005, 7.0710678),
])
def ct_siso(self, request):
name, systype, sysargs, refgpeak, reffpeak = request.param
return systype(*sysargs), refgpeak, reffpeak

@pytest.fixture(params=[
('underdamped', ct.tf, ([100],[1, 2*0.5*10, 100]), 1e-4, 1.1547005, 7.0710678),
])
def dt_siso(self, request):
name, systype, sysargs, dt, refgpeak, reffpeak = request.param
return ct.c2d(systype(*sysargs), dt), refgpeak, reffpeak

@slycotonly
def test_linfnorm_ct_siso(self, ct_siso):
sys, refgpeak, reffpeak = ct_siso
gpeak, fpeak = linfnorm(sys)
np.testing.assert_allclose(gpeak, refgpeak)
np.testing.assert_allclose(fpeak, reffpeak)

@slycotonly
def test_linfnorm_dt_siso(self, dt_siso):
sys, refgpeak, reffpeak = dt_siso
gpeak, fpeak = linfnorm(sys)
# c2d pole-mapping has round-off
np.testing.assert_allclose(gpeak, refgpeak)
np.testing.assert_allclose(fpeak, reffpeak)

@slycotonly
def test_linfnorm_ct_mimo(self, ct_siso):
siso, refgpeak, reffpeak = ct_siso
sys = ct.append(siso, siso)
gpeak, fpeak = linfnorm(sys)
np.testing.assert_allclose(gpeak, refgpeak)
np.testing.assert_allclose(fpeak, reffpeak)