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Original file line number	Diff line number	Diff line change
Expand Up		@@ -152,7 +152,7 @@ def describing_function(
		#
		# The describing function of a nonlinear function F() can be computed by
		# evaluating the nonlinearity over a sinusoid. The Fourier series for a
		# static nonlinear function evaluated on a sinusoid can be written as
		# nonlinear function evaluated on a sinusoid can be written as
		#
		# F(A\sin\omega t) = \sum_{k=1}^\infty M_k(A) \sin(k\omega t + \phi_k(A))
		#
Expand Down Expand Up		@@ -226,10 +226,10 @@ class DescribingFunctionResponse:
		"""Results of describing function analysis.

		Describing functions allow analysis of a linear I/O systems with a
		static nonlinear feedback function. The DescribingFunctionResponse
		class is used by the `describing_function_response`
		function to return the results of a describing function analysis. The
		response object can be used to obtain information about the describing
		nonlinear feedback function. The DescribingFunctionResponse class
		is used by the `describing_function_response` function to return
		the results of a describing function analysis. The response
		object can be used to obtain information about the describing
		function analysis or generate a Nyquist plot showing the frequency
		response of the linear systems and the describing function for the
		nonlinear element.
Expand Down Expand Up		@@ -283,16 +283,16 @@ def describing_function_response(
		"""Compute the describing function response of a system.

		This function uses describing function analysis to analyze a closed
		loop system consisting of a linear system with a static nonlinear
		function in the feedback path.
		loop system consisting of a linear system with a nonlinear function in
		the feedback path.

		Parameters
		----------
		H : LTI system
		Linear time-invariant (LTI) system (state space, transfer function,
		or FRD).
		F : static nonlinear function
		A static nonlinearity, either a scalar function or a single-input,
		F : nonlinear function
		Feedback nonlinearity, either a scalar function or a single-input,
		single-output, static input/output system.
		A : list
		List of amplitudes to be used for the describing function plot.
Expand Down Expand Up		@@ -405,8 +405,7 @@ def describing_function_plot(
		Nyquist plot with describing function for a nonlinear system.

		This function generates a Nyquist plot for a closed loop system
		consisting of a linear system with a static nonlinear function in the
		feedback path.
		consisting of a linear system with a nonlinearity in the feedback path.

		The function may be called in one of two forms:

Expand All		@@ -426,9 +425,9 @@ def describing_function_plot(
		H : LTI system
		Linear time-invariant (LTI) system (state space, transfer function,
		or FRD).
		F : static nonlinear function
		A static nonlinearity, either a scalar function or a single-input,
		single-output, static input/output system.
		F : nonlinear function
		Nonlinearity in the feedback path, either a scalar function or a
		single-input, single-output, static input/output system.
		A : list
		List of amplitudes to be used for the describing function plot.
		omega : list, optional
Expand Down

4 changes: 0 additions & 4 deletions 4 control/iosys.py

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Original file line number	Diff line number	Diff line change
Expand Up		@@ -757,10 +757,6 @@ def issiso(self):
		"""Check to see if a system is single input, single output."""
		return self.ninputs == 1 and self.noutputs == 1

		def _isstatic(self):
		"""Check to see if a system is a static system (no states)"""
		return self.nstates == 0


		# Test to see if a system is SISO
		def issiso(sys, strict=False):
Expand Down

29 changes: 17 additions & 12 deletions 29 control/nlsys.py

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Original file line number	Diff line number	Diff line change
Expand Up		@@ -181,7 +181,7 @@ def __call__(sys, u, params=None, squeeze=None):

		"""
		# Make sure the call makes sense
		if not sys._isstatic():
		if sys.nstates != 0:
		raise TypeError(
		"function evaluation is only supported for static "
		"input/output systems")
Expand All		@@ -199,7 +199,7 @@ def __call__(sys, u, params=None, squeeze=None):
		def __mul__(self, other):
		"""Multiply two input/output systems (series interconnection)"""
		# Convert 'other' to an I/O system if needed
		other = _convert_static_iosystem(other)
		other = _convert_to_iosystem(other)
		if not isinstance(other, InputOutputSystem):
		return NotImplemented

Expand Down Expand Up		@@ -231,7 +231,7 @@ def __mul__(self, other):
		def __rmul__(self, other):
		"""Pre-multiply an input/output systems by a scalar/matrix"""
		# Convert other to an I/O system if needed
		other = _convert_static_iosystem(other)
		other = _convert_to_iosystem(other)
		if not isinstance(other, InputOutputSystem):
		return NotImplemented

Expand Down Expand Up		@@ -263,7 +263,7 @@ def __rmul__(self, other):
		def __add__(self, other):
		"""Add two input/output systems (parallel interconnection)"""
		# Convert other to an I/O system if needed
		other = _convert_static_iosystem(other)
		other = _convert_to_iosystem(other)
		if not isinstance(other, InputOutputSystem):
		return NotImplemented

Expand All		@@ -284,7 +284,7 @@ def __add__(self, other):
		def __radd__(self, other):
		"""Parallel addition of input/output system to a compatible object."""
		# Convert other to an I/O system if needed
		other = _convert_static_iosystem(other)
		other = _convert_to_iosystem(other)
		if not isinstance(other, InputOutputSystem):
		return NotImplemented

Expand All		@@ -305,7 +305,7 @@ def __radd__(self, other):
		def __sub__(self, other):
		"""Subtract two input/output systems (parallel interconnection)"""
		# Convert other to an I/O system if needed
		other = _convert_static_iosystem(other)
		other = _convert_to_iosystem(other)
		if not isinstance(other, InputOutputSystem):
		return NotImplemented

Expand All		@@ -329,7 +329,7 @@ def __sub__(self, other):
		def __rsub__(self, other):
		"""Parallel subtraction of I/O system to a compatible object."""
		# Convert other to an I/O system if needed
		other = _convert_static_iosystem(other)
		other = _convert_to_iosystem(other)
		if not isinstance(other, InputOutputSystem):
		return NotImplemented
		return other - self
Expand All		@@ -355,6 +355,10 @@ def __truediv__(self, other):
		else:
		return NotImplemented

		# Determine if a system is static (memoryless)
		def _isstatic(self):
		return self.nstates == 0

		def _update_params(self, params):
		# Update the current parameter values
		self._current_params = self.params.copy()
Expand Down Expand Up		@@ -484,7 +488,7 @@ def feedback(self, other=1, sign=-1, params=None):

		"""
		# Convert sys2 to an I/O system if needed
		other = _convert_static_iosystem(other)
		other = _convert_to_iosystem(other)

		# Make sure systems can be interconnected
		if self.noutputs != other.ninputs or other.noutputs != self.ninputs:
Expand Down Expand Up		@@ -932,6 +936,7 @@ def _out(self, t, x, u):
		# Make the full set of subsystem outputs to system output
		return self.output_map @ ylist

		# Find steady state (static) inputs and outputs
		def _compute_static_io(self, t, x, u):
		# Figure out the total number of inputs and outputs
		(ninputs, noutputs) = self.connect_map.shape
Expand Down Expand Up		@@ -1711,8 +1716,8 @@ def ufun(t):
		dt = (t - T[idx-1]) / (T[idx] - T[idx-1])
		return U[..., idx-1] * (1. - dt) + U[..., idx] * dt

		# Check to make sure this is not a static function
		if nstates == 0: # No states => map input to output
		# Check to make sure see if this is a static function
		if sys.nstates == 0:
		# Make sure the user gave a time vector for evaluation (or 'T')
		if t_eval is None:
		# User overrode t_eval with None, but didn't give us the times...
Expand Down Expand Up		@@ -2924,8 +2929,8 @@ def _process_vector_argument(arg, name, size):
		return val, nelem


		# Utility function to create an I/O system from a static gain
		def _convert_static_iosystem(sys):
		# Utility function to create an I/O system (from number or array)
		def _convert_to_iosystem(sys):
		# If we were given an I/O system, do nothing
		if isinstance(sys, InputOutputSystem):
		return sys
Expand Down

7 changes: 5 additions & 2 deletions 7 control/statesp.py

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Original file line number	Diff line number	Diff line change
Expand Up		@@ -229,6 +229,9 @@ def __init__(self, args, *kwargs):
		self.C = C
		self.D = D

		# Determine if the system is static (memoryless)
		static = (A.size == 0)

		#
		# Process keyword arguments
		#
Expand All		@@ -242,7 +245,7 @@ def __init__(self, args, *kwargs):
		{'inputs': B.shape[1], 'outputs': C.shape[0],
		'states': A.shape[0]}
		name, inputs, outputs, states, dt = _process_iosys_keywords(
		kwargs, defaults, static=(A.size == 0))
		kwargs, defaults, static=static)

		# Create updfcn and outfcn
		updfcn = lambda t, x, u, params: \
Expand All		@@ -257,7 +260,7 @@ def __init__(self, args, *kwargs):
		states=states, dt=dt, **kwargs)

		# Reset shapes if the system is static
		if self._isstatic():
		if static:
		A.shape = (0, 0)
		B.shape = (0, self.ninputs)
		C.shape = (self.noutputs, 0)
Expand Down

20 changes: 10 additions & 10 deletions 20 control/xferfcn.py

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Original file line number	Diff line number	Diff line change
Expand Up		@@ -219,15 +219,22 @@ def __init__(self, args, *kwargs):
		raise ValueError("display_format must be 'poly' or 'zpk',"
		" got '%s'" % self.display_format)

		# Determine if the transfer function is static (needed for dt)
		#
		# Determine if the transfer function is static (memoryless)
		#
		# True if and only if all of the numerator and denominator
		# polynomials of the (MIMO) transfer function are zeroth order.
		#
		static = True
		for arr in [num, den]:
		# Iterate using refs_OK since num and den are ndarrays of ndarrays
		for poly_ in np.nditer(arr, flags=['refs_ok']):
		if poly_.item().size > 1:
		static = False
		break
		if not static:
		break
		self._static = static # retain for later usage

		defaults = args[0] if len(args) == 1 else \
		{'inputs': num.shape[1], 'outputs': num.shape[0]}
Expand Down Expand Up		@@ -1283,16 +1290,9 @@ def dcgain(self, warn_infinite=False):
		"""
		return self._dcgain(warn_infinite)

		# Determine if a system is static (memoryless)
		def _isstatic(self):
		"""returns True if and only if all of the numerator and denominator
		polynomials of the (possibly MIMO) transfer function are zeroth order,
		that is, if the system has no dynamics. """
		for list_of_polys in self.num, self.den:
		for row in list_of_polys:
		for poly_ in row:
		if len(poly_) > 1:
		return False
		return True
		return self._static # Check done at initialization

		# Attributes for differentiation and delay
		#
Expand Down

15 changes: 11 additions & 4 deletions 15 doc/descfcn.rst

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Original file line number	Diff line number	Diff line change
Expand Up		@@ -6,14 +6,21 @@ Describing Functions
		====================

		For nonlinear systems consisting of a feedback connection between a
		linear system and a static nonlinearity, it is possible to obtain a
		linear system and a nonlinearity, it is possible to obtain a
		generalization of Nyquist's stability criterion based on the idea of
		describing functions. The basic concept involves approximating the
		response of a static nonlinearity to an input :math:`u = A e^{j \omega
		t}` as an output :math:`y = N(A) (A e^{j \omega t})`, where :math:`N(A)
		\in \mathbb{C}` represents the (amplitude-dependent) gain and phase
		response of a nonlinearity to an input :math:`u = A e^{j \omega t}` as
		an output :math:`y = N(A) (A e^{j \omega t})`, where :math:`N(A) \in
		\mathbb{C}` represents the (amplitude-dependent) gain and phase
		associated with the nonlinearity.

		In the most common case, the nonlinearity will be a static,
		time-invariant nonlinear function :math:`y = h(u)`. However,
		describing functions can be defined for nonlinear input/output systems
		that have some internal memory, such as hysteresis or backlash. For
		simplicity, we take the nonlinearity to be static (memoryless) in the
		description below, unless otherwise specified.

		Stability analysis of a linear system :math:`H(s)` with a feedback
		nonlinearity :math:`F(x)` is done by looking for amplitudes :math:`A`
		and frequencies :math:`\omega` such that
Expand Down

9 changes: 8 additions & 1 deletion 9 doc/nlsys.rst

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Original file line number	Diff line number	Diff line change
Expand Up		@@ -23,6 +23,11 @@ Discrete time systems are also supported and have dynamics of the form
		x[t+1] &= f(t, x[t], u[t], \theta), \\
		y[t] &= h(t, x[t], u[t], \theta).

		A nonlinear input/output model is said to be "static" if the output
		:math:`y(t)` at any given time :math:`t` depends only on the input
		:math:`u(t)` at that same time :math:`t` and not on past or future
		values of :math:`u`.


		.. _sec-nonlinear-models:

Expand All		@@ -47,7 +52,9 @@ dynamics of the system can be in continuous or discrete time (use the
		The output function `outfcn` is used to specify the outputs of the
		system and has the same calling signature as `updfcn`. If it is not
		specified, then the output of the system is set equal to the system
		state. Otherwise, it should return an array of shape (p,).
		state. Otherwise, it should return an array of shape (p,). If a
		input/output system is static, the state `x` should still be passed to
		the output function, but the state is ignored.

		Note that the number of states, inputs, and outputs should generally
		be explicitly specified, although some operations can infer the
Expand Down

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