initial implementation of differential flatness module (python-contro…

…l#324)

No comments in 7 days, so assuming this is ok (adds optional functionality).

control/flatsys/__init__.py

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+# flatsys/__init__.py: flat systems package initialization file
+#
+# Copyright (c) 2019 by California Institute of Technology
+# All rights reserved.
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above copyright
+#    notice, this list of conditions and the following disclaimer in the
+#    documentation and/or other materials provided with the distribution.
+#
+# 3. Neither the name of the California Institute of Technology nor
+#    the names of its contributors may be used to endorse or promote
+#    products derived from this software without specific prior
+#    written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
+# FOR A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL CALTECH
+# OR THE CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
+# USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
+# ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
+# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
+# OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+# SUCH DAMAGE.
+#
+# Author: Richard M. Murray
+# Date: 1 Jul 2019
+
+r"""The :mod:`control.flatsys` package contains a set of classes and functions
+that can be used to compute trajectories for differentially flat systems.
+
+A differentially flat system is defined by creating an object using the
+:class:`~control.flatsys.FlatSystem` class, which has member functions for
+mapping the system state and input into and out of flat coordinates.  The
+:func:`~control.flatsys.point_to_point` function can be used to create a
+trajectory between two endpoints, written in terms of a set of basis functions
+defined using the :class:`~control.flatsys.BasisFamily` class.  The resulting
+trajectory is return as a :class:`~control.flatsys.SystemTrajectory` object
+and can be evaluated using the :func:`~control.flatsys.SystemTrajectory.eval`
+member function.
+
+"""
+
+# Basis function families
+from .basis import BasisFamily
+from .poly import PolyFamily
+
+# Classes
+from .systraj import SystemTrajectory
+from .flatsys import FlatSystem
+from .linflat import LinearFlatSystem
+
+# Package functions
+from .flatsys import point_to_point

control/flatsys/basis.py

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+# basis.py - BasisFamily class
+# RMM, 10 Nov 2012
+#
+# The BasisFamily class is used to specify a set of basis functions for
+# implementing differential flatness computations.
+#
+# Copyright (c) 2012 by California Institute of Technology
+# All rights reserved.
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above copyright
+#    notice, this list of conditions and the following disclaimer in the
+#    documentation and/or other materials provided with the distribution.
+#
+# 3. Neither the name of the California Institute of Technology nor
+#    the names of its contributors may be used to endorse or promote
+#    products derived from this software without specific prior
+#    written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
+# FOR A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL CALTECH
+# OR THE CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
+# USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
+# ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
+# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
+# OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+# SUCH DAMAGE.
+
+
+# Basis family class (for use as a base class)
+class BasisFamily:
+    """Base class for implementing basis functions for flat systems.
+
+    A BasisFamily object is used to construct trajectories for a flat system.
+    The class must implement a single function that computes the jth
+    derivative of the ith basis function at a time t:
+
+      :math:`z_i^{(q)}(t)` = basis.eval_deriv(self, i, j, t)
+
+    """
+    def __init__(self, N):
+        """Create a basis family of order N."""
+        self.N = N                    # save number of basis functions