diff --git a/control/canonical.py b/control/canonical.py index a62044322..7be7f88ad 100644 --- a/control/canonical.py +++ b/control/canonical.py @@ -205,7 +205,7 @@ def similarity_transform(xsys, T, timescale=1, inverse=False): timescale : float, optional If present, also rescale the time unit to tau = timescale * t inverse : bool, optional - If True (default), transform so z = T x. If False, transform + If False (default), transform so z = T x. If True, transform so x = T z. Returns diff --git a/control/modelsimp.py b/control/modelsimp.py index 966da1ce7..fe519b82d 100644 --- a/control/modelsimp.py +++ b/control/modelsimp.py @@ -1,4 +1,3 @@ -#! TODO: add module docstring # modelsimp.py - tools for model simplification # # Author: Steve Brunton, Kevin Chen, Lauren Padilla @@ -6,47 +5,16 @@ # # This file contains routines for obtaining reduced order models # -# Copyright (c) 2010 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. -# -# $Id$ + +import warnings # External packages and modules import numpy as np -import warnings -from .exception import ControlSlycot, ControlArgument, ControlDimension -from .iosys import isdtime, isctime -from .statesp import StateSpace + +from .exception import ControlArgument, ControlDimension, ControlSlycot +from .iosys import isctime, isdtime from .statefbk import gram +from .statesp import StateSpace from .timeresp import TimeResponseData __all__ = ['hankel_singular_values', 'balanced_reduction', 'model_reduction', @@ -108,89 +76,146 @@ def hankel_singular_values(sys): return hsv[::-1] -def model_reduction(sys, ELIM, method='matchdc'): - """Model reduction by state elimination. - - Model reduction of `sys` by eliminating the states in `ELIM` using a given - method. +def model_reduction( + sys, elim_states=None, method='matchdc', elim_inputs=None, + elim_outputs=None, keep_states=None, keep_inputs=None, + keep_outputs=None, warn_unstable=True): + """Model reduction by input, output, or state elimination. + + This function produces a reduced-order model of a system by eliminating + specified inputs, outputs, and/or states from the original system. The + specific states, inputs, or outputs that are eliminated can be + specified by either listing the states, inputs, or outputs to be + eliminated or those to be kept. + + Two methods of state reduction are possible: 'truncate' removes the + states marked for elimination, while 'matchdc' replaces the eliminated + states with their equilibrium values (thereby keeping the input/output + gain unchanged at zero frequency ["DC"]). Parameters ---------- sys : StateSpace Original system to reduce. - ELIM : array - Vector of states to eliminate. + elim_inputs, elim_outputs, elim_states : array of int or str, optional + Vector of inputs, outputs, or states to eliminate. Can be specified + either as an offset into the appropriate vector or as a signal name. + keep_inputs, keep_outputs, keep_states : array, optional + Vector of inputs, outputs, or states to keep. Can be specified + either as an offset into the appropriate vector or as a signal name. method : string - Method of removing states in `ELIM`: either 'truncate' or - 'matchdc'. + Method of removing states: either 'truncate' or 'matchdc' (default). + warn_unstable : bool, option + If `False`, don't warn if system is unstable. Returns ------- rsys : StateSpace - A reduced order model. + Reduced order model. Raises ------ ValueError - Raised under the following conditions: + If `method` is not either 'matchdc' or 'truncate'. + NotImplementedError + If the 'matchdc' method is used for a discrete time system. - * if `method` is not either ``'matchdc'`` or ``'truncate'`` - - * if eigenvalues of `sys.A` are not all in left half plane - (`sys` must be stable) + Warns + ----- + UserWarning + If eigenvalues of `sys.A` are not all stable. Examples -------- >>> G = ct.rss(4) - >>> Gr = ct.modred(G, [0, 2], method='matchdc') + >>> Gr = ct.model_reduction(G, [0, 2], method='matchdc') >>> Gr.nstates 2 - """ + See Also + -------- + balanced_reduction : Eliminate states using Hankel singular values. + minimal_realization : Eliminate unreachable or unobservable states. - # Check for ss system object, need a utility for this? + Notes + ----- + The model_reduction function issues a warning if the system has + unstable eigenvalues, since in those situations the stability of the + reduced order model may be different than the stability of the full + model. No other checking is done, so users must to be careful not to + render a system unobservable or unreachable. - # TODO: Check for continous or discrete, only continuous supported for now - # if isCont(): - # dico = 'C' - # elif isDisc(): - # dico = 'D' - # else: - if (isctime(sys)): - dico = 'C' - else: - raise NotImplementedError("Function not implemented in discrete time") + States, inputs, and outputs can be specified using integer offsets or + using signal names. Slices can also be specified, but must use the + Python ``slice()`` function. + + """ + if not isinstance(sys, StateSpace): + raise TypeError("system must be a StateSpace system") # Check system is stable - if np.any(np.linalg.eigvals(sys.A).real >= 0.0): - raise ValueError("Oops, the system is unstable!") - - ELIM = np.sort(ELIM) - # Create list of elements not to eliminate (NELIM) - NELIM = [i for i in range(len(sys.A)) if i not in ELIM] - # A1 is a matrix of all columns of sys.A not to eliminate - A1 = sys.A[:, NELIM[0]].reshape(-1, 1) - for i in NELIM[1:]: - A1 = np.hstack((A1, sys.A[:, i].reshape(-1, 1))) - A11 = A1[NELIM, :] - A21 = A1[ELIM, :] - # A2 is a matrix of all columns of sys.A to eliminate - A2 = sys.A[:, ELIM[0]].reshape(-1, 1) - for i in ELIM[1:]: - A2 = np.hstack((A2, sys.A[:, i].reshape(-1, 1))) - A12 = A2[NELIM, :] - A22 = A2[ELIM, :] - - C1 = sys.C[:, NELIM] - C2 = sys.C[:, ELIM] - B1 = sys.B[NELIM, :] - B2 = sys.B[ELIM, :] - - if method == 'matchdc': - # if matchdc, residualize + if warn_unstable: + if isctime(sys) and np.any(np.linalg.eigvals(sys.A).real >= 0.0) or \ + isdtime(sys) and np.any(np.abs(np.linalg.eigvals(sys.A)) >= 1): + warnings.warn("System is unstable; reduction may be meaningless") + + # Utility function to process keep/elim keywords + def _process_elim_or_keep(elim, keep, labels): + def _expand_key(key): + if key is None: + return [] + elif isinstance(key, str): + return labels.index(key) + elif isinstance(key, list): + return [_expand_key(k) for k in key] + elif isinstance(key, slice): + return range(len(labels))[key] + else: + return key + elim = np.atleast_1d(_expand_key(elim)) + keep = np.atleast_1d(_expand_key(keep)) + + if len(elim) > 0 and len(keep) > 0: + raise ValueError( + "can't provide both 'keep' and 'elim' for same variables") + elif len(keep) > 0: + keep = np.sort(keep).tolist() + elim = [i for i in range(len(labels)) if i not in keep] + else: + elim = [] if elim is None else np.sort(elim).tolist() + keep = [i for i in range(len(labels)) if i not in elim] + return elim, keep + + # Determine which states to keep + elim_states, keep_states = _process_elim_or_keep( + elim_states, keep_states, sys.state_labels) + elim_inputs, keep_inputs = _process_elim_or_keep( + elim_inputs, keep_inputs, sys.input_labels) + elim_outputs, keep_outputs = _process_elim_or_keep( + elim_outputs, keep_outputs, sys.output_labels) + + # Create submatrix of states we are keeping + A11 = sys.A[:, keep_states][keep_states, :] # states we are keeping + A12 = sys.A[:, elim_states][keep_states, :] # needed for 'matchdc' + A21 = sys.A[:, keep_states][elim_states, :] + A22 = sys.A[:, elim_states][elim_states, :] + + B1 = sys.B[keep_states, :] + B2 = sys.B[elim_states, :] + + C1 = sys.C[:, keep_states] + C2 = sys.C[:, elim_states] + + # Figure out the new state space system + if method == 'matchdc' and A22.size > 0: + if sys.isdtime(strict=True): + raise NotImplementedError( + "'matchdc' not (yet) supported for discrete time systems") + + # if matchdc, residualize # Check if the matrix A22 is invertible - if np.linalg.matrix_rank(A22) != len(ELIM): + if np.linalg.matrix_rank(A22) != len(elim_states): raise ValueError("Matrix A22 is singular to working precision.") # Now precompute A22\A21 and A22\B2 (A22I = inv(A22)) @@ -206,22 +231,29 @@ def model_reduction(sys, ELIM, method='matchdc'): Br = B1 - A12 @ A22I_B2 Cr = C1 - C2 @ A22I_A21 Dr = sys.D - C2 @ A22I_B2 - elif method == 'truncate': - # if truncate, simply discard state x2 + + elif method == 'truncate' or A22.size == 0: + # Get rid of unwanted states Ar = A11 Br = B1 Cr = C1 Dr = sys.D + else: raise ValueError("Oops, method is not supported!") + # Get rid of additional inputs and outputs + Br = Br[:, keep_inputs] + Cr = Cr[keep_outputs, :] + Dr = Dr[keep_outputs, :][:, keep_inputs] + rsys = StateSpace(Ar, Br, Cr, Dr) return rsys def balanced_reduction(sys, orders, method='truncate', alpha=None): """Balanced reduced order model of sys of a given order. - + States are eliminated based on Hankel singular value. If sys has unstable modes, they are removed, the balanced realization is done on the stable part, then @@ -276,7 +308,7 @@ def balanced_reduction(sys, orders, method='truncate', alpha=None): raise ValueError("supported methods are 'truncate' or 'matchdc'") elif method == 'truncate': try: - from slycot import ab09md, ab09ad + from slycot import ab09ad, ab09md except ImportError: raise ControlSlycot( "can't find slycot subroutine ab09md or ab09ad") @@ -346,7 +378,7 @@ def balanced_reduction(sys, orders, method='truncate', alpha=None): def minimal_realization(sys, tol=None, verbose=True): """ Eliminate uncontrollable or unobservable states. - + Eliminates uncontrollable or unobservable states in state-space models or cancelling pole-zero pairs in transfer functions. The output sysr has minimal order and the same response @@ -378,14 +410,14 @@ def _block_hankel(Y, m, n): """Create a block Hankel matrix from impulse response.""" q, p, _ = Y.shape YY = Y.transpose(0, 2, 1) # transpose for reshape - + H = np.zeros((q*m, p*n)) - + for r in range(m): # shift and add row to Hankel matrix new_row = YY[:, r:r+n, :] H[q*r:q*(r+1), :] = new_row.reshape((q, p*n)) - + return H @@ -431,7 +463,7 @@ def eigensys_realization(arg, r, m=None, n=None, dt=True, transpose=False): unit-area impulse response of python-control. Default is True. transpose : bool, optional Assume that input data is transposed relative to the standard - :ref:`time-series-convention`. For TimeResponseData this parameter + :ref:`time-series-convention`. For TimeResponseData this parameter is ignored. Default is False. Returns @@ -466,7 +498,7 @@ def eigensys_realization(arg, r, m=None, n=None, dt=True, transpose=False): YY = np.array(arg, ndmin=3) if transpose: YY = np.transpose(YY) - + q, p, l = YY.shape if m is None: @@ -476,14 +508,14 @@ def eigensys_realization(arg, r, m=None, n=None, dt=True, transpose=False): if m*q < r or n*p < r: raise ValueError("Hankel parameters are to small") - + if (l-1) < m+n: raise ValueError("not enough data for requested number of parameters") - + H = _block_hankel(YY[:, :, 1:], m, n+1) # Hankel matrix (q*m, p*(n+1)) Hf = H[:, :-p] # first p*n columns of H Hl = H[:, p:] # last p*n columns of H - + U,S,Vh = np.linalg.svd(Hf, True) Ur =U[:, 0:r] Vhr =Vh[0:r, :] @@ -500,7 +532,7 @@ def eigensys_realization(arg, r, m=None, n=None, dt=True, transpose=False): def markov(*args, m=None, transpose=False, dt=None, truncate=False): """markov(Y, U, [, m]) - + Calculate the first `m` Markov parameters [D CB CAB ...] from data. This function computes the Markov parameters for a discrete time @@ -579,7 +611,7 @@ def markov(*args, m=None, transpose=False, dt=None, truncate=False): # Get the system description if len(args) < 1: raise ControlArgument("not enough input arguments") - + if isinstance(args[0], TimeResponseData): data = args[0] Umat = np.array(data.inputs, ndmin=2) @@ -639,7 +671,7 @@ def markov(*args, m=None, transpose=False, dt=None, truncate=False): # This algorithm sets up the following problem and solves it for # the Markov parameters # - # (l,q) = (l,p*m) @ (p*m,q) + # (l,q) = (l,p*m) @ (p*m,q) # YY = UU @ H.T # # [ y(0) ] [ u(0) 0 0 ] [ D ] @@ -675,7 +707,7 @@ def markov(*args, m=None, transpose=False, dt=None, truncate=False): # Truncate first t=0 or t=m time steps, transpose the problem for lsq YY = Ymat[:, t:].T UU = UUT[:, t:].T - + # Solve for the Markov parameters from YY = UU @ H.T HT, _, _, _ = np.linalg.lstsq(UU, YY, rcond=None) H = HT.T/dt # scaling diff --git a/control/tests/docstrings_test.py b/control/tests/docstrings_test.py index 27ced105f..991ead3e5 100644 --- a/control/tests/docstrings_test.py +++ b/control/tests/docstrings_test.py @@ -39,7 +39,7 @@ control.ss2tf: '48ff25d22d28e7b396e686dd5eb58831', control.tf: '53a13f4a7f75a31c81800e10c88730ef', control.tf2ss: '086a3692659b7321c2af126f79f4bc11', - control.markov: 'eda7c4635bbb863ae6659e574285d356', + control.markov: 'a4199c54cb50f07c0163d3790739eafe', control.gangof4: '0e52eb6cf7ce024f9a41f3ae3ebf04f7', } diff --git a/control/tests/modelsimp_test.py b/control/tests/modelsimp_test.py index 616ef5f09..dead4eb75 100644 --- a/control/tests/modelsimp_test.py +++ b/control/tests/modelsimp_test.py @@ -3,11 +3,15 @@ RMM, 30 Mar 2011 (based on TestModelSimp from v0.4a) """ +import math +import warnings + import numpy as np import pytest +import control as ct from control import StateSpace, TimeResponseData, c2d, forced_response, \ - impulse_response, step_response, rss, tf + impulse_response, rss, step_response, tf from control.exception import ControlArgument, ControlDimension from control.modelsimp import balred, eigensys_realization, hsvd, markov, \ modred @@ -42,7 +46,7 @@ def testMarkovSignature(self): inputs=U, input_labels='u', ) - + # setup m = 3 Htrue = np.array([1., 0., 0.]) @@ -101,7 +105,6 @@ def testMarkovSignature(self): HT = markov(response, m) np.testing.assert_array_almost_equal(HT, np.transpose(Htrue)) response.transpose=False - # Test example from docstring # TODO: There is a problem here, last markov parameter does not fit @@ -212,7 +215,7 @@ def testMarkovResults(self, k, m, n): Mtrue = np.hstack([Hd.D] + [ Hd.C @ np.linalg.matrix_power(Hd.A, i) @ Hd.B for i in range(m-1)]) - + Mtrue = np.squeeze(Mtrue) # Generate input/output data @@ -238,8 +241,8 @@ def testMarkovResults(self, k, m, n): Mcomp_scaled = markov(response, m, dt=Ts) np.testing.assert_allclose(Mtrue, Mcomp, rtol=1e-6, atol=1e-8) - np.testing.assert_allclose(Mtrue_scaled, Mcomp_scaled, rtol=1e-6, atol=1e-8) - + np.testing.assert_allclose( + Mtrue_scaled, Mcomp_scaled, rtol=1e-6, atol=1e-8) def testERASignature(self): @@ -253,7 +256,7 @@ def testERASignature(self): B = np.array([[1.],[1.,]]) C = np.array([[1., 0.,]]) D = np.array([[0.,]]) - + T = np.arange(0,10,1) sysd_true = StateSpace(A,B,C,D,True) ir_true = impulse_response(sysd_true,T=T) @@ -262,7 +265,7 @@ def testERASignature(self): sysd_est, _ = eigensys_realization(ir_true,r=2) ir_est = impulse_response(sysd_est, T=T) _, H_est = ir_est - + np.testing.assert_allclose(H_true, H_est, rtol=1e-6, atol=1e-8) # test ndarray @@ -270,7 +273,7 @@ def testERASignature(self): sysd_est, _ = eigensys_realization(YY_true,r=2) ir_est = impulse_response(sysd_est, T=T) _, H_est = ir_est - + np.testing.assert_allclose(H_true, H_est, rtol=1e-6, atol=1e-8) # test mimo @@ -304,10 +307,10 @@ def testERASignature(self): step_true = step_response(sysd_true) step_est = step_response(sysd_est) - np.testing.assert_allclose(step_true.outputs, + np.testing.assert_allclose(step_true.outputs, step_est.outputs, rtol=1e-6, atol=1e-8) - + # test ndarray _, YY_true = ir_true sysd_est, _ = eigensys_realization(YY_true,r=4,dt=dt) @@ -315,7 +318,7 @@ def testERASignature(self): step_true = step_response(sysd_true, T=T) step_est = step_response(sysd_est, T=T) - np.testing.assert_allclose(step_true.outputs, + np.testing.assert_allclose(step_true.outputs, step_est.outputs, rtol=1e-6, atol=1e-8) @@ -343,7 +346,7 @@ def testModredMatchDC(self): np.testing.assert_array_almost_equal(rsys.D, Drtrue, decimal=2) def testModredUnstable(self): - """Check if an error is thrown when an unstable system is given""" + """Check if warning is issued when an unstable system is given""" A = np.array( [[4.5418, 3.3999, 5.0342, 4.3808], [0.3890, 0.3599, 0.4195, 0.1760], @@ -353,7 +356,16 @@ def testModredUnstable(self): C = np.array([[1.0, 2.0, 3.0, 4.0], [1.0, 2.0, 3.0, 4.0]]) D = np.array([[0.0, 0.0], [0.0, 0.0]]) sys = StateSpace(A, B, C, D) - np.testing.assert_raises(ValueError, modred, sys, [2, 3]) + + # Make sure we get a warning message + with pytest.warns(UserWarning, match="System is unstable"): + newsys1 = modred(sys, [2, 3]) + + # Make sure we can turn the warning off + with warnings.catch_warnings(): + warnings.simplefilter('error') + newsys2 = ct.model_reduction(sys, [2, 3], warn_unstable=False) + np.testing.assert_equal(newsys1.A, newsys2.A) def testModredTruncate(self): #balanced realization computed in matlab for the transfer function: @@ -391,7 +403,7 @@ def testBalredTruncate(self): B = np.array([[2.], [0.], [0.], [0.]]) C = np.array([[0.5, 0.6875, 0.7031, 0.5]]) D = np.array([[0.]]) - + sys = StateSpace(A, B, C, D) orders = 2 rsys = balred(sys, orders, method='truncate') @@ -412,7 +424,7 @@ def testBalredTruncate(self): # Apply a similarity transformation Ar, Br, Cr = T @ Ar @ T, T @ Br, Cr @ T break - + # Make sure we got the correct answer np.testing.assert_array_almost_equal(Ar, Artrue, decimal=2) np.testing.assert_array_almost_equal(Br, Brtrue, decimal=4) @@ -432,12 +444,12 @@ def testBalredMatchDC(self): B = np.array([[2.], [0.], [0.], [0.]]) C = np.array([[0.5, 0.6875, 0.7031, 0.5]]) D = np.array([[0.]]) - + sys = StateSpace(A, B, C, D) orders = 2 rsys = balred(sys,orders,method='matchdc') Ar, Br, Cr, Dr = rsys.A, rsys.B, rsys.C, rsys.D - + # Result from MATLAB Artrue = np.array( [[-4.43094773, -4.55232904], @@ -445,7 +457,7 @@ def testBalredMatchDC(self): Brtrue = np.array([[1.36235673], [1.03114388]]) Crtrue = np.array([[1.36235673, 1.03114388]]) Drtrue = np.array([[-0.08383902]]) - + # Look for possible changes in state in slycot T1 = np.array([[1, 0], [0, -1]]) T2 = np.array([[-1, 0], [0, 1]]) @@ -455,9 +467,46 @@ def testBalredMatchDC(self): # Apply a similarity transformation Ar, Br, Cr = T @ Ar @ T, T @ Br, Cr @ T break - + # Make sure we got the correct answer np.testing.assert_array_almost_equal(Ar, Artrue, decimal=2) np.testing.assert_array_almost_equal(Br, Brtrue, decimal=4) np.testing.assert_array_almost_equal(Cr, Crtrue, decimal=4) np.testing.assert_array_almost_equal(Dr, Drtrue, decimal=4) + + +@pytest.mark.parametrize("kwargs, nstates, noutputs, ninputs", [ + ({'elim_states': [1, 3]}, 3, 3, 3), + ({'elim_inputs': [1, 2], 'keep_states': [1, 3]}, 2, 3, 1), + ({'elim_outputs': [1, 2], 'keep_inputs': [0, 1],}, 5, 1, 2), + ({'keep_states': [2, 0], 'keep_outputs': [0, 1]}, 2, 2, 3), + ({'keep_states': slice(0, 4, 2), 'keep_outputs': slice(None, 2)}, 2, 2, 3), + ({'keep_states': ['x[0]', 'x[3]'], 'keep_inputs': 'u[0]'}, 2, 3, 1), + ({'elim_inputs': [0, 1, 2]}, 5, 3, 0), # no inputs + ({'elim_outputs': [0, 1, 2]}, 5, 0, 3), # no outputs + ({'elim_states': [0, 1, 2, 3, 4]}, 0, 3, 3), # no states + ({'elim_states': [0, 1], 'keep_states': [1, 2]}, None, None, None), +]) +@pytest.mark.parametrize("method", ['truncate', 'matchdc']) +def test_model_reduction(method, kwargs, nstates, noutputs, ninputs): + sys = ct.rss(5, 3, 3) + + if nstates is None: + # Arguments should generate an error + with pytest.raises(ValueError, match="can't provide both"): + red = ct.model_reduction(sys, **kwargs, method=method) + return + else: + red = ct.model_reduction(sys, **kwargs, method=method) + + assert red.nstates == nstates + assert red.ninputs == ninputs + assert red.noutputs == noutputs + + if method == 'matchdc': + # Define a new system with truncated inputs and outputs + # (assumes we always keep the initial inputs and outputs) + chk = ct.ss( + sys.A, sys.B[:, :ninputs], sys.C[:noutputs, :], + sys.D[:noutputs, :][:, :ninputs]) + np.testing.assert_allclose(red(0), chk(0)) pFad - 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