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Original file line number	Diff line number	Diff line change
Expand Up		@@ -220,20 +220,39 @@ def dlqe(args, *kwargs):
		Set the method used for computing the result. Current methods are
		'slycot' and 'scipy'. If set to None (default), try 'slycot'
		first and then 'scipy'.
		return_filter_form : bool, optional
		If True, return the Kalman filter gain
		:math:`P C^T (C P C^T + R_N)^{-1}` instead of the default predictor
		gain :math:`A P C^T (C P C^T + R_N)^{-1}`.

		Returns
		-------
		L : 2D array
		Kalman estimator gain.
		Kalman estimator gain. By default, this is the predictor-form gain.
		If `return_filter_form` is True, this is the filter-form gain.
		P : 2D array
		Solution to Riccati equation.
		Steady-state prediction error covariance (prior covariance) that
		solves the Riccati equation.

		.. math::

		A P + P A^T - (P C^T + G N) R^{-1} (C P + N^T G^T) + G Q G^T = 0
		P = A P A^T
		- A P C^T (C P C^T + R_N)^{-1} C P A^T
		+ G Q_N G^T

		E : 1D array
		Eigenvalues of estimator poles eig(A - L C).
		Eigenvalues of estimator poles. With the default predictor-form gain,
		these are `eig(A - L C)`. With `return_filter_form=True`, these are
		`eig(A @ (I - L C))`.

		Notes
		-----
		By default, `dlqe` returns the predictor-form gain
		:math:`A P C^T (C P C^T + R_N)^{-1}`. Use
		`return_filter_form=True` to return the filter-form gain
		:math:`P C^T (C P C^T + R_N)^{-1}` instead. The returned covariance
		`P` is the steady-state prediction error covariance used in both gain
		formulas.

		Examples
		--------
Expand All		@@ -250,8 +269,9 @@ def dlqe(args, *kwargs):
		# Process the arguments and figure out what inputs we received
		#

		# Get the method to use (if specified as a keyword)
		# Get keywords
		method = kwargs.pop('method', None)
		return_filter_form = kwargs.pop('return_filter_form', False)
		if kwargs:
		raise TypeError("unrecognized keyword(s): ", str(kwargs))

Expand Down Expand Up		@@ -300,6 +320,9 @@ def dlqe(args, *kwargs):
		# Compute the result (dimension and symmetry checking done in dare())
		P, E, LT = dare(A.T, C.T, G @ QN @ G.T, RN, method=method,
		_Bs="C", _Qs="QN", _Rs="RN", _Ss="NN")
		if return_filter_form:
		L = sp.linalg.solve((C @ P @ C.T + RN).T, (P @ C.T).T).T
		return L, P, E
		return LT.T, P, E


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44 changes: 44 additions & 0 deletions 44 control/tests/stochsys_test.py

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Original file line number	Diff line number	Diff line change
Expand Up		@@ -85,6 +85,39 @@ def test_DLQE(method):
		L, P, poles = dlqe(A, G, C, QN, RN, method=method)
		check_DLQE(L, P, poles, G, QN, RN)

		@pytest.mark.parametrize("method", [None,
		pytest.param('slycot', marks=pytest.mark.slycot),
		'scipy'])
		def test_DLQE_return_filter_form(method):
		A = np.array([[0., 1.], [0., 0.5]])
		G = np.eye(2)
		C = np.array([[1., 0.]])
		QN = np.eye(2)
		RN = np.array([[1.]])

		L_pred, P_pred, E_pred = dlqe(A, G, C, QN, RN, method=method)
		L_filter, P_filter, E_filter = dlqe(
		A, G, C, QN, RN, method=method, return_filter_form=True)
		L_expected = np.linalg.solve(
		(C @ P_pred @ C.T + RN).T, (P_pred @ C.T).T).T

		assert np.linalg.matrix_rank(A) < A.shape[0]
		assert not np.allclose(L_filter, L_pred)
		np.testing.assert_allclose(P_filter, P_pred)
		np.testing.assert_allclose(E_filter, E_pred)
		np.testing.assert_allclose(L_filter, L_expected)
		np.testing.assert_allclose(L_pred, A @ L_filter)
		np.testing.assert_allclose(
		np.sort_complex(E_pred),
		np.sort_complex(np.linalg.eigvals(A - L_pred @ C)), atol=1e-12)
		np.testing.assert_allclose(
		np.sort_complex(E_filter),
		np.sort_complex(np.linalg.eigvals(A @ (np.eye(2) - L_filter @ C))),
		atol=1e-12)

		L_pred_false, _, _ = dlqe(A, G, C, QN, RN, method=method, return_filter_form=False)
		np.testing.assert_allclose(L_pred, L_pred_false)

		def test_lqe_discrete():
		"""Test overloading of lqe operator for discrete-time systems"""
		csys = ct.rss(2, 1, 1)
Expand All		@@ -106,6 +139,17 @@ def test_lqe_discrete():
		np.testing.assert_almost_equal(S_lqe, S_dlqe)
		np.testing.assert_almost_equal(E_lqe, E_dlqe)

		# Optional dlqe() keywords should pass through lqe() for DT systems
		K_lqe, S_lqe, E_lqe = ct.lqe(dsys, Q, R, return_filter_form=True)
		K_dlqe, S_dlqe, E_dlqe = ct.dlqe(dsys, Q, R, return_filter_form=True)
		np.testing.assert_almost_equal(K_lqe, K_dlqe)
		np.testing.assert_almost_equal(S_lqe, S_dlqe)
		np.testing.assert_almost_equal(E_lqe, E_dlqe)

		# Calling lqe with return_filter_form=True for a continuous-time system should raise TypeError
		with pytest.raises(TypeError, match="unrecognized keyword"):
		ct.lqe(csys, Q, R, return_filter_form=True)

		# Calling lqe() with no timebase should call lqe()
		asys = ct.ss(csys.A, csys.B, csys.C, csys.D, dt=None)
		K_asys, S_asys, E_asys = ct.lqe(asys, Q, R)
Expand Down

9 changes: 7 additions & 2 deletions 9 doc/stochastic.rst

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Original file line number	Diff line number	Diff line change
Expand Up		@@ -183,9 +183,14 @@ called in several forms:
		where :code:`sys` is an :class:`LTI` object, and `A`, `G`, `C`, `QN`, `RN`,
		and `NN` are 2D arrays of appropriate dimension. If :code:`sys` is a
		discrete-time system, the first two forms will compute the discrete
		time optimal controller. For the second two forms, the :func:`dlqr`
		time optimal estimator. For the second two forms, the :func:`dlqe`
		function can be used. Additional arguments and details are given on
		the :func:`lqr` and :func:`dlqr` documentation pages.
		the :func:`lqe` and :func:`dlqe` documentation pages.

		For discrete-time systems, :func:`dlqe` returns the predictor-form gain
		:math:`A P C^T (C P C^T + R_N)^{-1}` by default. Use
		``return_filter_form=True`` to return the filter-form gain
		:math:`P C^T (C P C^T + R_N)^{-1}` instead.

		.. testsetup:: kalman

Expand Down

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