"""Matrix equation solver routines""" # Author: Jeffrey Armstrong # February 24, 2012 # Modified: Chad Fulton # June 19, 2014 # Modified: Ilhan Polat # September 13, 2016 import warnings import numpy as np from numpy.linalg import inv, LinAlgError, norm, cond, svd from scipy._lib._util import _apply_over_batch from ._basic import solve, solve_triangular, matrix_balance from .lapack import get_lapack_funcs from ._decomp_schur import schur from ._decomp_lu import lu from ._decomp_qr import qr from ._decomp_qz import ordqz from ._decomp import _asarray_validated from ._special_matrices import block_diag __all__ = ['solve_sylvester', 'solve_continuous_lyapunov', 'solve_discrete_lyapunov', 'solve_lyapunov', 'solve_continuous_are', 'solve_discrete_are'] @_apply_over_batch(('a', 2), ('b', 2), ('q', 2)) def solve_sylvester(a, b, q): """ Computes a solution (X) to the Sylvester equation :math:`AX + XB = Q`. Parameters ---------- a : (M, M) array_like Leading matrix of the Sylvester equation b : (N, N) array_like Trailing matrix of the Sylvester equation q : (M, N) array_like Right-hand side Returns ------- x : (M, N) ndarray The solution to the Sylvester equation. Raises ------ LinAlgError If solution was not found Notes ----- Computes a solution to the Sylvester matrix equation via the Bartels- Stewart algorithm. The A and B matrices first undergo Schur decompositions. The resulting matrices are used to construct an alternative Sylvester equation (``RY + YS^T = F``) where the R and S matrices are in quasi-triangular form (or, when R, S or F are complex, triangular form). The simplified equation is then solved using ``*TRSYL`` from LAPACK directly. .. versionadded:: 0.11.0 Examples -------- Given `a`, `b`, and `q` solve for `x`: >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[-3, -2, 0], [-1, -1, 3], [3, -5, -1]]) >>> b = np.array([[1]]) >>> q = np.array([[1],[2],[3]]) >>> x = linalg.solve_sylvester(a, b, q) >>> x array([[ 0.0625], [-0.5625], [ 0.6875]]) >>> np.allclose(a.dot(x) + x.dot(b), q) True """ # Accommodate empty a if a.size == 0 or b.size == 0: tdict = {'s': np.float32, 'd': np.float64, 'c': np.complex64, 'z': np.complex128} func, = get_lapack_funcs(('trsyl',), arrays=(a, b, q)) return np.empty(q.shape, dtype=tdict[func.typecode]) # Compute the Schur decomposition form of a r, u = schur(a, output='real') # Compute the Schur decomposition of b s, v = schur(b.conj().transpose(), output='real') # Construct f = u'*q*v f = np.dot(np.dot(u.conj().transpose(), q), v) # Call the Sylvester equation solver trsyl, = get_lapack_funcs(('trsyl',), (r, s, f)) if trsyl is None: raise RuntimeError('LAPACK implementation does not contain a proper ' 'Sylvester equation solver (TRSYL)') y, scale, info = trsyl(r, s, f, tranb='C') y = scale*y if info < 0: raise LinAlgError(f"Illegal value encountered in the {-info} term") return np.dot(np.dot(u, y), v.conj().transpose()) @_apply_over_batch(('a', 2), ('q', 2)) def solve_continuous_lyapunov(a, q): """ Solves the continuous Lyapunov equation :math:`AX + XA^H = Q`. Uses the Bartels-Stewart algorithm to find :math:`X`. Parameters ---------- a : array_like A square matrix q : array_like Right-hand side square matrix Returns ------- x : ndarray Solution to the continuous Lyapunov equation See Also -------- solve_discrete_lyapunov : computes the solution to the discrete-time Lyapunov equation solve_sylvester : computes the solution to the Sylvester equation Notes ----- The continuous Lyapunov equation is a special form of the Sylvester equation, hence this solver relies on LAPACK routine ?TRSYL. .. versionadded:: 0.11.0 Examples -------- Given `a` and `q` solve for `x`: >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[-3, -2, 0], [-1, -1, 0], [0, -5, -1]]) >>> b = np.array([2, 4, -1]) >>> q = np.eye(3) >>> x = linalg.solve_continuous_lyapunov(a, q) >>> x array([[ -0.75 , 0.875 , -3.75 ], [ 0.875 , -1.375 , 5.3125], [ -3.75 , 5.3125, -27.0625]]) >>> np.allclose(a.dot(x) + x.dot(a.T), q) True """ a = np.atleast_2d(_asarray_validated(a, check_finite=True)) q = np.atleast_2d(_asarray_validated(q, check_finite=True)) r_or_c = float for ind, _ in enumerate((a, q)): if np.iscomplexobj(_): r_or_c = complex if not np.equal(*_.shape): raise ValueError(f"Matrix {'aq'[ind]} should be square.") # Shape consistency check if a.shape != q.shape: raise ValueError("Matrix a and q should have the same shape.") # Accommodate empty array if a.size == 0: tdict = {'s': np.float32, 'd': np.float64, 'c': np.complex64, 'z': np.complex128} func, = get_lapack_funcs(('trsyl',), arrays=(a, q)) return np.empty(a.shape, dtype=tdict[func.typecode]) # Compute the Schur decomposition form of a r, u = schur(a, output='real') # Construct f = u'*q*u f = u.conj().T.dot(q.dot(u)) # Call the Sylvester equation solver trsyl = get_lapack_funcs('trsyl', (r, f)) dtype_string = 'T' if r_or_c is float else 'C' y, scale, info = trsyl(r, r, f, tranb=dtype_string) if info < 0: raise ValueError('?TRSYL exited with the internal error ' f'"illegal value in argument number {-info}.". See ' 'LAPACK documentation for the ?TRSYL error codes.') elif info == 1: warnings.warn('Input "a" has an eigenvalue pair whose sum is ' 'very close to or exactly zero. The solution is ' 'obtained via perturbing the coefficients.', RuntimeWarning, stacklevel=2) y *= scale return u.dot(y).dot(u.conj().T) # For backwards compatibility, keep the old name solve_lyapunov = solve_continuous_lyapunov def _solve_discrete_lyapunov_direct(a, q): """ Solves the discrete Lyapunov equation directly. This function is called by the `solve_discrete_lyapunov` function with `method=direct`. It is not supposed to be called directly. """ lhs = np.kron(a, a.conj()) lhs = np.eye(lhs.shape[0]) - lhs x = solve(lhs, q.flatten()) return np.reshape(x, q.shape) def _solve_discrete_lyapunov_bilinear(a, q): """ Solves the discrete Lyapunov equation using a bilinear transformation. This function is called by the `solve_discrete_lyapunov` function with `method=bilinear`. It is not supposed to be called directly. """ eye = np.eye(a.shape[0]) aH = a.conj().transpose() aHI_inv = inv(aH + eye) b = np.dot(aH - eye, aHI_inv) c = 2*np.dot(np.dot(inv(a + eye), q), aHI_inv) return solve_lyapunov(b.conj().transpose(), -c) @_apply_over_batch(('a', 2), ('q', 2)) def solve_discrete_lyapunov(a, q, method=None): """ Solves the discrete Lyapunov equation :math:`AXA^H - X + Q = 0`. Parameters ---------- a, q : (M, M) array_like Square matrices corresponding to A and Q in the equation above respectively. Must have the same shape. method : {'direct', 'bilinear'}, optional Type of solver. If not given, chosen to be ``direct`` if ``M`` is less than 10 and ``bilinear`` otherwise. Returns ------- x : ndarray Solution to the discrete Lyapunov equation See Also -------- solve_continuous_lyapunov : computes the solution to the continuous-time Lyapunov equation Notes ----- This section describes the available solvers that can be selected by the 'method' parameter. The default method is *direct* if ``M`` is less than 10 and ``bilinear`` otherwise. Method *direct* uses a direct analytical solution to the discrete Lyapunov equation. The algorithm is given in, for example, [1]_. However, it requires the linear solution of a system with dimension :math:`M^2` so that performance degrades rapidly for even moderately sized matrices. Method *bilinear* uses a bilinear transformation to convert the discrete Lyapunov equation to a continuous Lyapunov equation :math:`(BX+XB'=-C)` where :math:`B=(A-I)(A+I)^{-1}` and :math:`C=2(A' + I)^{-1} Q (A + I)^{-1}`. The continuous equation can be efficiently solved since it is a special case of a Sylvester equation. The transformation algorithm is from Popov (1964) as described in [2]_. .. versionadded:: 0.11.0 References ---------- .. [1] "Lyapunov equation", Wikipedia, https://en.wikipedia.org/wiki/Lyapunov_equation#Discrete_time .. [2] Gajic, Z., and M.T.J. Qureshi. 2008. Lyapunov Matrix Equation in System Stability and Control. Dover Books on Engineering Series. Dover Publications. Examples -------- Given `a` and `q` solve for `x`: >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[0.2, 0.5],[0.7, -0.9]]) >>> q = np.eye(2) >>> x = linalg.solve_discrete_lyapunov(a, q) >>> x array([[ 0.70872893, 1.43518822], [ 1.43518822, -2.4266315 ]]) >>> np.allclose(a.dot(x).dot(a.T)-x, -q) True """ a = np.asarray(a) q = np.asarray(q) if method is None: # Select automatically based on size of matrices if a.shape[0] >= 10: method = 'bilinear' else: method = 'direct' meth = method.lower() if meth == 'direct': x = _solve_discrete_lyapunov_direct(a, q) elif meth == 'bilinear': x = _solve_discrete_lyapunov_bilinear(a, q) else: raise ValueError(f'Unknown solver {method}') return x def solve_continuous_are(a, b, q, r, e=None, s=None, balanced=True): r""" Solves the continuous-time algebraic Riccati equation (CARE). The CARE is defined as .. math:: X A + A^H X - X B R^{-1} B^H X + Q = 0 The limitations for a solution to exist are : * All eigenvalues of :math:`A` on the right half plane, should be controllable. * The associated hamiltonian pencil (See Notes), should have eigenvalues sufficiently away from the imaginary axis. Moreover, if ``e`` or ``s`` is not precisely ``None``, then the generalized version of CARE .. math:: E^HXA + A^HXE - (E^HXB + S) R^{-1} (B^HXE + S^H) + Q = 0 is solved. When omitted, ``e`` is assumed to be the identity and ``s`` is assumed to be the zero matrix with sizes compatible with ``a`` and ``b``, respectively. The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- a : (M, M) array_like Square matrix b : (M, N) array_like Input q : (M, M) array_like Input r : (N, N) array_like Nonsingular square matrix e : (M, M) array_like, optional Nonsingular square matrix s : (M, N) array_like, optional Input balanced : bool, optional The boolean that indicates whether a balancing step is performed on the data. The default is set to True. Returns ------- x : (M, M) ndarray Solution to the continuous-time algebraic Riccati equation. Raises ------ LinAlgError For cases where the stable subspace of the pencil could not be isolated. See Notes section and the references for details. See Also -------- solve_discrete_are : Solves the discrete-time algebraic Riccati equation Notes ----- The equation is solved by forming the extended hamiltonian matrix pencil, as described in [1]_, :math:`H - \lambda J` given by the block matrices :: [ A 0 B ] [ E 0 0 ] [-Q -A^H -S ] - \lambda * [ 0 E^H 0 ] [ S^H B^H R ] [ 0 0 0 ] and using a QZ decomposition method. In this algorithm, the fail conditions are linked to the symmetry of the product :math:`U_2 U_1^{-1}` and condition number of :math:`U_1`. Here, :math:`U` is the 2m-by-m matrix that holds the eigenvectors spanning the stable subspace with 2-m rows and partitioned into two m-row matrices. See [1]_ and [2]_ for more details. In order to improve the QZ decomposition accuracy, the pencil goes through a balancing step where the sum of absolute values of :math:`H` and :math:`J` entries (after removing the diagonal entries of the sum) is balanced following the recipe given in [3]_. .. versionadded:: 0.11.0 References ---------- .. [1] P. van Dooren , "A Generalized Eigenvalue Approach For Solving Riccati Equations.", SIAM Journal on Scientific and Statistical Computing, Vol.2(2), :doi:`10.1137/0902010` .. [2] A.J. Laub, "A Schur Method for Solving Algebraic Riccati Equations.", Massachusetts Institute of Technology. Laboratory for Information and Decision Systems. LIDS-R ; 859. Available online : http://hdl.handle.net/1721.1/1301 .. [3] P. Benner, "Symplectic Balancing of Hamiltonian Matrices", 2001, SIAM J. Sci. Comput., 2001, Vol.22(5), :doi:`10.1137/S1064827500367993` Examples -------- Given `a`, `b`, `q`, and `r` solve for `x`: >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[4, 3], [-4.5, -3.5]]) >>> b = np.array([[1], [-1]]) >>> q = np.array([[9, 6], [6, 4.]]) >>> r = 1 >>> x = linalg.solve_continuous_are(a, b, q, r) >>> x array([[ 21.72792206, 14.48528137], [ 14.48528137, 9.65685425]]) >>> np.allclose(a.T.dot(x) + x.dot(a)-x.dot(b).dot(b.T).dot(x), -q) True """ # ensure that all arguments are present when using `_apply_over_batch` (gh-23336) return _solve_continuous_are(a, b, q, r, e, s, balanced) @_apply_over_batch(('a', 2), ('b', 2), ('q', 2), ('r', 2), ('e', 2), ('s', 2)) def _solve_continuous_are(a, b, q, r, e, s, balanced): # Validate input arguments a, b, q, r, e, s, m, n, r_or_c, gen_are = _are_validate_args( a, b, q, r, e, s, 'care') H = np.empty((2*m+n, 2*m+n), dtype=r_or_c) H[:m, :m] = a H[:m, m:2*m] = 0. H[:m, 2*m:] = b H[m:2*m, :m] = -q H[m:2*m, m:2*m] = -a.conj().T H[m:2*m, 2*m:] = 0. if s is None else -s H[2*m:, :m] = 0. if s is None else s.conj().T H[2*m:, m:2*m] = b.conj().T H[2*m:, 2*m:] = r if gen_are and e is not None: J = block_diag(e, e.conj().T, np.zeros_like(r, dtype=r_or_c)) else: J = block_diag(np.eye(2*m), np.zeros_like(r, dtype=r_or_c)) if balanced: # xGEBAL does not remove the diagonals before scaling. Also # to avoid destroying the Symplectic structure, we follow Ref.3 M = np.abs(H) + np.abs(J) np.fill_diagonal(M, 0.) _, (sca, _) = matrix_balance(M, separate=1, permute=0) # do we need to bother? if not np.allclose(sca, np.ones_like(sca)): # Now impose diag(D,inv(D)) from Benner where D is # square root of s_i/s_(n+i) for i=0,.... sca = np.log2(sca) # NOTE: Py3 uses "Bankers Rounding: round to the nearest even" !! s = np.round((sca[m:2*m] - sca[:m])/2) sca = 2 ** np.r_[s, -s, sca[2*m:]] # Elementwise multiplication via broadcasting. elwisescale = sca[:, None] * np.reciprocal(sca) H *= elwisescale J *= elwisescale # Deflate the pencil to 2m x 2m ala Ref.1, eq.(55) q, r = qr(H[:, -n:]) H = q[:, n:].conj().T.dot(H[:, :2*m]) J = q[:2*m, n:].conj().T.dot(J[:2*m, :2*m]) # Decide on which output type is needed for QZ out_str = 'real' if r_or_c is float else 'complex' _, _, _, _, _, u = ordqz(H, J, sort='lhp', overwrite_a=True, overwrite_b=True, check_finite=False, output=out_str) # Get the relevant parts of the stable subspace basis if e is not None: u, _ = qr(np.vstack((e.dot(u[:m, :m]), u[m:, :m]))) u00 = u[:m, :m] u10 = u[m:, :m] # Solve via back-substituion after checking the condition of u00 up, ul, uu = lu(u00) if 1/cond(uu) < np.spacing(1.): raise LinAlgError('Failed to find a finite solution.') # Exploit the triangular structure x = solve_triangular(ul.conj().T, solve_triangular(uu.conj().T, u10.conj().T, lower=True), unit_diagonal=True, ).conj().T.dot(up.conj().T) if balanced: x *= sca[:m, None] * sca[:m] # Check the deviation from symmetry for lack of success # See proof of Thm.5 item 3 in [2] u_sym = u00.conj().T.dot(u10) n_u_sym = norm(u_sym, 1) u_sym = u_sym - u_sym.conj().T sym_threshold = np.max([np.spacing(1000.), 0.1*n_u_sym]) if norm(u_sym, 1) > sym_threshold: raise LinAlgError('The associated Hamiltonian pencil has eigenvalues ' 'too close to the imaginary axis') return (x + x.conj().T)/2 def solve_discrete_are(a, b, q, r, e=None, s=None, balanced=True): r""" Solves the discrete-time algebraic Riccati equation (DARE). The DARE is defined as .. math:: A^HXA - X - (A^HXB) (R + B^HXB)^{-1} (B^HXA) + Q = 0 The limitations for a solution to exist are : * All eigenvalues of :math:`A` outside the unit disc, should be controllable. * The associated symplectic pencil (See Notes), should have eigenvalues sufficiently away from the unit circle. Moreover, if ``e`` and ``s`` are not both precisely ``None``, then the generalized version of DARE .. math:: A^HXA - E^HXE - (A^HXB+S) (R+B^HXB)^{-1} (B^HXA+S^H) + Q = 0 is solved. When omitted, ``e`` is assumed to be the identity and ``s`` is assumed to be the zero matrix. The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- a : (M, M) array_like Square matrix b : (M, N) array_like Input q : (M, M) array_like Input r : (N, N) array_like Square matrix e : (M, M) array_like, optional Nonsingular square matrix s : (M, N) array_like, optional Input balanced : bool The boolean that indicates whether a balancing step is performed on the data. The default is set to True. Returns ------- x : (M, M) ndarray Solution to the discrete algebraic Riccati equation. Raises ------ LinAlgError For cases where the stable subspace of the pencil could not be isolated. See Notes section and the references for details. See Also -------- solve_continuous_are : Solves the continuous algebraic Riccati equation Notes ----- The equation is solved by forming the extended symplectic matrix pencil, as described in [1]_, :math:`H - \lambda J` given by the block matrices :: [ A 0 B ] [ E 0 B ] [ -Q E^H -S ] - \lambda * [ 0 A^H 0 ] [ S^H 0 R ] [ 0 -B^H 0 ] and using a QZ decomposition method. In this algorithm, the fail conditions are linked to the symmetry of the product :math:`U_2 U_1^{-1}` and condition number of :math:`U_1`. Here, :math:`U` is the 2m-by-m matrix that holds the eigenvectors spanning the stable subspace with 2-m rows and partitioned into two m-row matrices. See [1]_ and [2]_ for more details. In order to improve the QZ decomposition accuracy, the pencil goes through a balancing step where the sum of absolute values of :math:`H` and :math:`J` rows/cols (after removing the diagonal entries) is balanced following the recipe given in [3]_. If the data has small numerical noise, balancing may amplify their effects and some clean up is required. .. versionadded:: 0.11.0 References ---------- .. [1] P. van Dooren , "A Generalized Eigenvalue Approach For Solving Riccati Equations.", SIAM Journal on Scientific and Statistical Computing, Vol.2(2), :doi:`10.1137/0902010` .. [2] A.J. Laub, "A Schur Method for Solving Algebraic Riccati Equations.", Massachusetts Institute of Technology. Laboratory for Information and Decision Systems. LIDS-R ; 859. Available online : http://hdl.handle.net/1721.1/1301 .. [3] P. Benner, "Symplectic Balancing of Hamiltonian Matrices", 2001, SIAM J. Sci. Comput., 2001, Vol.22(5), :doi:`10.1137/S1064827500367993` Examples -------- Given `a`, `b`, `q`, and `r` solve for `x`: >>> import numpy as np >>> from scipy import linalg as la >>> a = np.array([[0, 1], [0, -1]]) >>> b = np.array([[1, 0], [2, 1]]) >>> q = np.array([[-4, -4], [-4, 7]]) >>> r = np.array([[9, 3], [3, 1]]) >>> x = la.solve_discrete_are(a, b, q, r) >>> x array([[-4., -4.], [-4., 7.]]) >>> R = la.solve(r + b.T.dot(x).dot(b), b.T.dot(x).dot(a)) >>> np.allclose(a.T.dot(x).dot(a) - x - a.T.dot(x).dot(b).dot(R), -q) True """ # ensure that all arguments are present when using `_apply_over_batch` (gh-23336) return _solve_discrete_are(a, b, q, r, e, s, balanced) @_apply_over_batch(('a', 2), ('b', 2), ('q', 2), ('r', 2), ('e', 2), ('s', 2)) def _solve_discrete_are(a, b, q, r, e, s, balanced): # Validate input arguments a, b, q, r, e, s, m, n, r_or_c, gen_are = _are_validate_args( a, b, q, r, e, s, 'dare') # Form the matrix pencil H = np.zeros((2*m+n, 2*m+n), dtype=r_or_c) H[:m, :m] = a H[:m, 2*m:] = b H[m:2*m, :m] = -q H[m:2*m, m:2*m] = np.eye(m) if e is None else e.conj().T H[m:2*m, 2*m:] = 0. if s is None else -s H[2*m:, :m] = 0. if s is None else s.conj().T H[2*m:, 2*m:] = r J = np.zeros_like(H, dtype=r_or_c) J[:m, :m] = np.eye(m) if e is None else e J[m:2*m, m:2*m] = a.conj().T J[2*m:, m:2*m] = -b.conj().T if balanced: # xGEBAL does not remove the diagonals before scaling. Also # to avoid destroying the Symplectic structure, we follow Ref.3 M = np.abs(H) + np.abs(J) np.fill_diagonal(M, 0.) _, (sca, _) = matrix_balance(M, separate=1, permute=0) # do we need to bother? if not np.allclose(sca, np.ones_like(sca)): # Now impose diag(D,inv(D)) from Benner where D is # square root of s_i/s_(n+i) for i=0,.... sca = np.log2(sca) # NOTE: Py3 uses "Bankers Rounding: round to the nearest even" !! s = np.round((sca[m:2*m] - sca[:m])/2) sca = 2 ** np.r_[s, -s, sca[2*m:]] # Elementwise multiplication via broadcasting. elwisescale = sca[:, None] * np.reciprocal(sca) H *= elwisescale J *= elwisescale # Deflate the pencil by the R column ala Ref.1 q_of_qr, _ = qr(H[:, -n:]) H = q_of_qr[:, n:].conj().T.dot(H[:, :2*m]) J = q_of_qr[:, n:].conj().T.dot(J[:, :2*m]) # Decide on which output type is needed for QZ out_str = 'real' if r_or_c is float else 'complex' _, _, _, _, _, u = ordqz(H, J, sort='iuc', overwrite_a=True, overwrite_b=True, check_finite=False, output=out_str) # Get the relevant parts of the stable subspace basis if e is not None: u, _ = qr(np.vstack((e.dot(u[:m, :m]), u[m:, :m]))) u00 = u[:m, :m] u10 = u[m:, :m] # Solve via back-substituion after checking the condition of u00 up, ul, uu = lu(u00) if 1/cond(uu) < np.spacing(1.): raise LinAlgError('Failed to find a finite solution.') # Exploit the triangular structure x = solve_triangular(ul.conj().T, solve_triangular(uu.conj().T, u10.conj().T, lower=True), unit_diagonal=True, ).conj().T.dot(up.conj().T) if balanced: x *= sca[:m, None] * sca[:m] # Check the deviation from symmetry for lack of success # See proof of Thm.5 item 3 in [2] u_sym = u00.conj().T.dot(u10) n_u_sym = norm(u_sym, 1) u_sym = u_sym - u_sym.conj().T sym_threshold = np.max([np.spacing(1000.), 0.1*n_u_sym]) if norm(u_sym, 1) > sym_threshold: raise LinAlgError('The associated symplectic pencil has eigenvalues ' 'too close to the unit circle') return (x + x.conj().T)/2 def _are_validate_args(a, b, q, r, e, s, eq_type='care'): """ A helper function to validate the arguments supplied to the Riccati equation solvers. Any discrepancy found in the input matrices leads to a ``ValueError`` exception. Essentially, it performs: - a check whether the input is free of NaN and Infs - a pass for the data through ``numpy.atleast_2d()`` - squareness check of the relevant arrays - shape consistency check of the arrays - singularity check of the relevant arrays - symmetricity check of the relevant matrices - a check whether the regular or the generalized version is asked. This function is used by ``solve_continuous_are`` and ``solve_discrete_are``. Parameters ---------- a, b, q, r, e, s : array_like Input data eq_type : str Accepted arguments are 'care' and 'dare'. Returns ------- a, b, q, r, e, s : ndarray Regularized input data m, n : int shape of the problem r_or_c : type Data type of the problem, returns float or complex gen_or_not : bool Type of the equation, True for generalized and False for regular ARE. """ if eq_type.lower() not in ("dare", "care"): raise ValueError("Equation type unknown. " "Only 'care' and 'dare' is understood") a = np.atleast_2d(_asarray_validated(a, check_finite=True)) b = np.atleast_2d(_asarray_validated(b, check_finite=True)) q = np.atleast_2d(_asarray_validated(q, check_finite=True)) r = np.atleast_2d(_asarray_validated(r, check_finite=True)) # Get the correct data types otherwise NumPy complains # about pushing complex numbers into real arrays. r_or_c = complex if np.iscomplexobj(b) else float for ind, mat in enumerate((a, q, r)): if np.iscomplexobj(mat): r_or_c = complex if not np.equal(*mat.shape): raise ValueError(f"Matrix {'aqr'[ind]} should be square.") # Shape consistency checks m, n = b.shape if m != a.shape[0]: raise ValueError("Matrix a and b should have the same number of rows.") if m != q.shape[0]: raise ValueError("Matrix a and q should have the same shape.") if n != r.shape[0]: raise ValueError("Matrix b and r should have the same number of cols.") # Check if the data matrices q, r are (sufficiently) hermitian for ind, mat in enumerate((q, r)): if norm(mat - mat.conj().T, 1) > np.spacing(norm(mat, 1))*100: raise ValueError(f"Matrix {'qr'[ind]} should be symmetric/hermitian.") # Continuous time ARE should have a nonsingular r matrix. if eq_type == 'care': min_sv = svd(r, compute_uv=False)[-1] if min_sv == 0. or min_sv < np.spacing(1.)*norm(r, 1): raise ValueError('Matrix r is numerically singular.') # Check if the generalized case is required with omitted arguments # perform late shape checking etc. generalized_case = e is not None or s is not None if generalized_case: if e is not None: e = np.atleast_2d(_asarray_validated(e, check_finite=True)) if not np.equal(*e.shape): raise ValueError("Matrix e should be square.") if m != e.shape[0]: raise ValueError("Matrix a and e should have the same shape.") # numpy.linalg.cond doesn't check for exact zeros and # emits a runtime warning. Hence the following manual check. min_sv = svd(e, compute_uv=False)[-1] if min_sv == 0. or min_sv < np.spacing(1.) * norm(e, 1): raise ValueError('Matrix e is numerically singular.') if np.iscomplexobj(e): r_or_c = complex if s is not None: s = np.atleast_2d(_asarray_validated(s, check_finite=True)) if s.shape != b.shape: raise ValueError("Matrix b and s should have the same shape.") if np.iscomplexobj(s): r_or_c = complex return a, b, q, r, e, s, m, n, r_or_c, generalized_case