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aresolc.m
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% ARESOLC.F - MEX-function for solving algebraic Riccati equations
% using SLICOT routines SB02RD, SB02ND, SB02MT and SB02OD.
% SB02RD is a more accurate version of SB02MD, also
% computing condition numbers for Riccati equations.
%
% [X(,F),ev(,db),rcond1(,acc)] = aresolc(method,A,Q,R,B,L,flag)
% [X,ev(,db),rcond1(,acc)] = aresolc(method,A,Q,G,flag)
% [Z,scale,ev(,db),rcond1] = aresolc(method,A,Q,R,B,L,flag)
% [Z,scale,ev(,db),rcond1] = aresolc(method,A,Q,G,flag)
%
% method = 1: [X,F,ev,rcond1] = aresolc(1,A,Q,R,B,L,flag)
% [X,ev,rcond1,acc] = aresolc(1,A,Q,R,B,L,flag)
% [X,ev,rcond1,acc] = aresolc(1,A,Q,G,flag)
% [Z,scale,ev,rcond1] = aresolc(1,A,Q,R,B,L,flag)
% [Z,scale,ev,rcond1] = aresolc(1,A,Q,G,flag)
% method = 2: [X,F,ev,db,rcond1] = aresolc(2,A,Q,R,B,L,flag)
% [X,ev,db,rcond1] = aresolc(2,A,Q,R,B,L,flag)
% [X,ev,db,rcond1] = aresolc(2,A,Q,G,flag)
% [Z,scale,ev,db,rcond1] = aresolc(2,A,Q,R,B,L,flag)
% [Z,scale,ev,db,rcond1] = aresolc(2,A,Q,G,flag)
% method = 3: [X,F,ev,db,rcond1] = aresolc(3,A,Q,R,B,L,flag)
% [X,ev,db,rcond1] = aresolc(3,A,Q,R,B,L,flag)
% [Z,scale,ev,db,rcond1] = aresolc(3,A,Q,R,B,L,flag)
%
% ARESOLC solves the continuous-time algebraic Riccati equations
% -1
% 0 = Q + A'*X + X*A - (L + X*B)*R *(L + X*B)' , (1a)
%
% or the discrete-time algebraic Riccati equations
% -1
% X = A'*X*A - (L + A'*X*B)*(R + B'*X*B) (A'*X*B + L)' + Q; (1b)
%
% or, in alternate forms,
%
% 0 = Q + A'*X + X*A - X*G*X, (2a)
%
% or
% -1
% X = Q + A'*X*(I + G*X) *A. (2b)
%
% The function can also be used to compute an orthogonal basis of the
% subspace which generates the solution.
%
% When method = 1, the Schur method is used on Hamiltonian matrices
% _ _
% [ A -G ] [ A -G ]
% [ _ _ ] or [ ] (3a)
% [ -Q -A' ] [-Q -A']
%
% or symplectic matrices
%
% [ _ -1 _ -1 _ ] [ -1 -1 ]
% [ A A * G ] [ A A *G ]
% [ _ _-1 _ _ _ -1 _ ] or [ -1 -1 ] (3b)
% [ Q* A A' + Q*A *G ] [ Q*A A' + Q*A *G ]
%
% _ -1 _ -1 _ -1
% where A = A - B*R *L', Q = Q - L*R *L', G = B*R *B'.
%
% When method = 2, the generalized Schur method is used for discrete-
% time problems, on the pencil
% _ _
% [ A 0 ] [ I G ]
% a [ _ ] - b [ _ ], (4a)
% [-Q I ] [ 0 A' ]
% _ _ _
% where A, Q and G are defined above, or on the pencil
%
% [ A 0 ] [ I G ]
% a [ ] - b [ ], (4b)
% [-Q I ] [ 0 A' ]
%
% for (1b) or (2b).
%
% When method = 3, the generalized Schur method is used on the
% extended pencils
%
% [ A 0 B ] [ I 0 0 ]
% a [ Q A' L ] - b [ 0 -I 0 ], (5a)
% [ L' B' R ] [ 0 0 0 ]
% or
%
% [ A 0 B ] [ I 0 0 ]
% a [ Q -I L ] - b [ 0 -A' 0 ], (5b)
% [ L' 0 R ] [ 0 -B' 0 ]
%
% to solve the equations (1a) or (1b).
%
% Description of input parameters:
% method - integer option to indicate the method to be used:
% = 1 : use ordinary Schur method on matrix (3a) or (3b).
% = 2 : use generalized Schur method on pencil (4a) or (4b).
% = 3 : use generalized Schur method on pencil (5a) or (5b).
% A - real N-by-N system state matrix.
% Q - real symmetric N-by-N state weighting matrix.
% R - real symmetric M-by-M input weighting matrix.
% When method = 1, or 2, R must be nonsingular.
% B - real N-by-M input matrix.
% L - real N-by-M coupling matrix.
% G - real symmetric N-by-N matrix.
% flag - (optional) vector containing options and parameters.
% method = 1 : flag has length 4
% flag(1) = 0 : solve the continuous-time equation
% (1a) or (2a); otherwise, solve the
% discrete-time equation (1b) or (2b).
% flag(2) = 0 : compute the stabilizing solution;
% otherwise, compute the anti-stabilizing
% solution.
% flag(3) = 0 : use a scaling strategy;
% otherwise, do not use a scaling strategy.
% flag(4) = 0 : compute both the solution X and the
% feedback gain matrix F;
% = 1 : compute the solution X only;
% = 2 : compute the solution X and the accuracy
% estimates (separation, reciprocal
% condition number and forward error bound);
% < 0 : compute the invariant subspace.
% Default: flag(1:4) = [0,0,0,1].
%
% method = 2 : flag has length 3
% flag(1) = 0 : compute the stabilizing solution;
% otherwise, compute the anti-stabilizing
% solution.
% flag(2) = 0 : compute both the solution X and the
% feedback gain matrix F;
% > 0 : compute the solution X only;
% < 0 : compute the deflating subspace.
% flag(3) : tolerance to check the singularity of
% the matrix pencil. If tol <= 0, tol will
% be replaced by EPS, the machine epsilon.
% Default: flag(1:3) = [0,1,0].
%
% method = 3 : flag has length 4
% flag(1) = 0 : solve the continuous-time equation (1a);
% otherwise, solve the discrete-time
% equation (1b).
% flag(2) = 0 : compute the stabilizing solution;
% otherwise, compute the anti-stabilizing
% solution.
% flag(3) = 0 : compute both the solution X and the
% feedback gain matrix F;
% > 0 : compute the solution only;
% < 0 : compute the related subspace.
% flag(4) : tolerance to check the singularity of
% the matrix pencil. If tol <= 0, tol will
% be replaced by EPS, the machine epsilon.
% Default: flag(1:4) = [0,0,1,0].
%
% Description of output parameters:
% X - N-by-N real symmetric solution of Riccati equations (1)
% or (2).
% F - M-by-N real state feedback matrix, returned only when R
% and B are input arguments, NLHS >= 2 and X is computed.
% Z - 2N-by-N real matrix with orthonormal columns, the basis of
% the subspace related to the solution X.
% scale - the scaling factor used (internally), which should multiply
% the submatrix Z(N+1:2N,:) to recover X from Z.
% ev - complex vector of length N.
% method = 1 : the associated eigenvalues.
% method = 2,3 : the "a" part of the associated eigenvalues
% (see (4) or (5)).
% db - real vector of length N containing the "b" part of
% the associated eigenvalues (see (4) or (5)), i.e., the
% associated eigenvalues with respect to X are ev(k)/db(k),
% k=1,...,N.
% rcond1 - estimate of the reciprocal of the 1-norm condition number
% of the N-th order system of algebraic equations from which
% the solution matrix X is obtained.
% acc - real vector of length 3 containing estimates of the
% separation, reciprocal condition number and forward error
% bound, repectively. These results can be obtained only if
% meth = 1 and flag(4) = 2.
%
% Comments:
% For METH = 2 or 3, this function behaves as ARESOL.
%
% See also SLCAREGS, SLCARESC, SLDAREGS, SLDAREGSV, SLDARESC
% RELEASE 2.0 of SLICOT Basic Systems and Control Toolbox.
% Based on SLICOT RELEASE 5.7, Copyright (c) 2002-2020 NICONET e.V.
%
% Contributor:
% V. Sima, Katholieke Univ. Leuven, Belgium, Dec. 1999.
%
% Revisions:
% V. Sima, Research Institute for Informatics, Bucharest, Dec. 2002.
%