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Solve the Algebraic Riccati Equation
Inputs for identically dimensioned square matrices
n by n matrix;
n by n matrix or n by m matrix; in the latter case b is replaced by b:=b*b';
n by n matrix or p by m matrix; in the latter case c is replaced by c:=c'*c;
(optional argument; default = "B"
):
String option passed to balance
prior to ordered Schur decomposition.
Output
solution of the ARE.
Method Laub's Schur method (IEEE Transactions on Automatic Control, 1979) is applied to the appropriate Hamiltonian matrix.
Return the solution, x of the discrete-time algebraic Riccati equation
Inputs
n by n matrix;
n by m matrix;
n by n matrix, symmetric positive semidefinite, or a p by n matrix, In the latter case q:=q'*q is used;
m by m, symmetric positive definite (invertible);
(optional argument; default = "B"
):
String option passed to balance
prior to ordered QZ decomposition.
Output
solution of DARE.
Method Generalized eigenvalue approach (Van Dooren; SIAM J. Sci. Stat. Comput., Vol 2) applied to the appropriate symplectic pencil.
See also: Ran and Rodman, Stable Hermitian Solutions of Discrete Algebraic Riccati Equations, Mathematics of Control, Signals and Systems, Vol 5, no 2 (1992), pp 165-194.
Solve the differential Riccati equation for the LTI system sys. Solution of standard LTI state feedback optimization optimal input is Inputs
continuous time system data structure
state integral penalty
input integral penalty
state terminal penalty
limits on the integral
tolerance (used to select time samples; see below); default = 0.1
number of refinement iterations (default=10)
Outputs
time values at which p(t) is computed
list values of p(t); plist { i } is p(tvals(i))
tvals is selected so that: for every i between 2 and length(tvals).
Return controllability gramian of discrete time system
Inputs
n by n matrix
n by m matrix
Output
n by n matrix, satisfies
Solve the discrete-time Lyapunov equation
Inputs
n by n matrix;
Matrix: n by n, n by m, or p by n.
Output
matrix satisfying appropriate discrete time Lyapunov equation.
Options:
Method Uses Schur decomposition method as in Kitagawa, An Algorithm for Solving the Matrix Equation X = F X F' + S, International Journal of Control, Volume 25, Number 5, pages 745-753 (1977).
Column-by-column solution method as suggested in Hammarling, Numerical Solution of the Stable, Non-Negative Definite Lyapunov Equation, IMA Journal of Numerical Analysis, Volume 2, pages 303-323 (1982).
Return controllability gramian m of the continuous time system dx/dt = a x + b u.
m satisfies a m + m a' + b b' = 0.
Solve the Lyapunov (or Sylvester) equation via the Bartels-Stewart algorithm (Communications of the ACM, 1972).
If a, b, and c are specified, then lyap
returns
the solution of the Sylvester equation
If only (a, b)
are specified, then lyap
returns the
solution of the Lyapunov equation
If b is not square, then lyap
returns the solution of either
or
whichever is appropriate.
Solves by using the Bartels-Stewart algorithm (1972).
Compute generalized eigenvalues of the matrix pencil
a and b must be real matrices.
qzval
is obsolete; use qz
instead.
Compute product of zgep incidence matrix F with vector x.
Used by zgepbal
(in zgscal
) as part of generalized conjugate gradient
iteration.
Solve system of equations for dense zgep problem.
Construct right hand side vector zz
for the zero-computation generalized eigenvalue problem
balancing procedure. Called by zgepbal
.
Implementation of procedure REDUCE in (Emami-Naeini and Van Dooren, Automatica, # 1982).
Return nonz = number of rows of mat whose two norm exceeds meps, and zer = number of rows of mat whose two norm is less than meps.
Generalized conjugate gradient iteration to
solve zero-computation generalized eigenvalue problem balancing equation
fx=z; called by zgepbal
.
Apply givens rotation c,s to row vectors a, b. No longer used in zero-balancing (__zgpbal__); kept for backward compatibility.
Apply householder vector based on
to column vector y.
Called by zgfslv
.
References
Hodel, Computation of Zeros with Balancing, 1992, Linear Algebra and its Applications
Golub and Van Loan, Matrix Computations, 2nd ed 1989.
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