solves the discrete algebraic Riccati equation - Maple Help

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LinearAlgebra[DARE] - solves the discrete algebraic Riccati equation

 Calling Sequence DARE(A, B, Q, R, S, options, outopts)

Parameters

 A - Matrix(square) B - Matrix Q - Matrix R - Matrix(square) S - (optional) Matrix options - (optional) constructor options for the result objects and/or equation(s) of the form output = value outopts - (optional) equation(s) of the form outputoptions[o] = list, where o is one of X, L or G

Description

 • The DARE command solves the discrete algebraic Riccati equation,

${A}^{+}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}X\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}A-X-\left(S+{A}^{+}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}X\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\left({B}^{+}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}X\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B+R\right)}^{\mathrm{-1}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\left(S+{A}^{+}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}X\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B\right)}^{+}+Q=0$

 • The optional Matrix argument S defaults to the zero Matrix.
 • This routine operates in the real floating-point domain. Hence, the entries in the Matrix arguments must necessarily be of type numeric.

Examples

 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $a:=\mathrm{Matrix}\left(2,2,\left[\left[0,1\right],\left[0,0\right]\right]\right)$
 ${a}{:=}\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]$ (1)
 > $b:=\mathrm{Matrix}\left(2,2,\left[\left[0.5,0.7\right],\left[1,0\right]\right]\right)$
 ${b}{:=}\left[\begin{array}{cc}{0.5}& {0.7}\\ {1}& {0}\end{array}\right]$ (2)
 > $q:=\mathrm{Matrix}\left(3,2,\left[\left[1,0\right],\left[0,1\right],\left[0,0\right]\right]\right)$
 ${q}{:=}\left[\begin{array}{rr}{1}& {0}\\ {0}& {1}\\ {0}& {0}\end{array}\right]$ (3)
 > $r:=\mathrm{Matrix}\left(1,1,\left[\left[1\right]\right]\right)$
 ${r}{:=}\left[\begin{array}{r}{1}\end{array}\right]$ (4)
 > $\mathrm{DARE}\left(a,b,q,r\right)$
 $\left[\begin{array}{cc}{1.00000000000000}& {-}{4.13162735955822}{}{{10}}^{{-16}}\\ {-}{4.13162735955822}{}{{10}}^{{-16}}& {1.92116460960662}\end{array}\right]$ (5)
 > $\mathrm{DARE}\left(a,b,q,r,\mathrm{output}=X\right)$
 $\left[\begin{array}{cc}{1.00000000000000}& {-}{4.13162735955822}{}{{10}}^{{-16}}\\ {-}{4.13162735955822}{}{{10}}^{{-16}}& {1.92116460960662}\end{array}\right]$ (6)
 > $\mathrm{DARE}\left(a,b,q,r,⟨⟨1,0⟩|⟨0,1⟩⟩\right)$
 $\left[\begin{array}{cc}{1.00000000000000}& {-}{4.13162735955822}{}{{10}}^{{-16}}\\ {-}{4.13162735955822}{}{{10}}^{{-16}}& {1.92116460960662}\end{array}\right]$ (7)
 > $\mathrm{DARE}\left(a,b,q,r,\mathrm{output}=\left[X,L,G,\mathrm{rcond}\right]\right)$
 $\left[\begin{array}{cc}{1.00000000000000}& {-}{4.13162735955822}{}{{10}}^{{-16}}\\ {-}{4.13162735955822}{}{{10}}^{{-16}}& {1.92116460960662}\end{array}\right]{,}\left[\begin{array}{c}{3.38343745873938}{}{{10}}^{{-17}}{+}{0.}{}{I}\\ {-}{0.157670780786754}{+}{0.}{}{I}\end{array}\right]{,}\left[\begin{array}{cc}{0.}& {0.157670780786754}\end{array}\right]{,}{0.705446591923427402}$ (8)