The Riccati equation and its asymptotic behavior
Multi tool use
$begingroup$
Consider matrices $Ainmathbb{R}^{ntimes n},Binmathbb{R}^{ntimes m}$,
a positive semidefinite symmetric matrix $Qinmathbb{R}^{ntimes n}$
and a positive definite symmetric matrix $Rinmathbb{R}^{mtimes m}$.
Consider $G_{t}$ as solution of a discrete-time Riccati equation
as follows
begin{align*}
G_{N}= & Q,\
G_{t}= & A^{T}G_{t+1}A+Q-A^{T}G_{t+1}B(R+B^{T}G_{t+1}B)^{-1}B^{T}G_{t+1}A,,0leq tleq N-1.
end{align*}
Can we determine conditions under which the solution $G_{t}$ decreases,
i.e. conditions under which $G_{t}-G_{t-1}succeq0$ please? The notation
$succeq0$ means that the matrix $G_{t}-G_{t-1}$ is a positive semidefinite
symmetric matrix. Thanks.
linear-algebra optimal-control positive-semidefinite kalman-filter
asked Nov 26 '18 at 8:47
G. TravG. Trav
1529
$endgroup$
add a comment |
$begingroup$
Consider matrices $Ainmathbb{R}^{ntimes n},Binmathbb{R}^{ntimes m}$,
a positive semidefinite symmetric matrix $Qinmathbb{R}^{ntimes n}$
and a positive definite symmetric matrix $Rinmathbb{R}^{mtimes m}$.
Consider $G_{t}$ as solution of a discrete-time Riccati equation
as follows
begin{align*}
G_{N}= & Q,\
G_{t}= & A^{T}G_{t+1}A+Q-A^{T}G_{t+1}B(R+B^{T}G_{t+1}B)^{-1}B^{T}G_{t+1}A,,0leq tleq N-1.
end{align*}
Can we determine conditions under which the solution $G_{t}$ decreases,
i.e. conditions under which $G_{t}-G_{t-1}succeq0$ please? The notation
$succeq0$ means that the matrix $G_{t}-G_{t-1}$ is a positive semidefinite
symmetric matrix. Thanks.
linear-algebra optimal-control positive-semidefinite kalman-filter
asked Nov 26 '18 at 8:47
G. TravG. Trav
1529
$endgroup$
add a comment |
$begingroup$
Consider matrices $Ainmathbb{R}^{ntimes n},Binmathbb{R}^{ntimes m}$,
a positive semidefinite symmetric matrix $Qinmathbb{R}^{ntimes n}$
and a positive definite symmetric matrix $Rinmathbb{R}^{mtimes m}$.
Consider $G_{t}$ as solution of a discrete-time Riccati equation
as follows
begin{align*}
G_{N}= & Q,\
G_{t}= & A^{T}G_{t+1}A+Q-A^{T}G_{t+1}B(R+B^{T}G_{t+1}B)^{-1}B^{T}G_{t+1}A,,0leq tleq N-1.
end{align*}
Can we determine conditions under which the solution $G_{t}$ decreases,
i.e. conditions under which $G_{t}-G_{t-1}succeq0$ please? The notation
$succeq0$ means that the matrix $G_{t}-G_{t-1}$ is a positive semidefinite
symmetric matrix. Thanks.
linear-algebra optimal-control positive-semidefinite kalman-filter
asked Nov 26 '18 at 8:47
G. TravG. Trav
1529
$endgroup$
Consider matrices $Ainmathbb{R}^{ntimes n},Binmathbb{R}^{ntimes m}$,
a positive semidefinite symmetric matrix $Qinmathbb{R}^{ntimes n}$
and a positive definite symmetric matrix $Rinmathbb{R}^{mtimes m}$.
Consider $G_{t}$ as solution of a discrete-time Riccati equation
as follows
begin{align*}
G_{N}= & Q,\
G_{t}= & A^{T}G_{t+1}A+Q-A^{T}G_{t+1}B(R+B^{T}G_{t+1}B)^{-1}B^{T}G_{t+1}A,,0leq tleq N-1.
end{align*}
Can we determine conditions under which the solution $G_{t}$ decreases,
i.e. conditions under which $G_{t}-G_{t-1}succeq0$ please? The notation
$succeq0$ means that the matrix $G_{t}-G_{t-1}$ is a positive semidefinite
symmetric matrix. Thanks.
linear-algebra optimal-control positive-semidefinite kalman-filter
linear-algebra optimal-control positive-semidefinite kalman-filter
asked Nov 26 '18 at 8:47
G. TravG. Trav
1529
asked Nov 26 '18 at 8:47
G. TravG. Trav
1529
asked Nov 26 '18 at 8:47
G. TravG. Trav
1529
asked Nov 26 '18 at 8:47
G. TravG. Trav
1529
asked Nov 26 '18 at 8:47
G. TravG. Trav
1529
1529
add a comment |
add a comment |
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