The Riccati equation and its asymptotic behavior
$begingroup$
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014056%2fthe-riccati-equation-and-its-asymptotic-behavior%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014056%2fthe-riccati-equation-and-its-asymptotic-behavior%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
