Optimal control problem with a path constraint which involves controls at two distinct time points
I am faced with an optimal control problem in continuous time which includes a path constraint
which involves controls at two distinct points in time.
I do not know how to approach this problem. I do not even know what
to search for, since I am not sure about the correct terminology regarding
the constraint. I would be happy about any type of input regarding
how to approach this problem, as well as for references or terminology.
Here is more detail:
Let $mathbf{x}(t)inmathbb{R}^{n}$ be the state and $mathbf{u}(t)inmathbb{R}^{m}$
the control, $t_{0}$ the initial time and $t_{f}$ the final time.
The goal is to minimize
$$
int_{t_{0}}^{t_{f}}fleft(mathbf{x}(t)right)dt
$$
subject to laws of motion
$$
dot{mathbf{x}}(t)=g(mathbf{x}(t),mathbf{u}(t))
$$
an integral constraint
$$
int_{t_{0}}^{t_{f}}hleft(mathbf{x}(t),mathbf{u}(t)right)dt=0
$$
path constraints
$$
k(mathbf{x}(t),mathbf{u}(t))=0 forall tin[t_{0},t_{f}],
$$
and a path constraint of the type
$$
u_{1}(t)-u_{2}(taucdot t)=0 forall tinleft{ t:t_{0}leqtaucdot tleq t_{f}right} ,
$$
with $tau>0$ a scalar constant, and $u_{1}(t)$ and $u_{2}(t)$ are elements of $mathbf{u}(t)$. It is this final path constraint which troubles me.
dynamical-systems control-theory optimal-control
asked Dec 13 '18 at 10:41
U.T.U.T.
1062
add a comment |
I am faced with an optimal control problem in continuous time which includes a path constraint
which involves controls at two distinct points in time.
I do not know how to approach this problem. I do not even know what
to search for, since I am not sure about the correct terminology regarding
the constraint. I would be happy about any type of input regarding
how to approach this problem, as well as for references or terminology.
Here is more detail:
Let $mathbf{x}(t)inmathbb{R}^{n}$ be the state and $mathbf{u}(t)inmathbb{R}^{m}$
the control, $t_{0}$ the initial time and $t_{f}$ the final time.
The goal is to minimize
$$
int_{t_{0}}^{t_{f}}fleft(mathbf{x}(t)right)dt
$$
subject to laws of motion
$$
dot{mathbf{x}}(t)=g(mathbf{x}(t),mathbf{u}(t))
$$
an integral constraint
$$
int_{t_{0}}^{t_{f}}hleft(mathbf{x}(t),mathbf{u}(t)right)dt=0
$$
path constraints
$$
k(mathbf{x}(t),mathbf{u}(t))=0 forall tin[t_{0},t_{f}],
$$
and a path constraint of the type
$$
u_{1}(t)-u_{2}(taucdot t)=0 forall tinleft{ t:t_{0}leqtaucdot tleq t_{f}right} ,
$$
with $tau>0$ a scalar constant, and $u_{1}(t)$ and $u_{2}(t)$ are elements of $mathbf{u}(t)$. It is this final path constraint which troubles me.
dynamical-systems control-theory optimal-control
asked Dec 13 '18 at 10:41
U.T.U.T.
1062
Your formulation for the path constraint might have some ambiguity. For example when $t_0<0$ and $t_f>0$ then if $t<0$ then $t_0leqtau,t<0$, if $t=0$ then $tau,t=0$ and if $t>0$ then $0>tau,tleq t_f$. I am not sure if this is intended behavior, otherwise you could formulate it as: $$u_1(t) - u_2(tau)=0 forall,t,tauin{x:t_0leq xleq t_f}$$
– Kwin van der Veen
Dec 13 '18 at 10:55
Or is $tau$ a constant positive parameter?
– Kwin van der Veen
Dec 13 '18 at 12:55
Yes, edited to clarify.
– U.T.
Dec 13 '18 at 13:24
add a comment |
I am faced with an optimal control problem in continuous time which includes a path constraint
which involves controls at two distinct points in time.
I do not know how to approach this problem. I do not even know what
to search for, since I am not sure about the correct terminology regarding
the constraint. I would be happy about any type of input regarding
how to approach this problem, as well as for references or terminology.
Here is more detail:
Let $mathbf{x}(t)inmathbb{R}^{n}$ be the state and $mathbf{u}(t)inmathbb{R}^{m}$
the control, $t_{0}$ the initial time and $t_{f}$ the final time.
The goal is to minimize
$$
int_{t_{0}}^{t_{f}}fleft(mathbf{x}(t)right)dt
$$
subject to laws of motion
$$
dot{mathbf{x}}(t)=g(mathbf{x}(t),mathbf{u}(t))
$$
an integral constraint
$$
int_{t_{0}}^{t_{f}}hleft(mathbf{x}(t),mathbf{u}(t)right)dt=0
$$
path constraints
$$
k(mathbf{x}(t),mathbf{u}(t))=0 forall tin[t_{0},t_{f}],
$$
and a path constraint of the type
$$
u_{1}(t)-u_{2}(taucdot t)=0 forall tinleft{ t:t_{0}leqtaucdot tleq t_{f}right} ,
$$
with $tau>0$ a scalar constant, and $u_{1}(t)$ and $u_{2}(t)$ are elements of $mathbf{u}(t)$. It is this final path constraint which troubles me.
dynamical-systems control-theory optimal-control
asked Dec 13 '18 at 10:41
U.T.U.T.
1062
I am faced with an optimal control problem in continuous time which includes a path constraint
which involves controls at two distinct points in time.
I do not know how to approach this problem. I do not even know what
to search for, since I am not sure about the correct terminology regarding
the constraint. I would be happy about any type of input regarding
how to approach this problem, as well as for references or terminology.
Here is more detail:
Let $mathbf{x}(t)inmathbb{R}^{n}$ be the state and $mathbf{u}(t)inmathbb{R}^{m}$
the control, $t_{0}$ the initial time and $t_{f}$ the final time.
The goal is to minimize
$$
int_{t_{0}}^{t_{f}}fleft(mathbf{x}(t)right)dt
$$
subject to laws of motion
$$
dot{mathbf{x}}(t)=g(mathbf{x}(t),mathbf{u}(t))
$$
an integral constraint
$$
int_{t_{0}}^{t_{f}}hleft(mathbf{x}(t),mathbf{u}(t)right)dt=0
$$
path constraints
$$
k(mathbf{x}(t),mathbf{u}(t))=0 forall tin[t_{0},t_{f}],
$$
and a path constraint of the type
$$
u_{1}(t)-u_{2}(taucdot t)=0 forall tinleft{ t:t_{0}leqtaucdot tleq t_{f}right} ,
$$
with $tau>0$ a scalar constant, and $u_{1}(t)$ and $u_{2}(t)$ are elements of $mathbf{u}(t)$. It is this final path constraint which troubles me.
dynamical-systems control-theory optimal-control
dynamical-systems control-theory optimal-control
asked Dec 13 '18 at 10:41
U.T.U.T.
1062
asked Dec 13 '18 at 10:41
U.T.U.T.
1062
edited Dec 13 '18 at 13:23
U.T.
asked Dec 13 '18 at 10:41
U.T.U.T.
1062
asked Dec 13 '18 at 10:41
U.T.U.T.
1062
asked Dec 13 '18 at 10:41
U.T.U.T.
1062
1062
Your formulation for the path constraint might have some ambiguity. For example when $t_0<0$ and $t_f>0$ then if $t<0$ then $t_0leqtau,t<0$, if $t=0$ then $tau,t=0$ and if $t>0$ then $0>tau,tleq t_f$. I am not sure if this is intended behavior, otherwise you could formulate it as: $$u_1(t) - u_2(tau)=0 forall,t,tauin{x:t_0leq xleq t_f}$$
– Kwin van der Veen
Dec 13 '18 at 10:55
Or is $tau$ a constant positive parameter?
– Kwin van der Veen
Dec 13 '18 at 12:55
Yes, edited to clarify.
– U.T.
Dec 13 '18 at 13:24
add a comment |
Your formulation for the path constraint might have some ambiguity. For example when $t_0<0$ and $t_f>0$ then if $t<0$ then $t_0leqtau,t<0$, if $t=0$ then $tau,t=0$ and if $t>0$ then $0>tau,tleq t_f$. I am not sure if this is intended behavior, otherwise you could formulate it as: $$u_1(t) - u_2(tau)=0 forall,t,tauin{x:t_0leq xleq t_f}$$
– Kwin van der Veen
Dec 13 '18 at 10:55
Or is $tau$ a constant positive parameter?
– Kwin van der Veen
Dec 13 '18 at 12:55
Yes, edited to clarify.
– U.T.
Dec 13 '18 at 13:24
Your formulation for the path constraint might have some ambiguity. For example when $t_0<0$ and $t_f>0$ then if $t<0$ then $t_0leqtau,t<0$, if $t=0$ then $tau,t=0$ and if $t>0$ then $0>tau,tleq t_f$. I am not sure if this is intended behavior, otherwise you could formulate it as: $$u_1(t) - u_2(tau)=0 forall,t,tauin{x:t_0leq xleq t_f}$$
– Kwin van der Veen
Dec 13 '18 at 10:55
Your formulation for the path constraint might have some ambiguity. For example when $t_0<0$ and $t_f>0$ then if $t<0$ then $t_0leqtau,t<0$, if $t=0$ then $tau,t=0$ and if $t>0$ then $0>tau,tleq t_f$. I am not sure if this is intended behavior, otherwise you could formulate it as: $$u_1(t) - u_2(tau)=0 forall,t,tauin{x:t_0leq xleq t_f}$$
– Kwin van der Veen
Dec 13 '18 at 10:55
Or is $tau$ a constant positive parameter?
– Kwin van der Veen
Dec 13 '18 at 12:55
Or is $tau$ a constant positive parameter?
– Kwin van der Veen
Dec 13 '18 at 12:55
Yes, edited to clarify.
– U.T.
Dec 13 '18 at 13:24
Yes, edited to clarify.
– U.T.
Dec 13 '18 at 13:24
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%2f3037858%2foptimal-control-problem-with-a-path-constraint-which-involves-controls-at-two-di%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%2f3037858%2foptimal-control-problem-with-a-path-constraint-which-involves-controls-at-two-di%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
Your formulation for the path constraint might have some ambiguity. For example when $t_0<0$ and $t_f>0$ then if $t<0$ then $t_0leqtau,t<0$, if $t=0$ then $tau,t=0$ and if $t>0$ then $0>tau,tleq t_f$. I am not sure if this is intended behavior, otherwise you could formulate it as: $$u_1(t) - u_2(tau)=0 forall,t,tauin{x:t_0leq xleq t_f}$$
– Kwin van der Veen
Dec 13 '18 at 10:55
Or is $tau$ a constant positive parameter?
– Kwin van der Veen
Dec 13 '18 at 12:55
Yes, edited to clarify.
– U.T.
Dec 13 '18 at 13:24