Hankel norm and H infinity norm model reduction exam question. The Next CEO of Stack OverflowHow to derive thresholds from a pooled sample of valuesMathematical Biology and modellingmathematical biology (steady-states)mathematical biology1Lotka-Volterra First Integral and Fixed PointWhy is reducing this problem to a 2 dimension problem not feasibleNext generation matrix for virus dynamic modelFind a solution curve that connects a saddle point.unknown operator in homogenous model of heat application on plate.Galerkin projection model reduction exam question.
RigExpert AA-35 - Interpreting The Information
Rotate a column
Why is the US ranked as #45 in Press Freedom ratings, despite its extremely permissive free speech laws?
Is it okay to majorly distort historical facts while writing a fiction story?
How do I align (1) and (2)?
What flight has the highest ratio of timezone difference to flight time?
Can we say or write : "No, it'sn't"?
Why does standard notation not preserve intervals (visually)
Are police here, aren't itthey?
Is it professional to write unrelated content in an almost-empty email?
Why isn't the Mueller report being released completely and unredacted?
How to install OpenCV on Raspbian Stretch?
Why the difference in type-inference over the as-pattern in two similar function definitions?
Is the D&D universe the same as the Forgotten Realms universe?
Grabbing quick drinks
Can MTA send mail via a relay without being told so?
How I can get glyphs from a fraktur font and use them as identifiers?
The past simple of "gaslight" – "gaslighted" or "gaslit"?
Is micro rebar a better way to reinforce concrete than rebar?
Where do students learn to solve polynomial equations these days?
Yu-Gi-Oh cards in Python 3
Flying from Cape Town to England and return to another province
Does soap repel water?
I believe this to be a fraud - hired, then asked to cash check and send cash as Bitcoin
Hankel norm and H infinity norm model reduction exam question.
The Next CEO of Stack OverflowHow to derive thresholds from a pooled sample of valuesMathematical Biology and modellingmathematical biology (steady-states)mathematical biology1Lotka-Volterra First Integral and Fixed PointWhy is reducing this problem to a 2 dimension problem not feasibleNext generation matrix for virus dynamic modelFind a solution curve that connects a saddle point.unknown operator in homogenous model of heat application on plate.Galerkin projection model reduction exam question.
$begingroup$
Shown below is a question from a model reduction exam. I'm not sure how to answer the questions and I'm wondering if my approach is correct.
A continuous time system relates the inputs $u_1$ and $u_2$ to the output $y$ according to the differential equation.
$$doty+ rho y= u_1 + 2u_2$$
Where $rho$ is a real parameter.
a) $quad$ Determine for arbitrary $rho > 0$ the Hankel norm of this system.
b) $quad$ Determine for arbitrary $rho > 0$ the $H_infty$ norm of this system.
For the hankel norm we first must determine the state space representation. We assume $doty = dotx$. Which leads to:
$$dotx=-rho x+u_1+2u_2, quad y=1$$
So the state space form becomes:
$$dotx = beginbmatrix -rho endbmatrixx + beginbmatrix 1&2 endbmatrix beginbmatrix u_1\u_2 endbmatrix, quad y = beginbmatrix 1 endbmatrix x $$
So $A = beginbmatrix -rho endbmatrix, quad B = beginbmatrix 1&2 endbmatrix, quad C = beginbmatrix 1 endbmatrix$ and $D = 0$
Next, we need to determine the continuous time $infty$ horizon reachability and observability gramians using.
$$0 = AP+PA^top+BB^top$$
$$0 = A^topQ + QA +C^topC$$
This leads to $P = frac52 rho$ and $Q = frac12 rho$
The Hankel norm can then be determined using: $||Sigma||_H=sqrtlambda_max(PQ)= sqrtlambda_max(frac54 rho^2)$
The $H_infty$ norm can be determined using $||Sigma||_H_infty=sup
sigma_max(G(i omega))$
In which $G(i omega)=C(SI-A)^-1B+D$ But the matrix dimensions are incorrect to perform this calculation. So I don't have an idea on how to calculate the $||Sigma||_H_infty$ norm.
mathematical-modeling
asked Mar 19 at 12:29
user463102user463102
7913
$endgroup$
add a comment |
$begingroup$
Shown below is a question from a model reduction exam. I'm not sure how to answer the questions and I'm wondering if my approach is correct.
A continuous time system relates the inputs $u_1$ and $u_2$ to the output $y$ according to the differential equation.
$$doty+ rho y= u_1 + 2u_2$$
Where $rho$ is a real parameter.
a) $quad$ Determine for arbitrary $rho > 0$ the Hankel norm of this system.
b) $quad$ Determine for arbitrary $rho > 0$ the $H_infty$ norm of this system.
For the hankel norm we first must determine the state space representation. We assume $doty = dotx$. Which leads to:
$$dotx=-rho x+u_1+2u_2, quad y=1$$
So the state space form becomes:
$$dotx = beginbmatrix -rho endbmatrixx + beginbmatrix 1&2 endbmatrix beginbmatrix u_1\u_2 endbmatrix, quad y = beginbmatrix 1 endbmatrix x $$
So $A = beginbmatrix -rho endbmatrix, quad B = beginbmatrix 1&2 endbmatrix, quad C = beginbmatrix 1 endbmatrix$ and $D = 0$
Next, we need to determine the continuous time $infty$ horizon reachability and observability gramians using.
$$0 = AP+PA^top+BB^top$$
$$0 = A^topQ + QA +C^topC$$
This leads to $P = frac52 rho$ and $Q = frac12 rho$
The Hankel norm can then be determined using: $||Sigma||_H=sqrtlambda_max(PQ)= sqrtlambda_max(frac54 rho^2)$
The $H_infty$ norm can be determined using $||Sigma||_H_infty=sup
sigma_max(G(i omega))$
In which $G(i omega)=C(SI-A)^-1B+D$ But the matrix dimensions are incorrect to perform this calculation. So I don't have an idea on how to calculate the $||Sigma||_H_infty$ norm.
mathematical-modeling
asked Mar 19 at 12:29
user463102user463102
7913
$endgroup$
add a comment |
$begingroup$
Shown below is a question from a model reduction exam. I'm not sure how to answer the questions and I'm wondering if my approach is correct.
A continuous time system relates the inputs $u_1$ and $u_2$ to the output $y$ according to the differential equation.
$$doty+ rho y= u_1 + 2u_2$$
Where $rho$ is a real parameter.
a) $quad$ Determine for arbitrary $rho > 0$ the Hankel norm of this system.
b) $quad$ Determine for arbitrary $rho > 0$ the $H_infty$ norm of this system.
For the hankel norm we first must determine the state space representation. We assume $doty = dotx$. Which leads to:
$$dotx=-rho x+u_1+2u_2, quad y=1$$
So the state space form becomes:
$$dotx = beginbmatrix -rho endbmatrixx + beginbmatrix 1&2 endbmatrix beginbmatrix u_1\u_2 endbmatrix, quad y = beginbmatrix 1 endbmatrix x $$
So $A = beginbmatrix -rho endbmatrix, quad B = beginbmatrix 1&2 endbmatrix, quad C = beginbmatrix 1 endbmatrix$ and $D = 0$
Next, we need to determine the continuous time $infty$ horizon reachability and observability gramians using.
$$0 = AP+PA^top+BB^top$$
$$0 = A^topQ + QA +C^topC$$
This leads to $P = frac52 rho$ and $Q = frac12 rho$
The Hankel norm can then be determined using: $||Sigma||_H=sqrtlambda_max(PQ)= sqrtlambda_max(frac54 rho^2)$
The $H_infty$ norm can be determined using $||Sigma||_H_infty=sup
sigma_max(G(i omega))$
In which $G(i omega)=C(SI-A)^-1B+D$ But the matrix dimensions are incorrect to perform this calculation. So I don't have an idea on how to calculate the $||Sigma||_H_infty$ norm.
mathematical-modeling
asked Mar 19 at 12:29
user463102user463102
7913
$endgroup$
Shown below is a question from a model reduction exam. I'm not sure how to answer the questions and I'm wondering if my approach is correct.
A continuous time system relates the inputs $u_1$ and $u_2$ to the output $y$ according to the differential equation.
$$doty+ rho y= u_1 + 2u_2$$
Where $rho$ is a real parameter.
a) $quad$ Determine for arbitrary $rho > 0$ the Hankel norm of this system.
b) $quad$ Determine for arbitrary $rho > 0$ the $H_infty$ norm of this system.
For the hankel norm we first must determine the state space representation. We assume $doty = dotx$. Which leads to:
$$dotx=-rho x+u_1+2u_2, quad y=1$$
So the state space form becomes:
$$dotx = beginbmatrix -rho endbmatrixx + beginbmatrix 1&2 endbmatrix beginbmatrix u_1\u_2 endbmatrix, quad y = beginbmatrix 1 endbmatrix x $$
So $A = beginbmatrix -rho endbmatrix, quad B = beginbmatrix 1&2 endbmatrix, quad C = beginbmatrix 1 endbmatrix$ and $D = 0$
Next, we need to determine the continuous time $infty$ horizon reachability and observability gramians using.
$$0 = AP+PA^top+BB^top$$
$$0 = A^topQ + QA +C^topC$$
This leads to $P = frac52 rho$ and $Q = frac12 rho$
The Hankel norm can then be determined using: $||Sigma||_H=sqrtlambda_max(PQ)= sqrtlambda_max(frac54 rho^2)$
The $H_infty$ norm can be determined using $||Sigma||_H_infty=sup
sigma_max(G(i omega))$
In which $G(i omega)=C(SI-A)^-1B+D$ But the matrix dimensions are incorrect to perform this calculation. So I don't have an idea on how to calculate the $||Sigma||_H_infty$ norm.
mathematical-modeling
mathematical-modeling
asked Mar 19 at 12:29
user463102user463102
7913
asked Mar 19 at 12:29
user463102user463102
7913
asked Mar 19 at 12:29
user463102user463102
7913
asked Mar 19 at 12:29
user463102user463102
7913
asked Mar 19 at 12:29
user463102user463102
7913
7913
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%2f3154007%2fhankel-norm-and-h-infinity-norm-model-reduction-exam-question%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%2f3154007%2fhankel-norm-and-h-infinity-norm-model-reduction-exam-question%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
