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.
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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 |
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