Mixing
Extracting information from Singular Value Decomposition.
$begingroup$
I am currently working on a heat pump system. The problem involves multiple inputs and outputs.
During self study I came across the SVD technique, and learned that it can relate orthogonal inputs to outputs with a gain defined by the diagonal matrix.
My problem is that, because of the base changes, it appears to me that the relation between inputs and outputs and the physical variables defined in my system is no longer trivial.
I would greatly appreciate if someone with more experience could offer me a little guidance, as I have not been able to find answers anywhere.
Thanks
control-theory svd
asked Oct 5 '15 at 12:19
starskystarsky
63
$endgroup$
add a comment |
$begingroup$
I am currently working on a heat pump system. The problem involves multiple inputs and outputs.
During self study I came across the SVD technique, and learned that it can relate orthogonal inputs to outputs with a gain defined by the diagonal matrix.
My problem is that, because of the base changes, it appears to me that the relation between inputs and outputs and the physical variables defined in my system is no longer trivial.
I would greatly appreciate if someone with more experience could offer me a little guidance, as I have not been able to find answers anywhere.
Thanks
control-theory svd
asked Oct 5 '15 at 12:19
starskystarsky
63
$endgroup$
$begingroup$
What's your main issue? Do you want to decouple the system? Is it given as an LTI representation?
$endgroup$
– Max Herrmann
Oct 13 '15 at 9:43
$begingroup$
Thanks, Max Herrman. The system is a heap pump with highly non-linear behavior, but I do have a state space representation for it about an operating point. Though the system has thousands of variables, I have 3 actuating variables and 3 variables I am interested in controlling. The reason I started looking at SVD is to find clues about good decentralised control strategies. Thanks.
$endgroup$
– starsky
Oct 14 '15 at 12:37
$begingroup$
Thousands of states? Maybe you should think about model order reduction first. This would actually be an application example of the SVD. Apart from that, I would suggest an optimization-based approach to take care of the couplings implicitely.
$endgroup$
– Max Herrmann
Oct 14 '15 at 13:24
$begingroup$
Do you have a big matrix, say A, that describes (a linear approximation to) the behavior of your system ? If so, run SVD on it, and compare the top 3 inputs and outputs to the 3 you know about. If not, it's not clear to me what you're asking.
$endgroup$
– denis
Oct 27 '15 at 10:58
add a comment |
$begingroup$
I am currently working on a heat pump system. The problem involves multiple inputs and outputs.
During self study I came across the SVD technique, and learned that it can relate orthogonal inputs to outputs with a gain defined by the diagonal matrix.
My problem is that, because of the base changes, it appears to me that the relation between inputs and outputs and the physical variables defined in my system is no longer trivial.
I would greatly appreciate if someone with more experience could offer me a little guidance, as I have not been able to find answers anywhere.
Thanks
control-theory svd
asked Oct 5 '15 at 12:19
starskystarsky
63
$endgroup$
I am currently working on a heat pump system. The problem involves multiple inputs and outputs.
During self study I came across the SVD technique, and learned that it can relate orthogonal inputs to outputs with a gain defined by the diagonal matrix.
My problem is that, because of the base changes, it appears to me that the relation between inputs and outputs and the physical variables defined in my system is no longer trivial.
I would greatly appreciate if someone with more experience could offer me a little guidance, as I have not been able to find answers anywhere.
Thanks
control-theory svd
control-theory svd
asked Oct 5 '15 at 12:19
starskystarsky
63
asked Oct 5 '15 at 12:19
starskystarsky
63
asked Oct 5 '15 at 12:19
starskystarsky
63
asked Oct 5 '15 at 12:19
starskystarsky
63
asked Oct 5 '15 at 12:19
starskystarsky
63
63
$begingroup$
What's your main issue? Do you want to decouple the system? Is it given as an LTI representation?
$endgroup$
– Max Herrmann
Oct 13 '15 at 9:43
$begingroup$
Thanks, Max Herrman. The system is a heap pump with highly non-linear behavior, but I do have a state space representation for it about an operating point. Though the system has thousands of variables, I have 3 actuating variables and 3 variables I am interested in controlling. The reason I started looking at SVD is to find clues about good decentralised control strategies. Thanks.
$endgroup$
– starsky
Oct 14 '15 at 12:37
$begingroup$
Thousands of states? Maybe you should think about model order reduction first. This would actually be an application example of the SVD. Apart from that, I would suggest an optimization-based approach to take care of the couplings implicitely.
$endgroup$
– Max Herrmann
Oct 14 '15 at 13:24
$begingroup$
Do you have a big matrix, say A, that describes (a linear approximation to) the behavior of your system ? If so, run SVD on it, and compare the top 3 inputs and outputs to the 3 you know about. If not, it's not clear to me what you're asking.
$endgroup$
– denis
Oct 27 '15 at 10:58
add a comment |
$begingroup$
What's your main issue? Do you want to decouple the system? Is it given as an LTI representation?
$endgroup$
– Max Herrmann
Oct 13 '15 at 9:43
$begingroup$
Thanks, Max Herrman. The system is a heap pump with highly non-linear behavior, but I do have a state space representation for it about an operating point. Though the system has thousands of variables, I have 3 actuating variables and 3 variables I am interested in controlling. The reason I started looking at SVD is to find clues about good decentralised control strategies. Thanks.
$endgroup$
– starsky
Oct 14 '15 at 12:37
$begingroup$
Thousands of states? Maybe you should think about model order reduction first. This would actually be an application example of the SVD. Apart from that, I would suggest an optimization-based approach to take care of the couplings implicitely.
$endgroup$
– Max Herrmann
Oct 14 '15 at 13:24
$begingroup$
Do you have a big matrix, say A, that describes (a linear approximation to) the behavior of your system ? If so, run SVD on it, and compare the top 3 inputs and outputs to the 3 you know about. If not, it's not clear to me what you're asking.
$endgroup$
– denis
Oct 27 '15 at 10:58
$begingroup$
What's your main issue? Do you want to decouple the system? Is it given as an LTI representation?
$endgroup$
– Max Herrmann
Oct 13 '15 at 9:43
$begingroup$
What's your main issue? Do you want to decouple the system? Is it given as an LTI representation?
$endgroup$
– Max Herrmann
Oct 13 '15 at 9:43
$begingroup$
Thanks, Max Herrman. The system is a heap pump with highly non-linear behavior, but I do have a state space representation for it about an operating point. Though the system has thousands of variables, I have 3 actuating variables and 3 variables I am interested in controlling. The reason I started looking at SVD is to find clues about good decentralised control strategies. Thanks.
$endgroup$
– starsky
Oct 14 '15 at 12:37
$begingroup$
Thanks, Max Herrman. The system is a heap pump with highly non-linear behavior, but I do have a state space representation for it about an operating point. Though the system has thousands of variables, I have 3 actuating variables and 3 variables I am interested in controlling. The reason I started looking at SVD is to find clues about good decentralised control strategies. Thanks.
$endgroup$
– starsky
Oct 14 '15 at 12:37
$begingroup$
Thousands of states? Maybe you should think about model order reduction first. This would actually be an application example of the SVD. Apart from that, I would suggest an optimization-based approach to take care of the couplings implicitely.
$endgroup$
– Max Herrmann
Oct 14 '15 at 13:24
$begingroup$
Thousands of states? Maybe you should think about model order reduction first. This would actually be an application example of the SVD. Apart from that, I would suggest an optimization-based approach to take care of the couplings implicitely.
$endgroup$
– Max Herrmann
Oct 14 '15 at 13:24
$begingroup$
Do you have a big matrix, say A, that describes (a linear approximation to) the behavior of your system ? If so, run SVD on it, and compare the top 3 inputs and outputs to the 3 you know about. If not, it's not clear to me what you're asking.
$endgroup$
– denis
Oct 27 '15 at 10:58
$begingroup$
Do you have a big matrix, say A, that describes (a linear approximation to) the behavior of your system ? If so, run SVD on it, and compare the top 3 inputs and outputs to the 3 you know about. If not, it's not clear to me what you're asking.
$endgroup$
– denis
Oct 27 '15 at 10:58
add a comment |
1 Answer
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$begingroup$
I'm not sure if this answers your question but here is a few insights to SVD on systems.
Actually you need to calculate the SVD of the transfer function matrix for all the frequencies to find the gains of the inputs for different frequencies. So, let's say
$$G(s)=C(sI-A)^{-1}B+D$$
is the transfer function matrix. So the output vector can be calculated as
$$Y(s) = G(s) U(s)$$
where $Y(s)$ and $U(s)$ are the Laplace transform of the outputs and inputs respectively. Now you can use SVD on $G(j omega)$ for all $omega in (0,infty)$ to check the most amplified input vector for a specific frequency.
The maximum amplification over all frequencies is called the $H_infty$ norm of the system and $H_infty$ Optimal Control Theory deals with finding the controller for minimizing the $H_infty$ norm of the closed loop system.
answered Jan 11 at 17:01
obareeyobareey
3,01911128
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add a comment |
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1 Answer
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$begingroup$
I'm not sure if this answers your question but here is a few insights to SVD on systems.
Actually you need to calculate the SVD of the transfer function matrix for all the frequencies to find the gains of the inputs for different frequencies. So, let's say
$$G(s)=C(sI-A)^{-1}B+D$$
is the transfer function matrix. So the output vector can be calculated as
$$Y(s) = G(s) U(s)$$
where $Y(s)$ and $U(s)$ are the Laplace transform of the outputs and inputs respectively. Now you can use SVD on $G(j omega)$ for all $omega in (0,infty)$ to check the most amplified input vector for a specific frequency.
The maximum amplification over all frequencies is called the $H_infty$ norm of the system and $H_infty$ Optimal Control Theory deals with finding the controller for minimizing the $H_infty$ norm of the closed loop system.
answered Jan 11 at 17:01
obareeyobareey
3,01911128
$endgroup$
add a comment |
$begingroup$
I'm not sure if this answers your question but here is a few insights to SVD on systems.
Actually you need to calculate the SVD of the transfer function matrix for all the frequencies to find the gains of the inputs for different frequencies. So, let's say
$$G(s)=C(sI-A)^{-1}B+D$$
is the transfer function matrix. So the output vector can be calculated as
$$Y(s) = G(s) U(s)$$
where $Y(s)$ and $U(s)$ are the Laplace transform of the outputs and inputs respectively. Now you can use SVD on $G(j omega)$ for all $omega in (0,infty)$ to check the most amplified input vector for a specific frequency.
The maximum amplification over all frequencies is called the $H_infty$ norm of the system and $H_infty$ Optimal Control Theory deals with finding the controller for minimizing the $H_infty$ norm of the closed loop system.
answered Jan 11 at 17:01
obareeyobareey
3,01911128
$endgroup$
add a comment |
$begingroup$
I'm not sure if this answers your question but here is a few insights to SVD on systems.
Actually you need to calculate the SVD of the transfer function matrix for all the frequencies to find the gains of the inputs for different frequencies. So, let's say
$$G(s)=C(sI-A)^{-1}B+D$$
is the transfer function matrix. So the output vector can be calculated as
$$Y(s) = G(s) U(s)$$
where $Y(s)$ and $U(s)$ are the Laplace transform of the outputs and inputs respectively. Now you can use SVD on $G(j omega)$ for all $omega in (0,infty)$ to check the most amplified input vector for a specific frequency.
The maximum amplification over all frequencies is called the $H_infty$ norm of the system and $H_infty$ Optimal Control Theory deals with finding the controller for minimizing the $H_infty$ norm of the closed loop system.
answered Jan 11 at 17:01
obareeyobareey
3,01911128
$endgroup$
I'm not sure if this answers your question but here is a few insights to SVD on systems.
Actually you need to calculate the SVD of the transfer function matrix for all the frequencies to find the gains of the inputs for different frequencies. So, let's say
$$G(s)=C(sI-A)^{-1}B+D$$
is the transfer function matrix. So the output vector can be calculated as
$$Y(s) = G(s) U(s)$$
where $Y(s)$ and $U(s)$ are the Laplace transform of the outputs and inputs respectively. Now you can use SVD on $G(j omega)$ for all $omega in (0,infty)$ to check the most amplified input vector for a specific frequency.
The maximum amplification over all frequencies is called the $H_infty$ norm of the system and $H_infty$ Optimal Control Theory deals with finding the controller for minimizing the $H_infty$ norm of the closed loop system.
answered Jan 11 at 17:01
obareeyobareey
3,01911128
answered Jan 11 at 17:01
obareeyobareey
3,01911128
answered Jan 11 at 17:01
obareeyobareey
3,01911128
answered Jan 11 at 17:01
obareeyobareey
3,01911128
3,01911128
add a comment |
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$begingroup$
What's your main issue? Do you want to decouple the system? Is it given as an LTI representation?
$endgroup$
– Max Herrmann
Oct 13 '15 at 9:43
$begingroup$
Thanks, Max Herrman. The system is a heap pump with highly non-linear behavior, but I do have a state space representation for it about an operating point. Though the system has thousands of variables, I have 3 actuating variables and 3 variables I am interested in controlling. The reason I started looking at SVD is to find clues about good decentralised control strategies. Thanks.
$endgroup$
– starsky
Oct 14 '15 at 12:37
$begingroup$
Thousands of states? Maybe you should think about model order reduction first. This would actually be an application example of the SVD. Apart from that, I would suggest an optimization-based approach to take care of the couplings implicitely.
$endgroup$
– Max Herrmann
Oct 14 '15 at 13:24
$begingroup$
Do you have a big matrix, say A, that describes (a linear approximation to) the behavior of your system ? If so, run SVD on it, and compare the top 3 inputs and outputs to the 3 you know about. If not, it's not clear to me what you're asking.
$endgroup$
– denis
Oct 27 '15 at 10:58