How does state transition matrix indicate time-varying system, but $A$ matrix is constant?

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When I determine whether a linear dynamical system is time-varying or not by looking at the state transition matrix, I get confused by the following:

Normally, I look at the $A$ matrix to determine whether a system is time-varying. Obviously, if $A$ is a function of time $A(t)$, then it is time-varying, and if not, then it is not a time-varying system.

But, the state transition matrix looks like $phi(t) = mathcalL^-1[(sI-A)^-1]$. And then $phi(t)$ turns out to be a function of time.

What I'm trying to understand is how the state transition matrix is a function of time, but the $A$ matrix is not? I am intuitively trying to rationalize this.

The state transition matrix indicates time-varying system, but $A$ matrix is constant? So is the system $dot x(t)=phi(t)x(t)$ time-varying or not?

linear-algebra dynamical-systems transition-matrix steady-state

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asked Jul 31 at 5:16

Candic3

969

add a comment | 

up vote
0
down vote

favorite

When I determine whether a linear dynamical system is time-varying or not by looking at the state transition matrix, I get confused by the following:

Normally, I look at the $A$ matrix to determine whether a system is time-varying. Obviously, if $A$ is a function of time $A(t)$, then it is time-varying, and if not, then it is not a time-varying system.

But, the state transition matrix looks like $phi(t) = mathcalL^-1[(sI-A)^-1]$. And then $phi(t)$ turns out to be a function of time.

What I'm trying to understand is how the state transition matrix is a function of time, but the $A$ matrix is not? I am intuitively trying to rationalize this.

The state transition matrix indicates time-varying system, but $A$ matrix is constant? So is the system $dot x(t)=phi(t)x(t)$ time-varying or not?

linear-algebra dynamical-systems transition-matrix steady-state

share|cite|improve this question

asked Jul 31 at 5:16

Candic3

969

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

When I determine whether a linear dynamical system is time-varying or not by looking at the state transition matrix, I get confused by the following:

Normally, I look at the $A$ matrix to determine whether a system is time-varying. Obviously, if $A$ is a function of time $A(t)$, then it is time-varying, and if not, then it is not a time-varying system.

But, the state transition matrix looks like $phi(t) = mathcalL^-1[(sI-A)^-1]$. And then $phi(t)$ turns out to be a function of time.

What I'm trying to understand is how the state transition matrix is a function of time, but the $A$ matrix is not? I am intuitively trying to rationalize this.

The state transition matrix indicates time-varying system, but $A$ matrix is constant? So is the system $dot x(t)=phi(t)x(t)$ time-varying or not?

linear-algebra dynamical-systems transition-matrix steady-state

share|cite|improve this question

asked Jul 31 at 5:16

Candic3

969

When I determine whether a linear dynamical system is time-varying or not by looking at the state transition matrix, I get confused by the following:

Normally, I look at the $A$ matrix to determine whether a system is time-varying. Obviously, if $A$ is a function of time $A(t)$, then it is time-varying, and if not, then it is not a time-varying system.

But, the state transition matrix looks like $phi(t) = mathcalL^-1[(sI-A)^-1]$. And then $phi(t)$ turns out to be a function of time.

What I'm trying to understand is how the state transition matrix is a function of time, but the $A$ matrix is not? I am intuitively trying to rationalize this.

The state transition matrix indicates time-varying system, but $A$ matrix is constant? So is the system $dot x(t)=phi(t)x(t)$ time-varying or not?

linear-algebra dynamical-systems transition-matrix steady-state

share|cite|improve this question

asked Jul 31 at 5:16

Candic3

969

share|cite|improve this question

share|cite|improve this question

asked Jul 31 at 5:16

Candic3

969

asked Jul 31 at 5:16

Candic3

969

asked Jul 31 at 5:16

Candic3

969

969

add a comment | 

add a comment | 

1 Answer
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up vote
3
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Take a simple system $dotx = x$, then $A=1$ and $phi(t,t_0) = e^t-t_0$.
That is $A$ is constant, but the system response is time varying. In general this
will be the case unless $A=0$.

If you have $phi(t,t_0)$ in general you have $dotphi(t,t_0) = A(t) phi(t,t_0)$, or $A(t) = phi(t,t_0)^-1 dotphi(t,t_0) = phi(t_0,t) dotphi(t,t_0)$, so you can determine if $A$ is time varying or not.

Addendum: It is not too hard to show that the system is time invariant iff
$phi(t,t_0) = phi(t-t_0,0)$ for all $t,t_0$. A little more work shows that in this case $A$ is essentially constant.

share|cite|improve this answer

edited Aug 1 at 0:19

answered Jul 31 at 5:33

copper.hat

122k557155

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So basically you are saying that $A$ being a function of time, is a sufficient but not necessary condition for a system to be time-varying, right? If that's true, then I'm still wondering how to get some general guidelines of determining if system is time-varying, or LTI or neither.
  – Candic3
  Jul 31 at 15:37
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Not really. I am saying that even for a constant $A$ the system response will typically be time varying. I am just showing that for a simple time invariant system that the system response is time varying. If $A$ is constant, then the system will be time invariant, but the state response will (usually) be time varying.
  – copper.hat
  Jul 31 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  oh, i see. This is an important distinction then: the "System" being time-varying versus the state response being "time-varying". Those are different things, you seem to be implying.
  – Candic3
  Jul 31 at 16:04
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Well, yes, the dynamics are completely described by $A$ (which may be constant), but in general the responses will be time varying. Think of $A$ as the 'law describing motion'. The laws may be fixed but the motion need not be.
  – copper.hat
  Jul 31 at 16:10
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I added a little more info.
  – copper.hat
  Aug 1 at 0:20
  

  
  
  
  

 | 
show 3 more comments

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1 Answer
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1 Answer
1

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oldest

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oldest

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up vote
3
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Take a simple system $dotx = x$, then $A=1$ and $phi(t,t_0) = e^t-t_0$.
That is $A$ is constant, but the system response is time varying. In general this
will be the case unless $A=0$.

If you have $phi(t,t_0)$ in general you have $dotphi(t,t_0) = A(t) phi(t,t_0)$, or $A(t) = phi(t,t_0)^-1 dotphi(t,t_0) = phi(t_0,t) dotphi(t,t_0)$, so you can determine if $A$ is time varying or not.

Addendum: It is not too hard to show that the system is time invariant iff
$phi(t,t_0) = phi(t-t_0,0)$ for all $t,t_0$. A little more work shows that in this case $A$ is essentially constant.

share|cite|improve this answer

edited Aug 1 at 0:19

answered Jul 31 at 5:33

copper.hat

122k557155

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So basically you are saying that $A$ being a function of time, is a sufficient but not necessary condition for a system to be time-varying, right? If that's true, then I'm still wondering how to get some general guidelines of determining if system is time-varying, or LTI or neither.
  – Candic3
  Jul 31 at 15:37
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Not really. I am saying that even for a constant $A$ the system response will typically be time varying. I am just showing that for a simple time invariant system that the system response is time varying. If $A$ is constant, then the system will be time invariant, but the state response will (usually) be time varying.
  – copper.hat
  Jul 31 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  oh, i see. This is an important distinction then: the "System" being time-varying versus the state response being "time-varying". Those are different things, you seem to be implying.
  – Candic3
  Jul 31 at 16:04
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Well, yes, the dynamics are completely described by $A$ (which may be constant), but in general the responses will be time varying. Think of $A$ as the 'law describing motion'. The laws may be fixed but the motion need not be.
  – copper.hat
  Jul 31 at 16:10
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I added a little more info.
  – copper.hat
  Aug 1 at 0:20
  

  
  
  
  

 | 
show 3 more comments

up vote
3
down vote

Take a simple system $dotx = x$, then $A=1$ and $phi(t,t_0) = e^t-t_0$.
That is $A$ is constant, but the system response is time varying. In general this
will be the case unless $A=0$.

If you have $phi(t,t_0)$ in general you have $dotphi(t,t_0) = A(t) phi(t,t_0)$, or $A(t) = phi(t,t_0)^-1 dotphi(t,t_0) = phi(t_0,t) dotphi(t,t_0)$, so you can determine if $A$ is time varying or not.

Addendum: It is not too hard to show that the system is time invariant iff
$phi(t,t_0) = phi(t-t_0,0)$ for all $t,t_0$. A little more work shows that in this case $A$ is essentially constant.

share|cite|improve this answer

edited Aug 1 at 0:19

answered Jul 31 at 5:33

copper.hat

122k557155

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So basically you are saying that $A$ being a function of time, is a sufficient but not necessary condition for a system to be time-varying, right? If that's true, then I'm still wondering how to get some general guidelines of determining if system is time-varying, or LTI or neither.
  – Candic3
  Jul 31 at 15:37
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Not really. I am saying that even for a constant $A$ the system response will typically be time varying. I am just showing that for a simple time invariant system that the system response is time varying. If $A$ is constant, then the system will be time invariant, but the state response will (usually) be time varying.
  – copper.hat
  Jul 31 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  oh, i see. This is an important distinction then: the "System" being time-varying versus the state response being "time-varying". Those are different things, you seem to be implying.
  – Candic3
  Jul 31 at 16:04
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Well, yes, the dynamics are completely described by $A$ (which may be constant), but in general the responses will be time varying. Think of $A$ as the 'law describing motion'. The laws may be fixed but the motion need not be.
  – copper.hat
  Jul 31 at 16:10
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I added a little more info.
  – copper.hat
  Aug 1 at 0:20
  

  
  
  
  

 | 
show 3 more comments

up vote
3
down vote

up vote
3
down vote

Take a simple system $dotx = x$, then $A=1$ and $phi(t,t_0) = e^t-t_0$.
That is $A$ is constant, but the system response is time varying. In general this
will be the case unless $A=0$.

If you have $phi(t,t_0)$ in general you have $dotphi(t,t_0) = A(t) phi(t,t_0)$, or $A(t) = phi(t,t_0)^-1 dotphi(t,t_0) = phi(t_0,t) dotphi(t,t_0)$, so you can determine if $A$ is time varying or not.

Addendum: It is not too hard to show that the system is time invariant iff
$phi(t,t_0) = phi(t-t_0,0)$ for all $t,t_0$. A little more work shows that in this case $A$ is essentially constant.

share|cite|improve this answer

edited Aug 1 at 0:19

answered Jul 31 at 5:33

copper.hat

122k557155

Take a simple system $dotx = x$, then $A=1$ and $phi(t,t_0) = e^t-t_0$.
That is $A$ is constant, but the system response is time varying. In general this
will be the case unless $A=0$.

If you have $phi(t,t_0)$ in general you have $dotphi(t,t_0) = A(t) phi(t,t_0)$, or $A(t) = phi(t,t_0)^-1 dotphi(t,t_0) = phi(t_0,t) dotphi(t,t_0)$, so you can determine if $A$ is time varying or not.

Addendum: It is not too hard to show that the system is time invariant iff
$phi(t,t_0) = phi(t-t_0,0)$ for all $t,t_0$. A little more work shows that in this case $A$ is essentially constant.

share|cite|improve this answer

edited Aug 1 at 0:19

answered Jul 31 at 5:33

copper.hat

122k557155

share|cite|improve this answer

share|cite|improve this answer

edited Aug 1 at 0:19

edited Aug 1 at 0:19

edited Aug 1 at 0:19

answered Jul 31 at 5:33

copper.hat

122k557155

answered Jul 31 at 5:33

copper.hat

122k557155

answered Jul 31 at 5:33

copper.hat

122k557155

122k557155

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So basically you are saying that $A$ being a function of time, is a sufficient but not necessary condition for a system to be time-varying, right? If that's true, then I'm still wondering how to get some general guidelines of determining if system is time-varying, or LTI or neither.
  – Candic3
  Jul 31 at 15:37
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Not really. I am saying that even for a constant $A$ the system response will typically be time varying. I am just showing that for a simple time invariant system that the system response is time varying. If $A$ is constant, then the system will be time invariant, but the state response will (usually) be time varying.
  – copper.hat
  Jul 31 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  oh, i see. This is an important distinction then: the "System" being time-varying versus the state response being "time-varying". Those are different things, you seem to be implying.
  – Candic3
  Jul 31 at 16:04
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Well, yes, the dynamics are completely described by $A$ (which may be constant), but in general the responses will be time varying. Think of $A$ as the 'law describing motion'. The laws may be fixed but the motion need not be.
  – copper.hat
  Jul 31 at 16:10
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I added a little more info.
  – copper.hat
  Aug 1 at 0:20
  

  
  
  
  

 | 
show 3 more comments

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So basically you are saying that $A$ being a function of time, is a sufficient but not necessary condition for a system to be time-varying, right? If that's true, then I'm still wondering how to get some general guidelines of determining if system is time-varying, or LTI or neither.
  – Candic3
  Jul 31 at 15:37
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Not really. I am saying that even for a constant $A$ the system response will typically be time varying. I am just showing that for a simple time invariant system that the system response is time varying. If $A$ is constant, then the system will be time invariant, but the state response will (usually) be time varying.
  – copper.hat
  Jul 31 at 15:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  oh, i see. This is an important distinction then: the "System" being time-varying versus the state response being "time-varying". Those are different things, you seem to be implying.
  – Candic3
  Jul 31 at 16:04
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Well, yes, the dynamics are completely described by $A$ (which may be constant), but in general the responses will be time varying. Think of $A$ as the 'law describing motion'. The laws may be fixed but the motion need not be.
  – copper.hat
  Jul 31 at 16:10
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I added a little more info.
  – copper.hat
  Aug 1 at 0:20
  

  
  
  
  

So basically you are saying that $A$ being a function of time, is a sufficient but not necessary condition for a system to be time-varying, right? If that's true, then I'm still wondering how to get some general guidelines of determining if system is time-varying, or LTI or neither.
– Candic3
Jul 31 at 15:37

So basically you are saying that $A$ being a function of time, is a sufficient but not necessary condition for a system to be time-varying, right? If that's true, then I'm still wondering how to get some general guidelines of determining if system is time-varying, or LTI or neither.
– Candic3
Jul 31 at 15:37

1

1

Not really. I am saying that even for a constant $A$ the system response will typically be time varying. I am just showing that for a simple time invariant system that the system response is time varying. If $A$ is constant, then the system will be time invariant, but the state response will (usually) be time varying.
– copper.hat
Jul 31 at 15:41

Not really. I am saying that even for a constant $A$ the system response will typically be time varying. I am just showing that for a simple time invariant system that the system response is time varying. If $A$ is constant, then the system will be time invariant, but the state response will (usually) be time varying.
– copper.hat
Jul 31 at 15:41

oh, i see. This is an important distinction then: the "System" being time-varying versus the state response being "time-varying". Those are different things, you seem to be implying.
– Candic3
Jul 31 at 16:04

oh, i see. This is an important distinction then: the "System" being time-varying versus the state response being "time-varying". Those are different things, you seem to be implying.
– Candic3
Jul 31 at 16:04

1

1

Well, yes, the dynamics are completely described by $A$ (which may be constant), but in general the responses will be time varying. Think of $A$ as the 'law describing motion'. The laws may be fixed but the motion need not be.
– copper.hat
Jul 31 at 16:10

Well, yes, the dynamics are completely described by $A$ (which may be constant), but in general the responses will be time varying. Think of $A$ as the 'law describing motion'. The laws may be fixed but the motion need not be.
– copper.hat
Jul 31 at 16:10

1

1

I added a little more info.
– copper.hat
Aug 1 at 0:20

I added a little more info.
– copper.hat
Aug 1 at 0:20

 | 
show 3 more comments

 

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