Relationship between eigenfrequiencies and state space equations?

Discussion in 'Homework Help' started by cdummie, Feb 10, 2018.

1. cdummie Thread Starter Member

Feb 6, 2015
124
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First off, i am not quite sure is 'state space equations' correct term for what i meant, what i actually meant by 'state space equations' is the following:
set of differential equations with one derivative of a different variable in each of equations (those variables are called state variables), where those variables can be either a current through inductor or a voltage on capacitor. Now, the question is:

Let's say that we have the following set of "state space equations":

$\frac{dx_1}{dt}=ax_1 + bx_2 \\ \frac{dx_2}{dt}=cx_1 + dx_2$

Find the eigenfrequencies of the circuit represented by this equations?

I know how to determine eigenfrequencies when i have a circuit, firstly, i write input-output relationship for the circuit, and then i look for corresponding homogenous equation and then i can determine it's eigenfrequencies easily, on the other hand, finding equations of state is also relatively easy, but the problem here is that i don't see any relationship between these two procedures that could lead me to the answer of this questions. Any ideas or suggestions appreciated!

2. TheGoldenLamb New Member

Feb 9, 2018
12
1
I might be wrong on this, but from a purely linear algebra point of view, you've got a linear set of equations of the form

$X' = A\.X$

so what you call eigenfrequencies would just be related to the eigenvalues of the matrix, namely their imaginary part. This is just my input though, so feel free to take it with a grain of salt!

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3. cdummie Thread Starter Member

Feb 6, 2015
124
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That's interesting observation, if we write it in matrix form, it might be actually that eigenfrequencies are actually eigenvalues of the matrix $A$. But i am just not quite sure since i cannot clearly relate this to the input-output relationship of the circuit which should be second order linear differential equation since circuit described by this set of equations indeed is second order linear circuit, however, i like the idea, most likely it is a step towards solution to this problem.

4. MrAl AAC Fanatic!

Jun 17, 2014
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Hello,

To test your hypo, set up a second order system with say one inductor and one cap and a resistor or two, then solve the network. Compare it to a known solution.
2nd order systems are easy to solve so this should not be too difficult.

If you dont have a known solution, then try to figure out what they are asking.
From what i understand, an eigenfrequency would be a natural frequency of the system, so compare the system you solve to what it would take to solve the purely symbolic system.

BTW the system you show i think is formally called simply a set of ODE's while a state vector differential equation is a matrix form of the same thing. I dont hold anybody to this strict definition to often though because i myself dont see the forms as that important except when a reference to some technique does.

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5. cdummie Thread Starter Member

Feb 6, 2015
124
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I know how to solve second order linear differential equation i just need to sum up particular solution with solution to corresponding homogenous equation, but thing that is of interest for me here is only solution to homogenous equation which is in exponential form, because i can derive natural frequency of a circuit (i didn't knew if this was the correct term so i just translated it directly since word for natural frequency literary means eigenfrequency by analogy with eigenvalues in linear algebra) from it easily.

On the other hand, i can determine the set of state equations (again, not sure if that's completely correct term, but i believe you know what i meant) and actually in this case it is given to me so there's no need to find one, but it asks me to determine the eigenfrequencies (i think that there should be two of them) of a circuit when state equations are given.

6. Tesla23 AAC Fanatic!

May 10, 2009
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A simple exercise is to look at a parallel resonant circuit,
$i=-C\frac{dv}{dt}$
$v=L\frac{di}{dt}$

and look at the eigenvalues.

Given you are trying to find the natural frequencies of the system, it doesn't matter what coordinate system you use, and the one where the equations are diagonal is simpler.

Look at http://lpsa.swarthmore.edu/Representations/SysRepTransformations/TF2SS.html

which shows how to get from the state-space representation to transfer function:

and note that the denominator of the transfer function is $det(sI-A)$, so the poles of the transfer function are the zeroes of $det(A - \lambda I)$, the eigenvalues of the state space equations.

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7. cdummie Thread Starter Member

Feb 6, 2015
124
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This is what i was looking for, this is the way i can relate space state equation to the eigenfrequecies (natural frequencies of a circuit), i do have trouble understanding it, since i am not familiar with laplace transform that much, but i guess i'll manage to figure it out. Thanks!

8. MrAl AAC Fanatic!

Jun 17, 2014
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Hi,

I guess you did not understand what i said.

If you start with a circuit, you can compare your results to the results you would get with the system you have to work with now. It's a very simple concept. You'll get constants like a,b,c,d and then you can relate them to what you need in the system at hand.

The solution will be in the form:
sqrt(f(a,b,c,d))

and that will give you two solutions. But in EITHER system, the circuit or the one at hand, you'll see how the constants fit into the solution.
Start with virtually any second order system as i said.

Note the solution form shown above does not specify any values for a,b,c,d yet, just as your system at hand does not specify them yet.

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