State Space Model for a Circuit R, L, C

[Matt atkinson]

I have been struggling to solve the following problem for days.

I was given the state space model of a circuit and asked to determine the system elements. So basically, I think I have to find a suitable circuit which describes the state space equation. I am not really sure how to approach this question and every time I attempt it, it becomes really complicated.

Here's the task:
The system is to be implemented by a passive electrical circuit whose elements comprise of resistors R, capacitors X and/or inductors L. The input is a voltage source $v_i(t)$ and the output $v_o(t)$. Given, the system state variables are chosen as the voltages across capacitors and the currents through inductors, determine a model structure and the system elements, whether a R,C or L component which produces the given state model. Hence determine the matrix terms $a_{ij}, b_{i}$ and $c_{i}$ in terms of R, L and C.

Here's the state space model:
$\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \dot{x_3}\\ \dot{x_4}\\ \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & 0 & a_{24} \\ a_{31} & 0 & a_{33} & 0 \\ a_{41} & a_{42} & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1(t)\\x_2(t)\\x_3(t)\\x_4(t) \end{pmatrix} + \begin{pmatrix} b_1\\0\\0\\0 \end{pmatrix} v_i(t)$
$v_o(t)= \begin{pmatrix} 0& c_2 & 0& 0 \end{pmatrix} \begin{pmatrix} x_1(t)\\x_2(t)\\x_3(t)\\x_4(t) \end{pmatrix}$

So far, I have tried to make $x_1$ an inductor current and then I get stuck. Could anyone point me in the right direction?

 

[donpacino]

this is a fun little puzzle. Off the top of my head I don't KNOW of a way to solve it and I've never actually solved a problem like this. I do have a few Ideas

so first what do we know...

V = L di/dt ... V=L i_dot
and
I = X dv/dt ... I=X v_dot

we know that our output is component 2.

we know that there are four capacitors/inductors, as there are four statesHere comes the assumptions...

For an inductor current component (X_ k) k being (1,2,3 or 4), looking at the equation for voltage across inductors, you can assume that its governing equation is a sum of voltages.

V = L i_dot ... restructure as ... X_ k _dot = V_ k /L
we know from KVL that V1 + V2 + V3 ... + Vx = 0
so V_k = (-V2 -V3 -V4 -V5 -V6)
you can then extend this and solve for X_k_dot

but wait... isn't X_k a current in this example. well
V = I R, so
X_1_dot = A_11 * X_1

this means when X_k_dot is dependant on X_k, there is a resistor in series with the inductor. This occurs to make math and the physics work out.You can make similar (but slighty different) assumptions for the voltage across capacitor case. This should help you to piece together possible circuit architectures. Hint, the capacitor equation will likely form a node in the circuit.

another assumption, most of the time the input to a circuit is voltage.
So I would start out assuming X1 is the voltage across a cap.

I would take a blank sheet of paper and draw out what you think the architecture is.
once you make an assumption, using the rules you lay out it will define even more of the circuit. If an architecture becomes impossible (equations don't work with the circuit layout) then you made an incorrect assumption.

 

[milesyoung]

$\dot{x_1}$ and $b_1 v_i$ must have the same units.

$b_1$ either has the unit F^-1 or H^-1 and only one of these choices gives the right unit for $b_1 v_i$, which is V/s or A/s. You should be able to conclude that $x_1$ must have a unit of current.

Also, $v_o$ and $c_2 x_2$ must have the same units. You should be able to apply the same logic here to figure out the unit for $x_2$.

What else can you then conclude?

 

[donpacino]

$\dot{x_1}$ and $b_1 v_i$ must have the same units.

$b_1$ either has the unit F^-1 or H^-1 and only one of these choices gives the right unit for $b_1 v_i$, which is V/s or A/s. You should be able to conclude that $x_1$ must have a unit of current.

Also, $v_o$ and $c_2 x_2$ must have the same units. You should be able to apply the same logic here to figure out the unit for $x_2$.

What else can you then conclude?

I did not even notice that the input and outputs were given as voltages :(

 

[Matt atkinson]

Thank you for the help guys! sorry for the late reply I had internet issues and had to go see an old professor to sort it out he suggest pretty much the same thing as you donpacino so thankyou!