State Space Model for a Circuit R, L, C

In summary, I am 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. I have been trying to make the input and output voltages the same by using a resistor in series, but that doesn't seem to be working. I also tried to make the input and output currents the same, but that still isn't working. I have a few ideas, but I don't know if any of them are actually viable. In summary, I am struggling to solve
  • #1
Matt atkinson
116
1
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 [itex]v_o(t)[/itex]. 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 [itex]a_{ij}, b_{i}[/itex] and [itex]c_{i}[/itex] 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?
 
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  • #2
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.
 
  • #3
##\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?
 
  • #4
milesyoung said:
##\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 :(
 
  • #5
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!
 

1. What is a state space model?

A state space model is a mathematical representation of a dynamic system that describes the evolution of the system's internal state over time. It uses a set of variables, known as state variables, to represent the system's internal state and a set of equations to describe how the state variables change over time.

2. How does a state space model apply to circuits with R, L, and C components?

A state space model can be applied to circuits with R, L, and C components by using the state variables to represent the storage elements (inductors and capacitors) and the input/output variables to represent the resistive elements. The equations in the model will then describe how the state variables (current and voltage) change over time in response to the input (voltage) and output (current).

3. What are the advantages of using a state space model for circuit analysis?

Using a state space model for circuit analysis allows for a more comprehensive understanding of the circuit's behavior, as it takes into account the circuit's internal state variables. It also allows for the analysis of non-linear and time-varying circuits, which cannot be easily analyzed using traditional methods.

4. How are the state variables and equations determined for a circuit with R, L, and C components?

The state variables for a circuit with R, L, and C components are determined by identifying the storage elements (inductors and capacitors) in the circuit. The equations are then determined using Kirchhoff's laws and the circuit's component values, and they describe the relationships between the state variables and input/output variables.

5. Can a state space model be used for practical circuit analysis?

Yes, state space models can be used for practical circuit analysis. They are commonly used in control systems engineering and can be implemented in software tools such as MATLAB or SPICE to simulate circuit behavior and aid in circuit design and troubleshooting.

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