State Space model

[rfermkms]

Hello,

I'm trying to obtain a state space model of a non-linear non-stationary circuit:
s*X(s) = A(s) * X(s) + B(s) * U(s)
(Laplace form)

Where X is the state vector and U the input.

There are plenty of examples out there, but all of these use the voltage and current as input and thus obtain a matrix A containing inductances and capacitances relating the states. However, in my case, I have to control stubs (thus the L's and C's) and use these as my inputs. However, when doing this I can no longer obtain a matrix A that does not already include the L's and C's.

Example:

Ground-----Stub1-----V1
|
transmission line
|
Ground-----Stub2-----V2

Does anybody have any idea or experience with this kind of problem?

Thank you.

 

[magoo2]

Can you clarify what you mean by stubs? It's not clear to me what you mean.

 

[rfermkms]

Sorry if I wasn't clear. In my case, a stub simply refers to a transmission line with a specific length so that the impedance behaves capacitely or inductively. So in the given example, the admittance at a junction would be:

Ytotal = Ystub + Yrest with Ystub = -j*Y0*cot(beta*l)

where beta is the lossless wave propagation constant. By choosing l appropriately we can change the behaviour of the stub.

Thanks for taking the time to read my question. Any suggestions are welcome!

 

[QBplanner]

Are u sure what you are doing make sense. state space only represent linear models. if you want to represent non-linear one. you may have to linearize you system at certain operating point.
Fyi

 

[rfermkms]

Yes of course, linearization is used. The problem there is that the operating point varies in time. Anyway, I've changed the topolgy of my circuit (it's a matching circuit) and there might be an easier way of controlling the circuit.

Thanks to both of you for your reply.

 

[xnuke]

Have you looked at descriptor state space models?

E*dx/dt = A*x + B*u, y = C*x + D*u

The L and C terms come out of the A and B matrices and end up in the E matrix. When using inductor currents and capacitor voltages as the states, you typically solve the circuit equations for [di/dt; dv/dt], which means you have to divide by L and C. You don't divide by them in descriptor form, you factor them out into another matrix E that is square and diagonal with the L and C terms on the diagonal. For example, in a circuit with one L and one C, the left side of the equation will be [L, 0; 0, C]*[di/dt; dv/dt]. You can change the E matrix while you run simulations at any given time step.

xnuke
"Live and act within the limit of your knowledge and keep expanding it to the limit of your life." Ayn Rand, Atlas Shrugged.
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[rfermkms]

Thanks, xnuke! That is interesting: it would indeed remove the (L,C)-dependency of the A matrix. However, it would make the E matrix time-dependent and the control U will not contain the L's or C's, which is actually what is needed in the controller I'm trying to develop.

 

[rfermkms]

Correction: E doesn't become time dependent.
Btw, I'm a bit surprised that I had never heard of descriptor state space models before. I guess it can usually be reduced to the standard form, but still....Thanks for the tip!