# define the output - state space

Discussion in 'Math' started by kokkie_d, May 26, 2010.

1. ### kokkie_d Thread Starter Active Member

Jan 12, 2009
72
0
Hi,

I have the following schematic, see attachement.

I have created the input function as follows:
$
u_i(t) = L \frac{di_1(t)}{dt} + R_li_1(t) + U_c(t) + R_c(i_1(t) - i_2(t))

$

For the output function I have:

$

u_o(t) = R_oi_2(t)$

$
u_o(t) = U_c(t) + R_c(i_2 - i_1)

$

I am trying to set up for an Space State equation, but I am confused about the capacitor with resistor in series and that parallel to the output resistor.

I also have:
$
i = C\frac {dU_c(t)}{dt}
$

But which i is this then 1 or 2?

Cheers,

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2. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
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If you are familiar with the Laplace ['s' domain] transformation of circuits then you can do this rather more easily than by using differential equations. In any case the solution is quite tedious and some care is required in formulating the transfer function G(s) - which is then a good starting point for the ongoing State-Space formulation.

As a check for G(s) I get

$G(s)=\frac{U_{o}(s)}{U_{i}(s)}=\frac{R_{p}(s+\frac{1}{R_{c}C})}{L(s^{s}+( \frac{R_{L}}{L}+\frac{1}{(R_{o}+R_{c})C}+\frac{R_{p}}{L})s+\frac{R_{L}}{LC(R_{o}+R_{c})}+\frac{R_{p}}{LCR_{c}})}$

where

$R_{p}=\frac{R_{o}R_{c}}{(R_{o}+R_{c})}$

I didn't double check but the "form" of the equation looks reasonable.

3. ### kokkie_d Thread Starter Active Member

Jan 12, 2009
72
0
Hi T_N_K

As an answer that looks good. I have the answer on paper. But I am trying to reach that answer and I am struggling. I am familiar with laplace (maybe rusty).

Maybe I should reformulate my question:
1. Are my equations right?
I mean the output voltage is defined by a current through $R_o$ but it is also defined by the voltage drop over the capacitor in series with the current through the resistor $R_c$. And the capacitor is subject to two currents (I used mesh method to define the input and output equations. So, how do I define my output voltage (equation).
2. How do you get to $R_p$? It looks like a total resistance of parallel resistors but if that belongs to the output equation where has the capacitor gone?

Cheers,

4. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
784
I assumed you wanted the transfer function as I calculated it. Is that what you need to work out for your state-space formulation? Or was it the relationship of other variables that is of interest?

If it is indeed the network transfer G(s) you are seeking then you don't need to undertake any complex circuit analysis.

The relationship between Ui and Uo is a ratio (albeit complex) of impedances - it's really just a voltage divider when you reduce the main network branches to their equivalent impedances.

The value Rp comes out (in the maths) as a convenient expression of a component of the the effective impedance of the parallel combination of Ro and the (C + Rc) branch.

In any case if you intend to pursue your approach then your equations look OK.

The capacitor current, by your chosen conventions, would be i1(t)-i2(t).

In the end you need to eliminate the variables i1(t) and i2(t) if you want the simplest form of transfer function Uo/Ui. But I guess there's no obvious reason why you couldn't have the currents as intermediate state variables.

Last edited: May 27, 2010
kokkie_d likes this.
5. ### kokkie_d Thread Starter Active Member

Jan 12, 2009
72
0
Cheers T_N_K.

I have worked out the transfer function and I did not make a mistake, instead I contacted the author of the paper I was working from and he mentioned that they left out $R_l$ out of most part of the equation to make it simpler because it only had limited effect since it is very small. So, hence my confusion there.

Is there a book or good reference that explains about state space models?

6. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
784
There's some quite nice work done by Jonathan How in a lecture series on feedback control.

Search for "16.31 Feedback Control" - For instance topic #8 has a useful description of state-space formulation for different systems.