# initial conditions that make a system start its steady state at t=0

#### ABOU

Joined Nov 14, 2020
4
Hello,

say we have a system, it has a second order transfer function , and we know that the input function is a sum of some random sin cos and impulse functions , can we know weather it's possible to find certain initial conditions that would make the output of the system start its steady state at t = 0?

thanks!

#### ericgibbs

Joined Jan 29, 2010
16,018
hi ABOU.
Welcome to AAC.
Please post your best attempt at what you consider the answer is, we can then help.
E

#### ABOU

Joined Nov 14, 2020
4
well , I think that since the system's equation in laplace domain would be written as Y(s) = X(s) H(s) + C(s) where C(s) represents the initial conditions, I can maybe assume that C(s) = 0 then try to find Y(s) in Laplace domain , I should normally find and expression like this : Y(s) = Ytrans(s) + Ysteady(s) and then I can assume that C(s) = - Ytrans(s) to make sure that adding them would cancel out and leave only the steady component , but I find this solution too easy to be true , since I only have to prove if it's possible or not and I'm not supposed to calculate C(s)

#### MrAl

Joined Jun 17, 2014
9,350
Hello,

say we have a system, it has a second order transfer function , and we know that the input function is a sum of some random sin cos and impulse functions , can we know weather it's possible to find certain initial conditions that would make the output of the system start its steady state at t = 0?

thanks!
Hi,

Is this really homework? It seems like such a general question. A more specific question would be to find initial conditions of some particular system therefore i'll only answer this question in part for LTI systems.

Initial conditions are nothing more than the currents and voltages in all storage elements and sources at some particular time 't1', and if they are applied to the circuit artificially at time 't0', the circuit response must continue to evolve the same way it did at 't1'. Therefore if you apply all initial conditions to a circuit at time 't0' it will behave as if it was running for an infinitely long time even at 't0'.

To really answer this question you should try a simple system with maybe just two storage elements like an inductor and capacitor and maybe a resistor or two and a step source.
It is possible to measure the initial conditions after a long run time and those will be the initial conditions you will need to apply if you want the circuit to start at t=0 and show a steady state response. Often you can even solve for the initial conditions directly.
Once you work out how to do it with a simple system it will become obvious what you have to do with a more complex system.

Oh BTW sometimes it is as simple as removing the exponential part of the response but some examples would be better.

• ABOU

#### ABOU

Joined Nov 14, 2020
4
Hi,

Is this really homework? It seems like such a general question. A more specific question would be to find initial conditions of some particular system therefore i'll only answer this question in part for LTI systems.

Initial conditions are nothing more than the currents and voltages in all storage elements and sources at some particular time 't1', and if they are applied to the circuit artificially at time 't0', the circuit response must continue to evolve the same way it did at 't1'. Therefore if you apply all initial conditions to a circuit at time 't0' it will behave as if it was running for an infinitely long time even at 't0'.

To really answer this question you should try a simple system with maybe just two storage elements like an inductor and capacitor and maybe a resistor or two and a step source.
It is possible to measure the initial conditions after a long run time and those will be the initial conditions you will need to apply if you want the circuit to start at t=0 and show a steady state response. Often you can even solve for the initial conditions directly.
Once you work out how to do it with a simple system it will become obvious what you have to do with a more complex system.

Oh BTW sometimes it is as simple as removing the exponential part of the response but some examples would be better.
yeah that is exactly what I thought it's just too easy to be true .
Thanks a lot for your answer I learned how the concept of initial conditions work in real life so I apreciate it

#### MrAl

Joined Jun 17, 2014
9,350
yeah that is exactly what I thought it's just too easy to be true .
Thanks a lot for your answer I learned how the concept of initial conditions work in real life so I apreciate it
Oh ok sure you are welcome, but is is really interesting to take a simple 2nd order system and actually calculate the initial conditions without doing a simulation. Of course if you do a simulation you can let the time go to a large value and see a steady state response or very nearly steady state response and then simply measure all the currents and voltages in all the sources and storage elements and you have your initial conditions.

Another way of looking at it is that at any time 't' the final values at time 't' become the initial conditions at time 't+dt' where dt approaches zero, and this is exactly the principle that is used in switching circuits where the state of the switch changes abruptly. It also relates to the principle of the continuity of states.