# Modeling a second order system in time and frequency domain

For school I was asked to model the transfer function $$H(s)=\frac{3}{(s+4)(s+5)}$$ in both the time and frequency domains using initial conditions $$y(0)=2,y'(0)=3$$, a step input, and Simulink. I would appreciate any help in better understanding this problem and any real world example would also be appreciated as the coursework is not typically related to real applications.

TL;DR:

1. How do I pick which factor to apply each initial condition to in the frequency domain ($$s+4$$ vs. $$s+5$$)?
2. What changes should I make in my Simulink models to reach the same system response in both the time and frequency domain?
3. Do you have any words of wisdom to help me understand this topic more in depth?

The differential equation of this system is $$3(e^{-4t}-e^{-5t})$$ which I used to set up the frequency domain simulation, summing $$\frac{3}{s^2-9s-20}$$ with $$y(0)*3e^{-4t}$$ and $$y'(0)*3e^{-5t}$$.

My professor claimed the following response was accurate, but I have not been able to get the time domain representation using integrators to match.

Here is my attempt at the time domain model:

I passed $$y'(0)=3$$ into Integrator and $$y(0)=2$$ into Integrator1 as initial conditions, which yields the response:

It is my understanding the two responses should be the same. I can get a very similar response in the frequency domain by switching which expression to multiply with the initial conditions (that is swapping the constant blocks used as product inputs) but it is not exact.

• The first plot seems to have value -3 at time zero. Clearly, that doesn't match the specified initial condition. Why did your professor say that, that plot was correct?
– AJN
Commented Feb 16, 2022 at 14:14
• @AJN I'm not entirely sure he was listening to me, but the fact that he did say it was correct prompted me to post before swapping the initial conditions (which does get me a realistic response). That's what I'll do for my submission, but would you know why y(0) applies to the DE from s+5 over s+4?
– Red
Commented Feb 17, 2022 at 5:23
• The initial conditions are specified in terms of the time domain view of the system. I don't think it can be attributed to either s+4 or s+5 just like that. I have not seen the initial condition being apportioned to poles before. There might be some theory which I am not familiar with or which I can't recall now.
– AJN
Commented Feb 17, 2022 at 5:38

For the Simulink model, you can incorporate the initial conditions in the Simulink model. If $$y(t)$$ is the output and $$r(t)$$ is the input, then the differential equation for the transfer function is: $$\frac{\mathbb{d}^2 y(t)}{\mathbb{d}t} + 9\frac{\mathbb{d}y(t)}{\mathbb{d}t} +20y(t)=3 r(t)$$

This means that: $$Y(s)=\frac{(s+9)y(0) + y'(0)}{(s+4)(s+5)}+\frac{3}{(s+4)(s+5)}R(s)$$ Of course, the transfer function $$H(s)$$ is obtained when all initial conditions are zero: $$H(s)=\frac{Y(s)}{R(s)}=\frac{3}{(s+4)(s+5)}$$ Therefore you can simply model the following in simulink and use the values for $$y(0)=2$$ and $$y'(0)=3$$: $$Y(s)=\frac{2(s+9) + 3}{(s+4)(s+5)}+\frac{3}{(s+4)(s+5)}R(s)$$

Now, the time domain solution is: $$y(t)=\frac{49}{4} \mathrm{e}^{-4t}-\frac{52}{5}\mathrm{e}^{-5t}+\frac{3}{20}$$

The Simulink diagram is shown below:

The resulting plots are given below:

You can see that both solutions are identical.

Note: In modelling the initial conditions, we want the system response when the input $$r(t)$$ is zero. This is the zero input response which can be achieved by using an impulse response to the homogenous solution. And an impulse response can be generated by differentiating a step response. Hence the $$s$$ in the transfer functions that represent the initial conditions.

• I really appreciate you taking the time to work through this. This makes far more sense than the teachings in class and the note on using an impulse response for the initial conditions is extremely helpful.
– Red
Commented Mar 15, 2022 at 22:46

... model the transfer function $$H(s)=\frac{3}{(s+4)(s+5)}$$ in both the time and frequency domains using initial conditions $$y(0)=2,y'(0)=3 ...$$

Transfer functions are derived under the assumption of zero initial conditions AFAIK.

The differential equation of this system is $$3(e^{-4t}-e^{-5t})$$ which I used to set up the frequency domain simulation, summing $$\frac{3}{s^2-9s-20}$$ with $$y(0)*3e^{-4t}$$ and $$y'(0)*3e^{-5t}$$.

It is not an equation and it is not a differential equation.

Differential equation is something like $$y\prime\prime (t) + 9 y\prime (t) + 20 y(t) = 3 x(t)$$

In the "frequency domain approach", what you appear to be doing is computing the natural (?) and forced response (?) separately (I'm not sure).

... with $$y(0)*3e^{-4t}$$ and $$y'(0)*3e^{-5t}$$

I do not know how we can apportion each initial condition to a pole of the system. AFAIK, each pole would contribute to $$y(0)$$ as well as $$y^n\prime (0)$$

Do you have text book references or lecture notes where the initial conditions are paired to a pole in the system ?

• Your skepticism on everything is well founded. The models and equations I have been provided throughout the year are consistently "iffy" and I've just begun to work through the textbook on my own time. Thank you so much for the help!
– Red
Commented Mar 15, 2022 at 22:38