Converting a series connected transfer function to the state space model

[Master1022]

Homework Statement

Given transfer functions $G(s)$ and $C(s)$, find the state space models for those systems. Then find the state space model when they are connected in series

Relevant Equations

Transfer function

Hi,

I have a question about a homework problem: I am not sure why I do not seem to get the same answers when using different methods.

Question: Given transfer functions $G(s) = \frac{s - 1}{s + 4}$ and $C(s) = \frac{1}{s - 1}$, find the state space models for those systems. Then find the state space model when they are connected in series.

Attempt:
Intuitively I think that the following methods should both yield the same answer:
1) Find the state space models separately and then combine them in series
2) Combine the transfer functions in series and then convert to state space

however, these methods do not seem to give the same answer and I am unsure why.

Here is my attempt to convert $G(s)$ to a state space model.

and here is my working to convert $C(s)$. The bottom of the page uses a general formulation of two state-space models to derive a formula for a state-space model of the series system. The first state space model (representing $C(s)$) has no $d_1$ just like $C(s)$. I could have included it, but it would be 0 anyways.

The general formula derivation is continued and substituting values into the formula:

I am not sure why the two methods wouldn't give the same answer and any help would be greatly appreciated.

Thanks

 

[KataKoniK]

Thank you for your question. It is common to encounter discrepancies in results when using different methods to solve a problem. In this case, there are a few factors that could be contributing to the differences in your results.

Firstly, it is important to make sure that the state space models you are using are accurate representations of the transfer functions. It is possible that there may be a mistake in your conversion process, resulting in different state space models.

Secondly, when combining transfer functions in series, it is important to consider the order in which they are connected. This can affect the overall system and lead to different results.

Lastly, it is possible that there is a mistake in the algebra used to combine the state space models. I would suggest double checking your calculations and equations to ensure accuracy.

In general, it is always a good idea to check your work and try different methods to confirm your results. If you are still having trouble, I would recommend consulting with your instructor or a classmate for further assistance.

I hope this helps and good luck with your homework problem!