# Converting linear state space into a transfer function

• Std
In summary, the conversation discusses the steps of converting a given space to a transfer function and whether or not Laplace is needed in the process. It also mentions the differences in solving for state as x' instead of x(k), as well as how to obtain stability and feedback gain.
Std
Homework Statement
Need elaboration in solving steps of...

Given The state space X(k)= [ 0.5 -0.5; 0.5 0.5] X(k-1) + [0;1] u(k) and out y(k) = [ 1 0] x(k) It is requested to get the tansfer function, Stability... and to design state feedback gain [k1 k2] to place the system poles at the origin....
Relevant Equations
transfer function is = C* [SI-A]-1 * B
My questions are now... Do the steps of converting this space to transfer function include any laplace ? or just we do get [SI-A]-1 and then transfer function is = C* [SI-A]-1 * B As [1 0] * [s-1/det -0.5/det ; 0.5/det s-0.5/det] * [0; 1] = -0.5/s^2+s+0.5 I mean do we need any laplace after that and if yes ?? Why and when shall we use it?? Also what is the difference in steps of solving if the question given was descriping the state as x' not x(k)?

How to get stability ... and feedback gain??

I see this is a couple of months old. Where have these questions been? It also appears that @Std has been inactive since right after posting.

It has been awhile for me, so I'd have to look it up to get the steps to solve it. As I remember it, you should be able to stay in the s-domain solving for the poles and the gain, though. It's mostly algebra with the polynomials in s, if I remember correctly.

## What is a linear state space representation?

A linear state space representation is a mathematical model that describes the behavior of a system in terms of its state variables and inputs. It is commonly used in control systems engineering and allows for the analysis and design of a system's response to external stimuli.

## What is a transfer function?

A transfer function is a mathematical representation of the relationship between the input and output of a system. It is commonly used in control systems engineering to describe the response of a system to different inputs. It is typically represented as a ratio of two polynomials in the laplace domain.

## Why would someone want to convert a linear state space into a transfer function?

Converting a linear state space into a transfer function allows for easier analysis and design of a control system. Transfer functions are simpler to work with and can provide insights into the stability, performance, and sensitivity of a system.

## How do you convert a linear state space into a transfer function?

To convert a linear state space into a transfer function, you can use the Laplace transform. This involves taking the Laplace transform of the state equations and input equations, and then solving for the transfer function in terms of the state variables and input. Alternatively, you can also use the state space matrices to directly calculate the transfer function.

## What are the benefits of using a transfer function over a linear state space representation?

The main benefit of using a transfer function is that it simplifies the analysis and design of a control system. Transfer functions are easier to manipulate and can provide useful insights into the behavior of a system. Additionally, transfer functions can be used to calculate a system's response to different inputs, making it a powerful tool for control system design.

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