# Obtaining state space from transfer functions

• Liferider
In summary, the conversation discusses creating a state space model for a MIMO system with two inputs and two outputs. The transfer functions between inputs and outputs are provided and are all of 3rd degree. The transfer matrix is also mentioned, along with the equations for H11, H12, H21, and H22. The individual successfully solved the problem by creating two equations for the outputs and using them to create a state space model with 6 states. Finally, only two of the states were relevant and were used in the measurement equation.

#### Liferider

1. The problem statement
I have a MIMO system with two inputs and two outputs. I have all transfer functions between inputs and outputs. I want to obtain a state space model that describes this system. All transfer functions are of 3rd degree.

## Homework Equations

"Transfer matrix":
HP(s)=[H11, H12; H21, H22]

H11=$\frac{1}{100s^3+80s^2+17s+1}$

H12=$\frac{0.5}{100s^3+80s^2+17s+1}$

H21=$\frac{4.8}{5s^3+13.5s^2+7.5s+1 }$

H22=$\frac{16}{5s^3+13.5s^2+7.5s+1 }$

Last edited:
Solved it :-D

I created two equations for the outputs which resulted in two 3rd degree ODE's. From these two equations I created 6 states, where only two of them was of any interest to me, so I put those two states in the measurment equation of the state space model.

## 1. What is a state space representation?

A state space representation is a mathematical model used to describe the behavior of a dynamic system. It consists of a set of first-order differential equations that relate the system's input, output, and internal states.

## 2. Why is it important to obtain state space from transfer functions?

Obtaining state space from transfer functions allows us to analyze the behavior of a system in the time domain, rather than the frequency domain. This can provide more insight into the internal dynamics of the system and how it responds to different inputs.

## 3. How do you obtain state space from a transfer function?

To obtain state space from a transfer function, we can use a method called state space realization. This involves converting the transfer function into a set of first-order differential equations, which can then be expressed in matrix form to represent the system's state space.

## 4. What are the advantages of using state space representation?

State space representation has several advantages over other methods of system modeling, such as transfer function representation. It allows for more flexibility in modeling complex systems, it can handle nonlinear systems, and it provides a more intuitive understanding of the system's behavior.

## 5. Are there any limitations to obtaining state space from transfer functions?

One limitation of obtaining state space from transfer functions is that it may not always be possible for systems with multiple inputs or outputs. In addition, the state space representation may not accurately capture all of the system's dynamics, especially in highly complex systems.