# Control Systems/Realizations

## Realization

[edit | edit source]**Realization** is the process of taking a mathematical model of a system (either in the Laplace domain or the State-Space domain), and creating a physical system. Some systems are not realizable.

An important point to keep in mind is that the Laplace domain representation, and the state-space representations are equivalent, and both representations describe the same physical systems. We want, therefore, a way to convert between the two representations, because each one is well suited for particular methods of analysis.

The state-space representation, for instance, is preferable when it comes time to move the system design from the drawing board to a constructed physical device. For that reason, we call the process of converting a system from the Laplace representation to the state-space representation "realization".

## Realization Conditions

[edit | edit source]**Note:**

Discrete systems

*G(z)*are also realizable if these conditions are satisfied.

- A transfer function
*G(s)*is realizable if and only if the system can be described by a finite-dimensional state-space equation. *(A B C D)*, an ordered set of the four system matrices, is called a**realization**of the system*G(s)*. If the system can be expressed as such an ordered quadruple, the system is realizable.- A system
*G*is realizable if and only if the transfer matrix**G**(s) is a proper rational matrix. In other words, every entry in the matrix**G**(s) (only 1 for SISO systems) is a rational polynomial, and if the degree of the denominator is higher or equal to the degree of the numerator.

We've already covered the method for realizing a SISO system, the remainder of this chapter will talk about the general method of realizing a MIMO system.

## Realizing the Transfer Matrix

[edit | edit source]We can decompose a transfer matrix **G**(s) into a *strictly proper* transfer matrix:

Where G_{sp}(s) is a strictly proper transfer matrix. Also, we can use this to find the value of our *D* matrix:

We can define *d(s*) to be the lowest common denominator polynomial of all the entries in **G**(s):

*q*is the number of inputs,

*p*is the number of internal system states, and

*r*is the number of outputs.

Then we can define **G**_{sp} as:

Where

And the *N _{i}* are

*p × q*constant matrices.

If we remember our method for converting a transfer function to a state-space equation, we can follow the same general method, except that the new matrix *A* will be a block matrix, where each block is the size of the transfer matrix:

### Realizing System by Column

[edit | edit source]We can divide the **G(s)** into multiple column, realize them individually and join them back together later, for **G(s)**:

where we realize them and yield:

and the realization of the system will be: