# Control Systems/Realizations

## Realization

**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

-   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

We can decompose a transfer matrix **G**(s) into a *strictly proper*
transfer matrix:

$$\mathbf{G}(s) = \mathbf{G}(\infty) + \mathbf{G}_{sp}(s)$$

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

$$D = \mathbf{G}(\infty)$$

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

$$d(s) = s^r + a_1s^{r-1} + \cdots + a_{r-1}s + a_r$$

Then we can define **G**~sp~ as:

$$\mathbf{G}_{sp}(s) = \frac{1}{d(s)}N(s)$$

Where

$$N(s) = N_1s^{r-1} + \cdots + N_{r-1}s + N_r$$

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:

$$A = \begin{bmatrix}
                   -a_1I_p & -a_2I_p & \cdots & -a_{r-1}I_p & -a_rI_p \\
                   I_p     & 0       & \cdots & 0           & 0 \\
                   0       & I_p     & \cdots & 0           & 0 \\
                   \vdots  & \vdots  & \ddots & \vdots      & \vdots \\
                   0       & 0       & \cdots & I_p         & 0
           \end{bmatrix}$$

$$B = \begin{bmatrix}I_p \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$$

$$C = \begin{bmatrix}N_1 & N_2 & N_3 & \cdots & Nr\end{bmatrix}$$

### Realizing System by Column

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

$G(s) =\begin{bmatrix} G_1 & G_2 & G_3 &\dots & G_n \end{bmatrix}$

where we realize them and yield:

$G_i => (A_i,B_i,C_i,D_i)$

and the realization of the system will be:

$A  = \begin{bmatrix} 
A_1 & 0 & 0 & \dots &0\\ 
0 & A_2 & 0&&\vdots\\
0 & 0 & A_3\\
\vdots& & & \ddots &0\\
0&0&0&\dots &A_n
\end{bmatrix}$

$B  = \begin{bmatrix} 
B_1 & 0 & 0 & \dots &0\\ 
0 & B_2 & 0&&\vdots\\
0 & 0 & B_3\\
\vdots& & & \ddots &0\\
0&0&0&\dots &B_n
\end{bmatrix}$

$C = \begin{bmatrix} C_1 & C_2 & C_3 &\dots& C_n \end{bmatrix}$

$D = \begin{bmatrix} D_1 & D_2 & D_3 &\dots& D_n \end{bmatrix}$
