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 Gsp(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 Gsp 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 Ni are p &times; 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:

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

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

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

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