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

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

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

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

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

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

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

Remember, q is the number of inputs, p is the number of internal system states, and r is the number of outputs.

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

Then we can define G_sp as:

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

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:

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

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

$C = \begin{bmatrix}I_p & 0 & 0 & \cdots & 0\end{bmatrix}$

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