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:

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

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):

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

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

Then we can define G_sp as:

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

Where

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

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:

$A={\begin{bmatrix}-a_{1}I_{p}&-a_{2}I_{p}&\cdots &-a_{r-1}I_{p}&-a_{r}I_{p}\\I_{p}&0&\cdots &0&0\\0&I_{p}&\cdots &0&0\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&\cdots &I_{p}&0\end{bmatrix}}$

$B={\begin{bmatrix}I_{p}\\0\\0\\\vdots \\0\end{bmatrix}}$

$C={\begin{bmatrix}N_{1}&N_{2}&N_{3}&\cdots &Nr\end{bmatrix}}$

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:

$A={\begin{bmatrix}A_{1}&0&0&\dots &0\\0&A_{2}&0&&\vdots \\0&0&A_{3}\\\vdots &&&\ddots &0\\0&0&0&\dots &A_{n}\end{bmatrix}}$

$B={\begin{bmatrix}B_{1}&0&0&\dots &0\\0&B_{2}&0&&\vdots \\0&0&B_{3}\\\vdots &&&\ddots &0\\0&0&0&\dots &B_{n}\end{bmatrix}}$

$C={\begin{bmatrix}C_{1}&C_{2}&C_{3}&\dots &C_{n}\end{bmatrix}}$

$D={\begin{bmatrix}D_{1}&D_{2}&D_{3}&\dots &D_{n}\end{bmatrix}}$

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