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:

𝐆 ( s ) = 𝐆 ( ∞ ) + {𝐆}_{s p} ( s )

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

D = 𝐆 ( ∞ )

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} + ⋯ + {a}_{r − 1} s + {a}_{r}

Then we can define Gsp as:

{𝐆}_{s p} ( s ) = \frac{1}{d ( s )} N ( s )

Where

N ( s ) = {N}_{1} {s}^{r − 1} + ⋯ + {N}_{r − 1} s + {N}_{r}

And the Ni 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{matrix}− {a}_{1} {I}_{p} & − {a}_{2} {I}_{p} & ⋯ & − {a}_{r − 1} {I}_{p} & − {a}_{r} {I}_{p} \\ {I}_{p} & 0 & ⋯ & 0 & 0 \\ 0 & {I}_{p} & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & {I}_{p} & 0\end{matrix} ]

B = [ \begin{matrix}{I}_{p} \\ 0 \\ 0 \\ ⋮ \\ 0\end{matrix} ]

C = [ \begin{matrix}{N}_{1} & {N}_{2} & {N}_{3} & ⋯ & N r\end{matrix} ]

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{matrix}{G}_{1} & {G}_{2} & {G}_{3} & … & {G}_{n}\end{matrix} ]

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{matrix}{A}_{1} & 0 & 0 & … & 0 \\ 0 & {A}_{2} & 0 & & ⋮ \\ 0 & 0 & {A}_{3} \\ ⋮ & & & ⋱ & 0 \\ 0 & 0 & 0 & … & {A}_{n}\end{matrix} ]

B = [ \begin{matrix}{B}_{1} & 0 & 0 & … & 0 \\ 0 & {B}_{2} & 0 & & ⋮ \\ 0 & 0 & {B}_{3} \\ ⋮ & & & ⋱ & 0 \\ 0 & 0 & 0 & … & {B}_{n}\end{matrix} ]

C = [ \begin{matrix}{C}_{1} & {C}_{2} & {C}_{3} & … & {C}_{n}\end{matrix} ]

D = [ \begin{matrix}{D}_{1} & {D}_{2} & {D}_{3} & … & {D}_{n}\end{matrix} ]

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