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".
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:
Where Gsp(s) is a strictly proper transfer matrix. Also, we can use this to find the value of our D matrix:
We can define d(s) to be the lowest common denominator polynomial of all the entries in G(s):
Then we can define Gsp as:
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: