1. §3-3 realization for multivariable systems
1. Relationship among the controllability, observability and the transfer function matrices
Consider the dynamical equation
(3-36)
where A, B, C are n×n, n×p, q×n real constant matrices, respectively. The transfer function matrix of the Equation (3-36) is as follows:
(3-37)

2. where det(sIA) is called the pole polynomial of the system and G(s) is a strictly proper rational function matrix.

3. Theorem 3-7: System (A, B, C) is controllable and observable if and have no common factors.
Proof
Suppose (A, B, C) is uncontrollable or unobservable. Then, from Theorems (2-17) or (2-18), there exists a system with dimension , such that
The dimension of is , which implies that
and have common factors, a contradiction.
Q.E.D

4. Note that this is just a sufficient condition and can be illustrated by the following example.
Example 3-4: Consider the following dynamical equation:
It is easy to see that the system is controllable and observable. However, the transfer function matrix is

5. The pole polynomial of A has common factor (s-1) with
The pole polynomial of A has common factor (s-1) with Hence, the condition in the theorem is not necessary.
Definition 3-1: A transfer function matrix G(s) is proper, if is a constant matrix. G(s) is strictly proper if

6. Definition 3-2 the highest degree of s of the pole polynomial of G(s) is called the (McMillan) degree, and is denoted by G(s).
Example 3-6: Compute the McMillan degree of the following function matrices.
The pole polynomial of G1(s) and G2(s) are (s+1) and (s+1)2, respectively; hence

7. Example 3-7: Consider
It is easy to figure out the pole polynomial
Definition1-5: The least common denominator of all minors of G(s) is called the characteristic polynomial or pole polynomial.

8. Theorem 3-8 The system (3-32) is controllable and observable if and only if the pole polynomial of G(s) is equivalent to the characteristic polynomial of A,

9. Example 3-8:
The characteristic polynomial is

10. The corresponding transfer function matrix is
The pole polynomial of G(s) is
Hence, the system is controllable and observable.

11. 2. Realizations of Vector Transfer Functions
Example:

12. 2） Observable canonical form realization
Example: Find an irreducible realization of the following function:
Analysis: The least common denominator of the two elements of G(s) is (s+1)(s+2). Therefore,
where, y is a scalar, c and B are and matrices, respectively.

13. Hence, we have

14. Generalized form: Consider the strictly proper rational transfer function matrix as follows:
G(s) is a multi-input single output system and all its elements are irreducible. Let
be the least common denominator of all minors of G(s). Then the equation can be rewritten as:

15. where, are constant vectors.

16. Hence, the realization is as follows:

17. Example:

18. 2) Controllable canonical form realization
Example: Consider the following single-input multi-output transfer function matrix

20. In general, we consider the following transfer function matrix
are column vectors.
which is a single-input multi-output system, and can be realized as the controllable canonical form:

22. 3. Realization of transfer function matrix
Controllable realization
Take a two columns matrix as an example. Let the ith controllable-form realization be Ai , bi , Ci. We can obtain A, B, C :

23. Then, we have
where, and are of controllable canonical form, C1 and C2 have the following form:

24. The above realization is controllable and we can compute its transfer function matrix:

25. Generalized case
1) Rewrite the transfer function matrix in the following form:
2) Give the realization of , where
is controllable.
3) Construct (A, B, C), which is controllable:

26. 2) Observable-form realization
Let the ith observable-form be Ai, Bi, ci. We can obtain A, B, C as follows:
The realization is observable and its transfer function matrix is G(s).

28. Generalized case.
1) Write the transfer matrix :
2) Realize as , where
is observable-form.
3) Construct (A, B, C) as follows:

29. Example: Consider the rational transfer function matrix:
Find the controllable-form realization and observable-form realization of G(s), respectively.
The first subsystem
The second subsystem

30. from which we can obtain the observable-form realization as

31. It is easy to verify that the realization is observable but uncontrollable. We can figure out that δG(s)=3. Since the order of matrix A is 4, from Theorem 3-13, the realization is uncontrollable. If the controllable-form realization is needed, the controllability decomposition the Theorem 2-17 should be used.
The controllable-form realization is as follows:

32. Form (3-42), we obtain the following realization:

33. Example: Consider the following rational transfer function matrix:
Find an irreducible realization of G(s).
The common denominator of the minors of order 1 is s3. The common denominator of the minor of order 2 is s4; hence, the pole polynomial is s4.
(1)  G(s) = 4

34. (2) Observable-form realization
We can obtain the observable-form realization as

35. (3) Controllability decomposition.

37. (4) Finally, the irreducible realization is
(5) It can be checked that the transfer function matrix is

39. Theorem3-10: If the matrix can be written as
where, are distinct and are constant matrices. Then,
Proof: Let
Decompose as
where, and are and matrices.

40. Note that can be realized as
where

41. Then, we construct the matrices A, B and C as follows:
It is easy to verify that (A, B, C) is an irreducible
realization. Its dimension is Hence,
Q.E.D

42. Example:

43. An irreducible realization of G(s) is as follows: