1. §3-2 Realization of single variable systems
First of all, we study the relationship among controllability, observability and pole-zero cancellation of transfer functions.
1. Controllability, observability and pole-zero cancellation
Consider a single variable system whose dynamical equation is
The transfer function of (3-22) is

2. Theorem 3-6 Dynamical equation (3-22) is controllable and observable iff g(s) does not have pole-zero cancellation between D(s) and N(s).
Proof
Necessity: Suppose there exists a s=s0 such that N(s0)=0 and D(s0)=0:
Using the equation

3. Substituting s= s0 into the above equation, we have
Multiplying the left and right sides of Equation (1) by c and b, we have
Multiplying the left and right sides of Equation (1) by cA and b and noting the result of Equation (2), we have
…….., then we can obtain that

4. The set of equations can be rewritten as
Since the dynamical equation is observable, the above observability matrix is nonsingular. Then we have

5. Consider the equation (1-45), we have
However,
a contradiction.

6. Sufficiency: We shall prove that if N(s) and D(s) have no common factors and therefore, the Equation(3-30)
is controllable and observable. Suppose the system is uncontrollable or unobservable. Let (3-32) be uncontrollable. Then, by using controllability decomposition, it follows that

7. D(s) is an nth order polynomial, while D1(s) is an n1th order polynomial with n1 < n. It shows that N(s) and D(s) must have pole and zero cancellation, a contradiction.
Q.E.D

8. 2. Realization of rational functions
Consider the transfer function
d is the feed forward part of the following dynamical equation.
Hence, we only study the strictly proper rational part of Equation (3-30).

9. Problem: Consider a strictly proper rational function:
Find a (A, b, c) such that
We assume that the numerator and denominator of g(s) have no common factors.

10. 1) Irreducible realization of controllable canonical form
The realization is as follows:

11. Writing the corresponding system of (3-34) as

12. 3)

13. 2) Irreducible realization of observable canonical form
We can obtain the following observability canonical form:

14. Consider the following equation
From the Differential Theorem of Laplace transform,
Substituting them into Equation (1), we have

17. Finally, we consider . In fact, from (2), we have
Comparing the two equations, we have

19. 3) Jordan canonical form realization
Example:
Let

20. Because
hence,
From
we have

21. Finally, from
we have