§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

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

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

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

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

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

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

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).

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.

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

Writing the corresponding system of (3-34) as

3)

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

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

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

3) Jordan canonical form realization Example: Let

Because hence, From we have

Finally, from we have