§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)

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

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

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

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 .

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

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.

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,

Example 3-8: The characteristic polynomial is

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

2. Realizations of Vector Transfer Functions Example:

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.

Hence, we have

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:

where, are constant vectors.

Hence, the realization is as follows:

Example:

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

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:

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 :

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

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

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:

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).

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

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

from which we can obtain the observable-form realization as

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:

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

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

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

(3) Controllability decomposition.

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

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.

Note that can be realized as where

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

Example:

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