Modern Control Engineering

Section 9–2 / State-Space Representations of Transfer-Function Systems 655

Simplifying gives

(9–20)

Equation (9–20) is also a state equation that describes the same system as defined by Equation
(9–13).
The output equation, Equation (9–16), is modified to

y=CPz

or

(9–21)

Notice that the transformation matrix P, defined by Equation (9–18), modifies the coefficient
matrix of zinto the diagonal matrix. As is clearly seen from Equation (9–20), the three scalar state
equations are uncoupled. Notice also that the diagonal elements of the matrix P–1APin Equation
(9–19) are identical with the three eigenvalues of A. It is very important to note that the eigen-
values of Aand those of P–1APare identical. We shall prove this for a general case in what follows.

Invariance of Eigenvalues. To prove the invariance of the eigenvalues under a

linear transformation, we must show that the characteristic polynomials ∑lI-A∑and

@lI-P–1AP@are identical.

Since the determinant of a product is the product of the determinants, we obtain

Noting that the product of the determinants @P–1@and∑P∑is the determinant of the prod-

uct@P–1P@, we obtain

Thus, we have proved that the eigenvalues of A are invariant under a linear

transformation.

Nonuniqueness of a Set of State Variables. It has been stated that a set of state vari-

ables is not unique for a given system. Suppose that x 1 ,x 2 ,p,xnare a set of state variables.

=∑l I-A∑

@l I-P-^1 AP@ =@P-^1 P@@l I-A@

= @P-^1 @@P@@l I-A@

= @P-^1 @@l I-A@@P@

= @P-^1 (l I-A) P@

@l I-P-^1 AP@ = @l P-^1 P-P-^1 AP@

=[1 1 1]C

z 1

z 2

z 3

S

y =[1 0 0]C

1

- 1

1

1

- 2

4

1

- 3

9

SC

z 1

z 2

z 3

S

C

z# 1

z# 2

z

3

S =C

- 1

0

0

0

- 2

0

0

0

- 3

SC

z 1

z 2

z 3

S + C

3

- 6

3

Su