Modern Control Engineering

Section 9–6 / Controllability 677

Let us put

Then Equation (9–54) becomes

(9–55)

If the system is completely state controllable, then, given any initial state x(0),Equation

(9–55) must be satisfied. This requires that the rank of the n*nmatrix

ben.

From this analysis, we can state the condition for complete state controllability as fol-

lows: The system given by Equation (9–51) is completely state controllable if and only

if the vectors are linearly independent, or the n*nmatrix

is of rank n.

The result just obtained can be extended to the case where the control vector uis

r-dimensional. If the system is described by

whereuis an r-vector, then it can be proved that the condition for complete state

controllability is that the n*nrmatrix

be of rank n,or contain nlinearly independent column vectors. The matrix

is commonly called the controllability matrix.

EXAMPLE 9–10 Consider the system given by

Since

the system is not completely state controllable.

CBABD=B

1

0

1

0

R =singular

B

x# 1

x# 2

R = B

1

0

1

- 1

RB

x 1

x 2

R +B

1

0

Ru

CB  AB  p  An-^1 BD

CB  AB  p  An-^1 BD

x# =Ax+Bu

CB  AB  p  An-^1 BD

B, AB,p, An-^1 B

CB  AB  p  An-^1 BD

=-CBABpAn-^1 BDF

b 0

b 1







bn- 1

V

x(0)=-a

n- 1

k= 0

Ak Bbk

3

t 1

0

ak(t)u(t)dt=bk