Modern Control Engineering

Section 9–6 / Controllability 675

for some set of constants cj.This means that if xican be expressed as a linear combination

of the other vectors in the set, it is linearly dependent on them or it is not an independent

member of the set.

EXAMPLE 9–9 The vectors

are linearly dependent since

The vectors

are linearly independent since

implies that

Note that if an n*nmatrix is nonsingular (that is, the matrix is of rank nor the determinant

is nonzero) then ncolumn (or row) vectors are linearly independent. If the n*nmatrix is singular

(that is, the rank of the matrix is less than nor the determinant is zero), then ncolumn (or row)

vectors are linearly dependent. To demonstrate this, notice that

9–6 Controllability

Controllability and Observability. A system is said to be controllable at time t 0

if it is possible by means of an unconstrained control vector to transfer the system from

any initial state x(t 0 )to any other state in a finite interval of time.

A system is said to be observable at time t 0 if, with the system in state x(t 0 ),it is possible

to determine this state from the observation of the output over a finite time interval.

The concepts of controllability and observability were introduced by Kalman. They

play an important role in the design of control systems in state space. In fact, the

conditions of controllability and observability may govern the existence of a complete

solution to the control system design problem. The solution to this problem may not

Cy 1 y 2 y 3 D=C

1

2

3

1

0

1

2

2

2

S =nonsingular

Cx 1 x 2 x 3 D=C

1

2

3

1

0

1

2

2

4

S =singular

c 1 =c 2 =c 3 = 0

c 1 y 1 +c 2 y 2 +c 3 y 3 = 0

y 1 = C

1

2

3

S, y 2 = C

1

0

1

S, y 3 = C

2

2

2

S

x 1 +x 2 - x 3 = 0

x 1 = C

1

2

3

S, x 2 = C

1

0

1

S, x 3 = C

2

2

4

S