Modern Control Engineering

676 Chapter 9 / Control Systems Analysis in State Space

exist if the system considered is not controllable. Although most physical systems are

controllable and observable, corresponding mathematical models may not possess the

property of controllability and observability. Then it is necessary to know the conditions

under which a system is controllable and observable. This section deals with controlla-

bility and the next section discusses observability.

In what follows, we shall first derive the condition for complete state controllability.

Then we derive alternative forms of the condition for complete state controllability

followed by discussions of complete output controllability. Finally, we present the concept

of stabilizability.

Complete State Controllability of Continuous-Time Systems. Consider the

continuous-time system.

(9–51)

where

The system described by Equation (9–51) is said to be state controllable at t=t 0 if it is

possible to construct an unconstrained control signal that will transfer an initial state to

any final state in a finite time interval If every state is controllable, then the

system is said to be completely state controllable.

We shall now derive the condition for complete state controllability. Without loss of

generality, we can assume that the final state is the origin of the state space and that the

initial time is zero, or t 0 =0.

The solution of Equation (9–51) is

Applying the definition of complete state controllability just given, we have

or

(9–52)

Referring to Equation (9–48) or (9–50), can be written

(9–53)

Substituting Equation (9–53) into Equation (9–52) gives

x(0)=-a (9–54)

n- 1

k= 0

Ak B

3

t 1

0

ak(t)u(t)dt

e-At= a

n- 1

k= 0

ak(t) Ak

e-At

x(0)=-

3

t 1

0

e-At Bu(t)dt

xAt 1 B= 0 =eAt^1 x( 0 )+

3

t 1

0

eA(t^1 - t) Bu(t)dt

x(t)=eAt x(0) +

3

t

0

eA(t-t) Bu(t)dt

t 0 tt 1.

B=n* 1 matrix

A=n*n matrix

u =control signal (scalar)

x=state vector (n-vector)

x

=Ax+Bu

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