A First Course in FUZZY and NEURAL CONTROL

32 CHAPTER 2. MATHEMATICAL MODELS IN CONTROL

2.3 Controllability and observability

An importantfirst step in solving control problems is to determine whether the
desired objective can be achieved by manipulating the chosen control variables.
Intuitively, a control system should be designed so that the input can bring
it from any state to any other state in afinite time, and also so that all the
states can be determined from measuring the output variables. The concepts of
controllabilityandobservabilityformalize these ideas.
Aplantiscontrollableif at any given instant, it is possible to control each
state in the plant so that a desired outcome can be reached. In the case where
a mathematical model of the system is available in the form

x ̇(t)=F(x,u,t) (2.24)

the systemx ̇(t)is said to becompletely controllableif for anyt 0 , any initial
conditionx 0 =x(t 0 ),andanyfinal statexf,thereexistsafinite timeTand a
control functionu(t)definedontheinterval[t 0 ,T]such thatx(T)=xf.Note
thatx(T)is the solution of Equation 2.24 and clearly depends on the function
u(t). It can be shown that a linear, time-invariant system

x ̇(t)=Ax(t)+Bu(t) (2.25)

is completely controllable if and only if then◊nmcontrollability matrix

W=

£

BABA^2 B ∑∑∑ An−^1 B

§

(2.26)

has rankn,whereAisn◊nandBisn◊m. More generally, the system

x ̇(t)=A(t)x(t)+B(t)u(t) (2.27)

y(t)=C(t)x(t)

withAa continuousn◊nmatrix, is completely controllable if and only if the
n◊nsymmetriccontrollability matrix

W(t 0 ,t 1 )=

Zt 1

t 0

X(t)X−^1 (t 0 )B(t)BT(t)

°

X−^1

¢T

(t 0 )XT(t)dt (2.28)

is nonsingular, whereX(t)is the uniquen◊nmatrix satisfying

dX(t)

dt

=A(t)X(t),X(0) =I (2.29)

Other types of controllability can be defined. For example,output control-
labilityrequires attainment of arbitraryfinal output. The ability to control the
state gives rise to the notion that the output (response) of a system may also be
controllable, based on the assumption that if all the individual states in a sys-
tem are controllable, and the output is a linear combination of the states, then
the output must also be controllable. Generally, however, there is no guarantee