A First Course in FUZZY and NEURAL CONTROL

2.3. CONTROLLABILITY AND OBSERVABILITY 33

that a system that exhibits state controllability will also exhibit output con-
trollability. For example, if the output is a linear combination of two or more
states, and the states are not independent, then the system may not exhibit
output controllability.
We need to keep in mind that controllability is a black and white issue. A
model of a plant is either controllable in a given sense or it is not. Clearly,
to know that a plant is uncontrollable is a very useful piece of information.
However, to know that something is controllable really tells us nothing about
the degree of difficulty in achieving the desired objectives. From a practical point
of view, we would, of course, also like to know how to check the controllability
of a given system.
A plant isobservableif states can be determined from output observations.
Observability therefore, is concerned with the issue of what can be said about
the system state when one is given measurements of the plant output. In the
case where the mathematical model of the system is available in the form

x ̇(t)=F(x,u,t)

the system is said to becompletely observableif for any initial statex(0)
there is afinite timeT> 0 for whichx(0)can be uniquely deduced from the
outputy=G(x,u,t)and the inputu(t)over the interval[0,T].Measuringthe
response of an observable system allows one to formulate appropriate control
actions that can steer the system to its desired output. For many systems, some
system states cannot be determined by observing the output.
The outputy(t)of a system can be measured. The question is whether
we can reconstruct the initial condition from this measured output. It can be
shown that a time-invariant system

x ̇(t)=Ax(t)+Bu(t)

y(t)=Cx(t)

is completely observable if and only if thenr◊nobservability matrix

V=

£

CCACA^2 ∑∑∑ CAn−^1

§T

has rankn,whereAisn◊nandCisn◊r. A system

x ̇(t)=A(t)x(t)+B(t)u(t)

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

withA(t)continuous, is completely observable if and only if the symmetric
observability matrix

V(t 0 ,t 1 )=

Zt 1

t 0

X(τ)X−^1 (t 0 )CT(τ)Φ(τ,t 0 )dτ

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

dX

dt

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