A First Course in FUZZY and NEURAL CONTROL

2.2. STATE VARIABLES AND LINEAR SYSTEMS 29

2.2 Statevariablesandlinearsystems..................

We now begin to formalize the general framework of standard control, as exem-
plified by the two previous examples. The state of a system at a given timet
is described by a set of variablesxi(t),i=1, 2 ,...,n, calledstate variables.
These variables, that arefunctions, are usually written in the form of a vector
function
x(t)=(x 1 (t),x 2 (t),...,xn(t))

The standardmathematical modelof a control system is a system of differ-
ential equations
x ̇(t)=f(x(t),u(t),t)

involving the state variables and theinput(control)variables(also functions)

u(t)=(u 1 (t),u 2 (t),...,uk(t))

so that the future state of the system can be determined from it. These differ-
ential equations are calledstate equations.
In general, state variables cannot be measured directly, but instead, only
values of some other set of variables

y(t)=g(x(t),u(t)) = (y 1 (t),y 2 (t),...,ym(t))

calledoutput variablescan be measured. The equation

y=g(x,u)

is called theoutput equation.
A system whose performance obeys theprinciple of superpositionis de-
fined as alinear system. The principle states that the mathematical model
of a system islinearif, when the response to an inputuisg(x,u), then the
response to the linear combination

cu+dv

of inputs is that same linear combination

cg(x,u)+dg(x,v)

of the corresponding outputs. Here,canddare constants. Thefirst model of
a situation is often constructed to be linear because linear mathematics is very
well-developed and methods for control of linear systems are well-understood.
In practice, a linear system is an approximation of a nonlinear system near a
point, as is explained in Section 2.9. This leads to a piecewise linear system
and gives rise to simplified matrix algebra of the form discussed here. Most
systems in classical control theory aremodeled as piecewise linear systems, and
controllers are designed to control the approximated systems.