Nonlinear Control Sy.. - Free

4 CHAPTER 1. INTRODUCTION
This equation is referred to as the unforced state equation. Notice that, in general,
there is no difference between the unforced system with u = 0 or any other given
function u(x,t) (i.e., u is not an arbitrary variable). Substituting u = 'y(x, t) in
equation (1.4) eliminates u and yields the unforced state equation.
The second special case occurs when f (x, t) is not a function of time. In this case we
can write
x = f (x) (1.7)
in which case the system is said to be autonomous. Autonomous systems are invariant
to shifts in the time origin in the sense that changing the time variable from t to
T = t - a does not change the right-hand side of the state equation.
Throughout the rest of this chapter we will restrict our attention to autonomous systems.
Example 1.2 Consider again the mass spring system of Figure 1.1. According to Newton's
law
my = E forces
= f(t) - fk - fO-
In Example 1.1 we assumed linear properties for the spring. We now consider the more
realistic case of a hardening spring in which the force strengthens as y increases. We can
approximate this model by taking
fk = ky(1 + a2y2).
With this constant, the differential equation results in the following:
my+Qy+ky+ka2y3 = f(t).
Defining state variables xl = y, x2 = y results in the following state space realization
r xl = X2
(l x2 = -kM - La2X3- Ax2 + (e
t
which is of the form i = f (x, u). In particular, if u = 0, then
or i = f(x).
J it = x2
ll x2 = -m x1 - ma2xi
'M m X2