Nonlinear Control Sy.. - Free

6 CHAPTER 1. INTRODUCTION
case, which is that of first order (linear and nonlinear) autonomous systems. In other words,
we consider a system of the form
x = f (x)
(1.8)
where x(t) is a real-valued function of time. We also assume that f () is a continuous
function of x. A very special case of (1.8) is that of a first-order linear system. In this case,
f (x) = ax, and (1.8) takes the form
x = ax. (1.9)
It is immediately evident from (1.9) that the only equilibrium point of the first-order linear
system is the origin x = 0. The simplicity associated with the linear case originates in the
simple form of the differential equation (1.9). Indeed, the solution of this equation with an
arbitrary initial condition xo # 0 is given by
x(t) = eatxo (1.10)
A solution of the differential equation (1.8) or (1.9) starting at xo is called a trajectory.
According to (1.10), the trajectories of a first order linear system behave in one of two
possible ways:
Case (1), a < 0: Starting at x0, x(t) exponentially converges to the origin.
Case (2), a > 0: Starting at x0, x(t) diverges to infinity as t tends to infinity.
Thus, the equilibrium point of a first-order linear system can be either attractive or repelling.
Attractive equilibrium points are called stables, while repellers are called unstable.
Consider now the nonlinear system (1.8). Our analysis in the linear case was guided
by the luxury of knowing the solution of the differential equation (1.9). Unfortunately,
most nonlinear equations cannot be solved analytically2. Short of a solution, we look for
a qualitative understanding of the behavior of the trajectories. One way to do this is to
acknowledge the fact that the differential equation
x = f(x)
represents a vector field on the line; that is, at each x, f (x) dictates the "velocity vector"
x which determines how "fast" x is changing. Thus, representing x = f (x) in a twodimensional
plane with axis x and x, the sign of ± indicates the direction of the motion of
the trajectory x(t).
'See Chapter 3 for a more precise definition of the several notions of stability.
2This point is discussed in some detail in Chapter 2.