Nonlinear Control Sy.. - Free

1.5. SECOND-ORDER SYSTEMS: PHASE-PLANE ANALYSIS 9
case of first-order systems, this solution is called a trajectory from x0 and can be represented
graphically in the xl - x2 plane. Very often, when dealing with second-order systems, it is
useful to visualize the trajectories corresponding to various initial conditions in the xl - x2
plane. The technique is known as phase-plane analysis, and the xl - x2 plane is usually
referred to as the phase-plane.
From equation (1.14), we have that
1
±2
11 f2 (X)
= f().
The function f (x) is called a vector field on the state plane. This means that to each point
x' in the plane we can assign a vector with the amplitude and direction of f (x'). For
easy visualization we can represent f (x) as a vector based at x; that is, we assign to x the
directed line segment from x to x + f (x). Repeating this operation at every point in the
plane, we obtain a vector field diagram. Notice that if
x(t) = xl(t)
X2(t)
is a solution of the differential equation i = f (t) starting at a certain initial state x0i then
i = f (x) represents the tangent vector to the curve. Thus it is possible to construct the
trajectory starting at an arbitrary point x0 from the vector field diagram.
Example 1.6 Consider the second-order system
1 = x2
22 = -xi - x2.
Figure 1.4 shows a phase-plane diagram of trajectories of this system, along with the vector
field diagram. Given any initial condition x0 on the plane, from the phase diagram it is easy
to sketch the trajectories from x0. In this book we do not emphasize the manual construction
of these diagrams. Several computer packages can be used for this purpose. This plot, as
well as many similar ones presented throughout this book, was obtained using MAPLE 7.