CONTENTS CHAPTER PREVIOUS NEXT PREP FIND

PROBLEM 12 – 0452: Consider the control system shown in Fig. 1. The controller

is a nonlinear controller. Assuming that there is no input

signal and the system is subjected to only the initial

condition, determine the region of controllability for the

following two cases:

Case 1: The control signal can assume only two values,

u ± 1.

Case 2: The control signal can assume any value between

1 and – 1, or – 1 ≤ u ≤ 1.

Solution: The system equation is given by

ẍ + ẋ – 2x = u 1

Case 1: For u = 1 this equation becomes

ẍ + ẋ – 2x = 1

from which the equilibrium point is

x = – (1/2), ẋ = 0

Since the characteristic equation

λ₂ + λ – 2 = 0

has roots λ = 1 and λ = – 2, the equilibrium point [– (1/2), 0] is a saddle

point. The separatrices which pass through the saddle point have slopes

of – 2 and 1. Figure 2 is a state plane diagram showing the saddle point,

separatrices, and several trajectories.

For u = – 1, Eq. 1 becomes

ẍ + ẋ – 2x = – 1

The equilibrium point is now

x = (1/2), ẋ = 0

It is also a saddle point. The state plane diagram is identical to the

foregoing, except that it is shifted to the right by 1. Figure 3 is the

superposition of the two sets of diagrams corresponding to u = 1 and

u = – 1.

Graphically we see that the origin can be reached from any point

between separatrices I and II shown in Fig. 3. To reach the origin with

u = ± 1 it is necessary that the trajectory starting from a given initial point

intersects the curve S. (The curve S is the switching curve consisting of

the trajectory with u = – 1 in the upper half plane and that with u = 1 in the

lower half plane both passing through the origin). To reach the curve S

from points lying between separatrices I and II but to the left of S it is

necessary to use u = 1 until the curve S is reached. Similarly, for the

points lying between separatrices I and II but to the right of the curve S, it

is necessary to use u = – 1 until the curve S is reached.

Graphically, it is easily seen that it is not possible to enter the

region bounded by separatrices I and II from the exterior. Hence the

region bounded by two separatrices I and II is the domain of controllability

for u = ± 1.

Case 2: The domain of controllability in this case can be

determined similarly to Case 1. For u = constant and bounded by + 1 the

saddle points are

x = – (1/2)u, ẋ = 0 (– 1 ≤ u ≤ 1)

For – 1 ≤ u ≤ 1 (u is bounded but not necessarily constant), the maximum

width of the domain of controllability is obtained when |u| is maximum.

Hence for Case 2 the domain of controllability is the same as Case 1 and

is bounded by separatrices I and II shown in Fig. 3.