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CHAPTER
EXAMPLES OF NONLINEAR SYSTEMS
In the previous chapters, we have confined our attention to the behavior of linear systems or to the analysis of linearized equations representative of nonlinear systems in the vicinity of the steady-state condition. While much useful information can be obtained from such analysis, it frequently is desirable or necessary to consider nonlinearities in control system design. No real physical system is truly linear, particularly over a wide range of operating variables. Hence, to be complete, a control system design should allow for the possibility of a large deviation from steady-state behavior and resulting nonlinear behavior. The purpose of the next three chapters is to introduce some of the tools that can be used for this purpose and to indicate some of the complications that arise when nonlinear systems are considered.
DEFINITION OF A NONLINEAR SYSTEM
A nonlinear system is one for which the principle of superposition does not apply.
Thus, by superposition, the response of a linear system to the sum of two inputs is the same as the sum of the responses to the individual inputs. This behavior, which allows us to characterize completely a linear system by a transfer function, is not true of nonlinear systems. As an example, consider a liquid-level system. If the outflow is proportional to the square root of the tank level, superposition does not hold and the system is nonlinear. If the tank will always operate near the steady-state condition, the square-root behavior may be adequately represented by a straight line and super-
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NONLINEAR
CONTROL
Time __t
FIGURE 31-1 Distance-time plot for moon rocket.
position applied, as we have done before. On the other hand, if the tank level should fall to half the steady-state value, we would no longer expect the transfer function derived on the linearized basis to apply. The analysis becomes more complicated, as we shall see in our introduction to the study of nonlinear systems.
THE PHASE PLANE
The analysis of nonlinear dynamic systems may often be conceptually simplified by changing to a coordinate system known as phase space. In this coordinate system, time no longer appears explicitly, it being replaced by some other property of the system. For example, consider the flight of a rocket to the moon. In a grossly oversimplified manner, we may describe this motion by a plot of the distance of the rocket from the moon versus time. If all goes well, we would like such a plot to resemble Fig. 31.1. Note the initial acceleration during launch and the final deceleration at landing. We may, however, also represent this motion by a plot of rocket velocity versus distance from moon. This plot is shown in Fig. 31.2, where velocity is defined as d(distance from moon)/&. Figure 31.2 is called a phase diagram of the rocket motion. Time now appears merely as a parameter along the curve of the rocket motion. It has been replaced as a coordinate by the rocket velocity. Although in the present example Fig. 31.2 may not be of significant advantage over Fig. 3 1.1, we shall find phase diagrams very helpful in the analysis of certain nonlinear control systems. To begin our study of phase diagrams, we convert a linear motion studied previously in Chap. 8 to the phase plane. The linear motion will be that of the spring-mass-damper system.
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FIGURE 31-2 Velocity-distance plot for moon rocket.
EXAMPLES OF NONLINEAR SYSTEMS
PHASE-PLANE ANALYSIS OF DAMPED OSCILLATOR
The differential equation describing the motion of the system of Fig. 8.1 in response to a unit-step function is
,d2Y 7s
+2g+r= 1
(31.1)
Equation (31.1) has previously been solved to yield the motion in the form of Y(t) versus t as shown in Fig. 8.2. For phase analysis, however, we want the motion in terms of velocity versus position, Y versus Y, where the dot notation is used to indicate differentiation with respect to t. Hence, we rewrite Eq. (3 1.1) as
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