barcode reader integration with asp.net FIGURE 32-3 Phase portrait of lightly damped pendulum. in Software

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FIGURE 32-3 Phase portrait of lightly damped pendulum.
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492 NONLINEAR CONTROL
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to move away from the critical point. These conclusions agree with our physical understanding of the pendulum motion, since the steady condition at 8 = TT is easily destroyed. It is because of Liapunov s theorem that linear control theory is so successful in control system design. One really hopes to control the system so that it remains permanently in the vicinity of a particular point (i.e., a steady state). However, when serious upsets occur in an automatically controlled plant, moving it far from steady state, it is often necessary to return the plant to manual control until conditions are again close to steady state. This is because the controllers are designed for satisfactory operation in the linear range only. One of the great drawbacks of linear control theory is the fact that stability of the linearized equations guarantees stability of the nonlinear system only in some vicinity of the particular critical point. No information about the size of this vicinity or about the behavior outside this vicinity is obtained. If the linear vicinity is extremely small, then unknown to the designer who has used linear methods, almost any plant disturbance of practical size may result in control system failure. An example of this behavior will be given later.
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Limit Cycles
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The first major difference between linear and nonlinear motions is the possible existence of more than one critical point in the latter type. The second is the possible existence of limit cycles. A limit cycle is defined as a periodic oscillation whose amplitude and frequency depend only on the properties of the system and not on the initial state of the system (provided the initial state lies in a certain non-trivial region of the phase space). In the phase plane, stable limit cycles are recognized as closed curves which are approached asymptotically by all nearby trajectories. Unstable limit cycles are closed curves from which all nearby trajectories diverge. An example of a stable limit cycle is the steady-state behavior of a home heating system when controlled by a thermostat. A periodic oscillation in house temperature is always reached, and the amplitude and frequency of the oscillation am independent of the temperature that existed in the house at the time that the furnace was started. Unstable limit cycles can never be realized physically for any system by definition. However, as will be seen later, they divide the phase plane into regions of totally different dynamic behavior and hence are of considerable importance. It is important to distinguish between limit cycles and other closed curves which may occur. The linear system
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d2x T2-+x=o dt2
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phase-space
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solution x2 + T*(i)* = c2
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(32.8)
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METHODS OF PHASE-PLANE ANALYSIS
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where i = dxldt and the constant C depends on initial conditions. Equation (32.8) defines a family of concentric ellipses in the phase plane. However, these are not limit cycles, because the closed curve which is followed by the system depends on the initial state of the system through the constant C. In the next section, we shall study some limit cycles occurring in typical control systems.
OTHER ASPECTS. We have presented only those aspects of phase-plane analysis that will be of use in the examples to follow. This can be considered only as a brief introduction to the subject, and the intetested reader is referred to the references already cited for more information. Among the important subjects that have been omitted are graphical methods for determination of time along a trajectory, various aspects of phase-plane topology, and the mathematical aspects of stability.
EXAMPLES OF PHASE-PLANE ANALYSIS
In this section, we shall consider two different examples of the use of the phase plane to analyze nonlinear control systems. The first is a simple on-off control system for a stirred-tank heater. The second is the chemical teactor of Chap. 3 1. In both cases, the systems am second-order and autonomous, so that they ate ideal situations for use of the phase plane.
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