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3 STEADY-STATE OSCILLATIONS IN NONLINEAR SYSTEMS 3.0 INTRODUCTION The preceding chapter introduced the notion of a sinusoidal-input describing function (DF). Some of the implications of this type of linearization are discussed there. Here we apply the D F to the study of steady-state oscilla- tions. For D F utilization to be meaningful, certain conditions must be ful- filled by the nonlinearity, and also by the system in which the nonlinearity is present: 1. The nonlinear element is time4nvariant.l 2. No subharmonics are generated by the nonlinearity in response to a sinusoidal input. 3. The system filters nonlinearity output harmonics to the extent that only a trivial quantity is fed back. Condition 3 is the so-called$Iter hypothesis. We shall have a great deal to say about it in this chapter, for this heuristically motivated requirement is certainly a basic factor underlying DF success or failure. Certain periodically varying nonlinearities are also admissible.
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DETERMINATION OF LIMIT CYCLES Ill Three types of steady-state oscillations are of interest: 1. Forced oscillations 2. Conservative free oscillations 3. Limit cycles For our present purpose a forced oscillation is taken to be a systematic response whose frequency is precisely the forcing signal frequency and whose amplitude depends on the forcing signal amplitude. Forced oscilla- tions are encountered in the study of frequency response, in which connection we find a useful application for the DF. The next two oscillation types are behavioral modes of unforced systems. A conservative free oscillation is an initial condition-dependent periodic mode associated with nondissipative (conservative) systems. A continuous range of conservative-free-oscillation amplitudes and frequencies may be possible in a given system. Limit cycle denotes an initial condition-independent periodic mode occurring in dissipa- tive (nonconservative) systems. Only a discrete set of limit cycle amplitudes and frequencies may exist in a given system. We shall find the D F a most powerful tool for the study of both types of unforced behavior of nonlinear systems. 3.1 DETERMINATION OF LIMIT CYCLES The limit cycle phenomenon is deserving of special attention since it is apt to occur in any physical nonlinear system. A limit cycle can be desirable, for example, by providing the vibration (dither) which minimizes frictional effects in mechanical systems. In fact, it can be absolutely necessary for proper system performance, as we shall see when studying a certain adaptive control scheme in Chap. 6. On the other hand, a limit cycle can cause mechanical failure of a control system (destructive limit cycle) or operator discomfort and other undesirable effects, as in an aircraft autopilot.
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