gelb_ch4_ocr

# gelb_ch4_ocr - A N A L Y T I C DESCRIPTION O F TRANSIENT...

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ANALYTIC DESCRIPTION OF TRANSIENT OSCILLATIONS 213 If one of these amplifiers in the chain saturates at a small value of system error, the resulting loss of effective system gain results in a decrease in relative stability due to the phase lag at low frequencies introduced by the integral compensation. The consequence of this in high-performance servos, such as those which position the gimbals of an inertial guidance stable platform, is a violent oscillation when the servo is first turned on and the error is not within the linear "notch." This oscillation is damped, and the servo eventually settles into linear operation within the notch, but the nonlinear transient oscillation is an important characteristic of the servo, and an analytic description of this characteristic is of practical importance to the designer. Another illustration is the design of a feedback loop around a limit cycling system which regulates the amplitude of the limit cycle at some station. The design of the amplitude-regulating loop can be pursued in a rational manner only if one has a description of the dynamics relating a change in a system parameter, such as a forward gain, to the resulting change in the amplitude of the limit cycle. A transfer function which represents this dynamic effect will be derived in the following sections as a special case of the general study of transient oscillations in nonlinear systems. 4.1 ANALYTIC DESCRlPTlO N OF TRANSIENT OSCILLATIONS Consider the oscillatory performance of nonlinear systems which may be cast in the form of a single loop system with separable linear and nonlinear parts, as shown in Fig. 4.1-1. This is the same form as that treated in the preceding chapter on steady-state oscillations as shown in Fig. 3.3-2. further restriction that r = 0 must be made to permit practical solution of this problem. This does not rule out consideration of all forced responses, because many cases of common interest, such as a step-function response, can be given an equivalent description in terms of zero input with appropriate initial conditions on system variables. It is the response due to initial conditions which is calculated here. In the case of steady-state oscillations it is possible without undue labor to consider an input of the same form as that of the system output, namely, a steady-state oscillation. In this case, the inclusion of an input in the form of a transient oscillation is much more laborious, and would seem to be of little practical consequence. The linear parts of the system of Fig. 4.1-1 are time-invariant operators, and are considered to be given originally in terms of their transfer functions. The nonlinear part is characterized in this analysis by its DF. It is this approximation which limits the changing amplitude and frequency of the transient oscillation to slow changes; we shall later return to the question of
214 TRANSIENT OSCILLATIONS IN NONLINEAR SYSTEMS Figure 4.1-1 Form of system considered in study of transient oscillations.

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gelb_ch4_ocr - A N A L Y T I C DESCRIPTION O F TRANSIENT...

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