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By J. A. Richards

ISBN-10: 0387116893

ISBN-13: 9780387116891

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Extra info for Analysis of Periodically Time-Varying Systems: Communications and Control Engineering Series

Example text

192, Eq. (35). 22a) where (Jd = kd 3 and (Je = kc 3. For later reference, write this as ¢- (n, 0) nZ f'i M_ (k). 22b) In practice the parameters z and (J may not always be positive. Consideration of their definitions will show that both can be negative and, in addition, z can be zero. It is necessary therefore that the elements of the Wronskian matrix be suitably modified in those circumstances. Using the recurrence relations and identities found in McLachlan [5] the following special cases can be derived.

These are classical stability and short time stability. Briefly, a system is regarded as classically stable if its response remains bounded as time goes to infinity. On the other hand a system is said to be short time stable if its response remains within a certain specified bound during a given time of observation. Clearly a system can be classically stable, yet short time unstable, and vice versa. This is depicted in Fig. 1. In practice, short time stability is all that can be observed, although of course if an observation can be made over a long enough time and the system response can be measured, then classical stability can be implied by the 'short time' stability observed.

43) the state transition matrix over the time interval (0, t) is seen to be ¢(t,O) = L-1{sf - G}-l. Using Eq. 42) this can be evaluated for the second order constant coefficient system being considered: 48 Solutions to Periodic Differential Equations cjJ(t, 0) = L -1 [s C2 which should be compared to the state transition matrix derived in Sect. 2 for one interval of the rectangular waveform in the Meissner equation. The success of the Laplace transform in this derivation lies in the fact that the exponential kernel of the transformation is matched by the 'exponential' basis functions of the constant coefficient system.

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Analysis of Periodically Time-Varying Systems: Communications and Control Engineering Series by J. A. Richards

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