By J. A. Richards
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Extra info for Analysis of Periodically Time-Varying Systems: Communications and Control Engineering Series
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. Conversely, although practical systems should be treated in terms of short time stability considerations, classical stability is often a sufficiently accurate substitute. There are some exceptions to this in practice wherein the use of classical stability can lead to inaccurate results; in general however, parametric systems are described almost entirely in terms of classical stability, not the least reason for which is the difficulty in determining short time stability conditions.
Thus by differentiating (v - 1) times the relevant Wronskian matrix can be established and the state transition matrices ultimately determined. 14 Response to a Sinusoidal Forcing Function Although an analytical solution to a forced (inhomogeneous) Hill equation depends upon tractability of the homogeneous form, some general results of practical significance can be derived at this stage. The forced response of a general order periodic system, described for example 42 Solutions to Periodic Differential Equations by Eq.
1 of Willems [lJ and Chap. 8 of D'Angelo . Some definitions refer to the boundedness of forced response-if the output of a system is bounded Stability Theorems for Periodic Systems 51 time of observation ___ J:i~uQL ___ J time of observation - - -bound -------- ------1 I I I v v v " classically stable short - time unstable classically unstable short-time stable Fig. 1. Illustration of the concept of short time stability. Note that classically stable solutions can be short time unstable, and vice versa for a bounded input the system is said to be non-resonant or stable bounded-input, bounded output.
Analysis of Periodically Time-Varying Systems: Communications and Control Engineering Series by J. A. Richards