The analysis of sampled-data control systems with a periodically time-varying sampling rate

The z-transform is used to solve sampled-data systems which have a periodically time-varying sampling rate, i.e., systems which have a repetitive sampling pattern in which the time duration between the individual samples is not constant. Such systems are described by linear difference equations with periodic coefficients; however, the difference equation which describes the system at sampling instants corresponding to KN, where N is the period of the coefficients of the difference equation, and K = 0, 1, 2, …, is a linear difference equation with constant coefficients. Thus by forming a series of difference equations which individually describe the system at sampling instants corresponding to KN, KN+1, KN+2, …, (K+1)N−1, the time varying features of the system are in essence removed from the analysis and the z-transform can be used to solve the resulting constant coefficient difference equations. Also, the response between the sampling instants can be found using the solutions of these difference equations. The method presented is straightforward and can be used to analyze any linear sampled-data system with a periodic sampling pattern. Such a condition could occur, for example, when a computer is time shared by more than one system or in some telemetering devices which periodically give to control systems information on quantities being monitored but in which the desired information is not available at equally spaced intervals of time. This method can also be used to obtain an approximate solution for the output of any linear system which is excited by a periodic but nonsinusoidal forcing function and, because of the flexibility of the sampling pattern, should give more accurate results than an approximation which uses equally spaced samples. In this analysis, only periodicity of the sampling pattern is assumed, and no relationship between the individual sampling intervals is required. A few examples have been introduced to illustrate the analytical procedure and the features of the response of a system to sinusoidal inputs is indicated in one of the examples.