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Abstract
Most conventional control algorithms cause numerical problems where data is collected at sampling rates that are substantially higher than the dynamics of the equivalent continuous-time operation that is being implemented. This is of relevant interest in applications of digital control, in which high sample rates are routinely dictated by the system stability requirements rather than the signal processing needs. Digital control systems exhibit bandwidth limitations enforced by their closed-loop frequency requirements that demand very high sample rates. Considerable recent progress in reducing sample frequency requirements has been made through the use of non-uniform sampling schemes, so called alias-free signal processing. The approach prompts the simplification of complex systems and consequently enhances the numerical conditioning of the implementation algorithms that otherwise would require very high uniform sample rates. However, the control communities have not yet investigated the use of intentional non-uniform sampling. The purpose of this article is to address some algorithmic issues when using such regimes for digital control.
Notes
Notes
1. Where the sampling instances are known before hand.
2. That is why the use of such strategies is generally justified for high-frequency applications; i.e. to reduce the speed requirements of converters and, subsequently, the speed of calculations in the processor.
3. The sampled output will be a poor representation of the actual continuous-time response; inter-sample ripple (Goodwin and DeSouza Citation1984).
4. These can feed significant energies to high-frequency mechanical resonances.
5. In some cases the output might not even be available at every sampling time.
6. Such as lost data packets due to errors in the communications medium.
7. Also known as scheduling issues.
8. In addition, the δ-operator is often used instead of the z-operator since it improves the ‘robustness’ of closed-loop stability margins (Forsythe and Goodall Citation1991).
9. The idea has also been discussed briefly in Goodall (Citation2001).
10. The derivation of the generalised z-filter coefficient equations for non-uniform sampling can be found in Khan (Citation2010).
11. It is widely recognised that the canonical form of implementation has certain benefits since there are fewer stored variables and shift operations and hence is the most popular choice for implementation by control engineers (Forsythe and Goodall Citation1991).
12. By comparing the case of holding the digital filter coefficients constant (under non-uniform sampling conditions) also presents the motivation for using the non-uniform sampling algorithm. It is clear from the simulations that by not accounting for variations in the sampling regime can deteriorate the performance of the compensator.
13. This effect was expected. The reason for such an effect is due to the recursive nature of IIR filters. In the case of non-uniform sampling, the discrete-time filter coefficient values keep changing accordingly. The output result is then calculated based on incorrect internal variables that depend on different delays. Such variations hence introduce unexpected changes within the digital filter, causing disturbances to the output.
14. Such as a change in the coefficient values.
15. The derivation of the δ-filter coefficient equations for non-uniform sampling can be found in Khan (Citation2010).
16. The moving coil actuator has been provided by SMAC UK Ltd. (SMAC Citation2004).
17. The transfer function includes a notch filter that would be used to control a resonant mode in the system which in fact does not exist with the actuator chosen, but the more complex fourth-order controller is chosen anyway to demonstrate its applicability for a control algorithm with higher order controllers.
18. On the contrary, the response of a digital system to an input signal will consist of a sum of many sinusoids spaced at integer multiples of the sampling frequency.