Abstract
Periodic dynamical systems frequently arise in applications. Examples include batch processes that are taken through a periodic operating cycle, and systems where periodic or multirate sampling strategies are employed (Feuer & Goodwin 1996). A periodic system is time-varying in nature; however, by using time or frequency domain raising techniques, it is possible to reduce the analysis to that of a special LTI multivariable system.
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Notes and References
This chapter is mainly based on Goodwin & Gómez (1995).
Frequency Domain Raising
The origins of this tool can be related to early work of Zadeh (1950), who gave a frequency domain description of general time-varying systems. The modulation representation has been extensively used in the signal processing literature, see e.g., Shenoy, Bumside & Parks (1994), Vetterli (1987) and Vetterli (1989).
The modulated transfer matrix, when evaluated along the unit circle, is sometimes referred to as the alias component matrix (Smith & Barnwell 1987, Ramstad 1984, Vetterli 1989).
Time Domain Raising
For a description of time domain raising and its utility see e.g., Khargonekar, Poolla & Tannenbaum (1985), Meyer (1990), Ravi, Kharghonekar, Minto & Nett (1990) and Feuer & Goodwin (1996). This latter reference establishes the relation between time and frequency domain raising.
Transform Techniques
For a more detailed exposition of Fourier techniques see e.g., Feuer & Goodwin (1996). The double-sided Z-transform and its properties is studied, for example, in Franklin, Powell & Workman (1990).
Periodic Control of LTI Systems
Khargonekar et al. (1985) argue that, for an important class of robustness problems, discrete periodic compensators are superior to time-invariant ones. This superiority is explained in terms of improved gain and phase margins. Similar ideas are presented in Lee, Meerkov & Runolfsson (1987) for continuous-time systems, and in Francis & Georgiou (1988) for sampled-data systems. On the other hand, Khargonekar et al. (1985) showed that time-varying controllers offer no advantage over time-invariant ones in the problem of weighted sensitivity minimization. Furthermore, in Shamma & Dahleh (1991) it is argued that time-varying compensation does not improve optimal rejection of persistent bounded disturbances, and also it does not help in the bounded-input bounded-output robust stabilization of time-invariant plants with unstructured uncertainty.
An analysis of the use of periodic controllers, based on frequency domain arguments, has been given by Goodwin & Feuer (1992) and Feuer (1993). These works showed that the use of periodic control faces two problems: inherent presence of high frequency components, and sensitivity to high frequency system uncertainty.
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© 1997 Springer-Verlag London Limited
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Seron, M.M., Braslavsky, J.H., Goodwin, G.C. (1997). Extensions to Periodic Systems. In: Fundamental Limitations in Filtering and Control. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-0965-5_5
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DOI: https://doi.org/10.1007/978-1-4471-0965-5_5
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