This will simplify , i.e.,
Using the periodicity of , we may rewrite as
According to the following algebraic relationship,
where a, b, and c are vectors, (61) implies that
Therefore, we may specify a periodic parametric update law as
Recall that is the adaptation rate as defined in (49). For , becomes
With (60) and (64), we conclude that
The objective is achieved. The main results are summarized in the following theorem.
Theorem 4.1 Consider a spatial‐based nonlinear system (50) with spatially periodic parameters satisfying Assumption 3.2. The error dynamics described by (55) exists under Assumption 3.1. Assume that the control input is determined by (59) along with the periodic parametric adaptation law (63). Then, the tracking error will converge to 0 with the performance characteristics described by
Proof: Refer to .
Adaptive fuzzy control (AFC) has been investigated for coping with nonlinearities and uncertainties of unknown structures [38–40]. The major distinctions between AFC techniques and the ones described in Sections 2 and 3 are (a) time‐based (AFC) versus spatial‐based design (OFLRARC/OFBRARC) and (b) less information assumed on the nonlinearities/uncertainties (AFC) versus more information on the nonlinearities/uncertainties (OFLRARC/OFBRARC). Because, in spatial‐based design, a nonlinear coordinate transformation is conducted to change the independent variable from time to angular displacement, the systems under consideration in AFC and OFLRARC/OFBRARC are distinct. Next, AFC design techniques claim being able to tackle systems with a more generic class of nonlinearities/uncertainties, which relies on incorporating a fuzzy system to approximate those nonlinearities/uncertainties. It is not clear how to determine the required structure complexity of the fuzzy system (e.g., number of membership functions) to achieve desired control performance with reasonable control effort. Generally speaking, known characteristics of the uncertainties or disturbances should be incorporated as much as possible into the control design to improve performance, avoid conservativeness, and produce sensible control input. Therefore, instead of assuming the disturbances to be of generic type (as done by AFC), the methods presented in this chapter aim at a category of disturbances prevalent in rotary systems and explore the spatially periodic nature of the disturbances to design a specific control module and integrate into the overall control system.
The author gratefully acknowledges the support from the Ministry of Science and Technology, R.O.C. under grant MOST104-2221-E-005-043.
© 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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