Fractional calculus is a mathematical tool for augmenting conventional integrals and derivatives. When introduced to control theory, it poses new opportunities and challenges for engineers. In the literature, pioneers have revealed the benefits brought to some general control theory by fractional order (FO) modeling and control techniques. Yet, there has not

been a systematic study of such techniques for specific industrial processes. Therefore, this dissertation makes the efforts to fulfill the task. This research originates from the equipment control in the semiconductor manufacturing industry, and most problems under discussion are very practical. Newly developed methodologies for solving these problems are exhibited: for example, the relay feedback identification of FO models, auto-decoupling of FO multi-input-multi-output (MIMO) processes, relative gain array of FO MIMO processes, feedback linearization of nonlinear FO systems, and the FO sliding-mode based extreme seeking control for impedance matching, etc. In addition, comprehensive literature surveys on relevant topics are provided; and an extensive review and evaluation of existing numerical tools for fractional calculus and FO controls are conducted. Novel concepts, such as the pseudo frequency response, are promoted; and potential future research opportunities are identified. Through these efforts, fractional order modeling and control are expected to receive wider adoption so that this powerful tool can be used more broadly for the development of modern industry.

Fractional calculus is like a mutated gene fragment which generates varieties of research spices when it is grafted to any research breed. Beside the research in the scope of pure FO modeling and control, a cadenza chapter is provided in the end of this dissertation, in which some interesting thinking, experimental results and hypothesis on miscellaneous research topics are presented. These discussions involve topics related to either fractional calculus or controls, such as Arduino based control demo gadgets for education, EtherCAT timing jitter characterization, Levy distribution based random search, fractal analysis of the financial market, a Hurst exponent based technical indicator, etc.

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