Quantitative feedback theory

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Quantitative feedback theory (QFT), developed by Isaac Horowitz (Horowitz, 1963; Horowitz and Sidi, 1972), is a frequency domain technique utilising the Nichols chart (NC) in order to achieve a desired robust design over a specified region of plant uncertainty. Desired time-domain responses are translated into frequency domain tolerances, which lead to bounds (or constraints) on the loop transmission function. The design process is very transparent allowing a designer to see what trade-offs are necessary to achieve the desired performance.

Usually a system plant is represented by its Transform Function (Laplace in the continuos domain, Z-Transform in the discrete domain). In QFT every parameter of this function is represented as an interval of possible values, which have been obtained as a result of system identification process.

Therefore, the system is represented by a family of plants rather than by a standalone expression. After an analysis in the frequency domain, taking a finite number of frequencies, a set of templates is obtained which enclose the behaviour of the open loop system along the frequency.

QFT take care of the desired performance of system as a set of constraints represented in the frequency domain. Usually system performance is described as robustness to instability, rejection to input an output noise disturbances and tracking. All these considerations are summarized in a set of frequency constraints represented on the Nichols Chart (NC).

The controller design is undertaken on the NC with the frequency constraints and the nominal plant of the system, the plant which represents the frequency templates. At this point, designer begins to introduce controller functions and tune their parameters, a process called Loop Shaping, until the best possible controller is reached without violation of the frequency constraints.

Finally, the QFT design may be completed with a pre-filter design when it is required.