In control theory, 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 highly transparent, allowing a designer to see what trade-offs are necessary to achieve a desired performance level.
Plant templates
Usually any system can be represented by its Transfer Function (Laplace in continuous time domain), after getting the model of a system. As a result of experimental measurement, values of coefficients in the Transfer Function have a range of uncertainty. Therefore, in QFT every parameter of this function is included into an interval of possible values, and the system may be represented by a family of plants rather than by a standalone expression.
P ( s ) = { ∏ i ( s + z i ) ∏ j ( s + p j ) , ∀ z i ∈ [ z i , m i n , z i , m a x ] , p j ∈ [ p j , m i n , p j , m a x ] } {\displaystyle {\mathcal {P}}(s)=\left\lbrace {\dfrac {\prod _{i}(s+z_{i})}{\prod _{j}(s+p_{j})}},\forall z_{i}\in [z_{i,min},z_{i,max}],p_{j}\in [p_{j,min},p_{j,max}]\right\rbrace }
A frequency analysis is performed for a finite number of representative frequencies and a set of templates are obtained in the NC diagram which encloses the behaviour of the open loop system at each frequency.
Frequency bounds Usually system performance is described as robustness to instability (phase and gain margins), rejection to input and output noise disturbances and reference tracking. In the QFT design methodology these requirements on the system are represented as frequency constraints, conditions that the compensated system loop (controller and plant) could not break. With these considerations and the selection of the same set of frequencies used for the templates, the frequency constraints for the behaviour of the system loop are computed and represented on the Nichols Chart (NC) as curves. To achieve the problem requirements, a set of rules on the Open Loop Transfer Function, for the nominal plant L 0 ( s ) = G ( s ) P 0 ( s ) {\displaystyle L_{0}(s)=G(s)P_{0}(s)} may be found. That means the nominal loop is not allowed to have its frequency value below the constraint for the same frequency, and at high frequencies the loop should not cross the Ultra High Frequency Boundary (UHFB), which has an oval shape in the center of the NC.
Loop shaping The controller design is undertaken on the NC considering the frequency constraints and the nominal loop L 0 ( s ) {\displaystyle L_{0}(s)} of the system. At this point, the designer begins to introduce controller functions ( G ( s ) {\displaystyle G(s)} ) and tune their parameters, a process called Loop Shaping, until the best possible controller is reached without violation of the frequency constraints. The experience of the designer is an important factor in finding a satisfactory controller that not only complies with the frequency restrictions but with the possible realization, complexity, and quality. For this stage there currently exist different CAD (Computer Aided Design) packages to make the controller tuning easier.
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