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.
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.
l{P}(s)=\left\lbrace\dfrac{\prodi(s+zi)}{\prodj(s+pj)},\forallzi\in[zi,min,zi,max],pj\in[pj,min,pj,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.
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
L0(s)=G(s)P0(s)
The controller design is undertaken on the NC considering the frequency constraints and the nominal loop
L0(s)
G(s)
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.
Finally, the QFT design may be completed with a pre-filter (
F(s)
The QFT design methodology was originally developed for Single-Input Single-Output (SISO) and Linear Time Invariant Systems (LTI), with the design process being as described above. However, it has since been extended to weakly nonlinear systems, time varying systems, distributed parameter systems, multi-input multi-output (MIMO) systems (Horowitz, 1991), discrete systems (these using the Z-Transform as transfer function), and non minimum phase systems. The development of CAD tools has been an important, more recent development, which simplifies and automates much of the design procedure (Borghesani et al., 1994).
Traditionally, the pre-filter is designed by using the Bode-diagram magnitude information. The use of both phase and magnitude information for the design of pre-filter was first discussed in (Boje, 2003) for SISO systems. The method was then developed to MIMO problems in (Alavi et al., 2007).