Quantitative feedback 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 a system plant is represented by its Transform Function (Laplace in the continuous domain, Z-Transform in the discrete domain), after a process of system modelling an identification.

As a result of experimental measurements the values of coefficients in the Transform 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.

$mathcal\{P\}(s)\; =\; left\; lbrace\; dfrac\{prod\_\{i\}\; (s\; +\; z\_i)\}\{prod\_\{j\}\; (s\; +\; p\_j)\},\; forall\; z\_i\; in\; [z\_\{i,min\},\; z\_\{i,max\}]\; ,\; p\_j\; in\; [p\_\{j,min\},\; p\_\{j,max\}]\; ight\; brace$

A frequency analysis is performed for a finite number of frequencies and a set of "templates" is obtained in the NC diagram which encloses the behaviour of the open loop system at each frequency.

Frequency bounds

QFT takes 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 unstability, rejection to input an output noise disturbances and reference tracking. All these considerations are summarized in a set of "frequency constraints" represented on the Nichols Chart (NC) and a set of rules on the Open Loop Transfer Function $L(s)\; =\; G(s)P(s)$.

Loop shaping

The controller design is undertaken on the NC with the frequency constraints and the "nominal plant" $P\_0(s)$ of the system, the plant which represents the frequency templates. At this point, the designer begins to introduce controller functions ($G(s)$) and tune their parameters, a process called Loop Shaping, until the best possible controller is reached without violation of the frequency constraints.

For this stage there currently exist different CAD ("Computer Aided Design") packages to make easier the controller tuning, like the QFT package of MATLAB.

Prefilter design

Finally, the QFT design may be completed with a pre-filter ($F(s)$) design when it is required. Post design analysis is then performed to ensure the system response is satisfactory. In this case the Bode diagram is mainly used.

The QFT design methodology was originally developed for single-input single-output (SISO) 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, and to multi-input multi-output (MIMO) systems (Horowitz, 1991). 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).

References

* Horowitz, I., 1963, Synthesis of Feedback Systems, Academic Press, New York, 1963.
* Horowitz, I., and Sidi, M., 1972, “Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances,” International Journal of Control, 16(2), pp. 287-309.
* Horowitz, I., 1991, “Survey of Quantitative Feedback Theory (QFT),” International Journal of Control, 53(2), pp. 255-291.
* Borghesani, C., Chait, Y., and Yaniv, O., 1994, Quantitative Feedback Theory Toolbox Users Guide, The Math Works Inc., Natick, MA.
* Zolotas, A. (2005, June 8). " [http://cnx.rice.edu/content/m11109/latest/ QFT - Quantitative Feedback Theory] ". Connexions.

See also

* Control engineering
* Feedback
* Process control
* Robotic unicycle
* H infinity
* Optimal control
* Servomechanism
* Nonlinear control
* Adaptive control
* Robust control
* Intelligent control
* State space (controls)

External links

* [http://www.dept.ee.wits.ac.za/~pritchar/ControlII/phdmain.pdf Dr Charles J. Pritchard, Doctrate Thesis, University of the Witwatersrand, 1995]
* [http://www.mech.uq.edu.au/pgthesis/KERR,%20Murray.pdf Dr Murray Kerr, Doctrate Thesis, The University of Queensland, 2004]
* [http://www.sc.iitb.ac.in/~nataraj/ Professor P.S.V. Nataraj] , [http://www.sc.iitb.ac.in Interdisciplinary Programme in Systems and Control Engineering] , [http://www.iitb.ac.in Indian Institute of Technology Bombay] , Powai, Mumbai – 400 076, India.