Quantitative feedback theory
Encyclopedia
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 a system plant is represented by its Transform Function (Laplace in the continuous time 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.
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.
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
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.
of the system. At this point, the 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.
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.
) design when it is required. In the case of tracking conditions a shaping on the Bode diagram may be used. Post design analysis is then performed to ensure the system response is satisfactory according with the problem requirements.
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).
Isaac Horowitz
Isaac Horowitz was a notable scientist with significant contributions to automatic control theory. He developed and championed the Quantitative Feedback Theory which for the first time introduced a formal combination of the genuine frequency methodology founded by Hendrik Bode with plant...
(Horowitz, 1963; Horowitz and Sidi, 1972), is a frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
technique utilising the Nichols chart (NC) in order to achieve a desired robust design over a specified region of plant uncertainty. Desired time-domain
Time domain
Time domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...
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
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.
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
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 of the system. At this point, the 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.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.
Prefilter design
Finally, the QFT design may be completed with a pre-filter () design when it is required. In the case of tracking conditions a shaping on the Bode diagram may be used. Post design analysis is then performed to ensure the system response is satisfactory according with the problem requirements.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
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....
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).
See also
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- H infinity
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- Nonlinear controlNonlinear controlNonlinear control is the area of control engineering specifically involved with systems that are nonlinear, time-variant, or both. Many well-established analysis and design techniques exist for LTI systems ; however, one or both of the controller and the system under control in a general control...
- Adaptive controlAdaptive controlAdaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; a control law is needed that adapts itself...
- Robust controlRobust controlRobust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. Robust control methods are designed to function properly so long as uncertain parameters or disturbances are within some set...
- Intelligent controlIntelligent controlIntelligent control is a class of control techniques, that use various AI computing approaches like neural networks, Bayesian probability, fuzzy logic, machine learning, evolutionary computation and genetic algorithms.- Overview :...
- State space (controls)State space (controls)In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...
External links
- Dr Charles J. Pritchard, Doctorate Thesis, University of the Witwatersrand, 1995
- Dr Murray Kerr, Doctorate Thesis, The University of Queensland, 2004
- Professor P.S.V. Nataraj, Interdisciplinary Programme in Systems and Control Engineering, Indian Institute of Technology Bombay, Powai, Mumbai – 400 076, India.