Survey of quantitative feedback theory (QFT)

Abstract QFT is an engineering design theory devoted to the practical design of feedback control systems. The foundation of QFT is that feedback is needed in control only when plant ( P ), parameter and/or disturbance ( D ) uncertainties (sets 𝒫={ P }, 𝒟={ D }) exceed the acceptable ( A ) system performance uncertainty (set 𝒜={ A }). The principal properties of QFT are as follows. (1) The amount of feedback needed is tuned to the (𝒫, 𝒟, 𝒜) sets. If 𝒜 ‘exceeds’ (𝒫, 𝒟), feedback is not needed at all. (2) The simplest modelling is used: (a) command, disturbance and sensor noise inputs, and (b) the available sensing points and the defined outputs. No special controllability test is needed in either linear or non‐linear plants. It is inherent in the design procedure. There is no observability problem because uncertainty is included. The number of independent sensors determines the number of independent loop transmissions ( L i ), the functions which provide the benefits of feedback. (3) The simplest mathematical tools have been found most use ful—primarily frequency response. The uncertainties are expressed as sets in the complex plane. The need for the larger 𝒫, 𝒟 sets to be squeezed into the smaller 𝒜 set results in bounds on the L i ( jω ) in the complex plane. In the more complex systems a key problem is the division of the ‘feedback burden’ among the available L i ( jω ). Point‐by‐point frequency synthesis tremendously simplifies this problem. This is also true for highly uncertain non‐linear and time‐varying plants which are converted into rigorously equivalent linear time invariant plant sets and/or disturbance sets with respect to the acceptable output set 𝒜. Fixed point theory justifies the equivalence. (4) Design trade‐offs are highly transparent in the frequency domain: between design complexity and cost of feedback (primarily bandwidth), sensor noise levels, plant saturation levels, number of sensors needed, relative sizes of 𝒫, 𝒜 and cost of feedback. The designer sees the trade‐offs between these factors as he proceeds and can decide according to their relative importance in his particular situation. QFT design techniques with these properties have been developed step by step for: (i) highly uncertain linear time invariant (LTI) SISO single‐ and multiple‐loop systems, MIMO single‐loop matrix and multiple‐loop matrix systems; and (ii) non‐linear and time‐varying SISO and MIMO plants, and to a more limited extent for plants with distributed control inputs and sensors. QFT has also been developed for single‐ and multiple‐loop dithered non‐linear (adaptive) systems with LTI plants, and for a special class (FORE) of non‐linear compensation. New techniques have been found for handling non‐minimum‐phase (NMP) MIMO plants, plants with both zeros and poles in the right half‐plane and LTI plants with incidental hard non‐linearities such as saturation.