An Interval Analysis Algorithm for Automated Controller Synthesis in QFT Designs

In this paper, an interval analysis algorithm is proposed for the automatic synthesis of fixed structure controllers in quantitative feedback theory (QFT). The proposed algorithm is tested on several examples and compared with the controller designs given in the QFT literature. Compared to the existing methods for QFT controller synthesis, the proposed algorithm yields considerable improvement in the high frequency gain of the controller in all examples, and improvements in the cutoff frequency of the controller in all but one examples. Notation: $R$ denotes the field of real numbers, $Rn$ is the vector space of column vectors of length $n$ with real entries. A real closed nonempty interval is a one-dimensional box, i.e., a pair $x=[x̱,x¯]$ consisting of two real numbers $x̱$ and $x¯$ with $x̱⩽x¯$⁠. The set of all intervals is $IR$⁠. A box may be considered as an interval vector $x=(x1,…,xn)T$ with components $xk=[x̱k,x¯k]$⁠. A box $x$ can also be identified as a pair $x=[x̱,x¯]$ consisting of two real column vectors $x̱$ and $x¯$ of length $n$ with $x̱⩽x¯$⁠. A vector $x∊Rn$ is contained in a box $x,$ i.e., $x∊x$ iff $x̱⩽x⩽x¯$⁠. The set of all boxes of dimension $n$ is $IRn$⁠. The width of a box $x$ is wid $x=x¯−x̱$⁠. The range of a function $f:Rn→R$ over a box $x$ is $range(f,x)={f(x)∣x∊x}$⁠. A natural interval extension of $f$ on the box $x$ is obtained by replacing in the expression for $f$⁠, all occurrences of reals $xi$ with intervals $xi$ and all real operations with the corresponding interval operations. The natural interval evaluation of $f$ on $x$ is written as $f(x)$⁠. The interval function $f(x)$ is said to be of convergent of order $α$ if $widf(x)−wid{range(f,x)}⩽c{widx}α$⁠. By the inclusion property of interval arithmetic, range $(f,x)⊆f(x)$⁠.