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TECHNICAL PAPERS

# An Interval Analysis Algorithm for Automated Controller Synthesis in QFT Designs

[+] Author and Article Information
P. S. V. Nataraj

Systems and Control Engineering Group, Room 101, ACRE Building, Indian Institute of Technology, Mumbai 400076, Indianataraj@ee.iitb.ac.in

Sachin Tharewal

Systems and Control Engineering Group, Room 101, ACRE Building, Indian Institute of Technology, Mumbai 400076, Indiatharewal@iitb.ac.in

J. Dyn. Sys., Meas., Control 129(3), 311-321 (May 26, 2006) (11 pages) doi:10.1115/1.2397147 History: Received July 12, 2004; Revised May 26, 2006

## Abstract

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 $(f,x)⊆f(x)$.

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## Figures

Figure 4

Example 5.2. Plots of the nominal loop transmission functions (left subplot) and the Bode magnitude responses (right subplot) corresponding to (a) the controller synthesized by Zolotas and Halikias in Ref. 22, shown with dash-dot line and (b) the controller synthesized with the proposed interval analysis algorithm, shown with solid line.

Figure 1

The two degree-of-freedom structure used in QFT

Figure 2

Box feasibility conditions for single-valued upper bounds. With respect to the bound Bi, the L0 box at ωi is seen to be (a) feasible, (b) infeasible, and (c) ambiguous. Consequently, the corresponding parameter box z is feasible, infeasible, and ambiguous at ωi in (a), (b) and (c), respectively.

Figure 3

Example 5.1. Plots of the nominal loop transmission functions (left subplot) and the Bode magnitude responses (right subplot) corresponding to (a) the controller synthesized by Chen in Ref. 8, shown with dash-dot line and (b) the controller synthesized with the proposed interval analysis algorithm, shown with solid line.

Figure 5

Example 5.3. Plots of the nominal loop transmission functions (left subplot) and the Bode magnitude responses (right subplot) corresponding to (a) the controller synthesized by Bryant and Halikias in Ref. 3, shown with dash-dot line and (b) the controller synthesized with the proposed interval analysis algorithm, shown with solid line.

Figure 6

Example 5.4. Plots of the nominal loop transmission functions (left subplot) and the Bode magnitude responses (right subplot) corresponding to (a) the controller synthesized by Chait in Ref. 4, shown with dash-dot line and (b) the controller synthesized with the proposed interval analysis algorithm, shown with solid line.

Figure 7

Example 5.5. Plots of the nominal loop transmission functions (left subplot) and the Bode magnitude responses (right subplot) corresponding to (a) the controller synthesized by Thompson and Nwokah in Ref. 6, shown with dash-dot line and (b) the controller synthesized with the proposed interval analysis algorithm, shown with solid line.

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