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 $(f,x)⊆f(x)$.

1.
Horowitz
,
I. M.
, 1993,
Quantitative Feedback Design Theory (QFT)
,
QFT Publications
, Boulder, Co.
2.
Ballance
,
D. J.
, and
Gawthrop
,
P. J.
, 1991, “
Control Systems Design Via a QFT Approach
,”
Proceedings of the IEE Conference “Control 91
,” Edinburgh, UK, Vol.
1
,
476
481
.
3.
Bryant
,
G.
, and
Halikias
,
G.
, 1995, “
Optimal Loop-Shaping for Systems With Large Parameter Uncertainty Via Linear Programming
,”
Int. J. Control
0020-7179,
62
(
3
), pp.
557
568
.
4.
Chait
,
Y.
,
Chen
,
Q.
, and
Hollot
,
C. V.
, 1999, “
Automatic Loop-Shaping of QFT Controllers Via Linear Programming
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
121
, pp.
351
357
.
5.
Gera
,
A.
, and
Horowitz
,
J. M.
, 1992, “
Optimization of the Loop Transfer Function
,”
Int. J. Control
0020-7179,
31
, pp.
389
398
.
6.
Thompson
,
D. F.
, and
Nwokah
,
O. D. I.
, 1994, “
Analytical Loop Shaping Methods in Quantitative Feedback Theory
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
116
(
2
), pp.
169
177
.
7.
Borghesani
,
C.
,
Chait
,
Y.
, and
Yaniv
,
O.
,
The QFT Frequency Domain Design Toolbox for Use With MATLAB
,
The Math Works, Inc.
, Natick, MA.
8.
Chen
,
W.
,
Ballance
,
D. J.
, and
Li
,
Y.
, 1998,
Automatic Loop-Shaping in QFT Using Genetic Algorithms
,
Proceedings of 3rd Asia-Pacific Conference on Control and Measurement
, pp.
63
67
.
9.
Dallwig
,
S.
,
Neumaier
,
A.
, and
Schichl
,
H.
, 1997, “
Glopt—A Program for Constrained Global Optimization
,”
I.
Bomze
et al.
,
Developments in Global Optimization
,
Kluwer
, Dordrecht, pp.
19
36
.
10.
Floudas
,
C. A.
, and
Pardalos
,
P. M.
, 1992,
Recent Advances in Global Optimization
,
Princeton University Press
, Princeton, NJ.
11.
Floudas
,
C. A.
, and
Pardalos
,
P. M.
, 2000,
Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches
,
Kluwer
, Dordrecht.
12.
Horst
,
R.
, and
Pardalos
,
P. M.
, 1995,
Handbook of Global Optimization
,
, Dordrecht.
13.
Horst
,
R.
,
Pardalos
,
P. M.
, and
Thoai
,
N. V.
, 1995,
Introduction to Global Optimization
,
, Dordrecht.
14.
Moore
,
R. E.
,
Hansen
,
E.
, and
Leclerc
,
A.
, 1992, Rigourous Methods for Global “
Optimisation, Recent Advances in Global Optimization
,” eds.,
C. A.
Floudas
, and
P. M.
Pardalos
,
Princeton University Press
, Princeton, NJ.
15.
Pardalos
,
P. M.
, and
Rosen
,
J. B.
, 1987, “
Constrained Global Optimization: Algorithms and Applications
,” eds.,
Springer-Verlag
, New York.
16.
Hansen
,
E.
, 1992,
Global Optimization Using Interval Analysis
,
Marcel Dekker
, New York.
17.
Ichida
,
K.
, and
Fujii
,
Y.
, 1979, “
An Interval Arithmetic Method for Global Optimization
,”
Computing
0010-485X,
23
,
85
97
.
18.
Kearfott
,
R. B.
, 1992, “
An Interval Branch and Bound Algorithm for Bound Constrained Optimization Problems
,”
J. Global Optim.
0925-5001,
2
, pp.
259
280
.
19.
Kearfott
,
R. B.
, 1996,
Rigorous Global Search: Continuous Problems
,
, Dordrecht.
20.
Ratschek
,
H.
, and
Rokne
,
J.
, 1988,
New Computer Methods for Global Optimization
,
Wiley
, New York.
21.
Horowitz
,
I. M.
, 1973, “
Optimum Loop Transfer Function in Single-Loop Minimum-Phase Feedback Systems
,”
Int. J. Control
0020-7179,
18
, pp.
97
113
.
22.
Zolotas
,
A.
, and
Halikias
,
G.
, 1999, “
Optimal Design of PID Controllers Using the QFT Method
,”
IEE Proc.: Control Theory Appl.
1350-2379,
146
(
6
); pp.
585
589
.
23.
IMSL User Manual, 1987, Math. Library, FORTRAN Subroutine for Mathematical Applications, IMSL Problem-Solving Software Systems, V, 10.0, IMSL Inc., Houston, TX.
24.
Tharewal
,
S.
, 2005, “
Automated Synthesis of QFT Controllers and Prefilters Using Interval Global Optimization Techniques
,” PhD thesis, Indian Institute of Technology Bombay, Mumbai, India. Downloadable at http://www.sc.iitb.ac.in/theses/phd_theses/sachin_thesis.ziphttp://www.sc.iitb.ac.in/theses/phd_theses/sachin_thesis.zip
You do not currently have access to this content.