Frequency response finds a wide range of applications in many engineering sectors. When variations and uncertainties exist during the operation and lifetime of an engineering system, the calculation of frequency response for an uncertain dynamic system is required in order to assess the worst cases in terms of various criteria, for example gain and phase margins in control engineering, or peak magnitude at different modes in the finite element analysis of structures. This paper describes an analytical approach toward the identification of critical interior lines that possibly contribute to the boundary of the frequency response. It is allowed that uncertain parameters perturb transfer function coefficients in a nonlinear form. Conditions for critical interior lines contributing to the boundary of the frequency response are presented. An invariant property of these critical lines under open-loop and closed-loop configurations augmented by control systems or compensators is established, which greatly simplifies the analysis, design, and verification process when using frequency domain techniques. A procedure for computing frequency response and identifying the worst cases is then developed based on the combination of symbolic and numerical computation.

References

References
1.
Barmish
,
B. R.
,
Lagoa
,
C. M.
, and
Tempo
,
R.
, 1997, “
Radially Truncated Uniform Distributions for Probabilistic Robustness of Control Systems
,”
Proc. of American Control Conference
pp.
853
857
.
2.
Bai
,
E. W.
,
Tempo
,
R.
, and
Fu
,
M.
, 1997, “
Worst-Case Properties of the Uniform Distribution and Randomized Algorithms for Robustness Analysis
,”
Proc. of American Control Conference
pp.
861
865
.
3.
Barmish
,
B. R.
, and
Lagoa
,
C. M.
, 1997, “
The Uniform Distribution: A Rigorous Justification for Its Use in Robustness Analysis
,”
Math. Control, Signals, Syst.
,
10
, pp.
203
222
.
4.
Bailey
,
F. N.
, and
Hui
,
C. H.
, 1989, “
A Fast Algorithm for Computing Parametric Rational Functions
,”
IEEE Trans. on Autom. Control
,
34
(
11
), pp.
1209
1212
.
5.
Fu
,
M.
, 1990, “
Computing the Frequency Response of Linear Systems With Parametric Perturbation
,”
Syst. Control Lett.
,
15
, pp.
45
52
.
6.
Bartlett
,
A. C.
, 1993, “
Computation of the Frequency Response of Systems With Uncertain Parameters: A Simplification
,”
Int. J. Control
,
57
, pp.
1293
1309
.
7.
Tesi
,
A.
, and
Vicino
,
A.
, 1991, “
Kharitonov Segments Suffice for Frequency Response Analysis of Interval Plant-Controller Families
,”
In Control of Uncertain Dynamic Systems
,
CRC Press
,
Littleton, MA
, pp.
403
415
.
8.
Keel
,
L. H.
, and
Bhattacharyya
,
S. P.
, 1991, “
Frequency Domain Design of Interval Controllers
,”
In Control of Uncertain Dynamic Systems
,
CRC Press
,
Littleton, MA
, pp.
423
438
.
9.
Keel
,
L. H.
, and
Bhattacharyya
,
S. P.
, 1994, “
Robust Parametric Classical Control Design
,”
IEEE Trans. Autom. Control
,
39
, pp.
1524
1530
.
10.
Bhattacharyya
,
S. P.
,
Chapellat
,
H.
, and
Keel
,
L. H.
, 1995,
Robust Control: The Parametric Approach
.
Prentice Hall Inc.
11.
Chen
,
W.-H.
, and
Ballance
,
D. J.
, 1999, “
A Comparison of Methods for Computing Frequency Response of Uncertain Plants
,”
In Proceedings of 14th IFAC World Congress
,
Beijing, China
, 1999, Vol.
G
, pp.
7
12
.
12.
Jia
,
Y.
, 2002, “
Computing the Frequency Response of Systems Affinely Depending on Uncertain Parameters
,”
IEE Proc. Control Theory Appl.
,
149
(
4
), pp.
311
315
.
13.
Chen
,
W.-H.
, and
Ballance
,
D. J.
, 1999, “
Plant Template Generation of Uncertain Plants in Quantitative Feedback Theory
,”
ASME J. Dyn. Syst., Meas., Control
,
121
(
3
), pp.
358
364
.
14.
Moens
,
D.
, and
Vandepitte
,
D.
, 2005, “
A Survey of Non-Probabilistic Uncertainty Treatment in Finite Element Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
194
(
14–16
), pp.
1527
1555
.
15.
Moens
,
D.
, and
Vandepitte
,
D.
, 2004, “
An Interval Finite Element Approach for the Calculation of Envelope Frequency Response Functions
,”
Int. J. Numer. Methods Eng.
,
61
(
14
), pp.
2480
2507
.
16.
Moens
,
D.
, and
Vandepitte
,
D.
, 2007, “
Interval Sensitivity Theory and Its Application to Frequency Response Envelope Analysis of Uncertain Structures
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
21–24
), pp.
2486
2496
.
17.
Oberkampfa
,
W.
,
Heltonb
,
J.
,
Joslync
,
C.
,
Wojtkiewiczd
,
S.
, and
Fersone
,
S.
, 2004, “
Challenge Problems: Uncertainty in System Response Given Uncertain Parameters
,”
Reliab. Eng. Syst. Saf.
,
85
(
1
), pp.
11
19
.
18.
Fu
,
M.
, 1989, “
Polytopes of Polynomials With Zeros in a Specified Region: New Criterion and Algorithms
,” In Robustness in Identification and Control,
M.
Milanese
,
R.
Tempo
, and
A.
Vicino
, eds.
19.
Chockalingam
,
G.
, and
Dasgupta
,
S.
, 1993, “
Minimality, Stabilizability, and Strong Stablizability of Uncertain Plants
,”
IEEE Trans. Autom. Control
,
38
, pp.
1651
1661
.
20.
Ackermann
,
J.
, 1992, “
Does it Suffice to Check a Subset of a Multilinear Parameters in Robust Analysis
,”
IEEE Trans. Autom. Control
,
37
(
4
), pp.
487
488
.
21.
Ackermann
,
J.
,
Hu
,
H.
, and
Kaesbauer
,
D.
, 1990, “
Robust analysis: A Case Study
,”
IEEE Trans. Autom. Control
,
35
(
3
), pp.
352
356
.
22.
East
,
D. J.
, 1982, “
On the Determination of Plant Variation Bounds for Optimum Loop Synthesis
,”
Int. J. Control
,
35
(
5
), pp.
891
908
.
23.
Ballance
,
D. J.
, 1992, “
Comments on the Papers “A New Approach to Optimum Loop Synthesis” and “On the Determination of Plant Variation Bounds for Optimum Loop Synthesis
,”
Int. J. Control
,
55
(
1
), pp.
241
248
.
24.
Borghesani
,
C.
,
Chait
,
Y.
, and
Yaniv
,
O.
, 1995,
Quantitative Feedback Theory Toolbox User Manual
,
The Math Work Inc.
.
25.
Mora-Camino
,
F.
, and
Chaibou
,
A. K.
, 1993, “
Design of Guaranteed Performance Controllers for Systems With Varying Parameters
,”
J. Guid. Control Dyn.
,
16
, pp.
1185
1187
.
You do not currently have access to this content.