This paper addresses the issue of determining the most desirable “nominal closed-loop matrix” structure in linear state space systems, from stability robustness point of view, by combining the concepts of “quantitative robustness” and “qualitative robustness.” The qualitative robustness measure is based on the nature of interactions and interconnections of the system. The quantitative robustness is based on the nature of eigenvalue/eigenvector structure of the system. This type of analysis from both viewpoints sheds considerable insight on the desirable nominal system in engineering applications. Using these concepts, it is shown that three classes of quantitative matrices labeled “target sign stable (TSS) matrices,” “target pseudosymmetric (TPS) matrices,” and finally “quantitative ecological stable (QES) matrices” have features which qualify them as the most desirable nominal closed-loop system matrices. In this paper, we elaborate on the special features of these sets of matrices and justify why these classes of matrices are well suited to be the most desirable nominal closed-loop matrices in the linear state space framework. Establishment of this most desirable nominal closed-loop system matrix structure paves the way for designing controllers which qualify as robust controllers for linear systems with real parameter uncertainty. The proposed concepts are illustrated with many useful examples.

References

References
1.
Devarakonda
,
N.
, and
Yedavalli
,
R.
,
2011
, “
Qualitative Robustness and Its Role in Robust Control of Uncertain Engineering Systems
,”
ASME
Paper No. DSCC2011-6163.
2.
Yedavalli
,
R. K.
, and
Devarakonda
,
N.
,
2014
, “
Determination of Most Desirable Nominal Closed Loop State Space System Via Qualitative Ecological Principles
,”
ASME
Paper No. DSCC2014-6181.
3.
Dorato
,
P.
, and
Yedavalli
,
R. K.
,
1990
,
Recent Advances in Robust Control
,
IEEE Press
,
New York
.
4.
Yedavalli
,
R. K.
,
2014
,
Robust Control of Uncertain Dynamic Systems: A Linear State Space Approach
,
Springer
,
New York
.
5.
Yedavalli
,
R. K.
,
1993
, “
Flight Control Application of New Stability Robustness Bounds for Linear Uncertain Systems
,”
AIAA J. Guid., Control Dyn.
,
16
(
6
), pp.
1032
1037
.
6.
Patel
,
R. V.
, and
Toda
,
M.
,
1980
, “
Quantitative Measures of Robustness for Multivariable Systems
,”
Joint American Control Conference
.
7.
Edelstein-Keshet
,
L.
,
1988
,
Mathematical Models in Biology
,
McGraw-Hill
,
New York
.
8.
Hofbauer
,
J.
, and
Sigmund
,
K.
,
1988
, “
Growth Rates and Ecological Models: ABC on Ode
,”
The Theory of Evolutions and Dynamical Systems
,
Cambridge University Press
,
London
, pp.
29
59
.
9.
Hogben
,
L.
,
2007
,
Handbook of Linear Algebra
(Discrete Mathematics and Its Applications),
Chapman and Hall/CRC
,
New York
.
10.
Yedavalli
,
R. K.
, and
Devarakonda
,
N.
,
2009
, “
Qualitative Principles of Ecology and Their Implications in Quantitative Engineering Systems
,”
ASME
Paper No. DSCC2009-2621.
11.
Yedavalli
,
R. K.
, and
Devarakonda
,
N.
,
2010
, “
Sign Stability Concept of Ecology for Control Design With Aerospace Applications
,”
AIAA J. Guid. Control Dyn.
,
33
(
2
), pp.
333
346
.
12.
Devarakonda
,
N.
, and
Yedavalli
,
R. K.
,
2010
, “
Engineering Perspective of Ecological Sign Stability and Its Application in Control Design
,”
American Control Conference
, Baltimore, MD, June 30–July 2, pp.
5062
5067
.
13.
Yedavalli
,
R. K.
,
2006
, “
Qualitative Stability Concept From Ecology and Its Use in the Robust Control of Engineering Systems
,”
American Control Conference
, Minneapolis, MN, June 14–16, pp.
5097
5102
.
14.
Jeffries
,
C.
,
1974
, “
Qualitative Stability and Digraphs in Model Ecosystems
,”
Ecology
,
55
(
6
), pp.
1415
1419
.
15.
Quirk
,
J.
, and
Ruppert
,
R.
,
1965
, “
Qualitative Economics and the Stability of Equilibrium
,”
Rev. Econ. Studies
,
32
(
4
), pp.
311
326
.
16.
May
,
R.
,
1973
,
Stability and Complexity in Model Ecosystems
,
Princeton University Press
,
Princeton, NJ
.
17.
Allesina
,
S.
, and
Tang
,
S.
,
2012
, “
Stability Criteria for Complex Ecosystems
,”
Nature
,
483
(
7388
), pp.
205
208
.
18.
Allesina
,
S.
, and
Pascual
,
M.
,
2008
, “
Network Structure, Predator–Prey Modules, and Stability in Large Food Webs
,”
Theor. Ecol.
,
1
(
1
), pp.
55
64
.
19.
Brogan
,
W. L.
,
1974
,
Modern Control Theory
,
Prentice Hall
,
Englewood Cliffs, NJ
.
20.
Devarakonda
,
N.
, and
Yedavalli
,
R. K.
,
2010
, “
A New Robust Control Design Method for Linear Systems With Norm Bounded Time Varying Real Parameter Uncertainty
,”
ASME
Paper No. DSCC2010-4195.
21.
Kaszkurewicz
,
E.
, and
Bhaya
,
A.
,
2000
,
Matrix Diagonal Stability in Systems and Computation
,
Birkhauser
,
New York
.
22.
Franklin
,
G.
,
Powell
,
J. D.
, and
Naeini
,
A. E.
,
2006
,
Feedback Control of Dynamic Systems
,
Prentice Hall
,
Upper Saddle River, NJ
.
You do not currently have access to this content.