Efficient Computation of a
Guaranteed Lower Bound on the Robust Stability Margin of Uncertain
Systems
Efficient Computation of a Guaranteed Lower Bound on the Robust
Stability Margin of Uncertain Systems
V. Balakrishnan and F. Wang
In IEEE Trans. Aut. Contr., AC-44(11):2185-2190, November 1999
Abstract:
Sufficient conditions for the robust stability of
uncertain systems, with several different assumptions on
the structure and nature of the uncertainties, can be
derived in a unified framework with integral quadratic
constraints. These sufficient conditions, in turn, can be
used to derive lower bounds on the robust stability margin
for such systems. The lower bound is typically computed
with a bisection scheme, with each iteration requiring the
solution of a linear matrix inequality feasibility
problem. We show how this bisection can be avoided
altogether by reformulating the lower bound computation
problem as a single generalized eigenvalue minimization
problem, which can be solved very efficiently using
standard algorithms. We illustrate this with several
important, commonly-encountered special cases: Diagonal,
nonlinear uncertainties; diagonal, memoryless,
time-invariant sector-bounded (``Popov'') uncertainties;
structured dynamic uncertainties; and structured
parametric uncertainties. We also present a numerical
example that illustrates our approach.
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