In this paper we relate the stability radius that can be achieved for the closed-loop matrix $(A−BK)$ to the distance to unstabilizability of the pair $(A,B)$. In the paper we show that the closed-loop matrix $(A−BK)$ can achieve a stability radius of $γ$ with a real feedback matrix $K$ only if the distance to unstabilizability of $(A,B)$ is greater than $γ$. Thus the distance to the unstabilizability of $(A,B)$ provides an upper bound on the maximum stability radius that can be achieved by state feedback.

1.
Van Loan
,
C.
, 1985, “
How Near is a Stable Matrix to an Unstable Matrix?
,”
Contemp. Math.
0271-4132,
47
, pp.
465
477
.
2.
Hinrichsen
,
D.
, and
Pritchard
,
A. J.
, 1986, “
,”
Syst. Control Lett.
0167-6911,
7
, pp.
1
10
.
3.
Hinrichsen
,
D.
, and
Pritchard
,
A. J.
, 1986, “
Stability Radius for Structured Perturbations and the Algebraic Riccati Equation
,”
Syst. Control Lett.
0167-6911,
8
,
105
113
.
4.
Paige
,
C. C.
, 1981, “
Properties of Numerical Algorithms Related to Computing Controllability
,”
IEEE Trans. Electromagn. Compat.
0018-9375,
26
, pp.
130
138
.
5.
Eising
,
R.
, 1984, “
Between Controllable and Uncontrollable
,”
Syst. Control Lett.
0167-6911,
4
, pp.
263
264
.
6.
Boley
,
D. L.
, and
Lu
,
W. S.
, 1986, “
Measuring How Far a Controllable System is From an Uncontrollable One
,”
IEEE Trans. Autom. Control
0018-9286,
AC-31
, pp.
249
252
.
7.
Byers
,
R.
, 1988, “
A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices
,”
SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.
0196-5204,
9
, pp.
875
881
.
8.
Kenney
,
C.
, and
Laub
,
A. J.
, 1988, “
Controllability and Stability Radii For Companion Form Systems
,”
Math. Control, Signals, Syst.
0932-4194,
1
, pp.
239
256
.
9.
Calafiore
,
G.
, and
El Ghaoui
,
L.
, 2004, “
Ellipsoidal Bounds For Uncertain Linear Equations and Dynamical Systems
,”
Automatica
0005-1098,
40
, pp.
773
787
.
10.
Zhou
,
K.
, and
Doyle
,
J. C.
, 1997, “
Essentials of Robust Control
,” ISBN: 0135258332, Pearson Education.
11.
Doyle
,
J. C.
,
Glover
,
K.
,
Khargonekar
,
P.
, and
Francis
,
B. A.
, 1989, “
State-Space Solutions to Standard H2 and H∞ Control Problems
,”
IEEE Trans. Autom. Control
0018-9286,
34
, pp.
831
847
.
12.
Peterson
,
I. H.
, and
Hollot
,
C. V.
, 1986, “
A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems
,”
Automatica
0005-1098,
22
, pp.
397
411
.
13.
Boyd
,
S.
,
Ghaoui
,
L. E.
,
Feron
,
E.
, and
Balakrishnan
,
V.
, 1994, “
Linear Matrix Inequalities in System and Control Theory
,”
SIAM Studies in Applied Mathematics
, Vol.
15
.
14.
Byers
,
R.
, 1990, “
Detecting Nearly Uncontrollable Pairs
,”
Numerical Methods, Proceedings of the International Symposium
, MTNS-89, Amsterdam, The Netherlands,
M. A.
Kaashoek
,
J. H.
Van Schuppen
, and
A. C. M.
Ran
, eds., Vol.
III
, pp.
447
457
.