
92 Feedback Control in Systems Biology
for some domain D ⊂ R : x
e
∈ D. To find a matrix P w hich satisfies these
conditions, we first apply the asymptotic stability conditions to the linearised
system
δ ˙x = A δx, where δx = x − x
e
,
to get
V (δx) = δx
T
P δx > 0 (3.30a)
˙
V (δx) = δx
T
P δ ˙x + δ ˙xP δx
= δx
T
A
T
P δx + δx
T
P Aδx < 0 (3.30b)
Condition (3.30a) can be easily impos ed by requiring the matrix P to be
positive definite, that is
P > 0, (3.31a )
whereas condition (3.30b) is satisfied by any matrix P that satisfies the linear
matrix inequality
A
T
P + P A < 0. (3.31b)
The solution to the above set of L MIs can be calculated using standard opti-
misation packages (e.g. ...