Biomedical Engineering Reference
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Tabl e 2. 1 Example piecewise affine model and parameter inequalities
x c = κ c + κ c s ( x f f ) s + ( x c c ) s + ( x y y ) s + ( u, θ u )+ κ c s ( x f f ) − γ c x c
x y = κ y + κ y [1 − s + ( x c c ) s + ( x y y ) s + ( u, θ u )] − γ y x y
x f = κ f [1 − s + ( x c c ) s + ( x y y ) s + ( u, θ u )] s ( x f f )
+ κ f s + ( x g g ) s ( x t t ) s ( x f f ) × [1 − s + ( x c c ) s + ( x y y ) s + ( u, θ u )] − γ f x f
x g = κ g [1 − s + ( x g g ) s ( x t t )] s ( x f f ) − γ g x g
x t = κ t s + ( x g g ) s ( x t t ) s + ( x f f ) − γ t x t
x r = κ r s + ( x f f )+ κ r − γ r x r
0 c < κ c
γ c < κ c + κ c
c c < κ c + κ c
γ c
γ c
κ y
γ y y y <
κ y + κ y
γ y
0 y <
κ f
γ f f f f f <
κ f + κ f
γ f
0 f <
κ g
γ g
0 t t < κ t
γ t
0 g g <
Fig. 2.8 Asymptotic behavior of the PWA in the ( x f ,x g ) plane, for the case u =0 . Thick black
lines indicate sliding modes [ 24 ]
3. Damped oscillations around the point x g = θ g and x f = θ f . It is shown that all
trajectories will asymptotically converge to this point, which is an equilibrium in
the sense of Filippov;
4. x r ( t )
κ r + κ r
γ r
following the solution x f .
There are also sliding modes along the segments: x g = θ g with x f f and
x g g with x f = θ f (Fig. 2.8 ).
The PWA formalism allowed a more rigorous analysis of the complex network of
carbon starvation response in Escherichia Coli . Major participants were identified as
well as their roles in the presence or absence of nutritional stress. This PWA network
 
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