Structural identifiability

Source[2]

Consider a linear time-invariant system with the following state-space representation:

${\displaystyle {\begin{aligned}{\dot {x}}_{1}(t)&=-\theta _{1}x_{1},\\{\dot {x}}_{2}(t)&=\theta _{1}x_{1},\\y(t)&=\theta _{2}x_{2},\end{aligned}}}$

and with initial conditions given by ${\displaystyle x_{1}(0)=\theta _{3}}$ and ${\displaystyle x_{2}(0)=0}$. The solution of the output ${\displaystyle y}$ is

${\displaystyle y(t)=\theta _{2}\theta _{3}e^{-\theta _{1}t}\left(e^{\theta _{1}t}-1\right),}$

which implies that the parameters ${\displaystyle \theta _{2}}$ and ${\displaystyle \theta _{3}}$ are not structurally identifiable. For instance, the parameters ${\displaystyle \theta _{1}=1,\theta _{2}=1,\theta _{3}=1}$ generates the same output as the parameters ${\displaystyle \theta _{1}=1,\theta _{2}=2,\theta _{3}=0.5}$.

Source[3]

A model of a possible glucose homeostasis mechanism is given by the differential equations[6]

${\displaystyle {\begin{aligned}&{\dot {G}}=u(0)+u-(c+s_{\mathrm {i} }\,I)G,\\&{\dot {\beta }}=\beta \left({\frac {1.4583\cdot 10^{-5}}{1+\left({\frac {8.4}{G}}\right)^{1.7}}}-{\frac {1.7361\cdot 10^{-5}}{1+\left({\frac {G}{8.4}}\right)^{8.5}}}\right),\\&{\dot {I}}=p\,\beta \,{\frac {G^{2}}{\alpha ^{2}+G^{2}}}-\gamma \,I,\end{aligned}}}$

where (c, s_i, p, α, γ) are parameters of the system, and the states are the plasma glucose concentration G, the plasma insulin concentration I, and the beta-cell functional mass β. It is possible to show that the parameters p and s_i are not structurally identifiable: any numerical choice of parameters p and s_i that have the same product ps_i are indistinguishable.[3]