A dynamical interpretation of strong observability and detectability concepts for nonlinear systems with unknown inputs: application to biochemical processes

Abstract

Determination of the observability/detectability properties of a nonlinear system is fundamental to assess the possibility of constructing observers and the properties that can be assigned to them, as e.g., the assignability of the convergence rate. For linear systems this task can be solved by well-known techniques, for the case without perturbations as much as for the perturbed case. However, for nonlinear systems this study is usually a very hard task, in particular, when unknown inputs and/or perturbations are present. In this paper a general method to study these properties will be described, and its capabilities and feasibility will be assessed by means of a few case studies related to the culture of phytoplankton in the chemostat.

Proof

In this Appendix we provide a proof of the convergence of system (28), that is required at several places in “Application to a model of Phytoplankton culture with light influence”. Equations (28) can be written as

$$\begin{array}{l} \dot{\varepsilon}_3= - u_1 \varepsilon_3 + \phi \varepsilon_4 \\ \dot{\varepsilon}_4= - \left( \phi + u_1 \right) \varepsilon_4 \end{array} $$

(36)

where

$$\phi = \frac{ \rho_{\hbox{m}} C K_{\hbox{S}}}{\left(S+K_{\hbox{S}} \right) \left( S - \varepsilon_4+K_{\hbox{S}} \right)}.$$

Since the parameters ρ_m,  K _S and the variables C,  S and x ₄ are positive, it follows that S + K _S > 0 and $S-\varepsilon_4+K_S=x_4+K_S>0,$ and therefore ϕ > 0. Because of the positivity of the input u ₁ ≥ 0 the equilibrium point $\varepsilon_4=0$ of the second equation of (36) is asymptotically stable, i.e., $\varepsilon_4(t) \rightarrow 0$ as $t \rightarrow \infty$.

Adding the two equations in (36) one obtains

$$\frac{\hbox{d}}{\hbox{dt}}\left(\varepsilon_3 + \varepsilon_4\right) = -u_1 \left( \varepsilon_3 + \varepsilon_4 \right).$$

Under the assumption that u ₁ > 0 the variable $\varepsilon_3+\varepsilon_4$ converges asymptotically to zero, and since $\varepsilon_4 \rightarrow 0,$ then $\varepsilon_3 \rightarrow 0$ as $t \rightarrow \infty$.