Global Stability Analysis of the Model of Series/Parallel Connected CSTRs with Flow Exchange Subject to Persistent Perturbation on the Input Concentration

Abstract

In this paper, we study the convergence properties of a network model comprising three continuously stirred tank reactors (CSTRs) with the following features: (i) the first and second CSTRs are connected in series, whereas the second and third CSTRs are connected in parallel with flow exchange; (ii) the pollutant concentration in the inflow to the first CSTR is time varying but bounded; (iii) the states converge to a compact set instead of an equilibrium point, due to the time varying inflow concentration. The practical applicability of the arrangement of CSTRs is to provide a simpler model of pollution removal from wastewater treatment via constructed wetlands, generating a satisfactory description of experimental pollution values with a satisfactory transport dead time. We determine the bounds of the convergence regions, considering these features, and also: (i) we prove the asymptotic convergence of the states; (ii) we determine the effect of the presence of the side tank (third tank) on the transient value of all the system states, and we prove that it has no effect on the convergence regions; (iii) we determine the invariance of the convergence regions. The stability analysis is based on dead zone Lyapunov functions, and comprises: (i) definition of the dead zone quadratic form for each state, and determination of its properties; (ii) determination of the time derivatives of the quadratic forms and its properties. Finally, we illustrate the results obtained by simulation, showing the asymptotic convergence to the compact set.

Keywords

• global attractive set
• Lyapunov stability
• global asymptotic stability
• invariance
• diffuse flow modelling