Stability Conditions for Adaptive Inventory Control Systems

The main contribution of this work is three-fold. First we integrate an on-line adaptation with the inventory control framework using ideas from passivity based control. Second we derive the stability condition for adaptive inventory control system by connecting thermodynamic stability theory and passivity theory. Third we develop a case study to highlight the application of the theory.  The bioreactor case study is characterized by input multiplicity. It is shown that the choice of input output pairing provides the key to achieve input output stability of the control system, convergence of estimated parameters and stability of the zero-dynamics.

The main advantage of the proposed approach is that the adaptive mechanism is incorporated into the passivity based inventory control scheme so that the unknown parameters can be estimated. Passive input-output pairs are derived from mass balance of each chemical component in the bioreactor and unknown model parameters are adapted online, so that the effect of unknown process dynamics can be compensated for. Simulation and stability results show that the zero-dynamics are stable and that the online estimator guarantees stability of the closed-loop system. Input-output pair selection has a significant impact on the inventory control system. If the input-output is chosen then the stability of the zero-dynamics is guaranteed.

The global stability analysis is achieved using the composite energy function consisting of controlled dynamics, zero-dynamics and estimated dynamics. A composite Lyapunov function based on nonlinear control theory is used to represent the energy of the controlled dynamics and estimated dynamics. An entropy based function, so called the availability in classic thermodynamics, describes the energy of the zero-dynamics. The global stability analysis provides a theoretical proof to the different behaviors of bioreactor under adaptive-inventory controller due to various input-output pairs.