Lyapunov stability of economically oriented NMPC for cyclic processes

Several applications in process industries, such as simulated moving bed (SMB) separation and pressure swing adsorption (PSA), exhibit cyclic steady state behavior. Moreover, it is of economic interest to require energy intensive applications to take advantage of the periodically varying electricity price by changing the operating point frequently. Because traditional two-layer optimization methods are difficult to apply to these systems, we consider instead an economically oriented nonlinear model predictive control (NMPC) that directly considers system’s economic performance subject to the dynamic model. On the other hand, the commonly used Lyapunov framework to analyze the stability for the economically oriented NMPC cannot be applied directly. This work proposes two economically oriented NMPC formulations and proves nominal stability for both. We introduce transformed systems by subtracting the optimal cyclic steady state from the original system, for which the Lyapunov function can easily be established. Moreover, we show that the asymptotical stability of the transformed system is equivalent to that of the original system. Hence, the original systems are also nominally stable at the cyclic optimal solution. Finally, an industrial size air separation unit case study with periodic electricity cost is presented.