259079 Complementarity Formulations for the Nonlinear Model Predictive Control of Non-Smooth Systems
When addressing the optimal control of systems governed by differential equations, an implicit assumption is that the differential equations that describe the evolution of these systems are continuous, and differentiable. However, dynamic process models are frequently subject to switches and other non-smooth elements. Examples include relief valves, check valves, phase changes, flow reversals, piecewise smooth correlations and so on. Such 'switching systems' fall under the umbrella of hybrid dynamic systems, with both continuous dynamics and discrete events embedded in them. The discrete events are autonomous to the system, and arise when the state trajectory intersects a switching surface (event trigger), causing the underlying differential equations to change (mode switch).
The focus of this work is on computing both open-loop and closed-loop optimal control solutions for such switching systems. We focus on the simultaneous method to transcribe the optimal control problem into an NLP, through a full discretization of states and controls. We represent the switching events through complementarity conditions, which retain the NLP nature of the optimal control problem without introducing binary variables. We modify the simultaneous method by introducing the switching time as a decision variable. This allows us to locate the switching event accurately at a finite element boundary resulting in an accurate computation of both the states and gradients of the optimal control problem. We also extend the proposed method to a Nonlinear Model Predictive Control framework, which involves a closed-loop solution of the above dynamic optimization problem. Several examples are presented to highlight the efficacy and computational performance of the suggested approach.
 Baumrucker, B. T. and L.T. Biegler, “MPEC Strategies for Optimization of Hybrid Dynamic Systems,” Journal of Process Control, 19, 1248-1256 (2009)