Continuous Optimization for Control of Finite-State Machines with Cascaded Hysteresis via Time-Freezing

Control problems with Finite-State Machines (FSM) are often solved using integer variables, leading to a mixed-integer optimal control problem (MIOCP). This paper proposes an alternative method to describe a subclass of FSMs using complementarity constraints and time-freezing. The FSM from this subclass is built up by a sequence of states where a transition between the states is triggered by a single switching function. This can be looked at as a cascade of hysteresis loops where a memory effect is used to maintain the active state of the state machine. Based on the reformulation for hybrid systems with a hysteresis loop [13], a method is developed to reformulate this subclass in a similar fashion. The approach transforms the original problem into a Piecewise Smooth System (PSS), which can be discretized using the recently developed Finite Elements with Switch Detection [15], allowing for high-accuracy solutions. The reformulation is compared to a mixed-integer formulation from the literature on a time-optimal control problem. This work is a first step towards the general reformulation of FSMs into nonsmooth systems without integer states.