Principles for Hybrid Models of Physical Systems

The examples illustrate a number of characteristics specific to hybrid system models:

• When mode changes occur the system state vector has to be transferred correctly. This is complicated by discontinuous changes in state variables, which is governed by conservation of state [4], and the possibility that even the state vector itself may change.
• When a mode change occurs, a number of consecutive instantaneous changes may follow. This requires the final mode to be derived correctly and the state vector has to be computed across a number of intermediate modes governed by invariance of state.

To ensure physical consistency and prevent ill defined models, hybrid dynamic systems have to obey the following requirements:

• When a sequence of mode changes starts, it should not end up in a loop of instantaneous change, as this would prevent real time from progressing. This called the principle of divergence of time.
• Discontinuous changes in state variables may require computation of limit values in time. If limit values at switching time, , require future knowledge of system behavior, , the model is acausal, and, consequently, ill defined. This is called the principle of temporal evolution of state.
• The interval-point paradigm requires that the entire time-line is covered during the domain of operation. This implies that two types of jumps are possible when a discontinuity at occurs:
  1. 

     

     : moves from an interval, flow , to a point, pinnacle or flow . In this case, is transported to and is taken as .

  2. 

     

     : moves from a point, pinnacle or boundary point , to a flow, . In this case, is transported to and is taken as .

  A

  

  transition results in the activation of a flow, and the system evolves continuously before a new sequence of switches is initiated.

In this paper we address these issues in a formal mathematical framework derived from our systematic hybrid bond graph modeling approach [5].