## Hybrid Systems Modeling

Hybrid system is a class of systems which incorporates both discrete dynamics and continuous dynamics. In a hybrid system, the discrete dynamics and continuous dynamics are intertwined with each other. Normally, the continuous evolution influences the discrete evolution, by the definition of guard conditions. On the other hand, the continuous evolution is ruled by different discrete states. Compared with conventional models, the hybrid system is more general and more complex. This model can be applied to model the systems with switching control law, nonlinear dynamics, time-varying parameters, etc.

There are many different models for hybrid systems. Some models treat the transition between different discrete states as the transition of a Markov chain. Some models use deterministic guard conditions to describe the transition of discrete states. Our research focus on a special class of the stochastic hybrid system model, which is called the Stochastic Linear Hybrid System (SLHS). In the SLHS, the continuous state evolution is described by a set of stochastic linear differential or difference equations. To incorporate the uncertainties of the discrete state transitions into the guard condition, we consider a vector space which is spanned by the continuous state vector space and a random variable (vector) space. The mode transitions are described by a set of stochastic guards which are a set of disjoint polyhedral partitions of a vector space. We have shown the mathematical consistency of the model. Furthermore, the special structure of the SLHS model facilitates implementation of a computationally efficient hybrid estimation algorithm.

To illustrate the SLHS model, we consider the following aircraft tracking problem in Air Traffic Control (ATC). Figure 1 shows the trajectory of an aircraft flying along a Standard Instrument Departure (SID) route in Air Traffic Control. The flight route consists of two straight legs intersecting at a waypoint Wp. The aircraft dynamics is modeled as a stochastic hybrid system consisting of two flight modes: a Constant Velocity (CV) mode and a Coordinated Turn (CT) mode. Along the standard flight route, the aircraft would start to turn at the point FCP10. The corresponding mode transition from CV to CT can be modelled by a guard ds1 <= d1* as illsutrated in Figure 2. In other hybrid system models, it is assumed that the guard is deterministic, i.e. aircraft trajectories do not deviate from the SID route. In practice, due to aircraft's navigation uncertainties and unknown pilot's intents, the aircraft trajectories could deviate from the nominal one as illustrated in Figure 3. To model these uncertainties, we consider the parameter d1* as a random variable. Thus, we can use the SLHS to model this system. In the SLHS, its stochastic guard conditions include random variables to describe the uncertainties in the transition of the discrete states.

In the SLHS model, its guard conditions are in the form of linear inequalities. To expand the application of the SLHS model, we propos a new model of stochastic hybrid system whose guard condition is the quadratic form of the continuous state and the random variable. This model can be applied to the cases where linear inequalities are not enough to describe the transition accurately. Also, we derive a computationally efficient estimation algorithm for the stochastic system with quadratic guard conditions. We also investigate the SLHS model with unknown fault input other than noise disturbance in the continuous state evolution. In real applications, the unknown fault input can be regarded as the fault in actuators or fault in system dynamics. This model can be applied to Fault Detection and Isolation. For the hybrid system model, our future research will include more general case of stochastic hybrid systems. We will explore more complex continuous dynamics including nonlinear, non-Gaussian case, and more complex discrete dynamics including automaton and nonlinear guard conditions.

Figure 1: Planned Trajectory of an aircraft along an ATC flight route

Figure 2: Hybrid system model of aircraft trajectory in Figure 1

Figure 3. Typical trajectory which deviates from the planned flight path

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