## SDTHE Algorithm

Based on the IMM approach, we develop a hybrid estimation called the **State-Dependent-Transition Hybrid Estimation (SDTHE)** algorithm. Because the SDTHE algorithm utilizes the special structure of the SLHS (namely, the linear Gaussian continuous state dynamics within each mode, and the mode transitions described by polytopic guards), it is more computationally efficient than other hybrid estimation algorithms, such as the particle filters. Despite the additional complexity due to the continuous-state-dependent mode transitions, the SDTHE algorithm has about the same computational costs as the IMM algorithm. Figure 1 compares the performance of the SDTHE and the performance of IMM in an aircraft tracking scenario.

Figure 1: Monte Carlo simulation results of estimation performance of STDHE algorithm

(Model 1 and Model 2) vs. IMM algorithm (Model 3).

Based on the basic algorithm of STDHE, we propose another algorithm which considers the stochastic hybrid system model whose **guard condition is in the quadratic form**. This estimation algorithm can be applied to many cases, because in real applications, the quadratic form is suitable to describe the switching of a hybrid system. Figure 2 shows an aircraft tracking scenario where the two aircraft change their flight modes to solve a potential conflict between them when the distance between them is less than a certain value. In such cases, linear inequalities, which is computationally easy, cannot be used to approximate the quadratic guards. Avoiding use the Monte Carlo method to compute the transition probability, we derive a computationally efficient hybrid algorithm based on an analytic method of cumulative density function (cdf) expansion. Figure 3 compares the estimation result of our algorithm and the IMM algorithm.

Figure 2: Aircraft tracking scenario: guard condition in the quadratic form

Figure 3: Performance of the proposed algorithm vs. the IMM algorithm

(Monte Carlo simulation with 100 simulation runs).

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