Principles of Modern Radar - Volume 2 1891121537

688 CHAPTER 15 Multitarget, Multisensor TrackingAdaptive filters are fundamentally limited. The process noise is increased (to accommodatetrack maintenance in the presence of mis-matched target dynamics) at theexpense of track accuracy. These limitations drove the literature toward multiple modelapproaches, as depicted in the middle of Figure 15-8. These multiple model schemes runmultiple trackers in parallel; the goal is to provide a tracker for each likely mode of targetdynamics. At any given time, the mode yielding the smallest innovations is selected. Moreadvanced versions of this algorithm incorporate hysteresis, so the selection doesn’t bouncebetween modes on each sequential update.While multiple model schemes provide an improvement over adaptive filters (for manyapplications), they are not ideal. The biggest complaint lodged against these schemes isthat the overall filter output can contain sharp discontinuities over time as the selectedmode jumps from one tracker to another. Covariance consistency (over time) can alsosuffer in these schemes.Hence, interacting multiple model (IMM) estimators were eventually developed asa way of addressing the mis-match between target dynamics and the track filter [31].This idea (depicted on the right side of Figure 15-8) is similar to the multiple modeschemes, but rather than selecting the state and covariance from a single filter, the resultsof multiple filters are blended together. The weights, w i , for each track filter correspond tothe probability that each mode best represents the target dynamics. By incorporating theresults from all the parallel filters, the IMM estimator is able to provide more consistentcovariances and strike a better balance between track maintenance and track accuracy.Section 15.2.2.1 further discusses the IMM estimator; Section 15.2.2.2 briefly presents avariation on this scheme that addresses one of its key shortcomings.15.2.2.1 Interacting Multiple Model (IMM) EstimatorsFigure 15-9 gives a more detailed view of the IMM estimator [31]. First, previous statesand covariances from all contributing modes are mixed together, creating the mixed statesand covariances that will be propagated to the current time in the filters associated witheach mode. The mixed state and covariance for each mode, l, is accomplished viaandP m k−1|k−1 = N ∑l=1N xk−1|k−1 m = ∑˜x k−1|k−1 l μl,m (15.19)l=1[ ()() ] Tμ l,m ˜P k−1|k−1 l + ˜x k−1|k−1 l − x k−1|k−1m ˜x k−1|k−1 l − x k−1|k−1m(15.20)where ˜x k−1|k−1 l refers to the updated state of mode l at t k−1, projected into the state spaceof the minimum-order filter 1 , ˜P k−1|k−1 l refers to its covariance (also projected into the statespace of the minimum-order filter), and μ l,m is the probability of having been in mode l1 The modes in the VS-IMM may include states with different dimensions. For example, a constantvelocitymode may model 6-D states (including position and velocity in x, y, and z), while a constantaccelerationmode may model 9-D states (including position, velocity, and acceleration in x, y, and z).When mixing the states and covariances together, the larger-dimension states must be reduced to the sizeof the smaller-dimension ones. For example, acceleration is dropped from the 9-D state so that the statecan be added to the 6-D state. The projected states and covariances are denoted as ˜x and ˜P, respectively.