Saltation Matrices: The Essential Tool for Linearizing Hybrid Dynamical Systems

I Figure 1: An example 2 mode hybrid system where the domains are shown in black circles D, the dynamics are shown with gray arrows F, the guard for the current domain is shown in red dashed g, and the reset from the current mode to the next mode is shown in blue R. The saltation matrix relies on differentiating the guards B. Saltation matrix derivation and resets so they must be differentiable. Excluding Zeno In this section, the derivation of the saltation matrix (2) is conditions ensures we avoid computing infinite saltation matrices presented, following the geometric derivation from [10] with in finite time, which would clearly be unsound for the addition of reset maps. There are many alternate ways analysis. Transversality ensures that neighboring trajectories to derive (2): a derivation using the chain rule is included in impact the same guard unless the impact point lies on any Appendix A and a derivation using a double limit can be found other guard surface, in which case the Bouligand derivative in [96]. is the appropriate analysis tool [52, 114-117]. Transversality Suppose the nominal trajectory of interest is x(t) as shown also ensures the denominator in (2) does not approach zero. in Figure 1. The trajectory starts in mode I and goes through a In some cases, the saltation matrix for a hybrid transition hybrid transition to mode J at time t. The saltation matrix is a can become an identity transformation.