In this second part of the paper we exploit the framework of Partial difference Equations (PdEs) over graphs for analyzing the behavior of multi-agent systems equipped with potential field based control schemes. We consider agent dynamics affected by errors and model the collective dynamics through nonlinear PdEs. Hinging on the properties of the Laplacian operator on graph, discussed in Part I, we prove alignment and collision avoidance both in leaderless and leader-follower models.

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In this paper we consider the problem of computing sets of observable states for discrete-time, piecewise affine systems. When the maximal set of observable states is fulldimensional, we provide an algorithm for reconstructing it up to a zero measure set. The core of the method is a quantifier elimination procedure that, in view of basic results on piecewise linear algebra, can be performed via the projection of polytopes on subspaces. We also provide a necessary condition on the minimal length of the observability horizon in order to expect a full-dimensional set of observable states. Numerical experiments highlight that the new procedure is considerably faster than the one proposed in [1].

In this paper, the observability properties of automotive powertrains with backlash are analysed. We model the powertrain as a hybrid system in the piecewise affine form and use measurements of the torque and the angular speed of the engine for computing the maximal set of observable states. This set, that is usually non-convex and disconnected, captures in a precise way how the main variables and parameters of the driveline influence the possibility of estimating the shaft twist. Then, we show how to exploit the knowledge of observable states in order to build computationally efficient deadbeat observers for the reconstruction of the powertrain states.