Control of multivehicle systems in the presence of uncertain dynamics (2013)

AbstractIn this paper, we present a cooperative control architecture for high-order multivehicle systems having non-identical nonlinear uncertain dynamics. The proposed methodology consists of a local cooperative controller and a vehicle-level controller for each vehicle. The former controller receives the relative output measurements of the neighbouring vehicles in order to solve a containment problem formulated on a leader–follower framework. Specifically, the leaders generate trajectories in which the vehicles (followers) converge to the convex hull formed by those of the leaders. For a special case with one leader, this controller synchronises the output of the vehicles with the output of the leader. The latter controller receives the internal-state measurements for suppressing the nonlinear uncertain dynamics of the vehicle by using a decentralised adaptive control approach. The interaction topology between vehicles is described by undirected graphs and extensions to directed graphs are further discussed. The stability and convergence properties of the proposed cooperative control architecture are analysed by using the results from linear algebra and the Lyapunov theory. Several numerical examples are provided to demonstrate the efficacy of the proposed cooperative control architecture.Keywords: multivehicle systemsleader–follower networkscooperative controladaptive controlcontainment AcknowledgementsThis research was supported in part by the National Institute of Standards and Technology under Grant 70NANB10H013.NotesThroughout the paper, the movement of vehicles will not cause edges to appear or disappear.Unless otherwise noted, we assume that the graph is connected and undirected.The leaders are indexed first and the followers are indexed last in the original graph without loss of generality.That is, for each follower, there exists at least one leader that has a directed path to that follower (Li et al., Citation2011).Because, as we see later, we require Re(λi(F(·))) > 0 and the property that each row of −F(·)−1G(·) has a sum equal to 1 for the analysis, and both conditions hold either for a connected, undirected graph , or for a directed graph containing a directed spanning tree.Since Ar is Hurwitz, it follows from converse Lyapunov theory (Haddad & Chellaboina, Citation2008) that there exists a unique P satisfying (Equation24(24) ) for a given R.The modification term in (Equation22(22) ) with given by (Equation25(25) ) was first used in Yucelen and Haddad Citation(2013) for centralized adaptive control.This can be justified in the context of gradient minimization; that is, this modification minimizes the distance between and (Yucelen & Haddad, Citation2013).Since full state information exchange is not practical, for example, in most aircraft applications, here we require relative output information exchange between neighboring vehicles.To see this, consider the cost function . Then, its negative gradient with respect to θi yields . Similar to Yucelen Citation(2012) and Calise and Yucelen Citation(2012), adding this term to (Equation30(30) ) minimizes the distance between θi(t) and νi(t).The convergence properties of the overall multivehicle system subject to the proposed vehicle-level controller and local cooperative controller is discussed later.This is true as a direct consequence of Schur transformation.It follows from (Equation46(46) ) that , for all , and is radially unbounded.It follows from (Equation46(46) ), (Equation48(48) ), and (Equation50(50) ) that , for all , and is radially unbounded.Note that and can be satisfied by design and without loss of generality. Furthermore, since it is assumed that the state vector xi(t), i = 1, … , NF, is accessible, xi0 = xri0 is realizable (Han & Narendra, Citation2012).As a direct consequence of (Equation73(73) ) and (Equation78(78) ), one can further show that as t → ∞.