Design and Analysis of Cooperative Control Problems in Lyapunov framework


Recently there has been considerable interest in multiagent coordination or cooperative control [41]. This has led to the emergence of several interesting control problems. One such problem is the rendezvous problem. In a rendezvous problem, one desires to have several agents arrive at predefined destination points simultaneously. Cooperative strike and cooperative jamming are two examples of the rendezvous problem. In the first scenario, multiple strikes are executed within a time interval, from different agents firing from different distances and traveling at different speeds. In the second scenario, one or more agents need to start jamming slightly before the strike vehicle enters the danger zone and sustain jamming until strike vehicle exits. In both the scenarios, it is imperative that all the agents act simultaneously else the objective is not fulfilled. The idea of rendezvous extends beyond just convergence to a static set of destination points or the origin. 

Rendezvous can also entail formation flying or interception problems where the origin is effectively moving. Interception of incoming ballistic missiles is a rendezvous problem where the origin becomes a moving target and one of the agents is noncooperating. Formation flying is a type of rendezvous problem where multiple agents must coordinate position and velocity. The docking of two spacecraft is a rendezvous problem that involves the two spacecraft matching both position and velocity with the proper orientation. Airtoair refueling is another rendezvous problem. Additional applications arise in submersibles where robotic vehicles must converge upon a set location, either moving or stationary. 

fig 7. Cone as an attractor. 

In the dynamical systems literature, the problem of cooperation and competition has been addressed in the context of cone invariance. The cone is used to define a partial order on the system trajectories, which results in the cooperative or competitive behavior of the system. In the seminal work by Hirsch [56, 57, 58, 59, 60, 61] on systems of differential equations that are competitive or cooperative, he developed what is known as monotone dynamical systems theory [62]. He demonstrated that the generic solution of a cooperative and irreducible system of differential equations converges to a set of equilibria. Furthermore, the flow on a compact limit set of an ndimensional cooperative or competitive system of differential equations is shown to be topologically conjugate to the flow of an n − 1 dimensional system of differential equations, restricted to a compact invariant set. We analyze the problem of rendezvous in this framework and demonstrate that rendezvous of multiple agents is equivalent to the invariance of an appropriately defined cone. Figure 7 illustrates cone invariant trajectories for a three agent system. 

We analyze the rendezvous problem in the framework of cooperative and competitive dynamical systems that has had some remarkable applications to biological sciences. Cooperative and competitive dynamical systems are shown to generate monotone flows by the classical MullerKamke theorem [62], which are analyzed using set invariance theory. In our current work, we have established equivalence between the rendezvous problem and invariance of an appropriately defined cone and focus on mechanisms to generate feedback between the vehicles, a key part of the rendezvous problem. The problem of rendezvous is cast as a stabilization problem, with a set of constraints on the trajectories of the agents defined on the phase plane. The nagent rendezvous problem is formulated as an ellipsoidal cone invariance problem in the ndimensional phase space. Theoretical results based on set invariance theory and monotone dynamical systems have been developed. The necessary and sufficient conditions for rendezvous of linear systems are presented in form of linear matrix inequalities. These conditions are also interpreted in the Lyapunov framework using multiple Lyapunov functions, which provides certificates for stability and synchronization of multiple agents. 


Figure 8(a) illustrates rendezvous of three agents, modeled as Dubin’s car, in (x,y) plane. Figure 8(b), shows robustness with respect to uncertainty in the behaviour of one vehicle and the response of other vehicles. In fig. 8(b), agent one exhibits noncooperative dynamics from T=1 sec to T=3 sec. 
Observe the procrastination in the trajectories of agents two and three, as compared with their trajectories in fig.8(a). This framework for designing and analyzing cooperative control algorithms admits existing synthesis tools for linear and nonlinear control systems. Please see related publications for more details. 
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