Investigating Time-Delay Effects for Multivehicle Formation Control
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Advances incommunication, navigation, and computational systems have enabled greater autonomy in multivehicle systems. However, time-delay effects owing to measurement, actuation, communication, or operator delays are introduced as system complexity increases. In this paper, a straightforward method is presented to determine the maximum allowable time delays for a linear, n-dimensional system. Delay differential equations are one method to model systems with delay, and the theory presented here is based upon well-known stability results for a scalar, first-order delay differential equation. Modal-coordinate transformations are used to diagonalize, and thus, decouple closed-loop equations of motion to which the scalar, first-order delay-differential-equation results are applied in order to find optimal bounds on the time delay. Hence, stability bounds for a given closed-loop control form can be determined by solving an eigenvalue problem, which is a departure from other delay-differential-equation results that require solutions to linear matrix inequalities or a problem-specific Lyapunov function to prove stability. This theoretical development is applied to a formation-control problem with five vehicles. Control laws are developed for the vehicle formation using two different communication structures: leader-follower and bidirectional. Simulation results are used to demonstrate formation convergence, and robustness of the formations to time-delay effects are discussed.
