A Distributed Static Time-varying  Feedback for Formation Tracking Control of 2nd-order Nonholonomic Systems with Cooperative-Competitive Interactions

Abstract Formation tracking task is generally achieved by designing dynamic controllers, which yield additional inertial effects and may increase the difficulty of algorithm implementation. This paper presents a new static feedback to solve the formation tracking control problem of 2nd-order nonholonomic systems with cooperative-competitive interactions. Firstly, a time-varying position output is constructed to obtain double-integrator dynamics. Compared with the traditional hand position output, the constructed output makes it possible to achieve both small control inputs and small ultimate bound, and hence renders the resulting feedback more beneficial to practical applications. Then, the feedback is designed as the proportional derivative type with a correcting term, which overcomes the difficulty of controlling higher relative-degree system, and removes the dependence of control coefficients on spectral feature of interaction network. Rather than using external variables, the static feedback involves the special correcting term to eliminate the residual error caused by dynamic leader who is unavailable to some systems. Global uniform ultimate boundedness of the closed-loop double-integrators is strictly proved by applying Lyapunov method, graph theory and cascaded system theory. After this, a technical lemma about nonlinear system is developed to clarify the convergence of orientation and angular velocity under some persistently exciting condition on leader. Finally, simulations for six 2nd-order nonholonomic systems are carried out to demonstrate the formation tracking performance of the proposed static feedback controller.