Distributed continuous-time time-varying optimization for networked Lagrangian systems with quadratic cost functions

In this paper, the distributed time-varying optimization problem is investigated for networked Lagrangian systems with parametric uncertainties. Usually, in the literature, to address some distributed control problems for nonlinear systems, a networked virtual system is constructed, and a tracking algorithm is designed such that the agents’ physical states track the virtual states. It is worth pointing out that such an idea requires the exchange of the virtual states and hence necessitates communication among the group. In addition, due to the complexities of the Lagrangian dynamics and the distributed time-varying optimization problem, there exist significant challenges. This paper proposes distributed time-varying optimization algorithms that achieve zero optimum-tracking errors for the networked Lagrangian agents without the communication requirement. The main idea behind the proposed algorithms is to construct a reference system for each agent to generate a reference velocity using absolute and relative physical state measurements with no exchange of virtual states needed, and to design adaptive controllers for Lagrangian systems such that the physical states are able to track the reference velocities and hence the optimal trajectory. The algorithms introduce mutual feedback between the reference systems and the local controllers via physical states/measurements and are amenable to implementation via local onboard sensing in a communication unfriendly environment. Specifically, first, a base algorithm is proposed to solve the distributed time-varying optimization problem for networked Lagrangian systems under switching graphs. Then, based on the base algorithm, a continuous function is introduced to approximate the signum function, forming a continuous distributed optimization algorithm and hence removing the chattering. Such a continuous algorithm is convergent with bounded ultimate optimum-tracking errors, which are proportion to approximation errors. Finally, numerical simulations are provided to illustrate the validity of the proposed algorithms.