Optimized Control Gain Design For The Consensus Problem Of High-order Multi-agent Systems

The problem of multi-agent systems has attracted more and more attention in recent years. This paper investigates the problem of designing the consensus control gain of discrete-time multi-agent systems with process noise. Both state feedback consensus algorithm and output feedback consensus algorithm are considered in this paper.For state feedback systems, the performance is measured by the ultimate mean square deviation of the states of agents. The original control gain optimization problem takes a nonlinear matrix inequality form, which is difficult to handle. We propose an iterative method to solve the optimization problem. Each iteration starts from a given feasible control gain, which can guarantee the mean square bounded consensus of the system. By fixing the control gain, the original nonlinear problem becomes a linear matrix inequality (LMI) one and can be efficiently solved to yield some intermediate matrix variables. With the obtained matrix variables and the given feasible control gain, we introduce a perturbation method to approximate the original nonlinear problem with another LMI one. By solving that LMI problem, we generate a descent direction of the control gain. Moving the control gain along this descent direction, we can improve the system’s performance. By implementing a line search algorithm, we get a local optimal control gain. That achieved gain is of course feasible and can work as the starting point of the next iteration. For output feedback systems, an observer-based consensus algorithm is used and the performance is measured by the ultimate mean square deviation of the states of agents and the states of observers. The optimization problem also takes a nonlinear matrix inequality form. The iterative method we proposed for the state feedback system also works for the output feedback condition.Besides guaranteeing the mean square bounded consensus of multiple agents, our iterative method aims to optimize the performance of the system and can efficiently attenuate the effects of process noise. The validity of the iterative algorithm is confirmed by simulations.