Under practical environments, some certain noises may arise from the communication of agents due to electromagnetic interference, sensor error, and other disturbances, which are usually modeled by additive noise in the relative position state between an agent and its neighbors. In this article, a formation control strategy is considered for this kind of relative position state with the additive noise, where the unmeasured velocity information is estimated by an observer-based strategy. On the other hand, the considered problem is transformed into a convergence problem of infinite products of a sequence of general stochastic matrices. The general stochastic matrix means that a matrix has row sum one but its entries do not necessarily require non-negative. This article develops the convex analysis to cope with such infinite products. Then, a sufficient condition is obtained to ensure that the specified formation shape in a noisy environment can be achieved. Subsequently, an example is presented to show the effectiveness of the result.
- 0102 Applied Mathematics