Products of row stochastic matrices and their applications to cooperative control for autonomous mobile robots

In this paper, cooperative control of dynamic systems is formulated as the problem of choosing a linear feedback control law of the systems´ outputs and making the states of individual systems converge to the same steady state. As such, cooperative behavior of the overall system can be studied by investigating the convergence property of products of row stochastic matrices. Two new results on the convergence of matrix products are obtained, one on products of lower-triangular matrices and the other on products of lower-triangular matrices and general matrices. Neither of the two results requires that matrices be irreducible, and they can he used as the tools for the design and stability analysis of cooperative control. In particular, less-restrictive conditions on the design of cooperative control feedback matrices are established for a general class of MIMO dynamic systems of finite but arbitrary relative degree. The proposed design doesn´t require either the directed robot sensor graph being irreducible or a fixed leader. An example is provided to illustrate the proposed design method and new results.