Inverse optimality of cooperative control for networked systems

In this paper, inverse optimal control design is applied to quantify performance of linear cooperative systems. It is known that a cooperative control consists of an individual negative feedback and a network positive feedback and that, if the sensing/communication matrix sequence is sequentially complete (or the graph has a globally reachable node or a spanning tree), the corresponding network system is cooperatively asymptotically stable (or convergent to a consensus). It is shown that, for any given topology, the individual negative feedback control part by itself is inversely optimal and so are the pair of individual negative feedback and network positive feedback parts. It is also established that a cooperative system of varying topologies is inversely optimal with respect to a quadratic performance index. Since the optimal performance index depends upon future changes of network topology, an online algorithm is also proposed to estimate the cost incurred up to any instant of time. These results are useful to quantify performance of a cooperative networked system, and they could also be used to improve the choices of weights in the nonnegative row-stochastic network feedback matrix.