Spectral radius minimization for optimal average consensus and output feedback stabilization

Abstract

In this paper, we consider two problems which can be posed as spectral radius minimization problems. Firstly, we consider the fastest average agreement problem on multi-agent networks adopting a linear information exchange protocol. Mathematically, this problem can be cast as finding an optimal W ∈ R n × n such that x ( k + 1 ) = W x ( k ) , W 1 = 1 , 1 T W = 1 T and W ∈ S ( E ) . Here, x ( k ) ∈ R n is the value possessed by the agents at the k th time step, 1 ∈ R n is an all-one vector and S ( E ) is the set of real matrices in R n × n with zeros at the same positions specified by a network graph G ( V , E ) , where V is the set of agents and E is the set of communication links between agents. The optimal W is such that the spectral radius ρ ( W − 1 1 T / n ) is minimized. To this end, we consider two numerical solution schemes: one using the q th-order spectral norm (2-norm) minimization ( q -SNM) and the other gradient sampling (GS), inspired by the methods proposed in [Burke, J., Lewis, A., & Overton, M. (2002). Two numerical methods for optimizing matrix stability. Linear Algebra and its Applications, 351–352, 117–145; Xiao, L., & Boyd, S. (2004). Fast linear iterations for distributed averaging. Systems & Control Letters, 53(1), 65–78]. In this context, we theoretically show that when E is symmetric, i.e. no information flow from the i th to the j th agent implies no information flow from the j th to the i th agent, the solution W s ( 1 ) from the 1-SNM method can be chosen to be symmetric and W s ( 1 ) is a local minimum of the function ρ ( W − 1 1 T / n ) . Numerically, we show that the q -SNM method performs much better than the GS method when E is not symmetric. Secondly, we consider the famous static output feedback stabilization problem, which is considered to be a hard problem (some think NP-hard): for a given linear system ( A , B , C ) , find a stabilizing control gain K such that all the real parts of the eigenvalues of A + B K C are strictly negative. In spite of its computational complexity, we show numerically that q -SNM successfully yields stabilizing controllers for several benchmark problems with little effort.