Spectral radius minimization for optimal average consensus and output feedback stabilization

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∈Rn×n such that x(k+1)=Wx(k)x(k+1)=Wx(k), W1=1, 1TW=1T and W∈S(E)W∈S(E). Here, x(k)∈Rn is the value possessed by the agents at the kkth time step, 1∈Rn is an all-one vector and S(E)S(E) is the set of real matrices in Rn×n with zeros at the same positions specified by a network graph G(V,E)G(V,E), where VV is the set of agents and EE is the set of communication links between agents. The optimal WW is such that the spectral radius ρ(W−11T/n) is minimized. To this end, we consider two numerical solution schemes: one using the qqth-order spectral norm (2-norm) minimization (qq-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 EE is symmetric, i.e. no information flow from the iith to the jjth agent implies no information flow from the jjth to the iith agent, the solution Ws(1) from the 1-SNM method can be chosen to be symmetric and Ws(1) is a local minimum of the function ρ(W−11T/n). Numerically, we show that the qq-SNM method performs much better than the GS method when EE 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)(A,B,C), find a stabilizing control gain KK such that all the real parts of the eigenvalues of A+BKCA+BKC are strictly negative. In spite of its computational complexity, we show numerically that qq-SNM successfully yields stabilizing controllers for several benchmark problems with little effort.