Spectral radius minimization for optimal average consensus and output feedback stabilization
journal contributionposted on 03.11.2009, 16:30 by Yoonsoo Kim, Da-Wei Gu, Ian Postlethwaite
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 ε R n x n such that x(k+1)=Wx(k), W1 = 1 , 1TW 1T and W ε S (Ε). Here x(k)ε Rn is the value possessed by the agents at the kth time step, 1 ε Rn is an all-one vector and S (Ε) is the set of real matrices in R n x n with zeros at the same positions specified by a network graph G(ν, Ε) where ν is the set of agents and Ε is the set of communication links between agents. The optimal W is such that the spectral radius ρ(W - 11T/n) is minimized. To this end, we consider two numerical solution schemes: one using the qth-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 ε is symmetric, i.e. no information flow from the ith to the jth agent implies no information flow from the jth to the ith 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 q-SNM method performs much better than the GS method when ε 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+BKC 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.