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 is an element of R-nxn such that x(k + 1) = Wx(k), W1 = 1, 1(T)W = 1(T) and W is an element of s(E). Here, x(k) is an element of R-n is the value possessed by the agents at the kth time step, 1 is an element of R-n is an all-one vector and s(E) is the set of real matrices in R-nxn with zeros at the same positions specified by a network graph g(V, E), where V is the set of agents and 9 is the set of communication links between agents. The optimal W is such that the spectral radius rho(W - 11(T)/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 E 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 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 rho(W - 11(T)/n). Numerically, we show that the q-SNM method performs much better than the GS method when 9 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. (C) 2009 Elsevier Ltd. All rights reserved.