Distributed Stability Tests for Large-Scale Systems With Limited Model Information

Abstract

Privacy concerns spark the desire to analyze largescale interconnected systems in a distributed fashion, i.e. without a central entity having global model knowledge. Two different approaches are presented to analyze stability of interconnected linear time-invariant systems with limited model knowledge. The two algorithms implement sufficient stability conditions and require information exchange only with direct neighbors thus reducing the need to share model data widely and ensuring privacy. The first algorithm is based on an M-matrix condition, the second one on Lyapunov inequalities. Both algorithms rely on distributed optimization using a dual decomposition approach. Numerical investigations are used to validate both approaches.

Distributed control design with local model information and guaranteed stability

“…Note that Algorithm 2 only leads to an approximate solution because it is based on dual decomposition. However, the convergence of the above Algorithm follows with Theorem 3 in [Deroo et al 2014] where it is shown that the number of nec-essary iterations to achieve a desired accuracy ε of the approximate solution of problem (21) can be computed in advance. Remark 3.…”

“…Because Algorithm 1 runs using only information from neighbors it would be undesirable if more than neighborhood information would be necessary to determine a solution of (11). Hence, in this subsection we apply a method developed in [Deroo et al 2014] to find (P, δ) using only neighborhood information.…”

“…It can be shown that (11) has a feasible solution if and only if the optimal objective function value of problem (18) is negative (which then also implies δ > 0). For details, we refer to [Deroo et al 2014].…”

“…While some of the subproblems in Algorithm 2 may look challenging at first glance, there are actually small-scale closed form solutions for all of them. For instance, the solution for Y s,k+1 in step 1 is determined as follows [Deroo et al 2014]: Consider the spectral decomposition of the symmetric matrix X s Y given by…”

“…The main challenge involves the determination of a suitable terminal cost term that ensures stability. While this is usually done in a centralized fashion, we present a method to determine it in a distributed fashion by distributedly solving a structured LMI condition [Deroo et al 2014]. Afterwards, the gradient descent method from [Deroo et al 2012] is used to determine the actual feedback matrix.…”