This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach.Index Terms-Linear matrix inequalities, Lyapunov stability, piecewise linear systems.

No abstract

No abstract

The alternating direction method of multipliers (ADMM) has emerged as a powerful technique for largescale structured optimization. Despite many recent results on the convergence properties of ADMM, a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking.In this paper we find the optimal algorithm parameters that minimize the convergence factor of the ADMM iterates in the context of ℓ2-regularized minimization and constrained quadratic programming. Numerical examples show that our parameter selection rules significantly outperform existing alternatives in the literature.

No abstract

We present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to Nedić and Bertsekas [SIAM J. Optim., 12 (2001), pp. 109-138]. Our novel algorithm is particularly suitable for distributed implementation and execution, and possible applications include distributed optimization, e.g., parameter estimation in networks of tiny wireless sensors. The stochastic component in the algorithm is described by a Markov chain, which can be constructed in a distributed fashion using only local information. We provide a detailed convergence analysis of the proposed algorithm and compare it with existing, both deterministic and randomized, incremental subgradient methods.

Several stability conditions for hybrid systems are formulated as LMI problems. This includes the computations of continuous piecewise quadratic Lyapunov functions and multiple Lyapunov functions. The proposed methods are related to classical frequency domain methods, such as the Circle and Popov criteria. Finally, it is pointed out how controller design for hybrid systems can be formulated as an LMI problem. Several examples that highlight the proposed methods are included.

Abstract-In this paper we propose a subgradient method for solving coupled optimization problems in a distributed way given restrictions on the communication topology. The iterative procedure maintains local variables at each node and relies on local subgradient updates in combination with a consensus process. The local subgradient steps are applied simultaneously as opposed to the standard sequential or cyclic procedure. We study convergence properties of the proposed scheme using results from consensus theory and approximate subgradient methods. The framework is illustrated on an optimal distributed finite-time rendezvous problem.