S. Emre Tuna

Brief paper: Synchronizing linear systems via partial-state coupling

A basic result in the synchronization of linear systems via output coupling is presented. For identical discrete-time linear systems that are detectable from their outputs and neutrally stable, it is shown that a linear output feedback law exists under which the coupled systems globally asymptotically synchronize for all fixed connected (asymmetrical) network topologies. An algorithm is provided to compute such a feedback law based on individual system parameters.

Robust hybrid controllers for continuous-time systems with applications to obstacle avoidance and regulation to disconnected set of points

We give an elementary proof of the fact that, for continuous-time systems, it is impossible to use (even discontinuous) pure state feedback to achieve robust global asymptotic stabilization of a disconnected set of points or robust global regulation to a target while avoiding an obstacle. Indeed, we show that arbitrarily small, piecewise constant measurement noise can keep the trajectories away from the target. We give a constructive, Lyapunov-based hybrid state feedback that achieves robust regulation in the above mentioned settings

Global synchronization on the circle

Abstract The convexity arguments used in the consensus literature to prove synchronization in vector spaces can be applied to the circle only when all agents are initially located on a semicircle. Existing strategies for (almost-)global synchronization on the circle are either restricted to specific interconnection topologies or use auxiliary variables. The present paper first illustrates this problem by showing that weighted, directed interconnection topologies can be designed to make any reasonably chosen configuration of the agents on the circle a stable equilibrium of a basic continuous-time consensus algorithm. Then it proposes a so-called “gossip algorithm”, which achieves global asymptotic synchronization on the circle with probability 1 for a large class of interconnections, without using auxiliary variables, thanks to the introduction of randomness in the system.

Optimal regulation of homogeneous systems

We present an offline numerical algorithm to generate a discontinuous feedback law to robustly regulate the origin of a continuous-time homogeneous system through sample and hold. The proposed feedback comes out of the solution of an infinite horizon optimization problem in discrete time and is in the form of a look-up table. We show that when applied to either chained systems or systems in power form the algorithm results in a closed loop whose origin is globally exponentially stable.

Synchronization of nonlinearly coupled harmonic oscillators

Synchronization of coupled harmonic oscillators is investigated. Coupling considered here is pairwise, unidirectional, and described by a nonlinear function (whose graph resides in the first and third quadrants) of some projection of the relative distance (between the states of the pair being coupled) vector. Under the assumption that the interconnection topology defines a connected graph, it is shown that the synchronization manifold is semiglobally practically asymptotically stable in the frequency of oscillations.

Synchronization analysis of coupled Lienard-type oscillators by averaging

Sufficient conditions for the synchronization of coupled Lienard-type oscillators are investigated via averaging technique. The coupling considered here is fixed, nonsymmetric, and nonlinear. Under the assumption that the interconnection topology defines a connected graph, it is shown that the solutions of oscillators converge arbitrarily close to each other, starting from initial conditions arbitrarily far apart, provided that the frequency of oscillations is large enough and the initial phases of oscillators all lie in an open semicircle. It is also shown that the nearly-synchronized oscillations always take place around some fixed magnitude independent of the initial conditions and the coupling functions.

Sufficient Conditions on Observability Grammian for Synchronization in Arrays of Coupled Linear Time-Varying Systems

Synchronizability of stable, output-coupled, identical, linear time-varying systems is studied. It is shown that if the observability grammian satisfies a persistence of excitation condition, then there exists a bounded, linear time-varying feedback law that yields exponential synchronization for all fixed, asymmetrical interconnections with connected graphs. Also, a weaker condition on the grammian is given for asymptotic synchronization. No assumption is made on the strength of coupling between the systems.

Brief paper: Growth rate of switched homogeneous systems

We consider discrete-time homogeneous systems under arbitrary switching and study their growth rate, the analogue of joint spectral radius for switched linear systems. We show that a system is asymptotically stable if and only if its growth rate is less than unity. We also provide an approximation algorithm to compute growth rate with arbitrary accuracy.

Homogeneous hybrid systems and a converse Lyapunov theorem

In this paper we introduce homogeneity for hybrid systems (using generalized dilations) and provide basic implications of this property similar to that of continuous-time and discrete-time homogeneous systems. In our main result we state that stability of a hybrid system that is robust with respect to small perturbations implies the existence of a homogeneous Lyapunov function for the system. This converse Lyapunov theorem unifies the previous results

Synchronization under matrix-weighted Laplacian

Synchronization in a group of linear time-invariant systems is studied where the coupling between each pair of systems is characterized by a different output matrix. Simple methods are proposed to generate a (separate) linear coupling gain for each pair of systems, which ensures that all the solutions converge to a common trajectory.