Gerd S. Schmidt

Cooperative control of linear parameter-varying systems

The state consensus problem among groups of linear parameter-varying (LPV) systems is addressed. The systems are assumed to have an identical LPV structure. We consider both homogeneous groups, where the time-varying parameters are identical, and heterogeneous groups, where the parameters can be different for all agents. Motivated by relative sensing we assume static, possibly parameter-varying, diffusive couplings between neighboring agents. We provide necessary and sufficient conditions for state consensus and, in the homogeneous case, present design procedures for the coupling gains based on linear matrix inequalities.

Frequency synchronization and phase agreement in Kuramoto oscillator networks with delays

We investigate frequency synchronization and phase agreement in networks of non-identical Kuramoto oscillators with delayed coupling. We provide delay-dependent lower bounds for the coupling gain so that a network with all-to-all coupling and constant heterogeneous delays achieves synchronization. The given bounds are determined explicitly by the system parameters. Furthermore, we show that frequency synchronization and phase agreement are possible in general directed networks of non-identical oscillators which contain a spanning tree if the delays satisfy a self-consistency condition. Networks with delays differ at this point notably from networks without delays, because phase agreement is not possible in undelayed non-identical Kuramoto oscillator networks. In addition to the sufficient conditions for synchronization, we investigate the connection between phase agreement and the existence of solutions of a necessary self-consistency condition.

Phase synchronization through entrainment by a consensus input

The paper proposes a synchronization mechanism in a set of nonlinear oscillators interconnected through a communication network. In contrast to many existing results, we do not employ strong, diffusive couplings between the individual oscillators. Instead, each individual oscillator is weakly forced by a linear resonator system. The resonator systems are synchronized using results from consensus theory. The synchronized resonator systems force the frequencies of the nonlinear oscillators to a constant frequency and thereby yield synchronization of the oscillators. We prove this result using the theory of small forcings of stable oscillators. This synchronization scheme allows for synchronization of nonlinear oscillators over uniformly connected communication graphs.

Output synchronization of linear parameter-varying systems via dynamic couplings

We address the output synchronization problem for groups of linear parameter-varying (LPV) agents. The agents are assumed to have identical LPV structure depending on different local parameters which make the group heterogeneous. First, we show that homogeneous groups of LPV systems with static diffusive couplings synchronize over uniformly connected graphs. Then, based thereon, we propose a constructive and scalable solution to the output synchronization problem for groups of LPV systems with heterogeneous parameters. The results are illustrated in a numerical example.

A solution for a class of output regulation problems on SO(n)

The present work considers a special class of output regulation problems on SO(n) and gives an explicit solution with strong convergence properties to this class of problems.

On the Necessity of Diffusive Couplings in Linear Synchronization Problems With Quadratic Cost

We show that diffusive couplings are necessary for minimization of cost functionals integrating quadratic synchronization error and quadratic input signals. This holds for identical linear systems with eigenvalues either on the imaginary axis or in the open left half-plane, whilst for eigenvalues in the open right half-plane, we present a counterexample in which the strong solution to the associated algebraic Riccati equation is not diffusive. For nonidentical systems satisfying the internal model principle for synchronization, we show that a certain part of the coupling must be diffusive. For equally chosen weights in the cost functional, we show that the dimension of the associated algebraic Riccati equation can be reduced significantly.

Output Regulation for Control Systems on

This technical note considers a solution to an output regulation problem for control systems on the special Euclidean group SE(n). The proposed solution consists of a convergent observer together with a certainty equivalence implementation of an almost globally stable state feedback law. The design of the observer and the state feedback are carried out independently. We show that all closed loop solutions remain bounded, that the steady-state solution is contained in exactly one of a finite number of isolated sets which are all unstable except one, which is the desired equilibrium set and which is asymptotically stable.

SE(n)

This paper considers a certain class of output regulation problems for the rigid body equations in natural representation and gives an explicit solution with strong convergence properties.

: A Separation Principle Based Approach

Consensus in bistable and multistable multi-agent systems (MAS) is investigated. The considered MAS consists of agents with nonlinear, bistable or multistable dynamics and local coupling. For this MAS, consensus conditions are presented for arbitrary large networks depending on the algebraic connectivity of the underlying graph. In contrast to most publications in the consensus literature, the considered MAS has only a finite number of discrete consensus points instead of a consensus subspace. This introduces a new kind of consensus problem to the literature on MAS. The differences to classical consensus problems are discussed and the main result is illustrated in a simulation example.

Output regulation for attitude control: A global approach

Abstract A group coordination problem is studied, where different agents are modeled as double integrator systems. Each agent aims to minimize a convex optimization function on the velocity. The coordination objective, studied in this note, is to find distributed coordination laws, such that the agents achieve a common velocity, which is the minimizer of the sum of all objective functions, while achieving a certain inter-agent spacing. Two different fully distributed coordination algorithms are proposed and analyzed. The algorithms combine ideas from distributed optimization and group coordination.