In this paper, controllability of systems defined on graphs is discussed. We consider the problem of controllability of the network for a family of matrices carrying the structure of an underlying directed graph. A one-to-one correspondence between the set of leaders rendering the network controllable and zero forcing sets is established. To illustrate the proposed results, special cases including path, cycle, and complete graphs are discussed. Moreover, as shown for graphs with a tree structure, the proposed results of the present paper together with the existing results on the zero forcing sets lead to a minimal leader selection scheme in particular cases.
This paper deals with robust synchronization of uncertain multi-agent networks. Given a network with for each of the agents identical nominal linear dynamics, we allow uncertainty in the form of additive perturbations of the transfer matrices of the nominal dynamics. The perturbations are assumed to be stable and bounded in -norm by some a priori given desired tolerance. We derive state space formulas for observer based dynamic protocols that achieve synchronization for all perturbations bounded by this desired tolerance. It is shown that a protocol achieves robust synchronization if and only if each controller from a related finite set of feedback controllers robustly stabilizes a given, single linear system. Our protocols are expressed in terms of real symmetric solutions of certain algebraic Riccati equations and inequalities, and also involve weighting factors that depend on the eigenvalues of the graph Laplacian. For undirected network graphs we show that within the class of such dynamic protocols, a guaranteed achievable tolerance can be obtained that is proportional to the quotient of the second smallest and the largest eigenvalue of the Laplacian. We also extend our results to additive nonlinear perturbations with -gain bounded by a given tolerance. Index Terms-Laplacian matrix.Manuscript
In this paper we contribute a theoretical framework that sheds a new light on the problem of microgrid analysis and control. The starting point is an energy function comprising the "kinetic" energy associated with the elements that emulate the rotating machinery and terms taking into account the reactive power stored in the lines and dissipated on shunt elements. We then shape this energy function with the addition of an adjustable voltage-dependent term, and construct socalled Bregman storage functions satisfying suitable dissipation inequalities. Our choice of the voltage-dependent term depends on the voltage dynamics under investigation. Several microgrids dynamics that have similarities or coincide with dynamics already considered in the literature are captured in our incremental energy analysis framework. The twist with respect to existing results is that our incremental storage functions allow for a large signal analysis of the coupled microgrid obviating the need for simplifying linearization techniques and for the restrictive decoupling assumption in which the frequency dynamics is fully separated from the voltage one. A complete Lyapunov stability analysis of the various systems is carried out along with a discussion on their active and reactive power sharing properties.C. De Persis and N. Monshizadeh are with ENTEG and the J.C. Willems
Abstract-Network controllability is the ability to control the entire network, meaning that we can drive the network from any initial state to any desired final state in finite time by using appropriate inputs which are applied to a subset of nodes of the network. Despite obvious advantages, network controllability is not always feasible as it may ask for a considerable portion of the nodes to be controlled. Moreover, there are cases where controllability of the entire network is not of interest, but rather we are interested in controllability properties of certain parts of the network. This motivates us to investigate the so-called "targeted controllability" of the network, where controllability is only required for a subset of nodes in the network. Noting that targeted controllability can be treated as an output controllability problem, we investigate the (strong) structural output controllability properties of the network from a topological viewpoint. In addition, we examine the structural properties of the reachable subspace of the network. To this end, we use the notion of zero forcing sets, which has been recently exploited in the context of structural controllability.
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