Nima Monshizadeh

Robust Synchronization of Uncertain Linear Multi-Agent Systems

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 H∞-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 L2-gain bounded by a given tolerance.

Projection-Based Model Reduction of Multi-Agent Systems Using Graph Partitions

In this paper, we establish a projection-based model reduction method for multiagent systems defined on a graph. Reduced order models are obtained by clustering the vertices (agents) of the underlying communication graph by means of suitable graph partitions. In the reduction process, the spatial structure of the network is preserved and the reduced order models can again be realized as multiagent systems defined on a graph. The agents are assumed to have single-integrator dynamics and the communication graph of the original system is weighted and undirected. The proposed model reduction technique reduces the number of vertices of the graph (which is equal to the dynamic order of the original multi-agent system) and yields a reduced order multiagent system defined on a new graph with a reduced number of vertices. This new graph is a weighted symmetric directed graph. It is shown that if the original multiagent system reaches consensus, then so does the reduced order model. For the case that the clusters are chosen using an almost equitable partition (AEP) of the graph, we obtain an explicit formula for the H2-norm of the error system obtained by comparing the input-output behaviors of the original model and the reduced order model. We also prove that the error obtained by taking an arbitrary partition of the graph is bounded from below by the error obtained using the largest AEP finer than the given partition. The proposed results are illustrated by means of a running example.

Zero Forcing Sets and Controllability of Dynamical Systems Defined on Graphs

In this technical note, 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 technical note together with the existing results on the zero forcing sets lead to a minimal leader selection scheme in particular cases.

A Simultaneous Balanced Truncation Approach to Model Reduction of Switched Linear Systems

This paper deals with model reduction by balanced truncation of switched linear systems (SLS). We consider switched linear systems whose dynamics, depending on the switching signal, switches between finitely many linear systems with a common state space. These linear systems are called the modes of the SLS. The idea is to seek for conditions under which there exists a single state space transformation that brings all modes of the SLS in balanced coordinates. As a measure of reachability and observability of the state components of the SLS, we take the average of the diagonal gramians. We then perform balanced truncation by discarding the state components corresponding to the smallest diagonal elements of this average balanced gramian. In order to carry out this program, we derive necessary and sufficient conditions under which a finite collection of linear systems with common state space can be balanced by a single state space transformation. Among other things, we derive sufficient conditions under which global uniform exponential stability of the SLS is preserved under simultaneous balanced truncation. Likewise, we derive conditions for preservation of positive realness or bounded realness of the SLS. Finally, in case that the conditions for simultaneous balancing do not hold, or we simply do not want to check these conditions, we propose to compute a suitable state space transformation on the basis of minimization of an overall cost function associated with the modes of the SLS. We show that in case our conditions do hold, this transformation is in fact simultaneously balancing, bringing us back to the original method described in this paper.

Stability and synchronization preserving model reduction of multi-agent systems

Abstract In this paper, stability and synchronization preserving model reduction schemes are developed for linear multi-agent systems. The multi-agent systems that are considered here are composed of general, yet identical linear subsystems, and the communication topology is assumed to be time-independent. First, under the assumption that the agents have stable internal dynamics and the network is stable, the dynamic order of the agents is reduced such that the corresponding reduced order network is again stable. Then, starting from a synchronized network where agents are allowed to have unstable dynamics, a reduced order model for the network which preserves synchronization is obtained. In addition, model reduction error bounds are established to compare the behavior of the original network to that of the reduced order model. The proposed results are illustrated through a numerical example.

Output agreement in networks with unmatched disturbances and algebraic constraints

This paper considers a problem of output agreement in heterogeneous networks with dynamics on the nodes as well as on the edges. The control and disturbance signals entering the nodal dynamics are “unmatched” meaning that some nodes are only subject to disturbances, and are deprived of actuating signals. To further enrich our model, we accommodate (solvable) algebraic constraints in a subset of nodal dynamics. We show that appropriate dynamic feedback controllers achieve output agreement on a desired vector. We also investigate the case of an optimal steady-state control over the network. The proposed results are applied to a heterogeneous microgrid.

Bregman Storage Functions for Microgrid Control

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 so-called Bregman storage functions satisfying suitable dissipation inequalities. Our choice of the voltage-dependent term depends on the voltage dynamics under investigation. Several microgrid controllers 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. This obviates 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.

A Lyapunov approach to control of microgrids with a network-preserved differential-algebraic model

We provide sufficient conditions for asymptotic stability and optimal resource allocation for a network-preserved microgrid model with active and reactive power loads. The model considers explicitly the presence of constant-power loads as well as the coupling between the phase angle and voltage dynamics. The analysis of the resulting nonlinear differential algebraic equation (DAE) system is conducted by leveraging incremental Lyapunov functions, definiteness of the load flow Jacobian and the implicit function theorem.

A modular design of incremental Lyapunov functions for microgrid control with power sharing

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 incremental storage functions satisfying suitable dissipation inequalities. Our choice of the voltage-dependent term depends on the voltage dynamics/controller 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 an 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.

Disturbance decoupling problem for multi-agent systems: a graph topological approach

Abstract This paper studies the disturbance decoupling problem for multi-agent systems with single integrator dynamics and a directed communication graph. We are interested in topological conditions that imply the disturbance decoupling of the network, and more generally guarantee the existence of a state feedback rendering the system disturbance decoupled. In particular, we will develop a class of graph partitions, which can be described as a “topological translation” of controlled invariant subspaces in the context of dynamical networks. Then, we will derive sufficient conditions in terms of graph partitions such that the network is disturbance decoupled, as well as conditions guaranteeing solvability of the disturbance decoupling problem. The proposed results are illustrated by a numerical example.