Strong Structural Controllability Research Articles

The controllability and structural controllability of Laplacian dynamics

In this paper, classic controllability and structural controllability under two protocols are investigated. For classic controllability, the multiplicity of eigenvalue zero of general Laplacian matrix L ∗ is shown to be determined by the sum of the numbers of zero circles, identical nodes and opposite pairs; however, it is always simple for the Laplacian L with diagonal entries in absolute form. For a fixed structurally balanced topology, the controllable subspace is proved to be invariant even if the antagonistic weights are selected differently under the corresponding protocol with L. For a graph expanded from a star graph rooted from a single leader, the dimension of controllable subspace is two under the protocol associated with L ∗ . In addition, the system is structurally controllable under both protocols if and only if the topology without unaccessible nodes is connected. As a reinforcing case of structural controllability, strong structural controllability requires the system to be controllable for any choice of weights. The connection between parent nodes and child nodes affects strong structural controllability because it determines the linear relationship of the control information from parent nodes. This discovery is a major factor in establishing the sufficient conditions on strong structural controllability for multi-agent systems under both protocols, rather than for complex networks, about latter results are already abundant. Finally, we present a sufficient and necessary condition to identify the strong structural controllability. Moreover, a method to construct a strongly structurally controllable graph is proposed.

Strong Structural Controllability and Observability of Individual Nodes: A Graph-Theoretical Approach

Strong Structural Controllability and Observability of Individual Nodes: A Graph-Theoretical Approach

Strong structural controllability of structured networks with MIMO node systems

Strong structural controllability of structured networks with MIMO node systems

Strong Structural Controllability of Boolean Networks: Polynomial-Time Criteria, Minimal Node Control, and Distributed Pinning Strategies

In this paper, we initiate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strong structural controllability</i> of Boolean networks (BNs), coping with the trouble in identifying full nodal dynamics in practice, and utilize it to build up some existing results about BNs. The derived criteria for the strong structural controllability of BNs are polynomial-time checkable. As an interesting feature, controllability is shown to be equivalent to fixed-time controllability in this regard. We further consider the minimal strong structural controllability problem of BNs which, reduced from the minimal vertex cover problem of graphs, is turned out to be NP-hard. Proceeding forward, strong structural controllability provides useful insights to exclude the deficiencies of existing approaches. First, the network aggregation subject to the controllability of BNs is built by showing that the predecessor aggregated blocks can be dropped in turn if they are strongly structurally controllable. Second, the distributed pinning controllers are carried out to render arbitrary BN controllable through node-to-node message exchange, while traditional results only check the controllability with the preassigned controllers. Notably, the time complexity to design such controllers is only exponential with the maximal in-degree of pinned nodes rather than the node number of BNs. Third, in scenarios of BNs encountering probability disturbances, a basic theorem is put forward to elaborate the equivalence between several types of asymptotic stability. Together with this theorem, the criteria of strong structural controllability also facilitate the design of distributed controllers to asymptotically stabilize probabilistic BNs in probability.

Strong Structural Controllability Based on Leader-Follower Framework

Strong Structural Controllability Based on Leader-Follower Framework

Computation of the Distance-Based Bound on Strong Structural Controllability in Networks

In this article, we study the problem of computing a tight lower bound on the dimension of the strong structurally controllable subspace (SSCS) in networks with Laplacian dynamics. The bound is based on a sequence of vectors containing the distances between leaders (nodes with external inputs) and followers (remaining nodes) in the underlying network graph. Such vectors are referred to as the distance-to-leaders vectors. We give exact and approximate algorithms to compute the longest sequences of distance-to-leaders vectors, which directly provide distance-based bounds on the dimension of SSCS. The distance-based bound is known to outperform the other known bounds (for instance, based on zero-forcing sets), especially when the network is partially strong structurally controllable. Using these results, we discuss an application of the distance-based bound in solving the leader selection problem for strong structural controllability. Further, we characterize strong structural controllability in path and cycle graphs with a given set of leader nodes using sequences of distance-to-leaders vectors. Finally, we numerically evaluate our results on various graphs.

Strong Structural Controllability of Directed Graphs via Zero Forcing Sets

This paper presents several conditions for strong structural controllability of a directed structured network, which is determined by the non-zero/zero patterns of the weights of directed edges. First, we introduce the existing notion of zero forcing sets with the necessary and sufficient condition for strong structural controllability. Based on the zero forcing sets, we introduce the necessary condition that a strongly structurally controllable graph with the minimum number of m-inputs can be decomposed into m-disjoint controllable paths and bridge edges between them. With a disjoint controllable path as a basic graph, we explore the properties of a strongly structurally controllable graph. Then, we present several merging rules that can maintain the strong structural controllability based on the notion of zero forcing sets.

Strong structural controllability of networks: Comparison of bounds using distances and zero forcing

We study the strong structural controllability (SSC) of networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace (SSCS), which is equal to the smallest possible rank of controllability matrix under admissible coupling weights among the nodes In this paper, we compare two tight lower bounds on the dimension of SSCS: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We first show that each of these two bounds can be arbitrarily better than the other in some special cases. We then show that the distance-based lower bound is usually better than the zero-forcing-based bound when the value of the latter is less than the dimensionality of the overall network state, n. On the other hand, we also show that any set of leaders that makes the distance-based bound equal to n necessarily makes the zero-forcing-based bound equal to n (the converse is not true). These results indicate that while the zero-forcing-based approach may be preferable when the focus is only on verifying complete SSC (dimension of SSCS is equal to n), the distance-based approach usually yields a closer bound on the dimension of SSCS when the bounds are both smaller than n. Furthermore, we also present a novel bound based on combining these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two original bounds. Finally, we support our analysis with numerical results on various graphs.

Robust Strong Structural Controllability of Complex Power Systems

Ensuring the control of power systems is crucial for their safe operation. This paper analyses the robust controllability of complex power systems from the structural sighta structural point of view. Stressing the dominant role of generators in the control of power systems, we propose three kinds of controllable networks by generator nodes. Additionally, the satisfied conditions and the relevant proof of zero forcing set in the controllable networks by generator nodes and extra nodes are given. Besides, tThe satisfied conditions and the relevant proofs of the largest set of removable edges that have no -effect on the strong structural controllability in three kinds of controllable networks by generator nodes are also proposed. Finally, the robustness of strong structural controllability of IEEE 39 bus system and IEEE 14 bus system have been analyzed. The zero-forcing set and the largest set of removable edges of IEEE 39 bus system and IEEE 14 bus system are provided.

Scalable Controllability Analysis of Structured Networks

This article deals with strong structural controllability of structured networks. A structured network is a family of heterogeneous structured systems (called node systems) that are interconnected by means of a structured interconnection law, all given by pattern matrices. Here, we consider structured networks with single-input–single-output node systems. It is shown that such a structured network is strongly structurally controllable if and only if an associated structured system is. This structured system will, in general, have a very large state space dimension and, therefore, existing tests for verifying strong structural controllability are not tractable. The main result of this article circumvents this problem. We show that controllability can be tested by replacing the original network by a new network in which all original node systems have been replaced by (auxiliary) node systems with state space dimensions either 1 or 2. Hence, controllability of the original network can be verified by testing controllability of a structured system with state space dimension at most twice the number of node systems, regardless of the state space dimensions of the original node systems.

Strong targeted controllability of multi-agent systems with time-varying topologies over finite fields

In this paper, we study the strong targeted controllability of multi-agent systems with time-varying topologies over finite fields from a graph theoretic perspective. We consider leader–follower networks, where only some of the agents are governed by external control inputs, namely the leaders. We develop a graph-theoretic characterization for strong targeted controllability of multi-agent systems with time-varying topologies over finite fields. Specifically, we show that a time-varying multi-agent system is strongly targeted controllable over any finite field if the graphs of the system satisfy certain properties. Our main results are applicable to time-varying topologies with arbitrary fields and coupling weights. We also present a necessary and sufficient condition for the strong structural controllability of the time-invariant multi-agent systems.

Partial strong structural controllability

This paper introduces a new controllability notion, termed partial strong structural controllability (PSSC), on a structured system whose entries of system matrices are either fixed zero or indeterminate, which naturally extends the conventional strong structural controllability (SSC) and bridges the gap between structural controllability and SSC. Dividing the indeterminate entries into two categories, generic entries and unspecified entries, a system is PSSC, if for almost all values of the generic entries in the parameter space except for a set of measure zero, and any nonzero (complex) values of the unspecified entries, the corresponding system is controllable. We highlight that this notion generalizes the generic property embedded in the conventional structural controllability for single-input systems. We then give algebraic and (bipartite) graph-theoretic necessary and sufficient conditions for single-input systems to be PSSC, the latter of which can be verified in polynomial time. Conditions for multi-input systems are subsequently given for a special case. It is shown the established results can induce a new maximum matching based criterion for SSC over the system bipartite graph representations.

Edge Augmentation With Controllability Constraints in Directed Laplacian Networks

In this letter, we study the maximum edge augmentation problem in directed Laplacian networks to improve their robustness while preserving lower bounds on their strong structural controllability (SSC). Since adding edges could adversely impact network controllability, the main objective is to maximally densify a given network by selectively adding missing edges while ensuring that SSC of the network does not deteriorate beyond certain levels specified by the SSC bounds. We consider two widely used bounds: first is based on the notion of zero forcing (ZF), and the second relies on the distances between nodes in a graph. We provide an edge augmentation algorithm that adds the maximum number of edges in a graph while preserving the ZF-based bound, and also derive a closed-form expression for the exact number of edges added to the graph. Then, we examine the edge augmentation while preserving the distance-based bound and present a randomized algorithm that guarantees an α-approximate solution with high probability. Finally, we numerically evaluate and compare these edge augmentation solutions.

On Strong Structural Controllability of Temporal Networks

In this letter, we study strong structural controllability of linear time varying network systems that change network topology and edge weights with time. We derive graph based necessary and sufficient conditions for strong structural controllability of temporal networks. We provide a polynomial time algorithm to compute the dimension of the strong structurally controllable subspace of a temporal network. This allows us to verify whether a given set of inputs leads to a controllable system for all choices of numerical realizations. Next, we show that the problem of finding a minimum size input for strong structural controllability of temporal networks is an NP-hard problem. We propose a greedy heuristic algorithm to optimize the size of the input set for strong structural controllability of temporal networks and compare the performance of the greedy algorithm with the optimum solution through simulations.

Structural controllability analysis based on dilations under leader–follower framework

Abstract In this paper, the structural controllability of leader–follower systems over digraphs is discussed. Firstly, with the aid of leader–follower partition, we analyze the topological characteristics of systems without dilations. Moreover, we divide the digraph into the leader subgraph and the follower subgraph. By analyzing the topological characteristics of the follower subgraph, a sufficient condition to ensure the system structurally controllable is given. Furthermore, some special topologies of follower graphs are analyzed, such as unidirectional cycle graphs and directed paths, and the corresponding sufficient conditions of structural controllability are given. Meanwhile, a method to construct digraphs with structural controllability is presented. Finally, the strong structural controllability of directed tree graphs is considered, as well as the selection mechanism of leaders and the lower bound of the number of leaders under strong structural controllability are given. The above results reflect the effects of topologies on strong (or weak) structural controllability.

Strong structural controllability of colored structured systems

This paper deals with strong structural controllability of linear structured systems in which the system matrices are given by zero/nonzero/arbitrary pattern matrices. Instead of assuming that the nonzero and arbitrary entries of the system matrices can take their values completely independently, this paper allows equality constraints on these entries, in the sense that a priori given entries in the system matrices are restricted to take arbitrary but identical values. To formalize this general class of structured systems, we introduce the concepts of colored pattern matrices and colored structured systems. The main contribution of this paper is that it generalizes both the classical results on strong structural controllability of structured systems as well as recent results on controllability of systems defined on colored graphs. In this paper we will establish both algebraic and graph-theoretic conditions for strong structural controllability of this more general class of structured systems.

Strong Structural Controllability of Networks under Time-Invariant and Time-Varying Topological Perturbations

This paper investigates the robustness of strong structural controllability for linear time-invariant and linear time-varying directed networks with respect to structural perturbations, including edge deletions and additions. In this direction, we introduce a new construct referred to as a perfect graph associated with a network with a given set of control nodes. The tight upper bounds on the number of edges that can be added to, or removed from a network, while ensuring strong structural controllability, are then derived. Moreover, we obtain a characterization of critical edge-sets, the maximal sets of edges whose any subset can be respectively added to, or removed from a network, while preserving strong structural controllability. In addition, procedures for combining networks to obtain strongly structurally controllable network-of-networks are proposed. Finally, controllability conditions are proposed for networks whose edge weights, as well as their structures, can vary over time.

Tradeoff Between Controllability and Robustness in Diffusively Coupled Networks

In this article, we demonstrate a conflicting relationship between two crucial properties— controllability and robustness —in linear dynamical networks of diffusively coupled agents. In particular, for any given number of nodes $N$ and diameter $D$ , we identify networks that are maximally robust using the notion of Kirchhoff's index and then analyze their strong structural controllability. For this, we compute the minimum number of leaders, which are the nodes directly receiving external control inputs, needed to make such networks controllable under all feasible coupling weights between agents. Then, for any $N$ and $D$ , we obtain a sharp upper bound on the minimum number of leaders needed to design strong structurally controllable networks with $N$ nodes and $D$ diameter. We also discuss that the bound is best possible for arbitrary $N$ and $D$ . Moreover, we construct a family of graphs for any $N$ and $D$ such that the graphs have maximal edge sets (maximal robustness) while being strong structurally controllable with the number of leaders in the proposed sharp bound. We then analyze the robustness of this graph family. The results suggest that optimizing robustness increases the number of leaders needed for strong structural controllability. Our analysis is based on graph-theoretic methods and can be applied to exploit network structure to co-optimize robustness and controllability in networks.

A novel framework for modelling the 3D mesostructure of steel fibre reinforced concrete

This paper presents a novel method for modelling the three-dimensional four-phase mesostructure of steel fibre reinforced concrete (SFRC) consisting of coarse aggregates, mortar, interfacial transition zone and fibres, which provides several advanced features. It enables constructing various types of short discrete fibre like straight, spiral and hooked-end fibres with one- or three-dimensional elements while configuring their sizes and orientations. A key aspect of the approach is the capability of generating realistic-shaped aggregates with size- and shape-adjustability among fibres in a stochastic domain based on the relation of Voronoi and Delaunay structures. The mesostructure model is a powerful simulation tool for characterisation of such heterogenous materials because of the strong structural controllability, the algorithm’s flexibility, the low computational cost and the productivity for many random samples in a statistical framework. To show the feasibility, the models were successfully implemented in a finite element case study to evaluate the elastic modulus of SFRC.

A Unifying Framework for Strong Structural Controllability

This article deals with strong structural controllability of linear systems. In contrast to the existing work, the structured systems studied in this article have a so-called zero/nonzero/arbitrary structure, which means that some of the entries are equal to zero, some of the entries are arbitrary but nonzero, and the remaining entries are arbitrary (zero or nonzero). We formalize this in terms of pattern matrices, whose entries are either fixed zero, arbitrary nonzero, or arbitrary. We establish necessary and sufficient algebraic conditions for strong structural controllability in terms of full rank tests of certain pattern matrices. We also give a necessary and sufficient graph-theoretic condition for the full rank property of a given pattern matrix. This graph-theoretic condition makes use of a new color change rule that is introduced in this article. Based on these two results, we then establish a necessary and sufficient graph-theoretic condition for strong structural controllability. Moreover, we relate our results to those that exist in the literature and explain how our results generalize previous work.