Controllability Subspaces Research Articles

Controllability of Multiagent Networks With Antagonistic Interactions

This paper addresses the controllability of a class of antagonistic multiagent networks with both positive and negative edges. All the agents of the multiagent network run a consensus algorithm using a signed Laplacian. Based on the generalized equitable partition, we propose a graph-theoretic characterization of an upper bound on the controllable subspace. Then, we provide a necessary condition for the controllability of the system and give an algorithm to compute the partition. Furthermore, we prove that for a structurally balanced network, the controllability is equivalent to that of the corresponding all-positive network, if the leaders are chosen from the same vertex set. Several examples are given to illustrate these results.

Suboptimal hierarchical control strategy to improve energy efficiency of vapour-compression refrigeration systems

This paper deals with optimal control of vapour-compression refrigeration systems, where only the compressor speed and the expansion valve opening are considered as manipulable inputs. Any given cycle is completely defined by a three-variable set, thus it is an underactuated control problem. The controllability analysis presented by the authors in a previous work applying linear theory is extended to a pointwise nonlinear analysis based on the phase portrait method. It is concluded that there is no full controllability and only a two-dimensional subspace of the three-dimensional solution space can be explored. A suboptimal hierarchical control strategy is proposed, where an online optimizer explores the two-dimensional controllable subspace to generate the reference on the degree of superheating, which, along with the cooling demand set point and the uncontrolled state, defines the cycle in steady state. The uncontrolled state is, by definition, not manipulable and defines the maximum achievable efficiency. Some simulation results comparing the proposed control architecture with other strategies studied in the literature are included, regarding the energy efficiency achieved in steady state and the dynamic behaviour of the controlled variables, where the controllability issues are highlighted.

Controllability of multiagent systems based on path and cycle graphs

SummaryThis paper studies the controllability of multiagent systems based on path and cycle graphs. For paths or cycles, sufficient and necessary conditions are presented for determining the locations of leaders under which the controllability can be realized. Specifically, the controllability of a path is shown to be determined by a set generated only from its odd factors, and the controllability of a cycle is determined by whether the distance between 2 leaders belongs to a set generated from its even (odd) factors when the number of its nodes is even (odd). For both graphs, the dimension of the controllable subspace is also derived. Moreover, the technique used in the derivation of the above results is further used to get sufficient and necessary conditions for several different types of graphs generated from path and cycle topologies. These types of graphs can be regarded as typical topologies in the study of multiagent controllability, and accordingly the obtained results have meaningful enlightenment for the research in this field.

Controllable subspace of edge dynamics in complex networks

For the edge dynamics in some real networks, it may be neither feasible nor necessary to be fully controlled. An accompanying issue is that, when the external signal is applied to a few nodes or even a single node, how many edges can be controlled? In this paper, for the edge dynamics system, we propose a theoretical framework to determine the controllable subspace and calculate its generic dimension based on the integer linear programming. This framework allows us not only to analyze the control centrality, i.e., the ability of a node to control, but also to uncover the controllable centrality, i.e., the propensity of an edge to be controllable. The simulation results and analytic calculation show that dense and homogeneous networks tend to have larger control centrality of nodes and controllable centrality of edges, but the negatively correlated in- and out-degrees of nodes or edges can reduce the two centrality. The positive correlation between the control centrality of node and its out-degree leads to that the distribution of control centrality, instead of that of controllable centrality, is encoded by the out-degree distribution of networks. Meanwhile, the positive correlation indicates that the nodes with high out-degree tend to play more important roles in control.

Controllability of multi‐agent systems under directed topology

SummaryIn this paper, the controllability of multi‐agent systems is studied under leader‐follower framework, where the interconnection between agents is directed. The concept of leader‐follower connectedness is first introduced for directed graph to check controllability, and some graph‐theoretic conditions are derived for constructively designed topologies. Then, distance partition and almost equitable partition are employed to quantitatively study the controllable subspaces. Moreover, the relationship between the state invariance of multi‐agent systems and the existence of almost equitable partition is discussed. Copyright © 2017 John Wiley & Sons, Ltd.

On the fixed controllable subspace in linear structured systems

In this paper we consider interconnected networks that are described by means of structured linear systems with state and control variables. We represent these systems, whose matrices contain fixed zeros and free parameters, by means of directed graphs and study questions concerning controllability and the controllable subspace.We show in this paper that the controllable subspace can have a part that will be present for almost all values of the free parameters. It actually is a subspace of the controllable subspace and will be referred to as the fixed controllable subspace. The subspace can then be seen as a kind of robustly controllable part of the system. Indeed, it is a subspace in the state space with the generic property that states in it can be steered in an arbitrary way.We derive a characterization of the fixed controllable subspace using the graph representation. The obtained characterization makes use of well-known algorithms from optimization and networks theory. To get some more insight in the components in the fixed part, we also give a representation of the structured linear systems by means of bipartite graphs. Using the Dulmage–Mendelsohn decomposition, we are able to decompose our structured systems in such a way that in some special cases, the fixed controllable subspace can be obtained directly from the decomposition.

On the Kronecker Canonical Form of Singular Mixed Matrix Pencils

Consider a linear time-invariant dynamical system that can be described as $F\dot{x}(t) =A{x}(t)+B{u}(t)$, where $A$, $B$, and $F$ are mixed matrices, i.e., matrices having two kinds of nonzero coefficients: fixed constants that account for conservation laws and independent parameters that represent physical characteristics. The controllable subspace of the system is closely related to the Kronecker canonical form of the mixed matrix pencil $\begin{pmatrix}A-sF\mid B\end{pmatrix}$. Under a physically meaningful assumption justified by the dimensional analysis, we provide a combinatorial characterization of the sums of the minimal row/column indices of the Kronecker canonical form of mixed matrix pencils. The characterization leads to a matroid-theoretic algorithm for efficiently computing the dimension of the controllable subspace for the system with nonsingular $F$.

On controllability, near-controllability, controllable subspaces, and nearly-controllable subspaces of a class of discrete-time multi-input bilinear systems

This paper considers a class of discrete-time multi-input inhomogeneous bilinear systems. The structure of such systems is most close to linear time-invariant systems’ but they own a strong property. That is, if the systems are uncontrollable, they can still be nearly controllable. Necessary and sufficient conditions for controllability and near-controllability of the systems are established by using a classical decomposition. Furthermore, a geometric characterization is given for the systems such that controllable subspaces and nearly-controllable subspaces are derived and characterized. Similar results on controllability are also obtained for the continuous-time counterparts of the systems. Finally, examples are provided to demonstrate the conceptions and results of this paper.

Differentially Controllable Subspace for Linear Periodic Systems

It is well known that the concepts of controllability and differential controllability are coincident for linear time-varying systems with analytic coefficients. In this note, we discuss the difference between these concepts for linear periodic continuous-time systems with piecewise-analytic coefficients. We propose the concept of differentially controllable subspace in order to compare with the controllable subspace, and then, the formulae for computing the differentially controllable subspace are derived in both integral and differential forms. Reachability and differential reachability are also discussed in a similar way. The significance of assuming piecewise-analytic coefficients for computing the differentially controllable subspace is explained in detail by a counterexample to the previous work.

Graph Controllability Classes for the Laplacian Leader-Follower Dynamics

In this paper, we consider the problem of obtaining graph-theoretic characterizations of controllability for the Laplacian-based leader-follower dynamics. Our developments rely on the notion of graph controllability classes, namely, the classes of essentially controllable, completely uncontrollable, and conditionally controllable graphs. In addition to the topology of the underlying graph, the controllability classes rely on the specification of the control vectors; our particular focus is on the set of binary control vectors. The choice of binary control vectors is naturally adapted to the Laplacian dynamics, as it captures the case when the controller is unable to distinguish between the followers and, moreover, controllability properties are invariant under binary complements. We prove that the class of essentially controllable graphs is a strict subset of the class of asymmetric graphs and provide numerical results that suggests that the ratio of essentially controllable graphs to asymmetric graphs increases as the number of vertices increases. Although graph symmetries play an important role in graph-theoretic characterizations of controllability, we provide an explicit class of asymmetric graphs that are completely uncontrollable, namely the class of block graphs of Steiner triple systems. We prove that for graphs on four and five vertices, a repeated Laplacian eigenvalue is a necessary condition for complete uncontrollability but, however, show through explicit examples that for eight and nine vertices, a repeated eigenvalue is not necessary for complete uncontrollability. For the case of conditional controllability, we give an easily checkable necessary condition that identifies a class of binary control vectors that result in a two-dimensional controllable subspace. Several constructive examples demonstrate our results.

Placement optimization of actuators and sensors for gyroelastic body

Gyroelastic body refers to a flexible structure with a distribution of stored angular momentum provided by fly wheels or control moment gyroscopes. The angular momentum devices can exert active torques to the structure for vibration suppression or shape control. This article mainly focuses on the placement optimization issue of the actuators and sensors on the gyroelastic body. The control moment gyroscopes and angular rate sensors are adopted as actuators and sensors, respectively. The equations of motion of the gyroelastic body incorporating the detailed actuator dynamics are linearized to a loosely coupled state-space model. Two optimization approaches are developed for both constrained and unconstrained gyroelastic bodies. The first is based on the controllability and observability matrices of the system. It is only applicable to the collocated actuator and sensor pairs. The second criterion is formulated from the concept of controllable and observable subspaces. It is capable of handling the cases of both collocated and noncollocated actuator and sensor pairs. The illustrative examples of a cantilevered beam and an unconstrained plate demonstrate the clear physical meaning and rationality of the two proposed methods.

Strong structural control centrality of a complex network

We introduce the definition of strong structural control centrality, which represents the dimension of strong structurally controllable subspace or the capability of a single node to control an entire directed and weighted network in a strongly structural manner. A purely algebraic algorithm to calculate strong structural control centrality is proposed. Then we explore the quality of the strong structural control centrality. The relation between strong structural control centrality and the layer index in a directed tree inspires us to research (1) some fast methods to achieve strong structural control centrality and (2) the size relationship of strong structural control centralities of nodes belonging to different supernodes.

Structural controllability of temporal networks

The control of complex systems is an ongoing challenge of complexity research. Recent advances using concepts of structural control deduce a wide range of control related properties from the network representation of complex systems. Here, we examine the controllability of systems for which the timescale of the dynamics we control and the timescale of changes in the network are comparable. We provide analytical and computational tools to study controllability based on temporal network characteristics. We apply these results to investigate the controllable subnetwork using a single input. For a generic class of model networks, we witness a phase transition depending upon the density of the interactions, describing the emergence of a giant controllable subspace. We show the existence of the two phases in real-world networks. Using randomization procedures, we find that the overall activity and the degree distribution of the underlying network are the main features influencing controllability.

Controllability and observability analysis for vertex domination centrality in directed networks.

Topological centrality is a significant measure for characterising the relative importance of a node in a complex network. For directed networks that model dynamic processes, however, it is of more practical importance to quantify a vertex's ability to dominate (control or observe) the state of other vertices. In this paper, based on the determination of controllable and observable subspaces under the global minimum-cost condition, we introduce a novel direction-specific index, domination centrality, to assess the intervention capabilities of vertices in a directed network. Statistical studies demonstrate that the domination centrality is, to a great extent, encoded by the underlying network's degree distribution and that most network positions through which one can intervene in a system are vertices with high domination centrality rather than network hubs. To analyse the interaction and functional dependence between vertices when they are used to dominate a network, we define the domination similarity and detect significant functional modules in glossary and metabolic networks through clustering analysis. The experimental results provide strong evidence that our indices are effective and practical in accurately depicting the structure of directed networks.

Construction and minimality of coordinated linear systems

Coordinated linear systems are a particular class of hierarchical systems with a top-to-bottom information structure, consisting of a coordinator system and two or more subsystems. This paper deals with the construction and minimality of coordinated linear systems. Construction procedures are given to transform unstructured or interconnected systems into coordinated linear systems, using the geometric (i.e. basis-independent) concepts of observability and controllability subspaces. Several concepts of minimality for coordinated linear systems are suggested and characterized in order to identify decompositions which are ‘as decentralized as possible’. The extension of the developed methods and results to the broader class of hierarchical linear systems is discussed.

Preview suspension control for a full tracked vehicle

In this study, preview control algorithms for the active and semi-active suspension systems of a full tracked vehicle (FTV) are designed based on a 3-D.O.F model and evaluated. The main issue of this study is to make the ride comfort characteristics of a fast moving tracked vehicle better to keep an operator’s driving capability. Since road wheels almost trace the profiles of the road surface as long as the track doesn’t depart from the ground, the preview information can be obtained by measuring only the absolute position or velocity of the first road wheel. Simulation results show that the performance of the sky-hook suspension system almost follows that of full state feedback suspension system and the on-off semi-active system carries out remarkable performance with the combination of 12 on-off semi-active suspension units. The results simulated with 1st and 2nd weighting sets mean that the suspension system combined with the soft type of inner suspension and hard type of outer suspension can carry out better ride comfort characteristics than that with identical suspensions. The full tracked vehicle (FTV) system is uncontrollable and the system is split into controllable and uncontrollable subspace using singular value decomposition transformation. Frequency response curves to four types of inputs, such as heaving, pitching, rolling, and warping inputs, also demonstrate the merits of preview control in ride comfort. All the frequency characteristic responses confirm the continuous time results.

Upper and Lower Bounds for Controllable Subspaces of Networks of Diffusively Coupled Agents

This technical note studies the controllability of diffusively coupled networks where some agents, called leaders, are under the influence of external control inputs. First, we consider networks where agents have general linear dynamics. Then, we turn our attention to infer network controllability from its underlying graph topology. To do this, we consider networks with agents having single-integrator dynamics. For such networks, we provide lower and upper bounds for the controllable subspaces in terms of the distance partitions and the maximal almost equitable partitions, respectively. We also provide an algorithm for computing the maximal almost equitable partition for a given graph and a set of leaders.

On Near-Controllability, Nearly Controllable Subspaces, and Near-Controllability Index of a Class of Discrete-Time Bilinear Systems: A Root Locus Approach

This paper studies near-controllability of a class of discrete-time bilinear systems via a root locus approach. A necessary and sufficient criterion for the systems to be nearly controllable is given. In particular, by using the root locus approach, the control inputs which achieve the state transition for the nearly controllable systems can be computed. Furthermore, for the non-nearly controllable systems, nearly-controllable subspaces are derived and near-controllability index is defined. Accordingly, the controllability properties of such class of discrete-time bilinear systems are fully characterized. Finally, examples are provided to demonstrate the results of the paper.

Factorizing the Monodromy Matrix of Linear Periodic Systems

Abstract This note proposes a new approach to computing the Kalman canonical decomposition of finite-dimensional linear periodic continuous-time systems by extending the Floquet theory. Controllable and observable subspaces are characterized by factorizing the monodromy matrix. Then, the conditions for the existence of several periodic Kalman canonical decompositions are extensively studied. The relations to the Floquet factorization, the Floquet-like factorization, and the period-specific realization are also discussed.

On nearly-controllable subspaces of a class of discrete-time bilinear systems

If a linear time-invariant system is uncontrollable, then the state space can be decomposed as a direct sum of a controllable subspace and an uncontrollable subspace. In this paper, for a class of discrete-time bilinear systems which are uncontrollable but can be nearly controllable, by studying the nearly-controllable subspaces and defining the near-controllability index, the controllability properties of the systems are fully characterized. Examples are provided to illustrate the conceptions and results of the paper.