Elsevier

Automatica

Volume 49, Issue 11, November 2013, Pages 3322-3328
Automatica

Brief paper
Input addition and leader selection for the controllability of graph-based systems

https://doi.org/10.1016/j.automatica.2013.07.021Get rights and content

Abstract

In this paper, we consider dynamical graph-based models, which are well fitted for the structural analysis of complex systems. A significant amount of work has been devoted to the controllability of such graph based models, e.g. recently for multi-agent systems or complex networks. We study here the controllability through input addition in this framework. We present several variants of this problem depending on the freedom which is left to the designer on the additional inputs. We use a unified framework, which allows us to encompass the different applications and representations (large scale systems, complex communications networks, multi-agent systems, …) and provide convenient graph tools for their analysis. Our contribution is to characterize the structural modifications of the system resulting from an input addition (or a leader selection) and of the mechanisms which lead to controllability. We provide information on the possible location of additional inputs and on the minimal number of inputs to be added for controllability.

Introduction

The dynamical graph-based models became popular recently because of their potential application to various problems such as networked systems (Pajic, Sundaram, Pappas, & Mangharam, 2011), multi-agent systems and consensus methods (Jafari et al., 2011, Rahmani et al., 2009, Sundaram and Hadjicostis, 2011).

The graph representation is well fitted for the structural analysis. The structural properties of the system under investigation are revealed by the graph inspection which abstracts away the numerical system parameters. A significant amount of work has been recently devoted to the controllability of such graph based models, e.g. for multi-agent systems (Jafari et al., 2011, Rahmani et al., 2009) or complex networks (Liu, Slotine, & Barabasi, 2011). In these works, the choice of inputs (actuators or leaders) for controllability is a key point. In a dual way, for engineering systems, the sensor implementation for observability is an important practical problem (Boukhobza and Hamelin, 2009, Commault et al., 2005).

Here we address the input addition and leader selection problems for the controllability of graph-based systems. The main contributions of this paper are as follows:

  • We use a unified framework, which allows us to encompass the different applications and representations (large scale systems, complex communications networks, multi-agent systems, …) and provide convenient graph tools for their analysis.

  • We develop a deep analysis of the structural modifications of the system resulting from an input addition (or a leader selection) and of the mechanisms which lead to controllability.

  • We not only look for a specific solution but we are also interested in the structural characterization of all the solutions.

  • Furthermore, the use of adequate graph tools (strongly connected components and the DM-decomposition) leads to a significant reduction of the size of the problem.

  • We also give tight bounds on the minimal number of additional inputs (or leaders) necessary for controllability.

In this paper, our model will be linear structured systems which have been used for the study of a lot of properties of systems assuming only the knowledge of their structure (Dion et al., 2003, Lin, 1974). With these systems, one can associate graphs which allow us to study in a simple way generic or structural properties and to give solvability conditions for standard control problems such as decoupling or disturbance rejection (Dion et al., 2003). The same type of model has been used for the study of multi-agent systems (Jafari et al., 2011) or complex networks (Liu et al., 2011).

We study the controllability through input addition in this framework. We present several variants of this problem, depending on the freedom which is left to the designer on the additional inputs, and propose a unified structural analysis. Preliminary results were given in Commault and Dion (2013). In the context of multi-agent systems, the problem is different, we select among the agents those which will become leaders, and the problem is to make this choice in such a way that the other agents (which are called followers) are controllable (Rahmani et al., 2009).

It is well known that the system is structurally controllable if and only if the graph of the system is input connected and satisfies a rank condition (Lin, 1974). Using strongly connected components of the system associated graph, we give the minimal number and the location of states which have to be touched for ensuring input connection. Using classical tools of graph theory, we analyze the rank condition and prove that the Dulmage–Mendelsohn decomposition allows us to characterize the rank defect. The minimal number and the location of additional inputs to be implemented in order to satisfy the rank condition is then given. For each subproblem, the structural characterization of solutions is performed on a reduced graph. In the different variants of the problem, we give tight bounds on the minimal number of inputs to insure controllability.

The outline of this paper is as follows. Section  2 formulates the input addition problem for controllability and the leader selection problem for controllability of multi-agent systems. The linear structured systems and structural controllability are presented in Section  3. Section  4 studies the controllability with additional inputs for the different versions of the problem. Section  5 is devoted to the choice of leaders for controllability in a multi-agent system. Some concluding remarks end the paper.

Section snippets

Additional inputs for controllability

In this paper, we consider the linear system Σ defined by (1)Σ:ẋ(t)=Ax(t)+Bu(t), where x(t)Rn is the state vector and u(t)Rm the input vector. A and B are matrices of appropriate dimensions. We study the controllability of this system, i.e. the possibility to drive in finite time the state from the origin to any point in the state space. This is known to be equivalent to the condition rank[B,AB,,An1B]=n. When Condition (2) is not verified, we try to satisfy it by adding new inputs. We look

Linear structured systems

We consider a linear system with parameterized entries and denoted by ΣΛ: ΣΛ:ẋ(t)=AΛx(t)+BΛu(t), where x(t)Rn is the state vector and u(t)Rm the input vector. AΛ and BΛ are matrices of appropriate dimensions. This system is called a linear structured system if the entries of the composite matrix JΛ=[AΛ,BΛ] are either fixed zeros or independent parameters (not related by algebraic equations). Λ={λ1,λ2,,λk} denotes the vector of independent parameters of the composite matrix JΛ.

For such

Controllability with input addition

We study in this section the different versions of the input addition for controllability, depending on the freedom which is left for the additional inputs, as stated in Section  2. The structured extended system ΣEΛ is defined as: ΣEΛ:ẋ(t)=AΛx(t)+[BΛ,B̄Λ][u(t)T,ū(t)T]T. We denote by dcE the connection defect of ΣEΛ and drE the rank defect of ΣEΛ.

Leader selection for controllability in a multi-agent system

Here we start with a structured multi-agent system ΠΛ. The problem is to choose a set of vertices in the graph G(ΠΛ) which correspond to the leaders, transform them into inputs in the transformed graph G(ΠLΛ), and check that the structural controllability conditions of Theorem 3 are satisfied on G(ΠLΛ). The transformed graph G(ΠLΛ) is obtained from G(ΠΛ) by deleting the incoming edges for the leader vertices in G(ΠΛ).

State now, for the multi-agent case, the propositions corresponding to

Some algorithmic and complexity considerations

As seen in Theorem 3, checking structural controllability amounts to checking separately the input-connection and the rank conditions. It may happen that the satisfaction of one of these conditions directly follows from the nature of the problem. For example, in the multi-agent problems, it is natural to assume that an agent uses its own value for computing its state, which means that αii0 for i=1,,N in Eq. (4). This is reflected in a self-loop around each vertex in the associated graph. In

Concluding remarks

In this paper, we studied in a unified framework controllability through different input addition settings for dynamical graph-based systems. The paper provides an analysis of the structural mechanisms of controllability resulting from an input addition (or leader selection) under different assumptions, and gives a structural characterization of all the solutions. We gave information on the number and the location of additional inputs for controllability. We revisited some recent results on

Christian Commault was born in 1950 in Le Gouray, France. He received the Engineer degree, the Doctor-Engineer degree and the Docteur d’Etat degree from the Institut National Polytechnique de Grenoble in 1973, 1978 and 1983 respectively. He taught for two years in the Dakar Institute of Technology (Sénégal). He was a visiting researcher in the Department of Mathematics at the University of Groningen (The Netherlands) for one year. From 1986 to 1988 he worked in the Research Centre of Renault,

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Christian Commault was born in 1950 in Le Gouray, France. He received the Engineer degree, the Doctor-Engineer degree and the Docteur d’Etat degree from the Institut National Polytechnique de Grenoble in 1973, 1978 and 1983 respectively. He taught for two years in the Dakar Institute of Technology (Sénégal). He was a visiting researcher in the Department of Mathematics at the University of Groningen (The Netherlands) for one year. From 1986 to 1988 he worked in the Research Centre of Renault, Rueil-Malmaison.

He is currently a professor at the Ecole Nationale Supérieure De l’Energie, l’Eau et l’Environnement (ENSE3), Grenoble. He teaches basic control systems, multivariable systems, operations research and performance evaluation.

He is a researcher of GIPSA-Lab and is in charge of the doctoral program in Electrical Engineering. His main research interests are in performance evaluation of production systems and in linear multivariable systems (mainly in structured systems both for control and diagnosis).

Jean-Michel Dion was born in 1950 in La Tronche, France. He received the Ph.D. degree from the National Polytechnic Institute of Grenoble (INPG), France in 1977. He is a senior researcher at the French National Center for Scientific Research (CNRS) working at GIPSA-Lab Grenoble. He was successively head of the Laboratoire d’Automatique de Grenoble, vice president of the National Polytechnic Institute of Grenoble and Director of GIPSA-Lab. He held the position of Director of the Department of Physics and Engineering Sciences at the French Ministry of Research and Education and then of vice-general director in charge of French university research. He is now senior scientific advisor at the Commissariat Général à l’Investissement. He was a visiting professor at the National Polytechnic Institute of Mexico city and is doctor honoris causa of the Polytechnic University of Bucharest. His main research interests are in the fields of linear system theory, time delay systems and application to electromechanical systems. He has published more than 100 technical papers published in refereed journals, and 200 international conference papers. Jean-Michel Dion served as associate editor for Automatica, and was chairman of the IFAC Technical Committee Linear Systems. He was a Member of the IFAC Council. He received the IFAC outstanding service award and was awarded as a fellow of the IFAC.

The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Riccardo Scattolini under the direction of Editor Frank Allgöwer.

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