Decentralized state estimation for the control of network systems

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Abstract

The paper proposes a decentralized state estimation method for the control of network systems, where a cooperative objective has to be achieved. The nodes of the network are partitioned into independent nodes, providing the control inputs, and dependent nodes, controlled by local interaction laws. The proposed state estimation algorithm allows the independent nodes to estimate the state of the dependent nodes in a completely decentralized way. To do that, it is necessary for each independent node of the network to estimate the control input components computed by the other independent nodes, without requiring communication among the independent nodes. The decentralized state estimator, including an input estimator, is developed and the convergence properties are studied. Simulation results show the effectiveness of the proposed approach.

Introduction

The paper considers the control of network systems problem, where the goal is to achieve some desired dynamic cooperative behavior. Network systems include, as example, sensor networks [2], networks of autonomous agents (such as robots and UAVs) [3], biological networks [4], transportation networks [5], microgrids [6], social [7] and economic networks.

Decentralized control of network systems has been widely addressed in the last few years, mainly in the application fields of multi-robot systems [3], distributed sensor networks [2], and interconnected manufacturing equipments [8]. Controllability issues of network systems are analyzed in [9].

Generally speaking, the aim of decentralized control strategies is implementing local interaction rules to regulate the state of the overall system to some desired configuration. In fact, mainly investigated coordinated behaviors include aggregation, swarming, formation control, coverage and synchronization [3], [10], [11], [12], [13], [14], [15]. While they constitute fundamental basic low level objectives to be achieved in multi-agent systems, these coordinated behaviors are still far away from several interesting real world applications.

In this paper, we consider heterogeneous networks, composed of nodes that are assigned with different roles. In particular, the nodes are partitioned into independent nodes, providing control inputs, and dependent nodes, controlled through local interaction. In this paper, in order to control the state of the entire network, we propose a decentralized estimation method to let the independent nodes estimate the state of the dependent nodes. In particular, both the estimation and the control phases are computed locally at each node in a decentralized way.

As a motivating example, consider the problem analyzed in [16], [17], where a team of mobile robots is controlled to implement a cooperative dynamic behavior, modeled as the cooperative tracking of desired periodic trajectories. The motivation behind this work was to provide a model for cooperative operations, similar to those performed by groups of human operators, such as the production cycle for a certain object, or the construction of a building. The team of robots was partitioned into two groups: a (small) set of independent robots was used as control inputs for the (large) set of dependent robots, evolving according to local interaction. In order to achieve such a cooperative behavior, the input for the system depends on the state of the entire team.

Another example is represented by microgrids [6] for energy production and delivery. In these systems, distributed generators adapt their production rate (i.e. the input for the system) based on the need of the loads (i.e. the state of the system), which are connected by means of a network. Efficient management of microgrids requires knowledge of the state of the entire network.

In these application examples, independent nodes (i.e. the input points for the network system) have usually access only to a subset of the state variables of the system: namely, they can only measure the state of their neighboring nodes. It is worth noting that, if a connected communication network exists among the control nodes, information can be shared among them. However, this is not always feasible, nor reliable, and can cause drawbacks. In particular, when considering static nodes (e.g. the generators in a microgrid) having a connected communication network to exchange information among the nodes requires significant infrastructure, which may not be feasible, and may raise security and privacy issues. On the other hand, when considering mobile nodes (e.g. mobile robots), several strategies exist in the literature to guarantee connectivity preservation [18], [19], [20], [21], [22], but they introduce constraints on the admissible trajectories of the nodes, that may be undesirable.

To address this kind of problems, in this paper we introduce a completely decentralized estimation procedure: without requiring any communication among the independent nodes, the proposed method provides a reliable estimation of the entire state of the network system that can be used for control purposes.

The paper is organized as follows. A review of the relevant literature and the main contributions of the paper are provided in Section 2. In Section 3, the notation used throughout the paper is presented. The considered problem is introduced in Section 4. In Section 5 the decentralized state estimator is designed and the related estimation error is analyzed in Section 6. Then, some convergence conditions are derived in Section 7. The effectiveness of the proposed method is shown in simulation in Section 9. Some final remarks can be found in Section 10.

Section snippets

Related work

The research on distributed and decentralized state estimation is a fertile and extensive field, with a lot of remarkable contributions. For a survey on distributed estimation see, for example, [23]. Decentralized estimation is often exploited for control purposes, since it allows the implementation of control strategies based on global quantities relying only on locally available information [24], [25]. Two main different approaches have been proposed in the state of the art to the problem of

Notation and mathematical operators

In this section we define some symbols that will be used throughout the paper.

The symbol Iρ will be used to indicate the identity matrix in Rρ×ρ, while the symbols Oρ and Oρ,σ will be used to indicate a square and a rectangular zero matrix in Rρ×ρ and in Rρ×σ, respectively.

Moreover, we will use vi to denote the ith component of vector v, and Ψi to denote the ith block of a block diagonal matrix Ψ.

Given a list of vectors χiRρ, i=1,,σ, we use the symbol col( · ) to denote the vector χ¯Rρσ,

Problem formulation

Consider a set V of N nodes, whose interconnection is modeled by means of a connected undirected graph G.

Let xiRm be the state of the ith node and consider the following single-integrator kinematic model:x˙i=μiwhere μiRm is each node’s control input.

In order to keep the notation simple, in the following we will consider the scalar case, thus assuming xiR and that the results can be extended to the multi-dimensional case, considering each component separately.

Let us now divide the nodes into

The decentralized state observer

In order to allow each independent node to implement the control strategy in Eq. (11) exploiting only locally available information, it is necessary to derive a state estimator of the network system state xD. Moreover, since each independent node is not able to communicate to other independent nodes, it is necessary to estimate the control input components computed by the other independent nodes.

We assume that all the matrices (system and regulator matrices) are known and constant. Hence, each

Estimation error analysis

For analysis purposes, we consider an extended formulation of the state vector estimate x^E, collecting the state estimates of the independent robots, that isx^E=col(d^i,i=1,,NI).The dynamics of the extended estimator can be described asx^˙E=AEx^E+BEu^EKE(yDEx^E),where AE=INIA is a block matrix having non-null blocks only on the diagonal, equal to A; BE=INIB is defined in an analogous way; u^E can be computed similarly to x^E asu^E=col(u^i,i=1,,NI),KE is a NIND × NI block matrix having on

Convergence analysis

In this section we analyze the convergence properties of the proposed estimation scheme. In particular, the following Theorem provides a methodology to locally define matrix Ki in Eq. (13) in such a way that the estimation error asymptotically converges to zero.

Theorem 1

Consider the state estimation scheme defined in Eq. (13), and let λi, min be the minimum eigenvalue of A+kibi+BF, i=1,,NI. If, i,,NI, ki is defined so that the following holds:λi,min(INIB)F˜E,then the estimation error (24)

Measurement noise

For a more complete analysis, it is possible to consider the presence of measurement noises, by including an additive term in the output Eq. (9), thus obtainingyi=bixD+η[i],where η[i] is a random noise for the ith node with mean value equal to η¯i, and standard deviation equal to σ¯i. Therefore, we can rewrite the output equation using the extended formulation and we obtain: y=DExE+η, where η is a vector collecting all the components of η[i], i=1,,NI, ordered according to their index. The

The considered problem

As a case study, we consider a multi-agent system composed of N interconnected mobile robots, partitioned into two groups: a small group of independent robots, and a large group of dependent robots. The objective of the system is to solve a tracking problem: namely, a set of periodic setpoint trajectories is designed, and the independent robots are controlled in such a way that the dependent robots asymptotically track those trajectories.

For this purpose, we consider the following

Conclusions

This paper proposes a decentralized state estimation method. The purpose is the control of network systems in order to track arbitrary setpoint trajectories. The proposed state estimation algorithm is designed allowing each independent node to estimate the input of the other independent nodes and the state of the dependent nodes, without requiring communication among the independent nodes. Conditions are derived to formally guarantee that estimation error asymptotically converges to zero. The

Acknowledgments

The dissemination of this work is being supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 739551 (KIOS CoE). This work was also partially supported by the research project Stability and Control of Power Networks with Energy Storage (STABLE-NET) which is funded by the RCUK Energy Programme (contract no: EP/L014343/1).

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