Distributed Observers for LTI Systems
We consider the problem of distributed state estimation of a linear time-invariant (LTI) system by a network of sensors. We develop a distributed observer that guarantees asymptotic reconstruction of the state for the most general class of LTI systems, sensor network topologies and sensor measurement structures. Our analysis builds upon the following key observation - a given node can reconstruct a portion of the state solely by using its own measurements and constructing appropriate Luenberger observers; hence it only needs to exchange information with neighbors (via consensus dynamics) for estimating the portion of the state that is not locally detectable. This intuitive approach leads to a new class of distributed observers with several appealing features. Furthermore, by imposing additional constraints on the system dynamics and network topology, we show that it is possible to construct a simpler version of the proposed distributed observer that achieves the same objective while admitting a fully distributed design phase. Our general framework allows extensions to time-varying networks that result from communication losses, and scenarios including faults or attacks at the nodes.
In many applications involving large-scale complex systems (such as the power grid, transportation systems, industrial plants, etc.), the state of the system is monitored by a group of sensors spatially distributed over large sparse networks where the communication between sensors is limited (see [1, 2]). To model such a scenario, consider the discrete-time linear time-invariant dynamical system
where is the discrete-time index, is the state vector and is the system matrix. The state of the system is monitored by a network111The terms ‘network’ and ‘communication graph’ are used interchangeably throughout the paper. of sensors, each of which receives a partial measurement of the state at every time-step. Specifically, the -th sensor has access to a measurement of the state, given by
where and . We use to represent the collective measurement vector, and to denote the collection of the sensor observation matrices. These sensors are represented as nodes of an underlying directed communication graph which governs the information flow between the sensors.
Each node is capable of exchanging information with its neighbors and performing computational tasks. The goal of each node is to estimate the entire system state based on its respective (limited) state measurements and the information obtained from neighbors. This is known as the distributed state estimation problem.
In this paper, our objective is to design a distributed algorithm that guarantees asymptotic reconstruction of the entire state at each sensor node.222The problem is formally stated in Section 2.2. For much of the paper, we will focus on developing theory for linear time-invariant systems and time-invariant directed communication graphs. In Section 7, however, we shall establish that our proposed framework can be extended to account for certain types of time-varying networks that may arise as a consequence of intermittent communication link failures.
The paper is organized as follows. In Section 2, we formally describe the problem, discuss related work and summarize our contributions. Some preliminary ideas and terminology required for subsequent analysis are presented in Section 3. Section 4 highlights the key ideas of our distributed estimation scheme via a simple illustrative example. In Section 5, we solve the most general version of the problem whereas in Section 6 we provide a solution strategy for a simpler variant of the original problem that enjoys several implementation benefits. We discuss the extension of our framework to time-varying networks in Section 7 and provide a simulation example in Section 8. Conclusions and avenues for future work are presented in Section 9.
2 System Model
A directed graph is denoted by , where is the set of nodes and represents the edges. An edge from node to node , denoted by , implies that node can transmit information to node . The neighborhood of the -th node is defined as The notation is used to denote the cardinality of a set . Throughout the rest of this paper, we use the terms ‘nodes’ and ‘sensors’ interchangeably.
The set of all eigenvalues of a matrix is denoted by . The set of all marginally stable and unstable eigenvalues of a matrix is denoted by . For a matrix , we use and to denote the algebraic and geometric multiplicities, respectively, of an eigenvalue . An eigenvalue is said to be simple if . For a set of matrices, we use the notation to refer to a block diagonal matrix with the matrix as the -th block-diagonal entry. For a set , and a matrix , we define . We use the star notation to avoid writing matrices that are either unimportant or that can be inferred from context. We use to indicate an identity matrix of dimension .
2.2 Problem Formulation
Consider the LTI system given by (1), the measurement model specified by (2), and a predefined directed communication graph , where represents the set of nodes (or sensors). Each node maintains an estimate of the state of system (1), and updates such an estimate based on information received from its neighbors and its local measurements (if any). To formally define the problem under study, we use the following terminology.
Definition 1 (Distributed Observer).
A set of state estimate update and information exchange rules is called a distributed observer if , i.e., the state estimate maintained by each node asymptotically converges to the true state of the plant.
There are various technical challenges associated with constructing a distributed observer. First, if the pair is not detectable for some (or all) , then the corresponding nodes cannot estimate the true state of the plant based on their own local measurements, thereby dictating the need to exchange information with other nodes. Second, this exchange of information is restricted by the underlying communication graph . With these challenges in mind, we address and solve the following problem in this paper.
There are a variety of approaches to construct distributed observers (as defined in Definition 1) that have been proposed in the literature, which we will now review. After that, we will summarize how our approach differs from the existing approaches, before delving into the details of our construction.
2.3 Related Work
The papers [3, 4, 5] consider distributed estimation of scalar stochastic dynamical systems over general graphs; in these works, it is typically assumed that each node receives scalar local observations, leading to local observability at every node. The papers [6, 7] considered a version of this problem where the underlying communication graph is assumed to be complete. For more general stochastic systems, the Kalman filtering based approach to solving the distributed estimation problem has been explored by several researchers. The approach proposed in [8, 9, 10] relies on a two-step strategy - a Kalman filter based state estimate update rule, and a data fusion step based on average-consensus. The stability and performance issues of this method have been investigated in [11, 12]. A drawback of this method (and the ones in [4, 13, 14]), stems from the fact that they require a (theoretically) infinite number of data fusion iterations between two consecutive time steps of the plant dynamics in order to reach average consensus, thereby leading to a two-time-scale algorithm. More recently, finite-time data fusion has been studied in  and . Although an improvement over the infinite-time data fusion case, these methods still rely on a two-time-scale strategy. In , the authors employ an LMI-based approach for obtaining sub-optimal filter gains that minimize a quadratic cost function of the covariance matrices of the estimation errors.
In  and , the authors consider the distributed observer design problem for undirected graphs; in these works, they propose a scalar-gain estimator that runs on a single-time-scale.333By a single-time-scale algorithm, we imply an algorithm where each node operates at the same time-scale as the plant, and updates its estimate and transmits information to neighbors only once in each time-step. They introduce the notion of “Network Tracking Capacity” (NTC), a measure of the most unstable dynamics (in terms of the 2-norm of the state matrix) that can be estimated with bounded mean-squared error under their scheme. However, the tight coupling between the network and the plant dynamics typically limits the set of unstable eigenvalues that can be accommodated by their method without violating the constraints imposed upon the range of the scalar gain parameter. In , the author approaches the observer design problem from a geometric perspective and provides separate necessary and sufficient conditions for consensus-based distributed observer design. In [21, 22, 23, 24], the authors use single-time-scale algorithms, and work under the broadest assumptions, namely that the pair is detectable, where represents the system matrix, and is the collection of all the node observation matrices. In all of these works, the authors rely on state augmentation444In these works, some sensor nodes maintain observers of dimension larger than that of the state of the plant; hence, such observers are referred to as augmented observers, and the state they estimate is referred to as an augmented state. for casting the distributed estimation problem as a problem of designing a decentralized stabilizing controller for an LTI plant, using the notion of fixed modes [25, 26]. Specifically, in , the authors relate the distributed observer design problem for directed networks to the detectability of certain strongly connected clusters within the network, and provide a single necessary and sufficient condition for their scheme.
2.4 Summary of Contributions
In this paper, we provide a new approach to designing distributed observers for LTI dynamical systems. Specifically, we use the following simple, yet key observation - for each node, there may be certain portions of the state that the node can reconstruct using only its local measurements. The node thus does so. For the remaining portion of the state space, the node relies on a consensus-based update rule. The key is that those nodes that can reconstruct certain states on their own act as “root nodes” (or “leaders”) in the consensus dynamics, leading the rest of the nodes to asymptotically estimate those states as well. These ideas, in a nutshell, constitute the essence of our distributed estimation strategy.
We begin by considering the most general category of systems and graphs (taken together) for which a distributed observer can be constructed, and develop an estimation scheme that enjoys the following appealing features simultaneously, thereby differentiating our work from the existing literature discussed in Section 2.3: i) it provides theoretical guarantees regarding the design of asymptotically stable estimators; (ii) it results in a single-time-scale algorithm; (iii) it does not require any state augmentation; (iv) it requires only state estimates to be exchanged locally; and (v) it works under the broadest conditions on the system and communication graph. Subsequently, for a certain subclass of systems and communication graphs, we provide a simpler fully distributed estimation scheme (at both design- and run-time) for achieving asymptotic state reconstruction. Finally, we show that our proposed framework can be extended to guarantee asymptotic state reconstruction in the presence of communication losses that lead to time-varying networks.
Some of the results from Section 5 of the paper appeared (without proofs) in . This journal paper substantially expands upon the conference paper by providing full proofs of all results, a detailed analysis of classes of systems and graphs that allow efficient distributed implementations (at both the design- and run-time phases), an analysis of the robustness of our general framework to communication losses, and examples and simulations to complement the proposed theory.
Before we proceed with a formal analysis of the problem of designing a distributed observer, we first identify the main consideration that shall dictate our solution strategy, namely, the relationship between the measurement structure of the nodes and the underlying communication graph. To classify sets of systems and graphs based on this relationship, we need to first establish some notation. Accordingly, for each node , we denote the detectable and undetectable eigenvalues555Given a pair , an eigenvalue is said to be detectable if . Each stable eigenvalue of is by default considered to be detectable. of by the sets and , respectively. We define . Next, we introduce the notion of root nodes.
Definition 2 (Root nodes).
For each , the set of nodes that can detect is denoted by , and called the set of root nodes for .666Throughout the paper, for the sake of conciseness, we use the terminology ‘node can detect eigenvalue ’ to imply that .
We also recall the definition of a source component of a graph .
Definition 3 (Source Component).
Given a directed graph , a source component is defined as a strongly connected component of such that there are no edges from to .
Let there be source components of , denoted by . The subsystem associated with the -th source component is given by the pair . For the subsequent development, it should be noted that by a system , we refer to the matrix in equation (1), and the matrix containing each of the measurement matrices given by (2). Then, we classify systems and graphs based on the following two conditions.
A system and graph are said to satisfy Condition 1 if the sub-system associated with every source component is detectable, i.e., the pair is detectable .
A system and graph are said to satisfy Condition 2 if for each unstable or marginally stable eigenvalue of the plant, there exists at least one root node within each source component, i.e., for all and all , there exists , such that .777Note that given a source component, Condition does not necessarily imply the existence of a single node within such a component that can simultaneously detect all the unstable and marginally stable eigenvalues of the system via its own measurements.
It is trivial to see that if a system and graph satisfy Condition 2, they also satisfy Condition 1. To see that the converse is not true in general, consider the -node network in Figure 1, and the following model:
From Figure 1, we see that the network has two source components, namely, the strong component formed by nodes and (), and the isolated node (). Clearly, each of the pairs and are detectable. Thus, the system is detectable from each of the two source components. It follows that this system and graph satisfy Condition 1. However, neither node nor node can detect the eigenvalue based on just their own measurements, i.e., there does not exist a root node for within source component . Thus, this system and graph do not satisfy Condition 2.
In , the authors identified that a distributed observer cannot be constructed (regardless of the state update or exchange rules) if the system and graph do not satisfy Condition 1. They then designed a distributed observer for the class of systems and graphs satisfying Condition 1 by constructing augmented state observers (i.e., observers of dimension larger than that of the system) drawing upon connections to decentralized control theory. Here, we present an alternate and more direct design approach, and in the process, establish that it is possible to design a distributed observer without state augmentation for this (most general) class of systems and graphs.888The exact structure of our distributed observer presented in Section 5.5 illustrates that the dimension of the internal state/estimate maintained by a given node is equal to the dimension of the state . Before we delve into the specifics of the distributed observer design for systems and graphs satisfying Condition 1, we present a simple motivating example which serves to build intuition for the more complicated scenarios.999At this point, it is worth mentioning that although the distributed observer that we shall design for systems and graphs satisfying Condition 1 will also work for systems and graphs satisfying Condition 2, we will later propose an alternate scheme with various implementation benefits for the latter class of systems and graphs.
4 Illustrative Example
Consider a scalar unstable plant with dynamics given by . The plant is monitored by a network of nodes, as depicted by Figure 2. Node has a measurement given by , whereas nodes and have no measurements. Given this plant and network model, we wish to design a distributed observer. The commonly adopted approach in the literature is to develop a consensus-based state estimate update rule for each node in the network [21, 22, 23, 18, 19]. Here, we make the following observation: since node 1 can detect the eigenvalue of the plant based on its own measurements, it can run a Luenberger observer for estimating , without requiring information from its neighbors. Specifically, the following Luenberger observer allows node to estimate and predict the state:
Here, is the estimate of maintained by node at time-step . Now suppose nodes and update their respective estimates of as follows: . Since based on the Luenberger observer dynamics given by (4), it is easy to see that the estimates of nodes 2 and 3 also converge to the true state . This simple example illustrates the following key observations. (i) It is not necessary for every node in the network to run consensus dynamics for estimating the state. More generally, a node needs to run consensus for estimating only the portion of the state vector that is not locally detectable. The rest of the state can be estimated via appropriately designed Luenberger observers. (ii) An inspection of the observable subspace of each node guides the decision of participating (or not participating) in consensus for the example we considered. For more general system and measurement matrices, we shall rely on appropriate similarity transformations which shall reveal what a node can or cannot observe. (iii) Although node is in a position to receive information from node , it chooses not to listen to any of its neighbors. This pattern of information flow results in a special Directed Acyclic Graph (DAG) of the original network, rooted at node 1. In the DAG constructed in the illustrative example, node can be viewed as the source of information for the state , and the DAG structure can be viewed as the medium for transmitting information from the source to the rest of the network, without corrupting the source itself (this is achieved in this example by ignoring the edge from node to node ). Under this approach, note that every node maintains an observer of dimension , which is equal to the dimension of the state (i.e., there is no state augmentation). Based on these observations, we are now ready to extend the ideas conveyed by this simple example for tackling more general systems and networks.101010Notice that the original network in this illustrative example has only one source component comprised of the nodes and , and node is a root node for (node can detect ). Thus, the system and graph illustrated in this example satisfy Condition 2, and hence also Condition 1.
5 Estimation Scheme for systems and graphs satisfying Condition
In this section, we develop a distributed observer for systems and graphs satisfying Condition 1. For presenting the key ideas while reducing notational complexity, we shall make the following assumption.
The graph is strongly connected, i.e., there exists a directed path from any node to any other node , where .
Later, we shall argue that the development can be easily extended to any general directed network. For now, it suffices to say that any directed graph can be decomposed into strong components, some of which are source components (strong components with no incoming edges from the rest of the network); the strategy that we develop here for a strongly connected graph will be employed within each source component.
Since we are focusing on systems and graphs satisfying Condition 1, it follows that under Assumption 1, the pair is detectable (as a strongly connected graph is one single source component).
Note that under Condition with a strongly connected graph, one might consider the possibility of aggregating all the sensor measurements at a central node and constructing a centralized Luenberger observer, leveraging the fact that is detectable. However, for large networks, the routing of measurement information to and from such a central node via multiple hops would induce delays. A distributed approach (such as the one considered in this paper) alleviates such a difficulty. Additionally, as we discuss in Section 7, the general framework developed in this paper allows for extensions to communication losses and sensor failures, which are typical benefits expected of a distributed algorithm.
We are now in a position to detail the steps to be followed for designing a distributed observer for systems and graphs satisfying Condition 1. We start by providing a generalization of the Kalman observable canonical form to a setting with multiple sensors.
5.1 Multi-Sensor Observable Canonical Decomposition
Given a system matrix and a set of sensors where the -th sensor has an observation matrix given by , we introduce the notion of a multi-sensor observable canonical decomposition in this section. The basic philosophy underlying such a decomposition is as follows: given a list of indexed sensors, perform an observable canonical decomposition with respect to the first sensor. Then, identify the observable portion of the state space with respect to sensor 2 within the unobservable subspace of sensor 1, and repeat the process until the last sensor is reached. Thus, one needs to perform observable canonical decompositions, one for each sensor, with the last decomposition revealing the portions of the state space that can and cannot be observed using the cumulative measurements of all the sensors. The details of the multi-sensor observable canonical decomposition are captured by the proof of the following result (given in Appendix A).
Given a system matrix , and a set of sensor observation matrices , define . Then, there exists a similarity transformation matrix which transforms the pair to , such that
Furthermore, the following properties hold: (i) the pair is observable ; and (ii) the matrix describes the dynamics of the unobservable subspace of the pair .
Figure 3 illustrates the steps of the multi-sensor observable canonical decomposition for a sensor network with nodes. The first step involves an observable canonical decomposition of the pair () via the matrix . Next, reveals the portion of the unobservable subspace of () that can be observed using the observation matrix . Finally, reveals the portion of the unobservable subspace of () that can be observed using the observation matrix . For this example, we have . In the following section, we discuss how the multi-sensor observable canonical decomposition is applicable to the problem of designing a distributed observer for systems and graphs satisfying Condition 1.
Note that while describing the multi-sensor observable canonical decomposition, we did not specify any rule for indexing the sensors. This is precisely because the technique we propose solves Problem 1 regardless of the way the sensors are indexed, as long as the system and graph satisfy Condition 1. However, the question of appropriately ordering the sensors (or including redundancy) will become important when dealing with stochastic systems or with sensor failures. We discuss these issues later in the paper.
5.2 Observer Design
Using the matrix identified in Proposition 1, we perform the coordinate transformation to obtain
Here, is precisely the unobservable portion of the state , with respect to the pair . We call the -th sub-state, and the unobservable sub-state. Notice that based on the multi-sensor observable canonical decomposition, there is a one-to-one correspondence between a node and its associated sub-state . Accordingly, node is viewed as the source of information of its corresponding sub-state , and is tasked with the responsibility of estimating this sub-state. For each of the sub-states, we thus have a unique source of information (based on the initial labeling of the nodes). However, there is no unique source of information for the unobservable sub-state , as this portion of the state does not correspond to the observable subspace of any of the nodes in the network. Each node will thus maintain an estimate of , which it updates as a linear function of its own estimates of each of the sub-states .
It should be noted that a given sub-state in equation (7) might be of zero dimension (i.e., the sub-state can be empty). For instance, this can happen if its corresponding source of information, namely node , has no measurements, i.e., if .
where the matrices describe the coupling that exists between the unobservable sub-state and each of the sub-states . Define as the estimate of the -th sub-state maintained by the -th node. The estimation policy adopted by the -th node is as follows - it uses a Luenberger-style update rule for updating its associated sub-state estimate , and a consensus based scheme for updating its estimates of all other sub-states , where Based on the dynamics (8), the Luenberger observer at node is constructed as
where is a gain matrix which needs to be designed. For estimation of the -th sub-state, where , the -th node again mimics the first equation in (8), but this time relies on consensus dynamics of the form
where is the weight the -th node associates with the -th node, for the estimation of the -th sub-state. The weights are non-negative and satisfy
In equation (11), the first term is a standard consensus term, while the second term has been introduced specifically to account for the coupling that exists between a given sub-state and sub-states to (as given by (8)). Let denote the estimate of the unobservable sub-state maintained by the -th node. Mimicking equation (9), each node uses the following rule to update :
5.3 Error Dynamics at the -th Node
5.4 Analysis of the Estimation Scheme for Systems and Graphs Satisfying Condition
In this section, we present our main result, formally stated as follows.
Consider the composite error in estimation of sub-state by all of the nodes in , defined as
We will prove that converges to zero asymptotically (recall that there are precisely nodes in the network, each responsible for estimating a certain sub-state). We prove by induction on . Consider the base case , i.e., the estimation of the first sub-state. Let the index set represent a topological ordering111111Such an ordering results when a standard Breadth-First Search (BFS)  algorithm is applied to the graph , with node as the root node of the tree. Specifically, the order represents the order in which the nodes are added to the spanning tree when the BFS algorithm is implemented, i.e., node would be added first, followed by node and so on. This ordering naturally leads to a lower triangular adjacency matrix for the constructed spanning tree. consistent with a spanning tree rooted at node 1 (the source of information for sub-state 1). Note that based on Assumption 1, it is always possible to find such a spanning tree. Next, consider the composite error vector
where the entries of the weight matrix are populated by the appropriate weights defined by equation (15) (note that and is the first column of ). Notice that . By construction, the pair is observable. Thus, it is always possible to find a gain matrix such that is Schur stable. Next, we impose the constraint that for the estimation of sub-state 1, non-zero consensus weights are assigned to only the branches of the spanning tree consistent with the ordering , i.e., a node listens to only its parent in such a tree. In this way, becomes lower triangular with eigenvalues equal to zero, without violating the stochasticity condition imposed on by equation (12). We conclude that by an appropriate choice of consensus weights, we can achieve (even if .121212Here, we use the result that if and , then the eigenvalues of the Kronecker product are the numbers .
Thus, can be made Schur stable and hence , implying (one is just a permutation of the other). Thus, the base case is proven. Next, suppose that converges to zero asymptotically , where . Consider the following composite error vector for the -th sub-state:
where the index set represents a topological ordering of the nodes of to obtain a spanning tree rooted at node (the source of information for sub-state ), and From the error dynamics equations given by (14) and (15), we obtain
By following the same train of logic as the base case, one concludes that can be made Schur stable by appropriate choices of the observer gain matrix , and consensus weight matrix (note that and is the first column of ). Specifically, non-zero weights are assigned in only on the branches of the tree rooted at node , consistent with the topological ordering. Notice that is simply a permutation of the rows of (permuted to match the order of indices in ). Further, based on our induction hypothesis, converges to zero asymptotically (since ). Thus, by Input to State Stability (ISS), we conclude that and hence , converges to zero asymptotically. We have thus proven that the composite estimation error for every sub-state asymptotically approaches zero, i.e., .
Finally, consider the error in estimation of the unobservable sub-state (given by equation (16)). As the system and graph under consideration satisfy Condition 1 and Assumption 1, it must be that the pair is detectable. Thus, based on Proposition 1, the matrix in (16) must be stable. Invoking ISS, we have that . Thus, every node in the network can asymptotically estimate , and hence , as . ∎
5.5 A Compact Representation of the Proposed Observer
In this section, we combine the update equations (10), (11) and (13) to obtain a compact representation of our distributed observer. To do so, we need to first introduce some notation. Accordingly, let be the matrix that extracts the -th sub-state from the transformed state vector , i.e., . Similarly, let be such that . Define . Next, notice that the transformed system matrix in equation (5) can be written as , where , and is a block lower-triangular matrix given by . Let (where ) be the vector of weights node associates with a neighbor for the estimation of the transformed state . Based on our estimation scheme, note that at any given time-step , node does not use the estimates received from its neighbors at time-step for estimating and , and hence these weight vectors assume the following form: . Also, notice that the element is not present in the vector if the -th sub-state is empty (i.e., of dimension ). Similarly, let be a vector with a ‘’ in the elements corresponding to the -th sub-state and the unobservable sub-state , and zeroes at all other positions. Finally, defining , using equations (10), (11) and (13), and noting that , we obtain the following overall state estimate update rule at node :
where denotes the estimate of the state maintained by node , and
From the structure of our overall estimator at node , as represented by equation (24), it is easy to see that the estimator maintained at each node has dimension equal to (i.e., equal to that of the state). Thus, our approach alleviates the need to construct augmented observers such as those considered in [23, 24].
Note that all the transformation and gain matrices appearing in (24) can be computed offline during a centralized design phase. Thus, although the observer design and the subsequent analysis were done in the coordinate system, no inversion from to is necessary while implementing (24) during run-time, i.e., the nodes directly exchange their estimates of the actual state , and not .
5.6 Summary of the Estimation Scheme for Systems and Graphs Satisfying Condition
The proposed distributed observer scheme for systems and graphs satisfying Condition (under the assumption that the graph is strongly connected) can be broadly decomposed into two main phases, namely the design phase and the distributed estimation phase. For clarity, we briefly enumerate the steps associated with each of these phases.
Each node of the graph is assigned a unique integer between to . Based on this numbering, the multi-sensor observable canonical decomposition (as outlined in the proof of Proposition 1) is performed, yielding the state .
Based on this transformation, each node is associated with a sub-state of that it is responsible for estimating. Recall that there are precisely sub-states, one corresponding to each node in the network; some of these sub-states might be empty.
For the estimation of a given sub-state, we construct a spanning tree rooted at the specific node which acts as the source of information for that sub-state. The resulting spanning tree guides the construction of the consensus weight matrix to be used for the estimation of that particular sub-state. We construct one spanning tree for the estimation of each non-empty sub-state.
Based on the constructed consensus weight matrices, and the Luenberger observer gains , the matrices , and in (24) are computed for each node .
Estimation Phase (Run-time):
Each node employs a Luenberger observer for constructing an estimate of its corresponding sub-state, and runs consensus dynamics for estimating the sub-states corresponding to the remaining nodes in the network. Summarily, a node implements (24) for estimating .