Set-Membership Filtering-Based Leader-Follower Synchronization of Discrete-time Linear Multi-Agent Systems

Most of the studies in the existing literature regarding multi-agent synchronization assume perfect modeling of the system, i.e., the mathematical description of the system is assumed to be perfect (see, for example, Refs. [8, 10, 11, 16, 17, 12]). However, this is incompatible with real-world engineering problems where the dynamical system is subject to unknown input disturbances, parametric uncertainties, unmodeled dynamics, etc. Because of these phenomena, the state of the system can not be known precisely and should be considered uncertain. For an overview on synchronization in uncertain multi-agent systems, see Refs. [6, 7, 9, 18] and references therein. Among the abovementioned phenomena, we focus on unknown input disturbances in this paper. Apart from the perfect system modeling assumption, perfect information regarding the states of the agents (full-state feedback) are assumed to be available for synchronization protocol design in a large number of studies (see, for example, Refs. [16, 10, 8]). Again, this assumption does not hold for practical applications where measured outputs (some function of the states), subject to output disturbances, are available. Although observer-based approaches, without considering output disturbances, have been investigated in the literature (see, for example, Refs. [4, 17, 9]), a state estimation or filtering-based approach would be more suitable to address the effects of both input and output disturbances in the synchronization problem (see, for example, Ref. [5]).

The Kalman filter [19], which is one of the most widely-studied stochastic filtering techniques, assumes that the input and output disturbances are Gaussian noises with known statistical properties. However, this assumption is difficult to validate in practice and might not hold for real-world systems. Therefore, it seems more realistic to assume the disturbances to be unknown-but-bounded [20, 21]. This approach leads to the concept of set-membership or set-valued state estimation (or filtering) (see, for example, Refs. [22, 23, 20, 24, 25, 21, 26]), which is deterministic and more suited to several practical applications [24]. Although the set-theoretic or set-valued concepts for synchronization have been investigated in the existing literature (see, for example, Refs. [27, 28, 29, 30]), studies explicitly utilizing set-membership or set-valued estimation techniques for the multi-agent synchronization problem are relatively rare (see, for instance, Refs. [13, 14]), despite the practical significance of this class of estimators/filters. Recently, a leader-follower synchronization protocol using set-membership estimation techniques was put forward in Ref. [31]. The synchronization objective in Ref. [31] was to construct ellipsoids centered at the leader’s state trajectory that contained states of the agents. This, however, is different from the concept of conventional leader-follower synchronization where the objective is to make the states of the agents converge to the leader’s state trajectory. To the best of the authors’ knowledge, set-membership estimation techniques have not been employed for the conventional leader-follower synchronization problem in the existing literature.

The technical approach and contributions of the paper are summarized in the following list.

• $\bullet$

  We develop a set-membership estimation-based (conventional) leader-follower synchronization protocol for high-order discrete-time linear multi-agent systems with the agents subject to unknown-but-bounded input and output disturbances. To the best of our knowledge, this is a novel contribution.

• $\bullet$

  Specifically, we focus on the ellipsoidal state estimation problem and adopt the terminology set-membership filter (SMF). Based on the approaches given in Refs. [23, 25, 24, 26], we convert the set estimation problem into a recursive algorithm that requires solutions to two semi-definite programs (SDPs) at each time-step.

• $\bullet$

  We consider each agent to be equipped with the SMF that estimates the state of the agent. Further, we assume that the agents are able to share the state estimate information with the neighbors locally and that information is utilized in the local control synthesis for synchronization. The local controller for each agent is chosen based on an $H_{2}$ type Riccati-based approach [16]. We show that the global error system is input-to-state stable (ISS) with respect to the input disturbances and estimation errors. Sufficient conditions for input-to-state stability are provided in terms of the system matrices of the agents, the Riccati design, and the interaction graph.

• $\bullet$

  Further, we calculate an upper bound on the norm of the global disagreement error and show that it decreases monotonically, converging to a limit as time goes to infinity.

The rest of this paper is organized as follows. Section 2 describes the preliminaries required for the SMF design. The formulation of the SMF is given in Section 3. The control input synthesis and related results for synchronization are given in Section 4. Finally, Section 5 includes the simulation examples and Section 6 presents the concluding remarks.

At each time-step $k\in\mathbb{Z}_{\star}$, upon receiving the measured output $\bm{y}_{k}$ with $\bm{v}_{k}\in\mathcal{E}(\bm{0}_{\bar{v}},\bm{R}_{k})$ and given $\bm{x}_{k}\in\mathcal{E}(\hat{\bm{x}}_{k|k-1},\bm{P}_{k|k-1})$, the objective is to find the optimal correction ellipsoid, characterized by $\hat{\bm{x}}_{k|k}$ and $\bm{P}_{k|k}$, such that $\bm{x}_{k}\in\mathcal{E}(\hat{\bm{x}}_{k|k},\bm{P}_{k|k})$. The corrected state estimate is given by

where $\bm{L}_{k}$ is the filter gain. Since $\bm{x}_{k}\in\mathcal{E}(\hat{\bm{x}}_{k|k-1},\bm{P}_{k|k-1})$, there exists a $\bm{z}_{k|k-1}\in\mathbb{R}^{\bar{n}}$ with $|\bm{z}_{k|k-1}|\leq 1$ such that

where $\bm{E}_{k|k-1}$ is the Cholesky factorization of $\bm{P}_{k|k-1}$, i.e., $\bm{P}_{k|k-1}=\bm{E}_{k|k-1}\bm{E}_{k|k-1}^{\textnormal{T}}$ [23, 24].

At each time-step $k\in\mathbb{Z}_{\star}$, given $\bm{x}_{k}\in\mathcal{E}(\hat{\bm{x}}_{k|k},\bm{P}_{k|k})$ and $\bm{w}_{k}\in\mathcal{E}(\bm{0}_{\bar{w}},\bm{Q}_{k})$, the objective is to find the optimal prediction ellipsoid, characterized by $\hat{\bm{x}}_{k+1|k}$ and $\bm{P}_{k+1|k}$, such that $\bm{x}_{k+1}\in\mathcal{E}(\hat{\bm{x}}_{k+1|k},\bm{P}_{k+1|k})$ where the predicted state estimate is given by

Again, since $\bm{x}_{k}\in\mathcal{E}(\hat{\bm{x}}_{k|k},\bm{P}_{k|k})$, we have

where $\bm{P}_{k|k}=\bm{E}_{k|k}\bm{E}_{k|k}^{\textnormal{T}}$ and $|\bm{z}_{k|k}|\leq 1$. Initialization is provided by $\hat{\bm{x}}_{0|-1}=\hat{\bm{x}}_{0}$ and $\bm{P}_{0|-1}=\bm{P}_{0}$ [19].

Consider a multi-agent system consisting of $N$ agents. The communication topology of the multi-agent system can be represented by a graph $\mathscr{G}=(\mathscr{V},\mathscr{E})$ where $\mathscr{V}=\{1,2,\dots,N\}$ is a nonempty node set and $\mathscr{E}\subseteq\mathscr{V}\times\mathscr{V}$ is an edge set of ordered pairs of nodes, called edges. Node $i$ in the graph represents agent $i$. We consider simple, directed graphs in this paper. The edge $(i,j)$ in the edge set of a directed graph denotes that node $j$ can obtain information from node $i$, but not necessarily vice versa. If an edge $(i,j)\in\mathscr{E}$, then node $i$ is a neighbor of node $j$. The set of neighbors of node $i$ is denoted as $\mathscr{N}_{i}$.

The adjacency matrix $\boldsymbol{\mathscr{A}}=[a_{ij}]\in\mathbb{R}^{N\times N}$ of a directed graph $(\mathscr{V},\mathscr{E})$ is defined such that $a_{ij}$ is a positive weight if $(j,i)\in\mathscr{E}$, and $a_{ij}=0$ otherwise. The graph Laplacian matrix $\boldsymbol{\mathscr{L}}$ is defined as $\boldsymbol{\mathscr{L}}=\boldsymbol{\mathscr{D}}-\boldsymbol{\mathscr{A}}$ where $\boldsymbol{\mathscr{D}}=[d_{ij}]\in\mathbb{R}^{N\times N}$ is the in-degree matrix with $d_{ij}=0,i\neq j$, and $d_{ii}=\sum_{j=1}^{N}a_{ij},i=1,2,\dots,N$. A directed path is a sequence of edges in a directed graph of the form ($i_{1}$, $i_{2}$), ($i_{2}$, $i_{3}$), $\dots$. The graph $\mathscr{G}$ contains a (directed) spanning tree if there exists a node, called the root node, such that every other node in $\mathscr{V}$ can be connected by a directed path starting from that node.

We consider $N$ agents connected via a directed graph and a leader. Agent $i$ ($i=1,2,\dots,N$) is a dynamical system of the form

where $\bm{x}^{(i)}_{k}\in\mathbb{R}^{n}$, $\bm{u}_{k}^{(i)}\in\mathbb{R}^{m}$, $\bm{y}^{(i)}_{k}\in\mathbb{R}^{p}$, $\bm{w}_{k}^{(i)}\in\mathbb{R}^{w}$, $\bm{v}_{k}^{(i)}\in\mathbb{R}^{v}$ are the state, control input, measured output, input and output disturbances for agent $i$, respectively. Clearly, the system described by Eq. (16) is in the form of the system described by Eq. (1), with the time-varying matrices replaced by the constant matrices. Next, we modify Assumption 1 and impose the following assumptions on the dynamics of agent $i$ ($i=1,2,\dots,N$).

Under this assumption, agent $i$ ($i=1,2,\dots,N$) employs the SMF in Algorithm 1 to estimate its own state. Now, we introduce the following assumption on the system matrices of the agents.

We consider the leader to be a system of the form

where $\bm{x}^{(0)}_{k}\in\mathbb{R}^{n}$ is the leader’s state and $\bm{y}^{(0)}_{k}$ is the output. Note that the leader is a virtual system that generates the reference trajectory for the agents $i=1,2,\dots,N$ to track. We define the local neighborhood tracking errors, using the corrected state estimates of the agents, as

where $g_{i}\geq 0$ are the pinning gains, $\hat{\bm{x}}^{(i)}_{k|k}$ and $\hat{\bm{x}}^{(j)}_{k|k}$ are the corrected state estimates of agent $i$ and $j$, respectively. If agent $i$ is pinned to the leader, we take $g_{i}>0$. Now, we choose the control input of agent $i$ as [16]

where $c>0$ is a coupling gain and $\bm{K}$ is a control gain matrix to be discussed subsequently. Hence, the global dynamics of the $N$ agents can be expressed as

with $\bm{x}^{(g)}_{k}=\text{col}[\bm{x}^{(1)}_{k},\dots,\bm{x}^{(N)}_{k}]$, $\bm{w}^{(g)}_{k}=\text{col}[\bm{w}^{(1)}_{k},\dots,\bm{w}^{(N)}_{k}]$, and

where $\boldsymbol{\mathcal{G}}=\text{diag}(g_{1},g_{2},....,g_{N})$ is the matrix of pinning gains, $\hat{\bm{x}}^{(g)}_{k|k}=\text{col}[\hat{\bm{x}}^{(1)}_{k|k},\dots,\hat{\bm{x}}^{(N)}_{k|k}]$, and $\bar{\bm{x}}^{(0)}_{k}=\left(\bm{1}_{N}\otimes\bm{x}^{(0)}_{k}\right)$. Note that the superscript $(g)$ is utilized to denote the global variables. Now, using Eq. (5) for each agent’s corrected state estimates, we can express $\bm{x}^{(g)}_{k}$ as

where $\bm{E}^{(g)}_{k|k}=\text{diag}(\bm{E}^{(1)}_{k|k},\dots,\bm{E}^{(N)}_{k|k})$, $\bm{z}^{(g)}_{k|k}=\text{col}[\bm{z}^{(1)}_{k|k},\dots,\bm{z}^{(N)}_{k|k}]$. Note that $\bm{E}^{(i)}_{k|k}\left(\bm{E}^{(i)}_{k|k}\right)^{\text{T}}=\bm{P}^{(i)}_{k|k}$ where $\bm{P}^{(i)}_{k|k}$ is the correction ellipsoid shape matrix for agent $i$ and $|\bm{z}^{(i)}_{k|k}|\leq 1$ for $i=1,\dots,N$. Our next assumption is regarding the interaction graph.

The global disagreement error [16] is defined as $\bm{\delta}^{(g)}_{k}=\bm{x}^{(g)}_{k}-\bar{\bm{x}}^{(0)}_{k}$. Utilizing Eqs. (18)-(20), we express the global error system as

where

with $\bm{\Gamma}=(\bm{I}_{N}+\boldsymbol{\mathscr{D}}+\boldsymbol{\mathcal{G}})^{-1}(\boldsymbol{\mathscr{L}}+\boldsymbol{\mathcal{G}})$. Now, we recall the following technical result from Ref. [16].

If the matrix $\bm{A}$ is unstable or marginally stable, then Lemma 4.5 requires Assumption 4 with the pair $(\bm{A},\bm{B})$ stabilizable [16]. Using Theorem 2 in Ref. [16], $c$ and $\bm{K}$ are chosen such that $\rho(\bm{A}_{c})<1$. To this end, we state the following result.

If $\bm{B}$ is full column rank, Eq. (23) has a positive definite solution $\boldsymbol{\mathcal{P}}$ only if the pair $(\bm{A},\bm{B})$ is stabilizable [16]. In this regard, Assumption 3 is pertinent. Next, we introduce the notion of input-to-state stability in the following definition.

We state the main result of this section in the next theorem.

Theorem 4.9 implies that the global disagreement error remains bounded under the proposed synchronization protocol.

Next, we state the following result based on Theorem 4.9 where $p_{0}$ and $\bar{q}$ are as in Assumption 2.