(SMF). Based on the approaches given in Refs. [23][24][25][26], we convert the set estimation problem into a recursive algorithm that requires solutions to two semidefinite programs (SDPs) at each time-step. 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 Riccatibased 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. 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 Sec. 3. The control input synthesis and related results for synchronization are given in Sec. 4. Finally, Sec. 5 includes the simulation example, and Sec. 6 presents the concluding remarks.

Notations and Definitions. The symbol Z ? denotes the set of non-negative integers. For a square matrix X, the notation X > 0 (respectively, X ! 0) means X is symmetric and positive definite (respectively, positive semidefinite). Similarly, X < 0 (respectively, X 0) means X is symmetric and negative definite (respectively, negative semidefinite). Furthermore, q X ðÞdenotes the spectral radius of a square matrix X. For any matrix Y, r max ðYÞ stands for the maximum singular value of Y. C (a, b) denotes an open circle of radius b in the complex plane, centered at a 2 R. Notations diagðÁÞ; I n ; O n ; 1 n , and 0 n denote blockdiagonal matrices, the n  n identity matrix, the n  n null matrix, and the vector of ones and zeros of dimension n, respectively. For vectors x 1 ; x 2 ; …; x M , we have col½x 1 ; x 2 ; …; x M ¼½x T 1 x T 2 … x T M T . The symbol jÁj denotes standard Euclidean norm for vectors and induced matrix norm for matrices. For any function h : Z ? ! R n , we have jjhjj ¼ supfjh k j : k 2 Z ? g. This is the standard l 1 norm for a bounded h. Ellipsoids are denoted by Eðc; PÞ¼fx 2 R n : ðx À cÞ T P À1 ðx À cÞ 1g, where c 2 R n is the center of the ellipsoid and P > 0 is the shape matrix that characterizes the orientation and "size" of the ellipsoid in R n . Notations traceðÁÞ and rankðÁÞ denote trace and rank of a matrix, respectively, and denotes the Kronecker product. The superscript T means vector or matrix transpose.

DEFINITION 1 [ 32,33].

for each fixed t ! 0, the function bðÁ; tÞ is a class K function and for each fixed s ! 0, the function bðs; ÁÞ is decreasing and bðs; tÞ!0 as t !1.

Consider the discrete-time dynamical systems of the form

where

v is the output disturbance. Also, A, B, G, C, and D are system matrices of appropriate dimensions. Following are the assumptions for systems of the form given in Eq. ( 1). ASSUMPTION 1.

(1.1) The initial state x 0 is unknown. However, it satisfies x 0 2Eðx 0 ; P 0 Þ, where x0 is a given initial estimate and P 0 is known. (1.2) w k and v k are unknown-but-bounded for all k 2 Z ? . Also,

We intend to develop an SMF for systems of the form in Eq. ( 1), having a correction-prediction structure similar to the Kalman filter variants (see, for example, Ref. [19]). Note that the SMF design in this paper is motivated by the SMF developed by the authors in Ref. [34]. The filtering objectives are as follows, where the corrected and predicted state estimates at time-step k are denoted by xkjk and xkþ1jk , respectively [34].

Step. At each time-step k 2 Z ? , upon receiving the measured output y k with v k 2Eð0 v ; R k Þ and given x k 2Eðx kjkÀ1 ; P kjkÀ1 Þ, the objective is to find the optimal correction ellipsoid, characterized by xkjk and P kjk , such that x k 2Eðx kjk ; P kjk Þ. The corrected state estimate is given by

where L k is the filter gain. Since x k 2Eðx kjkÀ1 ; P kjkÀ1 Þ, there exists a z kjkÀ1 2 R n with jz kjkÀ1 j 1 such that

where E kjkÀ1 is the Cholesky factorization of P kjkÀ1 , i.e., P kjkÀ1 ¼ E kjkÀ1 E T kjkÀ1 [23,24].

Step. At each time-step k 2 Z ? , given x k 2 Eðx kjk ; P kjk Þ and w k 2Eð0 w ; Q k Þ, the objective is to find the optimal prediction ellipsoid, characterized by xkþ1jk and P kþ1jk , such that x kþ1 2Eðx kþ1jk ; P kþ1jk Þ where the predicted state estimate is given by

Again, since x k 2Eðx kjk ; P kjk Þ, we have

where P kjk ¼ E kjk E T kjk and jz kjk j 1. Initialization is provided by x0jÀ1 ¼ x0 and P 0jÀ1 ¼ P 0 [19].

Remark 1. As mentioned in the filtering objectives, we are interested in finding the optimal ellipsoids, i.e., the minimum-"size" ellipsoids, at each time-step. There are two criteria for the "size" of an ellipsoid in terms of its shape matrix: trace criterion and log-determinant criterion [23]. In this paper, we have considered the trace criterion (see Theorems 1 and 2), which represents the sum of squared lengths of semi-axes of an ellipsoid [23]. As a result, the corresponding optimization problems are convex (see the SDPs in Eqs. ( 6) and ( 8)). Alternatively, for minimum-volume ellipsoids, one can consider the log-determinant criterion. However, this would render the optimization problems nonconvex, and additional modifications might be required to restore convexity (see, for example, Ref. [23]).

In this section, we formulate the SDPs to be solved at each time-step for the SMF. As the SMF design is motivated by the one in Ref. [34], we have adopted the notations and relevant statements provided in Ref. [34]. First, we state the result that summarizes the filtering problem at the correction step. THEOREM 1. Consider the system in Eq. (1) where P kjkÀ1 and Hðs 1 ; s 2 Þ are given by

Furthermore, the center of the correction ellipsoid is given by the corrected state estimate in Eq. ( 2). Proof. Follows from the proof of Theorem 1 in Ref. [34].

Next, we state the technical result for the prediction step. THEOREM 2. Consider the system in Eq. ( 1) under the Assumption 1 with the state x k in the correction ellipsoid Eðx kjk ; P kjk Þ and

Then, the successor state x kþ1 belongs to the optimal prediction ellipsoid Eðx kþ1jk ; P kþ1jk Þ if there exist P kþ1jk > 0; s i ! 0; i ¼ 3; 4 as solutions to the following SDP: min

where P kjk and Wðs 3 ; s 4 Þ are given by

Furthermore, the center of the prediction ellipsoid is given by the predicted state estimate in Eq. ( 4). Proof. Follows from the proof of Theorem 2 in Ref. [34].

Interior point methods can be implemented to efficiently solve the SDPs in Eqs. ( 6) and ( 8) [35]. The recursive SMF algorithm is summarized in Algorithm 1.

This section describes local control input synthesis for the leader-follower synchronization. Results presented in this section are based on the results given in Ref. [16], and, to be consistent, we have adopted some of the terminologies and notations used in Ref. [16].

4.1 Graph-Related Preliminaries. Consider a multi-agent system consisting of N agents [1]. The communication topology of the multi-agent system can be represented by a graph G ¼ðV; EÞ, where V ¼f1; 2; …; Ng is a nonempty node set and E V  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 ði; jÞ2E, then node i is a neighbor of node j. The set of neighbors of node i is denoted as N i .

The adjacency matrix A ¼½a ij 2R NÂN of a directed graph ðV; EÞ is defined such that a ij is a positive weight if ðj; iÞ2E, and a ij ¼ 0 otherwise. The graph Laplacian matrix L is defined as

A directed path is a sequence of edges in a directed graph of the form (i 1 , i 2 ), (i 2 , i 3 ), …. The graph G contains a (directed) spanning tree if there exists a node, called the root node, such that every other node in V can be connected by a directed path starting from that node.

4.2 Synchronization: Formulation and Results. We consider N agents connected via a directed graph and a leader. Agent i (i ¼ 1; 2; …; N) is a dynamical system of the form

where

are the state, control input, measured output, input, and output disturbances for agent i, respectively. Clearly, the system described by Eq. ( 9) is in the form of the system described by Eq. ( 1). Next, we modify Assumption 1 and impose the following assumptions on the dynamics of agent i (i ¼ 1; 2; …; N). ASSUMPTION q and jR ðiÞ k j r for all k 2 Z ? with some q; r > 0. Under this assumption, agent i (i ¼ 1; 2; …; 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. ASSUMPTION 3. B is full column rank with the pair ðA; BÞ stabilizable.

We consider the leader to be a system of the form

where x ð0Þ k 2 R n are the leader's state and y ð0Þ k is the output. Note that the leader is a virtual system that generates the reference trajectory for the agents i ¼ 1; 2; …; N to track. We define the local neighborhood tracking errors, using the corrected state estimates of the agents, as

where g i ! 0 are the pinning gains, xðiÞ kjk and xðjÞ kjk 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] Algorithm 1 The SMF Algorithm

2: Find P kjk and L k by solving the SDP in Eq. ( 6). 3: Calculate xkjk using Eq. ( 2). Also, calculate E kjk using P kjk ¼ E kjk E T kjk . 4: Solve the SDP in Eq. ( 8) to obtain P kþ1jk . 5: Calculate xkþ1jk using Eq. (4). Compute E kþ1jk using

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

with

where G ¼ diagðg 1 ; g 2 ; …:; g N Þ is the matrix of pinning gains, ASSUMPTION 4 [ 16]. The interaction graph contains a spanning tree with at least one nonzero pinning gain that connects the leader and the root node.

The global disagreement error [16] is defined as

k . Utilizing Eqs. ( 11)-( 13), we express the global error system as

where

with

GÞ. Now, we recall the following technical result from Ref. [16]. Lemma 1 [16].

If the matrix A is unstable or marginally stable, then Lemma 1 requires Assumption 4 with the pair ðA; BÞ stabilizable [16]. Using Theorem 2 in Ref. [16], c and K are chosen such that qðA c Þ < 1. To this end, we state the following result.

Lemma 2 [16]. Let Assumption 4 holds, and let P be a positive definite solution to the discrete-time Riccati-like equation

for some Q > 0. Define r ¼½r max ðQ À0:5 A T PBðB T PBÞ À1 B T PA Q À0:5 Þ À0:5

Furthermore, let there exists a Cðc 0 ; r 0 Þ containing all the eigenvalues

If B is a full column rank, Eq. ( 16) has a positive definite solution P only if the pair ðA; BÞ is stabilizable [16]. In this regard, Assumption 3 is pertinent. Next, we introduce the notion of inputto-state stability in the following definition. DEFINITION 2. A discrete-time system of the form x kþ1 ¼ /ðx k ; u 1k ; u 2k Þ; k 2 Z ? with u 1 : Z ? ! R m1 ; u 2 : Z ? ! R m2 ; /ð0 n ; 0 m1 ; 0 m2 Þ¼0 n is (globally) ISS if there exist a class KL function b and two class K functions c 1 ; c 2 such that, for each pair of inputs u 1 2 l m1 1 ; u 2 2 l m2 1 and each x 0 2 R n , it holds that jx k j bðjx 0 j; kÞþc 1 ðjju 1 jjÞ þ c 2 ðjju 2 jjÞ (17

for each k 2 Z ? . Remark 2. Definition 2 is adopted from Definition 3.1 in Ref. [32] and has been suitably modified for systems with two inputs using Definition IV.3 in Ref. [33].

We state the main result of this section in Theorem 3. THEOREM 3. Suppose the following conditions are satisfied: (i) Under Assumption 2, agent i (i ¼ 1; 2; …; N) employs the SMF in Algorithm 1 to estimate its own state; (ii) Assumptions 3 and 4 hold; (iii) c and K are chosen using Lemma 2. Then, the global error system in Eq. ( 14) is ISS.

Proof. The proof is inspired by Example 3.4 in Ref. [32]. Let us denote e ðgÞ k ¼ col½e 14) becomes

where e ðgÞ : Z ? ! R nN and w ðgÞ : Z ? ! R wN are the inputs. It is understood that e ðgÞ 2 l nN 1 and w ðgÞ 2 l wN 1 . Due to the choices of c and K along with Assumptions 3 and 4, we have qðA c Þ < 1. Hence, there exist constants a > 0 and l 2½0; 1Þ, such that jA k c j al k ; k 2 Z ? [32]. Then, the ISS property in Eq. ( 17) holds for the system in Eq. ( 14) with

where jðI N GÞj ¼ jI N jjGj¼jGj is utilized. Thus, along the trajectories of the system in Eq. ( 14), for each k 2 Z ? , it holds that jd ðgÞ k j bðjd ðgÞ 0 j; kÞþc 1 ðjje ðgÞ jjÞ þ c 2 ðjjw ðgÞ jjÞ (20) where the functions b; c 1 ; and c 2 are as in Eq. ( 19) with s ¼jd ðgÞ 0 j; s 1 ¼ jje ðgÞ jj; and s 2 ¼ jjw ðgÞ jj. Theorem 3 implies that the global disagreement error remains bounded under the proposed synchronization protocol.

Remark 3. Objective of the SMF-based synchronization in Ref. [31] was to contain the states of the agents in a confidence ellipsoid that might not be small in general. Thus, the approach outlined in Ref. [31] may lead to conservative results where the states of the agents might not converge to a neighborhood of the leader's state trajectory. On the other hand, we have shown that, under appropriate conditions, the global error system is ISS with respect to the input disturbances and estimation errors. Since an ISS system admits the "converging-input converging-state" property (see, Refs. [32] and [36] for details), jd ðgÞ k j would eventually converge to a neighborhood of zero as the estimation errors of the agents decrease. Thus, the agents would converge to a neighborhood of the leader's state trajectory. To this end, it is understood that jjw ðgÞ jj is relatively small (compared to jd ðgÞ 0 j and jje ðgÞ jj) as the input disturbances satisfy Assumption 2.2.

Next, we state the following result based on Theorem 3, where p 0 and q are as in Assumption 2.

for each k 2 Z ? with l ¼ða ffiffiffi ffi N p =ð1 À lÞÞ. Hence, the proof is completed by taking the limit in Eq. ( 21 22) is monotonically decreasing. The estimate given in Eq. ( 21) is a conservative one as we have utilized jje ðgÞ jj ffiffiffiffiffiffiffiffi p 0 N p and jjw ðgÞ jj ffiffiffiffiffiffi ffi qN p . Also, the bound jR ðiÞ k j r does not appear in Eqs. ( 21) and ( 22) as a result of utilizing jje ðgÞ jj je Remark 5. For a given multi-agent system (with the number of agents N, the matrices A; B; C; D; and G, and the interaction graph specified), we have jB c j and jGj fixed once c and K are properly chosen using Lemma 2. Thus, the conservatism of the bound in Eq. ( 21) can be reduced if the available upper bounds (i.e., p 0 and q) are sufficiently small.

A simulation example is provided in this section to illustrate the effectiveness of the proposed SMF-based leader-follower synchronization protocol. All the simulations are carried out on a desktop computer with a 16.00 GB RAM and a 3.40 GHz Intel (R) Xeon (R) E-2124 G processor running MATLAB R2019a. The SDPs in Eqs. ( 6) and ( 8) are solved utilizing "YALMIP" [37] with the "SDPT3" solver in the MATLAB framework. Since the disturbances are only assumed to be unknown-but-bounded, different kinds of disturbance realizations are possible, which satisfy the ellipsoidal assumptions (Assumptions 1.2 and 2.2), for example, periodic disturbances with time-varying or constant frequencies and amplitudes, random disturbances with each element being uniformly distributed in an interval, and so on.

We consider four agents, i.e., N ¼ 4. Matrices related to the dynamics of the leader and the agents are

where A is marginally stable. Also, Assumption 3 is satisfied with the above choices of G ¼ diagð1; 0; 0; 0Þ; D ¼ diagð1; 1; 1; 1Þ. With regards to Lemma 2, we choose Q ¼ 0:1I 2 ; c 0 ¼ð2=3Þ; r 0 ¼ 0:6. Clearly, Cðc 0 ; r 0 Þ contains all the eigenvalues of C, as shown in Fig. 1(b). Also, we have r ¼ 1 and ðr 0 =c 0 Þ¼0:9 < r. Hence, the conditions for Lemma 2 are satisfied, and we take c ¼ð1=c 0 Þ¼1:5; K ¼ðB T PBÞ À1 B T PA.

The synchronization results are shown in Figs. 2(a) and 2(b). Figure 2(a) shows that the trajectories of the agents converge close to that of the leader. As a result, the normalized global disagreement error norm converges to a neighborhood of zero (Fig. 2(b)). The dotted line in Fig. 2(b) denotes the conservative bound in Eq. ( 21) for which we have utilized p 0 ¼ 2 and q ¼ 0:1. Also, for l, we have taken a ¼ 1:1 and l ¼ 0:9. For this choice of a and l, jA k c j al k is satisfied, as shown in Fig. 2(c). With the above values, the conservative upper bound is equal to 2.462, which is shown using the dotted line in Fig. 2 (i ¼ 1; 2; 3; 4). Figures 3 and4 show that the SMFs for the agents perform adequately as the estimation errors remain in a neighborhood of zero and the error bounds decrease from the respective initial values. Also, the estimation errors are contained within the error bounds, which mean the SMFs of the agents are able to contain the respective true states inside the respective correction ellipsoids. The estimation error norms, shown in Fig. 5, further illustrate the effectiveness of the SMFs and demonstrate that the SMFs are able to reduce the estimation errors from the respective initial values, starting from the correction step at k ¼ 0. The results in Fig. 5 essentially verify our earlier use of jje ðgÞ jj je ðgÞ 0 j in deriving the conservative bound in Eq. ( 21). The trace of correction ellipsoid shape matrices for the SMFs of the agents is shown in Fig. 6, where P ðiÞ kjk (i ¼ 1; 2; 3; 4) denote the shape matrices of agent i's correction ellipsoids. Clearly, SMFs of the agents are able to reduce the trace from the initial values and construct optimal (minimum trace) correction ellipsoids at each time-step (starting from k ¼ 0). Quantitatively, the trace of these shape matrices converges approximately to 1.5 (see Fig. 6), which is approximately a 2.667-fold decrease with respect to the initial trace of 4. The trends shown in Fig. 6 for all the agents are roughly the same, as the same set of ellipsoidal parameters is utilized for the SMFs of all the agents and the agents have identical dynamics. 1 where the following two comparison metrics are used: (i) : root-meansquare value of j d k j. Also, w ðiÞ k is chosen randomly (uniform distribution) between Àa w 1 2 and a w 1 2 , and v ðiÞ k is chosen randomly (uniform distribution) between Àa v and a v . Thus, the first row in Table 1 corresponds to the result in Fig. 2(b). We observe that both the metrics in Table 1 are comparable among the three cases studied, despite the higher magnitudes of disturbances considered for the two cases in second and third rows of Table 1. Therefore, the j d k j trends for these two cases with higher disturbance magnitudes would be qualitatively similar to the one shown in Fig. 2(b).

A set-membership filtering-based leader-follower synchronization protocol for high-order discrete-time linear multi-agent systems has been put forward for which the global error system is shown to be input-to-state stable with respect to the input disturbances and estimation errors. A monotonically decreasing upper bound on the norm of the global disagreement error vector is calculated. Our future work would involve extending the proposed formulation for discrete-time nonlinear dynamical systems and switching network topologies. Also, we would extend the results in this paper by considering a control input for the leader or the leader to be any bounded reference trajectory.