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(2022)
Adaptive distributed observer for general linear leader systems over periodic switching digraphs
In this paper, we present a sufficient condition for the exponential stability of a class of linear switched systems. As an application of this stability result, we establish an output-based adaptive distributed observer for a general linear leader system over a periodic jointly connected switching communication network, which extends the applicability of the output-based adaptive distributed observer from a marginally stable linear leader system to any linear leader system and from an undirected switching graph to a directed switching graph. This output-based adaptive distributed observer will be applied to solve the leader-following consensus problem for multiple double-integrator systems. Keywords: Distributed observer, Multi-agent systems, Stability, Switched systems 1 Introduction drawback was overcome in [14] by proposing the so-called The distributed observer approach is an effective tool for adaptive distributed observer, which can estimate both the state and the system matrix of a neutrally stable leader sys- dealing with a variety of cooperative control problems tem over a jointly connected switching network. The adap- of multi-agent systems, such as the leader-following con- sensus problem [1–3], the containment control problem tive distributed observer in [14] was strengthened to han- [4], the cooperative output regulation problem [5–8], the dle a general linear leader system provided that the switch- leader-following output synchronization problem [9–11], ing signal is periodic in [15]. The second drawback was addressed in [16] by proposing the so-called output-based and the rendezvous problem [12, 13]. distributed observer which only makes use of the output A distributed observer for the leader system is a dis- of the leader system. By combining the ideas of the adap- tributed dynamic compensator that asymptotically esti- tive distributed observer in [14] and the output-based dis- mates the state of the leader system over a communica- tributed observer in [16], reference [17]further developed tion network. The distributed observer for linear leader the output-based adaptive distributed observer such that systems was first established over a connected static net- it can estimate the state of a leader system over a jointly work in [7] and then over a jointly connected switching connected switching network based on the output of the network in [8]. However, the distributed observers in [7, 8] leader system only and without requiring all followers to have two drawbacks. First, they assume all followers know know the system matrix of the leader system. Nevertheless, the system matrix of the leader system. Second, they as- the existing output-based adaptive distributed observer sume the state of the leader system is available. The first over a jointly connected switching network still requires that the system matrix of the leader system be marginally Correspondence: jhuang@mae.cuhk.edu.hk stable [17]. This requirement is restrictive as it cannot even Shenzhen Research Institute and Department of Mechanical and Automation handle the frequently encountered double-integrator sys- Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China tems. For this reason, in this paper, we will explore the © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. He and Huang Autonomous Intelligent Systems (2023) 3:1 Page 2 of 8 possibility of establishing an output-based adaptive dis- the switching signal takes the form tributed observer for a general linear leader system over a jointly connected switching network. Indeed, by estab- ⎪ 1, if sT ≤ t <(s + ω )T, ⎪ 2 lishing a key stability result of a class of linear periodic 2, if (s + ω )T ≤ t <(s + ω )T, 1 p p=1 switched systems, we manage to ascertain the existence of σ (t)= (1) ⎪ . an output-based adaptive distributed observer for a gen- . ρ–1 eral linear leader system over a periodic jointly connected ρ,if(s + ω )T ≤ t <(s +1)T, p=1 switching network. The rest of this paper is organized as follows. We sum- where s = 0,1,... , and ω , p = 1,..., ρ, are positive con- marize some useful stability results for periodic linear stants satisfying ω = 1. Under Assumptions 1 and 2, p=1 switched systems and establish a key stability result of a without loss of generality, we may assume that, for any k = i –1 k+1 class of linear switched systems in Sect. 2.InSect. 3,we ¯ ¯ 0,1,... , the union graph G = G .Aswitch- σ (t ) p j=i j p=1 apply the stability result to develop an output-based adap- ing digraph G satisfying Assumptions 1 and 2 is said to σ (t) tive distributed observer for a general linear leader system. be periodic jointly connected. In Sect. 4, we apply the output-based adaptive distributed observer to a leader-following consensus problem for mul- N ×N Remark 2.2 Let H ∈ R be obtained from the Lapla- σ (t) tiple double-integrator systems. Section 5 closes this paper ¯ ¯ cian matrix L of G by removing the first row and the σ (t) σ (t) with some remarks. first column of L . Under Assumption 1, by Corollary 4 σ (t) of [8], the origin of the linear switched system Notation R denotes the set of real numbers. (·)denotes the real part of a complex number. x denotes the Eu- φ =–μ(H ⊗ I )φ,(2) σ (t) q clidean norm of vector x and A denotes the induced Nq norm of matrix A by the Euclidean norm. For a square ma- where μ is an arbitrary positive real number and φ ∈ R , trix A, λ (A) denotes an eigenvalue of A with the small- min is exponentially stable. Moreover, under Assumptions 1 est real part and λ (A) denotes an eigenvalue of A with max and 2, consider the switching signal σ (t)defined in (1). the greatest real part. 1 denotes the N-dimensional col- From the proof of Lemma 2.2 of [15], all the eigenvalues p ×m i ρ umn vector with all elements being 1. For A ∈ R , of the matrix ω H have positive real parts. p p p=1 i = 1,..., n, col(A ,..., A )= A ··· A . ⊗ denotes 1 n 1 n the Kronecker product of matrices. block diag(A ,..., A ) Let 1 n denotes a block diagonal matrix whose diagonal block el- (λ (G)) max ements are A ,..., A . 1 n μ ¯ = .(3) (λ ( ω H )) min p p p=1 2Preliminaries Let σ (t):[0, ∞) → P = {1,2,..., ρ} be a piecewise constant The following lemma is rephrased from Lemma 2.2 of switching signal with dwell time τ > 0. Consider a switch- [15]. ¯ ¯ ¯ ¯ ing digraph G =(V, E )where V = {0,1,..., N } and σ (t) σ (t) ¯ ¯ ¯ E ⊆ V × V for t ≥ 0. Assume that the switching digraph Lemma 2.1 Consider the linear switched system as follows: σ (t) G satisfies the following assumptions. σ (t) ϑ = I ⊗ G – μ(H ⊗ I ) ϑ,(4) N σ (t) n Assumption 1 There exists a subsequence {i |k = 0,1,...} Nn n×n where σ (t) is defined in (1); ϑ ∈ R is the state; G ∈ R of {i|i = 0,1,...} with t – t < ν for some ν >0, such that i i k+1 k i –1 k+1 is a constant matrix; and μ ∈ R is a positive real number. the union graph G has the property that there is σ (t ) j=i j Under Assumptions 1 and 2, there exists a positive constant a path from node 0 to every other node. ¯ ¯ T such that, for any μ > μ ¯ and any 0< T < T , the origin 0 0 0 of system (4) is exponentially stable. Assumption 2 The switching signal σ (t) is periodic with period T. In what follows, we establish a stability result of the fol- lowing class of linear switched systems: Remark 2.1 Assumption 1 is called the jointly connected condition which allows the switching digraph to be dis- x ˙ = I ⊗ S – μ H ⊗ (LC) x, (5) N σ (t) connected at any time instant. As noted in [15], under As- sumption 2, we can assume without loss of generality that where N is a positive integer; μ is a positive real number; Nq q×q p×q x ∈ R is the state; S ∈ R and C ∈ R are constant q×p See Sect. 1.2 of [17] for an overview on digraphs. matrices; and L ∈ R is againmatrixtobedesigned. He and Huang Autonomous Intelligent Systems (2023) 3:1 Page 3 of 8 Assumption 3 The pair (C, S)is detectable. results in [16, 17] from a marginally stable matrix S to any matrix S and from an undirected graph to a directed graph Remark 2.3 Under Assumption 3 and the assumptions provided that the switching signal σ (t)satisfiesAssump- N ×N that S is marginally stable and the matrices H ∈ R , tion 2 with the period T being smaller than some thresh- p ∈ P with P being the switching index set, are positive old. semi-definite, the asymptotic stability of system (5)was es- tablished in [16] and the exponential stability of system (5) 3 Output-based adaptive distributed observer was established in [17]. In this section, we further apply Lemma 2.2 to establish the output-based adaptive distributed observer over a jointly Remark 2.4 Under Assumption 3, the following algebraic connected graph for a general linear leader system of the Riccati equation: following form: T T SP + PS – PC CP + I =0 (6) v ˙ = Sv, y = Cv,(9) q×q q p q×q p×q admits a unique positive definite solution P ∈ R [18]. where v ∈ R , y ∈ R ,and S ∈ R and C ∈ R are con- stant matrices. Let us first establish the exponential stability of system As in [17], for i = 1,..., N, consider a distributed dy- (5)with S being a general square matrix and σ (t)being a namic compensator as follows: periodic signal. Lemma 2.2 Under Assumptions 1, 2, and 3, let λ ,..., λ S = μ a (t)(S – S ), (10a) 1 N i 1 ij j i 1 –1 be the N eigenvalues of the matrix ω H , let μ ¯ = δ j=0 p p 1 p=1 where δ = min {(λ )}. Then, there exists a positive i=1,...,N i ¯ ¯ constant T such that, for any μ ≥¯ μ and any 0< T < T , 1 1 1 ˙ C = μ a (t)(C – C ), (10b) i 2 ij j i the origin of system (5) with L = PC is exponentially sta- j=0 ble. L = μ a (t)(L – L ), (10c) i 3 ij j i Proof Let j=0 M = I ⊗ S – μ H ⊗ PC C , p = 1,..., ρ, p N p ξ = S ξ + μ L a (t)(C ξ – C ξ ), (10d) i i i 4 i ij j j i i (7) j=0 M = ω M . c p p p=1 q×p where S = S, C = C, L = L ∈ R is againmatrixtobe 0 0 0 designed, and ξ = v; μ to μ are positive real numbers 0 1 4 Under Assumptions 1 and 2,byRemark 2.2,the N eigen- q×q p×q to be specified; and, for i = 1,..., N, S ∈ R , C ∈ R , i i values λ ,..., λ of the matrix ω H have positive 1 N p p p=1 q×p q N ×N L ∈ R , ξ ∈ R ,and a (t) is the element of the weighted i i ij real parts. Let Q ∈ R be a nonsingular matrix such that –1 N ×N adjacency matrix. J = Q( ω H )Q where J ∈ R is the Jordan form p p p=1 ρ ρ Given the leader system (9)and agroup of N dynamic of ω H .Since ω =1, M can be rewritten as p p p c p=1 p=1 compensators (10a)–(10d), we can define a switching di- ¯ ¯ ¯ ¯ ¯ graph G =(V, E )where V = {0,1,..., N } and E ⊆ σ (t) σ (t) σ (t) –1 T M = Q ⊗ I I ⊗ S – μ J ⊗ PC C (Q ⊗ I ). c q N q ¯ ¯ ¯ V × V for t ≥ 0. In the set V, node 0 is associated with (8) the leader system (9) and node i, i = 1,..., N,isassociated with the ith dynamic compensator of (10a)–(10d). For i = The block triangular structure of J implies that the eigen- 1,..., N, j = 0,1,..., N, i = j,(j, i) ∈ E if and only if the ith σ (t) values of M coincide with the eigenvalues of S– μλ PC C, c i dynamic compensator of (10a)–(10d)can make useofthe –1 i = 1,..., N. By Lemma 2.12 of [17], for any μ ≥ δ information of agent j at time t.The weighted adjacency with δ = min {(λ )},the matrices S – μλ PC C, i = i=1,...,N i i ¯ ¯ matrix of the digraph G is a nonnegative matrix A = 1 –1 σ (t) σ (t) 1,..., N,are Hurwitz. Hence, forany μ ≥ δ , M is Hur- 2 N (N+1)×(N+1) [a (t)] ∈ R ,where,for t ≥ 0, a (t)=0, and, 1 –1 ij ii i,j=0 witz. Invoking Lemma 3.22 of [19]with μ ≥¯ μ = δ for i = j, a (t)>0 ⇔ (j, i) ∈ E .Let H =[h (t)] ∈ ij σ (t) σ (t) ij i,j=1 completes the proof. N ×N R ,where h (t)= a (t)and h (t)=–a (t)if i = j, ii ij ij ij j=0 Remark 2.5 Since Lemma 2.2 imposes no restriction on then H is the matrix obtained from the Laplacian matrix σ (t) ¯ ¯ the eigenvalues of the matrix S and does not require H , L of G by removing the first row and the first col- p σ (t) σ (t) p = 1,..., ρ,tobesymmetric,ithas partially extended the umn of L .Let G =(V, E )with V = {1,..., N } and σ (t) σ (t) σ (t) He and Huang Autonomous Intelligent Systems (2023) 3:1 Page 4 of 8 E ⊆ V × V be the subgraph obtained from the switch- the solutions of systems (9) and (10a)–(10d) with L = PC σ (t) ing digraph G by removing all the edges between node exist for all t ≥ 0 and satisfy σ (t) 0and the nodes in V. lim S (t)– S = 0, (14a) t→∞ Assumption 4 The subgraph G is undirected for all t ≥ σ (t) lim C (t)– C = 0, (14b) 0. i t→∞ lim L (t)– L = 0, (14c) Assumption 5 The matrix S is marginally stable, that is, t→∞ there exists a positive definite matrix P satisfying P S + 0 0 lim ξ (t)– v(t) =0, i = 1,..., N (14d) SP ≤ 0. t→∞ all exponentially. The dynamic compensator (10a)–(10d) is a so-called output-based adaptive distributed observer for the leader ˜ ˜ ˜ Proof Let S = S – S, C = C – C, L = L – L,and v ˜ = ξ – v, i i i i i i i i system (9) due to the following lemma rephrased from ˜ ˜ ˜ ˜ ˜ ˜ i = 1,..., N.Also,let S = col(S ,..., S ), C = col(C ,..., C ), 1 N 1 N Theorem 4.7 of [17]. ˜ ˜ ˜ ˜ ˜ L = col(L ,..., L ), v ˜ = col(v ˜ ,..., v ˜ ), S =block diag(S , 1 N 1 N d 1 ˜ ˜ ˜ ˜ ˜ ˜ ..., S ), C =block diag(C ,..., C ), L =block diag(L , N d 1 N d 1 Lemma 3.1 Consider systems (9) and (10a)–(10d). Under ..., L ), and L =block diag(L ,..., L ). Then, from (9) N d 1 N Assumptions 1, 3, 4, and 5, there exists a constant matrix ˜ ˜ ˜ q×p and (10a)–(10d), as shown in Sect. 4.4.3 of [17], S, C, L, L ∈ R such that, for any μ , μ , μ , μ >0 and any initial 1 2 3 4 and v ˜ are governed by conditions v(0), S (0), C (0), L (0), and ξ (0), i = 1,..., N, the i i i i solutions of systems (9) and (10a)–(10d) exist for all t ≥ 0 ˜ ˜ and satisfy S =–μ (H ⊗ I )S, (15a) 1 σ (t) q ˜ ˜ C =–μ (H ⊗ I )C, (15b) 2 σ (t) p lim S (t)– S = 0, (11a) t→∞ ˜ ˜ L =–μ (H ⊗ I )L, (15c) 3 σ (t) q C (t)– C = 0, (11b) lim t→∞ v ˜ = A + M (t) v ˜ + F(t), (15d) σ (t) d lim L (t)– L = 0, (11c) t→∞ where lim ξ (t)– v(t) =0, i = 1,..., N (11d) t→∞ A = I ⊗ S – μ H ⊗ (LC) , σ (t) N 4 σ (t) all exponentially. ˜ ˜ M (t)= S – μ L (H ⊗ C) d d 4 d σ (t) (16) However, Lemma 3.1 is quite restrictive since it only ap- – μ L (H ⊗ I )C , 4 d σ (t) p d plies to a marginally stable leader system and requires the ˜ ˜ graph to be undirected. Thus, it is desirable to establish an F(t)= S – μ L (H ⊗ I )C (1 ⊗ v). d 4 d σ (t) p d N output-based adaptive distributed observer for the leader Nq Np system (9) without these restrictive assumptions, provided For i = 1,..., q,let s ˜ ∈ R and c ˜ ∈ R be the ith col- i i Nq that the switching signal σ (t) satisfies Assumption 2 with ˜ ˜ umn of S and C,respectively. For j = 1,..., p,let l ∈ R be the period T being smaller than some threshold. the jth column of L.Then, (15a), (15b), and (15c)can be written as Theorem 3.1 Consider systems (9) and (10a)–(10d). Un- der Assumptions 1, 2, and 3, let λ > (λ (S)) be some pos- max s ˜ =–μ (H ⊗ I )s ˜ , (17a) i 1 σ (t) q i itive real number and let c ˜ =–μ (H ⊗ I )c ˜ , i = 1,..., q, (17b) i 2 σ (t) p i μ ¯ = , (12) ρ ˜ ˜ ¯ l =–μ (H ⊗ I )l , j = 1,..., p. (17c) j 3 σ (t) q j (λ ( ω H )) min p p p=1 1 1 Under Assumption 1,byRemark 2.2, the origin of sys- μ ¯ = × . (13) (λ ( ω H )) min p p tem (17a)–(17c) is exponentially stable for any μ , μ , μ > p=1 1 2 3 ˜ ˜ 0. Thus, we have lim S (t)=0, lim C (t)=0, and t→∞ d t→∞ d ¯ ˜ Then there exists a positive constant T , such that, if lim L (t) = 0 all exponentially, which implies (14a), 2 t→∞ d μ , μ , μ > μ ¯ , μ ≥¯ μ , and 0< T < T , then, for any ini- (14b), and (14c). Since (14c)implies that L (t)is bounded 1 2 3 2 4 3 2 d tial conditions v(0), S (0), C (0), L (0), and ξ (0), i = 1,..., N, over [0, ∞), we have lim M (t) = 0 exponentially. i i i i t→∞ d He and Huang Autonomous Intelligent Systems (2023) 3:1 Page 5 of 8 In what follows, we show (14d). For this purpose, we first which is obtained from system (15d) by setting M (t)and show that there exists a positive constant T ,suchthat, for F(t) tozero. UnderAssumptions 1, 2,and 3, by Lemma 2.2, ¯ ¯ any μ , μ , μ > μ ¯ and any 0 < T < T , lim F(t)=0 ex- there exists a positive constant T ,suchthat, forany μ ≥ 1 2 3 2 3 t→∞ 4 4 ¯ ¯ ponentially, where F(t)is defined in (16). Let S = I ⊗ λ, μ ¯ and any 0 < T < T , the origin of system (26)isexpo- q 3 4 ¯ ˆ ¯ ¯ ¯ St St ˆ nentially stable. Let T = min{T , T }. Applying Lemma 1 S = I ⊗ λ, ζ =(I ⊗ e )s ˜ ,and γ =(I ⊗ e )c ˜ , i = 1,..., q. 2 3 4 p i N i i N i of [20]tosystem(15d) completes the proof. Then, ζ is governed by ¯ ¯ St St ˙ Remark 3.1 The exponential stability of the linear switch- ˙ ¯ ˜ ˜ ζ = I ⊗ Se s + I ⊗ e s i N i N i ed system (26) with a general square matrix S is cru- ¯ ¯ St St =(I ⊗ S) I ⊗ e s ˜ – μ I ⊗ e (H ⊗ I )s ˜ cial in establishing the output-based adaptive distributed N N i 1 N σ (t) q i observer (10a)–(10d). This stability result of (26)can- ¯ ¯ St St =(I ⊗ S) I ⊗ e s ˜ – μ (H ⊗ I ) I ⊗ e s ˜ N N i 1 σ (t) q N i not be obtained by any existing approach. Theorem 3.1 has made the output-based adaptive distributed observer = I ⊗ S – μ (H ⊗ I ) ζ , i = 1,..., q. (18) N 1 σ (t) q i (10a)–(10d) applicable to any linear leader system over a directed switching graph G that satisfies Assumption 2 σ (t) Similarly, γ is governed by with the period T being smaller than a threshold. γ˙ = I ⊗ S – μ (H ⊗ I ) γ , i = 1,..., q. (19) i N 2 σ (t) p i 4Example For all i = 1,..., q,system(18)(respectively,system(19)) In this section, we apply the output-based adaptive dis- tributed observer (10a)–(10d) to solve the leader-following is in the form of (4)with G = S, μ = μ ,and ϑ = ζ (re- 1 i ˆ ¯ ¯ consensus and disturbance rejection problem of four spectively, G = S, μ = μ ,and ϑ = γ ). Since (λ (S)) = 2 i max ¯ ˆ (λ (S)) = λ, under Assumptions 1 and 2, by Lemma 2.1, double-integrator systems as follows: max there exists a positive constant T ,suchthat, forany μ , μ > μ ¯ and any 0 < T < T , the origins of systems (18) x ˙ = x , i1 i2 1 2 2 3 and (19) are exponentially stable. Thus, for i = 1,..., N,we x ˙ = u + d , i2 i i have (27) x v i1 1 St λt e = – , ˜ ˜ lim e S (t) = lim e S (t) = 0, (20) i i x v i2 2 t→∞ t→∞ St λt y = x , i = 1,2,3,4, mi i1 ˜ ˜ lim e C (t) = lim e C (t) = 0 (21) i i t→∞ t→∞ where, for i = 1,2,3,4, x , x ∈ R; u ∈ R is the control in- i1 i2 i St λt exponentially. Since e ≤ βe for some positive real put to be designed; d = i × v ∈ R is the external distur- i 3 number β,from(20)and (21), for i = 1,..., N,wehave 2 bance; e ∈ R is the tracking error; y ∈ R is the measure- i mi ment output; and v = col(v , v , v ) ∈ R is the exogenous 1 2 3 St ˜ ˜ lim S (t)v(t) = lim S (t)e v(0) i i signal generated by the following leader system: t→∞ t→∞ λt ˜ ⎡ ⎤ ≤ lim βe S (t) v(0) = 0, (22) t→∞ 01 0 ⎣ ⎦ –1 0 0 v ˙ = Sv = v, St ˜ ˜ lim C (t)v(t) = lim C (t)e v(0) i i (28) t→∞ t→∞ 01 1 λt ≤ lim βe C (t) v(0) = 0 (23) y = Cv = 101 v. t→∞ exponentially. Thus, it holds that Hence, the four double-integrator systems are subject to exponentially growing disturbances. Since the pair (C, S) lim S (t) 1 ⊗ v(t) = 0, (24) is observable, Assumption 3 is satisfied. d N t→∞ We can interpret the above problem as a cooperative lim C (t) 1 ⊗ v(t) = 0 (25) d N output regulation problem (see Definition 1 of [8]) by t→∞ rewriting the four double-integrator systems in the follow- ing standard form: exponentially. Since L (t) is bounded over [0, ∞), from (16), (24), and (25), we have lim F(t) = 0 exponentially. t→∞ x ˙ = A x + B u + E v, Now, we are ready to show (14d). For this purpose, con- i i i i i i sider the following system: y = C x + D u + F v, (29) mi mi i mi i mi v ˜ = I ⊗ S – μ H ⊗ (LC) v ˜ (26) e = C x + D u + F v, i = 1,2,3,4, N 4 σ (t) i ri i ri i ri He and Huang Autonomous Intelligent Systems (2023) 3:1 Page 6 of 8 ¯ ¯ ¯ Figure 1 Communication graphs: (a) G ,(b) G ,and (c) G 1 2 3 where, for i = 1,2,3,4, x = col(x , x ), i i1 i2 01 0 000 A = , B = , E = , i i i 00 1 00 i Figure 2 Estimation errors of matrix S 10 0 C = , D = , ri ri 01 0 –100 F = , ri 0–1 0 C = 10 , D =0, F = 000 . mi mi mi Then it can be verified that the pair (A , B ) is controllable; i i the pair (C , A ) is observable; and the solution of the reg- mi i ulator equation X S = A X + B U + E , i i i i i i i = 1,2,3,4 (30) 0= C X + D U + F , ri i ri i ri 10 0 is given by X = and U =[–1 0 – i]. i i 01 0 Consider the following periodic switching signal: Figure 3 Estimation errors of matrix C 1, if sT ≤ t <(s + )T, 1 1 σ (t)= 2, if (s + )T ≤ t <(s + )T, (31) 4 2 and 3, if (s + )T ≤ t <(s +1)T, ⎡ ⎤ 00 00 ⎢ ⎥ where T =1and s = 0,1,... . Thus, σ (t)isin the form of (1) 00 00 ⎢ ⎥ H = . 1 1 1 3 ⎣ ⎦ with ω = , ω = ,and ω = . The three digraphs G , i = 1 2 3 i 0 01–1 4 4 2 1, 2, 3, associated with σ (t)are shown in Fig. 1.Although –100 1 the switching digraph G is disconnected at every time σ (t) instant t ≥ 0, it can be seen that Assumptions 1 and 2 are The four eigenvalues of the matrix ω H are λ =0.5, p p 1 p=1 ¯ 3 satisfied. We let a (t)=1 whenever (j, i) ∈ E . ij σ (t) λ =0.5, λ =0.25,and λ =0.25,thus (λ ( ω H )) = 2 3 4 min p p p=1 The three matrices associated with the three digraphs G , i min {(λ )} =0.25. From (13), μ ¯ =2. From (28), i=1,2,3,4 i 3 i = 1,2,3, are given by (λ (S)) = 1. Then we let λ =1.2 which satisfies λ > max (λ (S)). From (12), we have μ ¯ =4.8. max 2 ⎡ ⎤ ⎡ ⎤ By Theorem 3.1, we can design an output-based adaptive 0000 1–100 distributed observer of the form (10a)–(10d)for (28)with ⎢ ⎥ ⎢ ⎥ 0100 000 0 ⎢ ⎥ ⎢ ⎥ H = , H = , 1 2 μ =10, μ =10, μ =10, μ =10,and L = col(–0.2, 1.4, 4.2). ⎣ ⎦ ⎣ ⎦ 1 2 3 4 0000 000 0 As shown in [8], the cooperative output regulation prob- 0000 000 0 He and Huang Autonomous Intelligent Systems (2023) 3:1 Page 7 of 8 Figure 6 Tracking errors e =col(e , e ) i i1 i2 Figure 4 Estimation errors of matrix L observer are shown in Figs. 2 to 5. The tracking errors of the followers are shown in Fig. 6. As expected, satisfactory performance is observed. 5Conclusion In this paper, having established an exponential stability property for a class of linear switched systems, we have further developed an output-based adaptive distributed observer for a general linear leader system over a peri- odic jointly connected switching communication network, which extends the applicability of the output-based adap- tive distributed observer from a marginally stable linear leader system to a general linear leader system and from an undirected switching graph to a directed switching graph. Acknowledgements The authors would like to thank the anonymous reviewers for their valuable Figure 5 Estimation errors of state v comments and suggestions. Funding This work was supported in part by the Research Grants Council of the Hong lem of the multi-agent system composed of (28)and (29) Kong Special Administrative Region under Grant Nos. 14202619 and PDFS2223-4S02, and in part by the National Natural Science Foundation can be solved by the following dynamic measurement out- (NSFC) of China under Grant No 61973260. put feedback control law: Availability of data and materials Not applicable. u = K η + K ξ , (32a) i 1i i 2i i η ˙ =A η + B u + E ξ i i i i i i i Declarations (32b) + L (C η + D u + F ξ – y ), mi mi i mi i mi i mi Competing interests The authors declare that they have no competing interests. where, for i = 1,2,3,4, ξ ∈ R is generated by (10a)–(10d), Author contribution L = col(–22, –120), K =[–12 –7],and K = U –K X = CH and JH participated in the framework design and manuscript writing. JH is mi 1i 2i i 1i i the supervisor of CH at The Chinese University of Hong Kong. All authors read [11 7 –i]. In particular, for i = 1,2,3,4, K and L are such 1i mi and approved the final manuscript. that A + B K and A + L C are Hurwitz. i i 1i i mi mi Our simulation is conducted with v(0) = col(1,1,1) and Publisher’s Note other initial conditions randomly generated within the Springer Nature remains neutral with regard to jurisdictional claims in range [–1.5, 1.5]. The estimation errors of the distributed published maps and institutional affiliations. He and Huang Autonomous Intelligent Systems (2023) 3:1 Page 8 of 8 Received: 6 October 2022 Revised: 3 January 2023 Accepted: 7 February 2023 References 1. A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003) 2. W. Ni, D. Cheng, Leader-following consensus of multi-agent systems under fixed and switching topologies. Syst. Control Lett. 59(3), 209–217 (2010) 3. Y. Dong, Z. 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Autonomous Intelligent Systems – Springer Journals
Published: Feb 20, 2023
Keywords: Distributed observer; Multi-agent systems; Stability; Switched systems
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