This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CYBERNETICS
1
Adjustable Parameter-Based Distributed Fault Estimation Observer Design for Multiagent Systems With Directed Graphs Ke Zhang, Member, IEEE, Bin Jiang, Senior Member, IEEE, and Peng Shi, Fellow, IEEE
Abstract—In this paper, a novel adjustable parameter (AP)-based distributed fault estimation observer (DFEO) is proposed for multiagent systems (MASs) with the directed communication topology. First, a relative output estimation error is defined based on the communication topology of MASs. Then a DFEO with AP is constructed with the purpose of improving the accuracy of fault estimation. Based on H∞ and H2 with pole placement, multiconstrained design is given to calculate the gain of DFEO. Finally, simulation results are presented to illustrate the feasibility and effectiveness of the proposed DFEO design with AP. Index Terms—Adjustable parameter (AP), distributed diagnosis, multiagent systems (MASs), robust fault estimation.
I. I NTRODUCTION VER the past two decades, distributed control of multiagent systems (MASs) has attracted much attention from many scientific communities because of its wide application in various fields, such that consensus, formation control of vehicles, distributed control of multiple robotics, etc. [1]–[9]. Designing distributed protocols based on the relative information guarantees that the states of all agents reach an agreement, known as the consensus problem. All agents need to interact with each other and eventually reach an agreement. There is an increasing demand on safety and reliability of modern control systems. A fault occurred the system might lead to the loss of the system performance or stability. Fault diagnosis (FD) scheme is responsible for providing timely information of any faults within the system being monitored. In recent years, the design and analysis
O
Manuscript received July 21, 2015; revised November 12, 2015; accepted December 26, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61490703, Grant 61304112, and Grant 61573112, in part by the Natural Science Foundation of Jiangsu Province under Grant BK20131364, in part by the Australian Research Council under Grant DP140102180, Grant LP140100471, and Grant LE150100079, and in part by the 111 Project under Grant B12018. This paper was recommended by Associate Editor G.-P. Liu. K. Zhang and B. Jiang are with the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail: [email protected]; [email protected]). P. Shi is with the College of Automation, Harbin Engineering University, Harbin 150001, China, also with the School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, SA 5005, Australia, and also with the College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2513748
of FD using model-based approaches have obtained fruitful results [10]–[13]. The model-based FD scheme is made based on the generated residual via comparing the measurable output of a system with the output generated through the system’s mathematical model. In past few years, FD of MASs has attract attention and become a very hot topic [14]. Compared with the centralized architecture, the FD study of MASs is more complex because of the information exchanges among all agents. The fault occurred in a certain agent would be propagated to other ones through the communication graph and affects fault-free agents’ behavior, which results in performance degradation or even catastrophic accidents for the whole MASs. Therefore, the issue of FD is very critical for MASs to enhance the system safety. In [15], the problem of distributed fault detection and isolation for large-scale interconnected systems with respect to different fault models was studied. In [16], an adaptive neural network-based distributed fault detection and isolation approach was discussed for a class of interconnected uncertain nonlinear systems. In [17], the design and analysis of actuator fault detection and isolation filters for a network of unmanned vehicles was investigated. Jiang et al. [18] considered the distributed fault detection for a class of MASs with networked-induced delays and packet dropouts. The problem of distributed fault detection and isolation for a class of second-order discrete-time MASs was studied by using an optimal robust observer approach [19]. But most of these works only dealt with fault detection and isolation, and few of these results addressed the problem of on-line fault estimation, which is a challenging issue. For a class of MASs, a robust fault estimation method based on sliding mode observers was proposed for a collection of agents, but only undirected graphs were considered [20]. In [21], a consensus-tracking-based distributed fault estimation and distributed fault tolerant control problem for a multiagent system were proposed, but the studied plant was a special class of power systems. In this paper, our objective is to propose a distributed fault estimation observer (DFEO) with adjustable parameter (AP) aimed at improving fault estimation performance for a class of MASs subject to directed graphs. Main novelties of this paper are as follows. 1) A DFEO with AP of MASs is constructed using relative output estimation errors obtained from the communication topology.
c 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2
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2) The proposed AP-based technique can use the current output information to enhance the accuracy of fault estimation, and the existence condition of the proposed observer design is wider than some existing methods. 3) H∞ and H2 with pole placement (PP)-based multiconstrained approaches are given to calculate the gain of DFEO with AP and it is possible to get a smaller H∞ or H2 performance index. The proposed results are convex optimization and can be readily calculated by linear matrix inequalities (LMIs). The rest of this paper is organized as follows. System description is presented in Section II. In Section III, a novel DFEO design with AP is provided for MASs with directed communication graphs to improve the accuracy of fault estimation. Section IV presents simulation results of an airplane model to illustrate the feasibility and effectiveness of the proposed DFEO with AP. Section V draws the conclusions.
II. S YSTEM D ESCRIPTION
For each agent, the dynamic system (1) can be rewritten in the following form: ⎧ x˙ i (t) B A E xi (t) ⎪ ⎪ + ui (t) ⎪ ⎪ f˙i (t) = 0 0 (t) f ⎪ i ⎪ 0 ⎨ D 0 ωi (t) + f˙i (t) ⎪ ⎪ 0 I ⎪ ⎪
xi (t) ⎪ ⎪ ⎩ yi (t) = C 0 fi (t) where f˙i (t) is the derivative of the fault of the ith agent. 0 and I, respectively, denote a zero matrix and an identity matrix with appropriate dimensions. Let us denote augmented vectors and matrices below ωi (t) A E xi (t) ¯ , νi (t) = ˙ , A= x¯ i (t) = fi (t) 0 0 fi (t)
¯B = B , C¯ = C 0 , D ¯ = D 0 0 0 I then it follows:
A. Preliminaries Consider a directed graph G = (V, E, A) with a nonempty finite set of N nodes V = (v1 , v2 , . . . , vN ), a set of edges or arcs E ⊂ V × V, and the associated adjacency matrix A = [aij ] ∈ RN×N . In this paper, the graph is assumed to be time-invariant, i.e., A is constant. An edge rooted at node vj and ended at node vi is denoted by (vj , vi ), which means information can flows from node vj to node vi . aij is the weight of edge (vj , vi ) and aij = 1 if (vj , vi ) ∈ E, otherwise aij = 0. Node vj is called a neighbor of node vi if (vj , vi ) ∈ E. The set of neighbors of node vi is denoted as Ni = { j|(vj , vi ) ∈ E}. Define the in-degree matrix is D = diag{di } ∈ RN×N with di = j∈Ni aij and the Laplacian matrix as L = D − A. It is obvious that L1N = 0, where IN is a vector with dimension N × 1. The edges in the form of (vi , vi ) are called loops. G = diag{gi } ∈ RN×N is denoted as a loop matrix and has at least one diagonal item being 1. A graph with loops is called a multigraph, otherwise it is a simple graph. B. System Description Let us consider a group of N agents modeled under a directed communication graph and each agent with fault is described by a state-space model x˙ i (t) = Axi (t) + Bui (t) + Efi (t) + Dωi (t) (1) yi (t) = Cxi (t) Rn ,
Rm ,
Rp ,
where xi (t) ∈ ui (t) ∈ and yi (t) ∈ respectively, are the state, the input, and the output of the ith agent. ωi (t) ∈ Rd is the disturbance. fi (t) ∈ Rr represents the system component or actuator fault, which is bounded. A–E are constant real matrices of appropriate dimensions. It is supposed that matrices C and E are of full rank and the pair (A, C) is observable.
¯ xi (t) + Bu ¯ i (t) + Dν ¯ i (t) x¯˙ i (t) = A¯ ¯ yi (t) = C¯xi (t)
(2)
which is equivalent to the system dynamics (1). Design Objective: In this paper, for a class of MASs connected by a directed communication graph, a new DFEO with AP is proposed to enhance fault estimation performance by quantitative analysis. Before presenting main results, we first recall the following lemma, which will be used in the sequel. Lemma 1 [22]: The eigenvalues of a given matrix A ∈ Rn×n belong to the circular region D(α, τ ) with center α + j0 and radius τ if and only if there exists a symmetric positive definite matrix P ∈ Rn×n satisfying −P P(A − αIn ) < 0. (3) ∗ −τ 2 P III. M AIN R ESULTS A. DFEO With AP Since all agents are connected based on the communication topology, the fault estimation observer of MASs should consider the information exchange among agents. First, the relative output estimation error of the ith node is defined as follows:
aij yˆ i (t) − yi (t) − yˆ j (t) − yj (t) ζi (t) = j∈Ni
+ gi yˆ i (t) − yi (t)
where aij and gi , respectively, are the element of Laplace matrix L and loop-graph matrix G. Remark 1: Different from centralized structure, the DFEO design of MASs needs to use the relative output estimation error ζi (t), which is based on information exchanges from neighboring nodes.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHANG et al.: AP-BASED DFEO DESIGN FOR MASs WITH DIRECTED GRAPHS
3
For the augmented system (2), the following DFEO with AP for the ith agent is constructed using the relative output estimation error: ¯ i (t) − Hζ ¯ i (t) − θ F¯ ζ˙i (t) x˙ˆ¯ i (t) = A¯ xˆ¯ i (t) + Bu (4) ¯ ˆ yˆ i (t) = Cx¯ i (t)
Furthermore, it obtains
I + (L + G) ⊗ θ F¯ C¯ e˙¯ x (t)
¯ C¯ e¯ x (t) − IN ⊗ D ¯ ν(t). = IN ⊗ A¯ − (L + G) ⊗ H (8)
¯ , since F¯ C¯ = 0 C 0 = For I + (L + G) ⊗ θ (F¯ C) F 0 0 ¯ , it is easy to check that matrix [I +(L+G)⊗θ (F¯ C)] FC 0 ¯ is nonsingular. Let us denote a nonsingular matrix M ¯ = I + (L + G) ⊗ θ F¯ C¯ M
where xˆ¯ i (t) ∈ Rn and yˆ i (t) ∈ Rp are state and output the H 1 ¯ = (H1 ∈ Rn×p and of the observer, respectively. H H2 For the H2 ∈ Rr×p ) is observer gain matrix to be designed. 0 ˙ added term θ F¯ ζi (t), the scalar θ and F¯ = (F ∈ Rr×p ) F are APs to be determined. Remark 2: In DFEO with AP (4), the term θ F¯ ζ˙i (t) is added based on the observer design in [23], whose purpose is to improve fault estimation performance and make the observer (4) more reliable and flexible. This fact will be verified in Section IV. If the scalar θ is set as 0, the DFEO with AP converts into the normal one shown in [23]. For the ith node, the error vectors and a matrix are defined e¯ xi (t) = I¯r =
xˆ¯ i (t) − x¯ i (t),
0 Ir
efi (t) = fˆi (t) − fi (t)
then we can get the local error dynamics e˙¯ xi (t) = x˙ˆ¯ i (t) − x˙¯ i (t) ¯ exi (t) − Dν ¯ i (t) − Hζ ¯ i (t) − θ F¯ ζ˙i (t) = A¯ ¯ exi (t) − Dν ¯ i (t) = A¯ ⎡ ⎤
¯ xj (t) + gj C¯ ¯ exi (t) − Ce ¯ exi (t) ⎦ ¯⎣ −H aij C¯ j∈Ni
⎡
− θ F¯ ⎣
(5)
Ir e¯ xi (t) = I¯r e¯ xi (t).
(6)
j∈Ni
and
efi (t) = 0
In order to consider the fault estimation problem from the overall perspective, the global error vectors are defined
T e¯ x (t) = e¯ Tx1 (t), e¯ Tx2 (t), · · · , e¯ TxN (t) T ef (t) = eTf 1 (t), eTf 2 (t), · · · , eTf N (t)
T ν(t) = ν1T (t), ν2T (t), · · · , νNT (t) then we get the global error dynamics
¯ ν(t) e˙¯ x (t) = IN ⊗ A¯ e¯ x (t) − IN ⊗ D
¯ (L + G) ⊗ C¯ e¯ x (t) − IN ⊗ H
− IN ⊗ θ F¯ (L + G) ⊗ C¯ e˙¯ x (t)
¯ ν(t) = IN ⊗ A¯ e¯ x (t) − IN ⊗ D
¯ C¯ e¯ x (t) − (L + G) ⊗ θ F¯ C¯ e˙¯ x (t) − (L + G) ⊗ H
¯ C¯ e¯ x (t) = IN ⊗ A¯ − (L + G) ⊗ H
¯ ν(t) − (L + G) ⊗ θ F¯ C¯ e¯˙ x (t) − IN ⊗ D (7) where ⊗ represents the Kronecker product.
And the global fault estimation error is
ef (t) = IN ⊗ I¯r e¯ x (t).
(9)
(10)
For the error dynamics (9) and (10), after choosing param¯ we need to calculate observer gain H. ¯ Next, eters θ, F, we propose H∞ and H2 performances with PP to calcu¯ where H∞ and H2 constraints are presented late matrix H, to restrain the influence of terms ν(t) with respect to the fault estimation error and the purpose of introducing PP is to guarantee the system transient performance of fault estimation. B. Observer Gain Calculation Based on H∞ Performance With PP
⎤
aij C¯ e˙¯ xi (t) − C¯ e˙¯ xj (t) + gj C¯ e˙¯ xi (t) ⎦
and it follows:
¯ −1 IN ⊗ A¯ − (L + G) ⊗ H ¯ C¯ e¯ x (t) e˙¯ x (t) = M
¯ −1 IN ⊗ D ¯ ν(t). −M
Theorem 1: Let an H∞ performance level γ and a circular ¯ if there region D(α, τ ) be given. For a prescribed matrix M, exist a symmetric positive definite matrix P¯ ∈ R(n+r)×(n+r) and a matrix Y¯ ∈ R(n+r)×p satisfying
⎡ ⎤ ¯ T IN ⊗ P¯ D ¯ φ −M IN ⊗ I¯rT ⎣∗ −γ 0 ⎦
1
Adjustable Parameter-Based Distributed Fault Estimation Observer Design for Multiagent Systems With Directed Graphs Ke Zhang, Member, IEEE, Bin Jiang, Senior Member, IEEE, and Peng Shi, Fellow, IEEE
Abstract—In this paper, a novel adjustable parameter (AP)-based distributed fault estimation observer (DFEO) is proposed for multiagent systems (MASs) with the directed communication topology. First, a relative output estimation error is defined based on the communication topology of MASs. Then a DFEO with AP is constructed with the purpose of improving the accuracy of fault estimation. Based on H∞ and H2 with pole placement, multiconstrained design is given to calculate the gain of DFEO. Finally, simulation results are presented to illustrate the feasibility and effectiveness of the proposed DFEO design with AP. Index Terms—Adjustable parameter (AP), distributed diagnosis, multiagent systems (MASs), robust fault estimation.
I. I NTRODUCTION VER the past two decades, distributed control of multiagent systems (MASs) has attracted much attention from many scientific communities because of its wide application in various fields, such that consensus, formation control of vehicles, distributed control of multiple robotics, etc. [1]–[9]. Designing distributed protocols based on the relative information guarantees that the states of all agents reach an agreement, known as the consensus problem. All agents need to interact with each other and eventually reach an agreement. There is an increasing demand on safety and reliability of modern control systems. A fault occurred the system might lead to the loss of the system performance or stability. Fault diagnosis (FD) scheme is responsible for providing timely information of any faults within the system being monitored. In recent years, the design and analysis
O
Manuscript received July 21, 2015; revised November 12, 2015; accepted December 26, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61490703, Grant 61304112, and Grant 61573112, in part by the Natural Science Foundation of Jiangsu Province under Grant BK20131364, in part by the Australian Research Council under Grant DP140102180, Grant LP140100471, and Grant LE150100079, and in part by the 111 Project under Grant B12018. This paper was recommended by Associate Editor G.-P. Liu. K. Zhang and B. Jiang are with the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail: [email protected]; [email protected]). P. Shi is with the College of Automation, Harbin Engineering University, Harbin 150001, China, also with the School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, SA 5005, Australia, and also with the College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2513748
of FD using model-based approaches have obtained fruitful results [10]–[13]. The model-based FD scheme is made based on the generated residual via comparing the measurable output of a system with the output generated through the system’s mathematical model. In past few years, FD of MASs has attract attention and become a very hot topic [14]. Compared with the centralized architecture, the FD study of MASs is more complex because of the information exchanges among all agents. The fault occurred in a certain agent would be propagated to other ones through the communication graph and affects fault-free agents’ behavior, which results in performance degradation or even catastrophic accidents for the whole MASs. Therefore, the issue of FD is very critical for MASs to enhance the system safety. In [15], the problem of distributed fault detection and isolation for large-scale interconnected systems with respect to different fault models was studied. In [16], an adaptive neural network-based distributed fault detection and isolation approach was discussed for a class of interconnected uncertain nonlinear systems. In [17], the design and analysis of actuator fault detection and isolation filters for a network of unmanned vehicles was investigated. Jiang et al. [18] considered the distributed fault detection for a class of MASs with networked-induced delays and packet dropouts. The problem of distributed fault detection and isolation for a class of second-order discrete-time MASs was studied by using an optimal robust observer approach [19]. But most of these works only dealt with fault detection and isolation, and few of these results addressed the problem of on-line fault estimation, which is a challenging issue. For a class of MASs, a robust fault estimation method based on sliding mode observers was proposed for a collection of agents, but only undirected graphs were considered [20]. In [21], a consensus-tracking-based distributed fault estimation and distributed fault tolerant control problem for a multiagent system were proposed, but the studied plant was a special class of power systems. In this paper, our objective is to propose a distributed fault estimation observer (DFEO) with adjustable parameter (AP) aimed at improving fault estimation performance for a class of MASs subject to directed graphs. Main novelties of this paper are as follows. 1) A DFEO with AP of MASs is constructed using relative output estimation errors obtained from the communication topology.
c 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2
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2) The proposed AP-based technique can use the current output information to enhance the accuracy of fault estimation, and the existence condition of the proposed observer design is wider than some existing methods. 3) H∞ and H2 with pole placement (PP)-based multiconstrained approaches are given to calculate the gain of DFEO with AP and it is possible to get a smaller H∞ or H2 performance index. The proposed results are convex optimization and can be readily calculated by linear matrix inequalities (LMIs). The rest of this paper is organized as follows. System description is presented in Section II. In Section III, a novel DFEO design with AP is provided for MASs with directed communication graphs to improve the accuracy of fault estimation. Section IV presents simulation results of an airplane model to illustrate the feasibility and effectiveness of the proposed DFEO with AP. Section V draws the conclusions.
II. S YSTEM D ESCRIPTION
For each agent, the dynamic system (1) can be rewritten in the following form: ⎧ x˙ i (t) B A E xi (t) ⎪ ⎪ + ui (t) ⎪ ⎪ f˙i (t) = 0 0 (t) f ⎪ i ⎪ 0 ⎨ D 0 ωi (t) + f˙i (t) ⎪ ⎪ 0 I ⎪ ⎪
xi (t) ⎪ ⎪ ⎩ yi (t) = C 0 fi (t) where f˙i (t) is the derivative of the fault of the ith agent. 0 and I, respectively, denote a zero matrix and an identity matrix with appropriate dimensions. Let us denote augmented vectors and matrices below ωi (t) A E xi (t) ¯ , νi (t) = ˙ , A= x¯ i (t) = fi (t) 0 0 fi (t)
¯B = B , C¯ = C 0 , D ¯ = D 0 0 0 I then it follows:
A. Preliminaries Consider a directed graph G = (V, E, A) with a nonempty finite set of N nodes V = (v1 , v2 , . . . , vN ), a set of edges or arcs E ⊂ V × V, and the associated adjacency matrix A = [aij ] ∈ RN×N . In this paper, the graph is assumed to be time-invariant, i.e., A is constant. An edge rooted at node vj and ended at node vi is denoted by (vj , vi ), which means information can flows from node vj to node vi . aij is the weight of edge (vj , vi ) and aij = 1 if (vj , vi ) ∈ E, otherwise aij = 0. Node vj is called a neighbor of node vi if (vj , vi ) ∈ E. The set of neighbors of node vi is denoted as Ni = { j|(vj , vi ) ∈ E}. Define the in-degree matrix is D = diag{di } ∈ RN×N with di = j∈Ni aij and the Laplacian matrix as L = D − A. It is obvious that L1N = 0, where IN is a vector with dimension N × 1. The edges in the form of (vi , vi ) are called loops. G = diag{gi } ∈ RN×N is denoted as a loop matrix and has at least one diagonal item being 1. A graph with loops is called a multigraph, otherwise it is a simple graph. B. System Description Let us consider a group of N agents modeled under a directed communication graph and each agent with fault is described by a state-space model x˙ i (t) = Axi (t) + Bui (t) + Efi (t) + Dωi (t) (1) yi (t) = Cxi (t) Rn ,
Rm ,
Rp ,
where xi (t) ∈ ui (t) ∈ and yi (t) ∈ respectively, are the state, the input, and the output of the ith agent. ωi (t) ∈ Rd is the disturbance. fi (t) ∈ Rr represents the system component or actuator fault, which is bounded. A–E are constant real matrices of appropriate dimensions. It is supposed that matrices C and E are of full rank and the pair (A, C) is observable.
¯ xi (t) + Bu ¯ i (t) + Dν ¯ i (t) x¯˙ i (t) = A¯ ¯ yi (t) = C¯xi (t)
(2)
which is equivalent to the system dynamics (1). Design Objective: In this paper, for a class of MASs connected by a directed communication graph, a new DFEO with AP is proposed to enhance fault estimation performance by quantitative analysis. Before presenting main results, we first recall the following lemma, which will be used in the sequel. Lemma 1 [22]: The eigenvalues of a given matrix A ∈ Rn×n belong to the circular region D(α, τ ) with center α + j0 and radius τ if and only if there exists a symmetric positive definite matrix P ∈ Rn×n satisfying −P P(A − αIn ) < 0. (3) ∗ −τ 2 P III. M AIN R ESULTS A. DFEO With AP Since all agents are connected based on the communication topology, the fault estimation observer of MASs should consider the information exchange among agents. First, the relative output estimation error of the ith node is defined as follows:
aij yˆ i (t) − yi (t) − yˆ j (t) − yj (t) ζi (t) = j∈Ni
+ gi yˆ i (t) − yi (t)
where aij and gi , respectively, are the element of Laplace matrix L and loop-graph matrix G. Remark 1: Different from centralized structure, the DFEO design of MASs needs to use the relative output estimation error ζi (t), which is based on information exchanges from neighboring nodes.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. ZHANG et al.: AP-BASED DFEO DESIGN FOR MASs WITH DIRECTED GRAPHS
3
For the augmented system (2), the following DFEO with AP for the ith agent is constructed using the relative output estimation error: ¯ i (t) − Hζ ¯ i (t) − θ F¯ ζ˙i (t) x˙ˆ¯ i (t) = A¯ xˆ¯ i (t) + Bu (4) ¯ ˆ yˆ i (t) = Cx¯ i (t)
Furthermore, it obtains
I + (L + G) ⊗ θ F¯ C¯ e˙¯ x (t)
¯ C¯ e¯ x (t) − IN ⊗ D ¯ ν(t). = IN ⊗ A¯ − (L + G) ⊗ H (8)
¯ , since F¯ C¯ = 0 C 0 = For I + (L + G) ⊗ θ (F¯ C) F 0 0 ¯ , it is easy to check that matrix [I +(L+G)⊗θ (F¯ C)] FC 0 ¯ is nonsingular. Let us denote a nonsingular matrix M ¯ = I + (L + G) ⊗ θ F¯ C¯ M
where xˆ¯ i (t) ∈ Rn and yˆ i (t) ∈ Rp are state and output the H 1 ¯ = (H1 ∈ Rn×p and of the observer, respectively. H H2 For the H2 ∈ Rr×p ) is observer gain matrix to be designed. 0 ˙ added term θ F¯ ζi (t), the scalar θ and F¯ = (F ∈ Rr×p ) F are APs to be determined. Remark 2: In DFEO with AP (4), the term θ F¯ ζ˙i (t) is added based on the observer design in [23], whose purpose is to improve fault estimation performance and make the observer (4) more reliable and flexible. This fact will be verified in Section IV. If the scalar θ is set as 0, the DFEO with AP converts into the normal one shown in [23]. For the ith node, the error vectors and a matrix are defined e¯ xi (t) = I¯r =
xˆ¯ i (t) − x¯ i (t),
0 Ir
efi (t) = fˆi (t) − fi (t)
then we can get the local error dynamics e˙¯ xi (t) = x˙ˆ¯ i (t) − x˙¯ i (t) ¯ exi (t) − Dν ¯ i (t) − Hζ ¯ i (t) − θ F¯ ζ˙i (t) = A¯ ¯ exi (t) − Dν ¯ i (t) = A¯ ⎡ ⎤
¯ xj (t) + gj C¯ ¯ exi (t) − Ce ¯ exi (t) ⎦ ¯⎣ −H aij C¯ j∈Ni
⎡
− θ F¯ ⎣
(5)
Ir e¯ xi (t) = I¯r e¯ xi (t).
(6)
j∈Ni
and
efi (t) = 0
In order to consider the fault estimation problem from the overall perspective, the global error vectors are defined
T e¯ x (t) = e¯ Tx1 (t), e¯ Tx2 (t), · · · , e¯ TxN (t) T ef (t) = eTf 1 (t), eTf 2 (t), · · · , eTf N (t)
T ν(t) = ν1T (t), ν2T (t), · · · , νNT (t) then we get the global error dynamics
¯ ν(t) e˙¯ x (t) = IN ⊗ A¯ e¯ x (t) − IN ⊗ D
¯ (L + G) ⊗ C¯ e¯ x (t) − IN ⊗ H
− IN ⊗ θ F¯ (L + G) ⊗ C¯ e˙¯ x (t)
¯ ν(t) = IN ⊗ A¯ e¯ x (t) − IN ⊗ D
¯ C¯ e¯ x (t) − (L + G) ⊗ θ F¯ C¯ e˙¯ x (t) − (L + G) ⊗ H
¯ C¯ e¯ x (t) = IN ⊗ A¯ − (L + G) ⊗ H
¯ ν(t) − (L + G) ⊗ θ F¯ C¯ e¯˙ x (t) − IN ⊗ D (7) where ⊗ represents the Kronecker product.
And the global fault estimation error is
ef (t) = IN ⊗ I¯r e¯ x (t).
(9)
(10)
For the error dynamics (9) and (10), after choosing param¯ we need to calculate observer gain H. ¯ Next, eters θ, F, we propose H∞ and H2 performances with PP to calcu¯ where H∞ and H2 constraints are presented late matrix H, to restrain the influence of terms ν(t) with respect to the fault estimation error and the purpose of introducing PP is to guarantee the system transient performance of fault estimation. B. Observer Gain Calculation Based on H∞ Performance With PP
⎤
aij C¯ e˙¯ xi (t) − C¯ e˙¯ xj (t) + gj C¯ e˙¯ xi (t) ⎦
and it follows:
¯ −1 IN ⊗ A¯ − (L + G) ⊗ H ¯ C¯ e¯ x (t) e˙¯ x (t) = M
¯ −1 IN ⊗ D ¯ ν(t). −M
Theorem 1: Let an H∞ performance level γ and a circular ¯ if there region D(α, τ ) be given. For a prescribed matrix M, exist a symmetric positive definite matrix P¯ ∈ R(n+r)×(n+r) and a matrix Y¯ ∈ R(n+r)×p satisfying
⎡ ⎤ ¯ T IN ⊗ P¯ D ¯ φ −M IN ⊗ I¯rT ⎣∗ −γ 0 ⎦
