www.ietdl.org Published in IET Control Theory and Applications Received on 3rd January 2014 Revised on 21st July 2014 Accepted on 11th August 2014 doi: 10.1049/iet-cta.2014.0006
ISSN 1751-8644
Distributed H∞ filtering over sensor networks with heterogeneous Markovian coupling intercommunication delays Xiaohua Ge1 , Qing-Long Han1,2 , Xiefu Jiang1,3 1 Centre
for Intelligent and Networked Systems, Central Queensland University, Rockhampton, QLD 4702, Australia School of Engineering, Griffith University, Gold Coast Campus, QLD 4222, Australia 3 School of Automation, Hangzhou Dianzi University, Zhejiang 310018, People’s Republic of China E-mail:
[email protected] 2 Griffith
Abstract: This study is concerned with distributed H∞ filtering for continuous-time linear systems over sensor networks with heterogeneous Markovian coupling intercommunication delays. The set of sensor nodes forms a sensing and communication network whose topology is modelled by a directed graph that describes the measurement exchanged among neighbouring sensor nodes. Heterogeneous random coupling delays modelled by a Markov process are considered in the intercommunication between interacting sensor nodes. A refined two-step decoupling technique is presented to deal with the complicated coupling of exchanged measurement in the presence of the delays. A sufficient condition on the existence of desired distributed H∞ filters is derived such that the resultant filtering error system is mean square exponentially stable with prescribed weighting average H∞ performance. Two illustrative examples are given to show the effectiveness of the proposed results.
1
Introduction
In the context of distributed coordination information processing, a fundamental problem is to develop a distributed filtering algorithm to estimate an unavailable state signal through noisy measurement and a disturbed plant. In recent years, considerable research efforts have been made on distributed filtering, see for example, [1–3] on distributed Kalman filtering, [4, 5] on distributed state estimation and [6–9] on distributed H∞ filtering. Notice that although various results on distributed filtering over sensor networks have been reported in the literature, the majority of them neglect the effects of coupling network-induced delays between intercommunicating sensor nodes on filtering performance. When signals are transmitted over sensor networks, coupling network-induced delays inevitably occur in the intercommunication among interacting sensor nodes because of limited network bandwidth and congested network traffic. Furthermore, these coupling network-induced delays possess their own characteristics. For example, they are inherently random and time-varying [10]; and they are heterogeneous [11], that is, the delays induced in the intercommunication of two different sensor nodes are not identical. These delays are usually regarded as one of the main sources degrading system performance or even jeopardising the divergence of filtering algorithms. Therefore the effects of coupling networkinduced delays should be taken into account when dealing with the distributed filtering problem over sensor networks. Different from traditional filtering algorithms in the case of only single sensor node, see for example, [12–16], two 82 © The Institution of Engineering and Technology 2014
issues in solving distributed filtering problems over sensor networks are needed to be addressed. One issue is how to deal with the complicated coupling measurement between each sensor node and its neighbouring nodes in accordance with a given network topology. The other issue is how to deal with coupling network-induced delays in the intercommunication among interacting sensor nodes. It is worth stressing that the heterogeneous coupling characteristic of such delays renders the distributed filtering problem a challenging issue. Therefore it is desirable and necessary to develop an effective decoupling technique to deal with complicated coupling information in the presence of heterogeneous coupling intercommunication delays. To the best of the authors’ knowledge, there is no published result available in the existing literature on distributed H∞ filtering for systems over sensor networks subject to heterogeneous random coupling intercommunication delays, which motivates the present study. In this paper, a new distributed H∞ filtering algorithm for a continuous-time linear system subject to heterogeneous Markovian coupling intercommunication delays is presented. The set of sensor nodes forms a sensing and communication network. Each sensor node is allowed to effectively communicate with neighbouring nodes in accordance with a given network topology, which is modelled by a directed graph that describes measurement exchanged among intercommunicating sensor nodes. Heterogeneous random coupling delays, experienced in the intercommunication among interacting sensor nodes, are incorporated and modelled by a Markov process. Combining Kronecker product, a refined IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
www.ietdl.org two-step technique is proposed to realise the decoupling of the exchanged measurement in the presence of the delays. A sufficient condition on the existence of desired distributed H∞ filters is derived such that the resultant filtering error system is mean square exponentially stable with prescribed weighting average H∞ performance. An aircraft engine system and a numerical example are presented to illustrate the effectiveness of the proposed filter design method. Notation: The term diagN {·} stands for a block-diagonal matrix with N blocks. The term diagiN {Ui } stands for a block-diagonal matrix with N blocks indexed by i. The term vecN {·} (or colN {·}) denotes a row (or column) vector with N blocks. The term veciN {Ui } (or coliN {Ui }) denotes a row (or column) vector with N blocks indexed by i. The symbol ⊗ is used to denote the Kronecker product for matrices. The term sym(UV ) denotes the symmetrised expression UV + V T U T . Other notations in this paper are quite standard.
2 Problem statement and a refined decoupling technique 2.1
Fig. 1 Schematic diagram of distributed H∞ filtering over a sensor network
Plant
Consider the plant, which is described by the following continuous-time linear system of the form x˙ (t) = Ax(t) + Bv(t), x(0) = x0 (1) z(t) = Ex(t) + Fv(t) where x(t) ∈ Rnx is the state; z(t) ∈ Rnz is the objective output to be estimated; v(t) ∈ Rnv denotes the exogenous disturbance input belonging to L2 [0, ∞); A, B, E and F are known constant matrices with appropriate dimensions; and x0 is an initial condition. In this paper, we assume that the system matrix A is Hurwitz stable. A sensor network with N cooperative sensor nodes which are dispersedly deployed in a sensor field is used to monitor the measurement from the plant. Let V = {1, 2, . . . , N } be the index set of sensor nodes, E ⊆ V × V be the edge set of paired sensor nodes, and A = [aij ] ∈ RN ×N be the weighted adjacency matrix. The directed graph G = (V, E, A) represents the network topology and describes measurement exchanged among neighbouring sensor nodes. An edge of G is denoted by (i, j). The adjacency elements aij associated with the edges of the graph are positive, that is, aij > 0 ⇔ (i, j) ∈ E; otherwise aij = 0. Self-loops are allowed in the graph and (i, i) can be regarded as an additional edge. The set of neighbours of the node i plus the node itself is denoted by Ni = {j ∈ V : (i, j) ∈ E}. The measurement output model on each sensor node is given by yi (t) = Ci x(t) + Di v(t),
i∈V
(2)
where yi (t) ∈ R is the measurement output received by the sensor node i from the plant; Ci and Di , i ∈ V, are known constant matrices with appropriate dimensions. ny
2.2 Markovian coupling intercommunication delays In the sequel, we assume that the set of sensor nodes forms a sensing and communication network, as shown in Fig. 1. Each sensor node collects measurement not only from itself but also from its all underlying neighbouring nodes i1 , . . . , is ∈ Ni through the network. Owing to limited IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
bandwidth, network channel quality and network congestion, random network-induced delays inevitably occur during the intercommunication among those interacting sensor nodes. To emphasise the random and time-varying characteristics of the coupling network-induced delays, we consider the mode-dependent time-varying delays τij (r(t)), (i, j) ∈ E, which are governed by a continuous-time homogeneous Markov process {r(t), t ≥ 0} taking values in a finite set M = {1, 2, . . . , M } and with right-continuous trajectories. The transition probability matrix of {r(t), t ≥ 0} is given by = [πmn ]M ×M with π δ + o(δ), m = n Pr{r(t + δ) = n|r(t) = m} = mn 1 + πmm δ + o(δ), m = n where δ > 0, limδ→0 (o(δ)/δ) = 0, and πmn ≥ 0, for m = n, is the transition rate from mode m at time t to mode n at time t + δ, and πmm = − M n=1,n=m πmn . Without loss of generality, we assume that 0 < τ (m) ≤ τij(m) (t) ≤ τ (m) < ∞, r(t) = m ∈ M
τ˙ij(m) (t) ≤ μ(m) < ∞, (3)
where τ (m) , τ (m) , and μ(m) , ∀ m ∈ M, are known constants. Denote τ = minm∈M {τ (m) } and τ = maxm∈M {τ (m) }. Remark 1: The motivation of modelling coupling intercommunication delays by a continuous-time homogeneous Markov process {r(t), t ≥ 0} stems from practical network environments. To model phenomena such as varying network queues and varying network loads, the network model usually needs to have a memory, or a mode [17]. One way to model dependence between different working modes is by letting the network model be governed by the mode of an underlying Markov process. Then the effects of varying network loads can be modelled by a transition from one mode to another mode of the Markov process. For example, in Fig. 2, the network model has three typical modes, one for low network load, one for medium network load and one for high network load. The transitions between different modes in the communication network can be modelled by a Markov 83 © The Institution of Engineering and Technology 2014
www.ietdl.org 2.4
Fig. 2 Example of a Markov process modelling the mode in a communication network with multiple nodes and network-induced intercommunication delays
process with the transition probabilities πmn , m, n ∈ {1, 2, 3}, which model how frequent changes of the network modes are. Together with each network mode m, a corresponding coupling intercommunication delay τij (t) occurs during the intercommunication from node j and node i. In a realistic model, the intercommunication delays would probably be smaller for low loads because the network is often idle, and larger for high loads because network congestion may occur when multiple nodes try to transmit at the same time [17]. In this sense, the distribution of the coupling intercommunication delays may also possess Markovian characteristics and depend on each network mode m, that is, τij(m) (t). This is particularly the case when signals are transmitted over sensor networks where multiple sensor nodes with different physical constraints are dispersedly deployed in a wide region. For example, in [18], a sensor network is shown to have jumping behaviour because of network’s working environments (normal or hazardous) and the mobility of sensor nodes. Therefore the coupling intercommunication delays among interacting sensor nodes in essence possess Markovian characteristics. 2.3
Distributed H∞ filters
Consider the following full-order mode-dependent distributed H∞ filters x˙ˆ i (t) = Aˆ i (r(t))ˆxi (t) + Bˆ i (r(t))ˆyi (t) (4) zˆi (t) = Cˆ i (r(t))ˆxi (t) where xˆ i (t) ∈ R is the state of the filter i; zˆi (t) ∈ R is the output of the filter i and represents an estimation of z(t). The input of the filter i is given by aij yj (t − τij (r(t))) (5) yˆ i (t) = nx
nz
j∈Ni
which is collected from both the sensor node i itself and its all underlying neighbours through the network. For all ˆ (m) and Cˆ i(m) are the parameter matrii ∈ V; m ∈ M, Aˆ (m) i , Bi ces of the filters to be determined. The initial condition of the filter (4) with (5) is supplemented as xˆ i (θ ) = ηˆ i (θ ), ∀ θ ∈ [−τ , 0], where ηˆ i (θ ) is a continuous function with ηˆ i (0) = xˆ 0i . 84 © The Institution of Engineering and Technology 2014
Refined two-step decoupling technique
From the formulation (5), network-induced delays affecting the measurement of the sensor node i itself and its neighbours are taken into account. Furthermore, the delays under consideration are ‘heterogeneous’, which means that the delay in the communication link from the sensor node j to the sensor node i differs from the one from the sensor node i to the sensor node j. Note that Kronecker product [7, 8] and sparse matrices [6, 9, 19] provide two natural ways to realise the decoupling among the interacting nodes ‘without’ delays affecting the communication links. When it comes to heterogeneous coupling intercommunication delays, however, the decoupling becomes much more involved. We now propose a refined decoupling technique consisting of two steps. The first step is to introduce some equivalent delays τl(m) (t), l ∈ L = {1, 2, . . . , L}, to represent the coupling intercommunication delays τij(m) (t), (i, j) ∈ E. More specifically, we define some auxiliary delays τl(m) (t) satisfying τl(m) (t) ∈ [τ (m) , τ (m) ] ⊆ [τ , τ ], τ˙l(m) (t) ≤ μ(m) , m ∈ M, and a mapping function f to establish the relationship from E to L. We assume that L is ‘not greater than’ the cardinality of E, that is, L ≤ Ni=1 Ni , where Ni represents the number of neighbours of the sensor node i. Let Al = [aij(l) ]N ×N with its elements given by a , if l = f (i, j), i.e. τij(m) (t) = τl(m) (t) aij(l) = ij 0, otherwise then we have that A = Ll=1 Al . The second step of the decoupling technique is to use Kronecker product to reformulate the filtering error system, which will be demonstrated in the next subsection. Remark 2: The auxiliary delays τl(m) (t) can be deemed as some equivalent delays with respect to the mode-dependent time-varying delays τij(m) (t). The relationship between τl(m) (t) and τij(m) (t) is established by a mapping function f . The adjacency matrix A is reformulated by Al and the reformulation is closely related to the scalar L. In particular, if L is less than the cardinality of E, the mapping is a surjection f : E → L satisfying the link index l = f (i, j). In the worst case, when L is equal to the cardinality of E, the mapping is a bijection f : E → L satisfying the link index l = f (i, j) and it leads to Ni=1 Ni matrices Al which contain only one adjacency element aij . However, this worst case will inevitably increase the computational requirements of the filter design algorithms. 2.5
Filtering error system
Define an estimation error signal ei (t) = z(t) − zˆi (t) on each node. Let xˆ (t) = coliN {ˆxi (t)}, x¯ (t) = colN {x(t)}, e(t) = coliN {ei (t)}, w(t) = col2 {v(t), collL {v(t − τl(m) (t))}}, A¯ = diagN {A}, B¯ = colN {B}, C¯ = vecN {coliN {Ci }, 0, . . . , 0}, ¯ = coliN {Di }, E¯ = diagN {E}, F¯ = colN {F}, Aˆ m = diagiN {Aˆ (m) D i }, Bˆ m = diagiN {Bˆ i(m) }, Cˆ m = diagiN {Cˆ i(m) } and I = vec2 {I , 0}. Setting x˜ (t) = col2 {¯x(t), xˆ (t)}, combining (1), (2) and (4), and using Kronecker product, we obtain the following augmented filtering error system x(t − τl(m) (t)) + B˜ m w(t) x˙˜ (t) = A˜ m x˜ (t) + Ll=1 Aˇ (m) l I˜ ˜ e(t) = E˜ m x˜ (t) + Fw(t) (6) IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
www.ietdl.org for all r(t) = m ∈ M, where
where
0 A¯ 0 ˇ (m) , A = , A˜ m = l Bˆ m (Al ⊗ I )C¯ 0 Aˆ m B¯ vecL {0} B˜ m = ¯ 0 veclL {Bˆ m (Al ⊗ I )D} E˜ m = [E¯ − Cˆ m ],
V1 (t, x˜ (t), m) = x˜ T (t)Pm x˜ (t) L t V2 (t, x˜ (t), m) = x˜ T (s)IT Ql I˜x(s) ds (m)
V3 (t, x˜ (t), m) = κ
F˜ = [F¯ vecL {0}]
(i) The resultant augmented filtering error system (6) is mean square exponentially stable, that is, the state trajectory x˜ (t) of system (6) with v(t) = 0 satisfies E{˜x(t)2r(0)∈M } ≤
exp(−σ t) sup−τ ≤θ ≤0 η(θ )2 , where · denotes the usual L2 [0, ∞) norm, > 0 and σ > 0 are known constant scalars. (ii) Under the zero initial condition, the estimation error ei (t), ∀ i ∈ V, satisfies the following weighting average H∞ performance N1 Ni=1 ei (t)E2 < βγ 2 v(t)2 + (1 − β)γ 2 Ll=1 v(t − τl(m) (t))2 for all non-zero v(t) ∈ ∞ L2 [0, ∞), where ei (t)2E = E{ 0 eiT (t)ei (t) dt}. Remark 3: By defining some auxiliary delays τl(m) (t), l ∈ L, m ∈ M, a mapping f : E → L, and combining Kronecker product, a refined technique is provided to realise the decoupling of exchanged measurement between an individual sensor node and its neighbouring nodes in the presence of heterogeneous random coupling intercommunication delays. From the formulation (6), one can see that the distributed H∞ filtering problem has been transformed into an H∞ optimisation problem for a continuous-time Markov jump stochastic system with finite time-varying mode-dependent delays τl(m) (t), m ∈ M; l ∈ L, which can be solved by the linear matrix inequality (LMI) technique presented in Section 3. Remark 4: Similar to [14], the positive scalar β in (ii) is referred to as a prescribed weighting factor. It explicitly explains how the non-delayed disturbance and the delayed disturbance affect system performance separately at a different weighting rate. The introduction of the weighting factor β may increase the flexible dimensions in the solution space for the H∞ optimisation problem formulated above, which will be illustrated through an example in Section 5.
IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
˜ x(s) ds dθ x˜ T (s)IT QI˜
t+θ
−τ
x˙˜ T (s)IT RIx˙˜ (s) ds dθ
t+θ
˜ = Ll=1 Ql and δ = τ − τ . with κ = maxm∈M {|πmm |}, Q Then, we have the following result. Theorem 1: For given scalars γ > 0, τ ≥ τ > 0, μ(m) , β ∈ ˆ (m) and Cˆ i(m) , the (0, 1), and given filter parameters Aˆ (m) i , Bi resultant filtering error system (6) is mean square exponentially stable with the prescribed weighting average H∞ performance, if there exist real matrices Pm > 0, Ql > 0, R > 0 and Sl,m , m ∈ M; l ∈ L, of appropriate dimensions such that ⎡
mT R −R δLτ ∗
m
⎢ ⎢ ∗ ⎣ ∗
mT
⎤
⎥ 0 ⎥ ⎦ < 0, −NI
R ∗
Sl,m ≥ 0, m ∈ M; l ∈ L R (8)
where ˜ ˜ veclL {IAˇ (m) m = vec5 {IA, l }, IBm , 0, 0}, ˜ 0, 0} m = vec5 {E˜ m , vecL {0}, F, ⎤ ⎡ δL T (11) (12) (13)
I R 0 m m ⎥ ⎢ m τ ⎥ ⎢ ⎥ ⎢ l l T ⎥ ⎢ ∗ (22) 0 col {R + S } col {R + S } l,m L L m l,m ⎥ ⎢
m = ⎢ ∗ ⎥ ∗ m(33) 0 0 ⎥ ⎢ L T ⎥ δL ⎢ ⎢ ∗ ∗ ∗ − + L R − l=1 Sl,m ⎥ ⎦ ⎣ τ ∗ ∗ ∗ ∗ −LR with ˜ −
m(11) = P˜ m + sym(Pm A˜ m ) + (1 + κδ)IT QI
δL T I RI τ
(13)
m(12) = veclL {Pm Aˇ (m) = Pm B˜ m l }, m
m(22) = diaglL {−2R − sym(Sl,m ) − (1 − μ(m) )Ql }
m(33) = diag2 {−βγ 2 I , diagL {−(1 − β)γ 2 I }}, M
πmn Pn
Proof: See the Appendix.
In this section, an H∞ performance analysis criterion for the resultant filtering error system (6) is presented. To begin with, we choose the stochastic Lyapunov–Krasovskii functional candidate as Vh (t, x˜ (t), m)
(t)
n=1
3 Distributed H∞ filtering performance analysis
h=1
−τ
V4 (t, x˜ (t), m) = δL
P˜ m =
V (t, x˜ (t), m) =
−τ t
0 t
The initial condition of the augmented filtering error system (6) is supplemented as x˜ (θ ) = η(θ ), ∀ θ ∈ [−τ , 0], where η(θ) is a continuous function with η(0) = col2 {colN {x0 }, coliN {ˆx0i }}. ‘The distributed H∞ filtering problem’ to be addressed is stated as follows: Given a prescribed level of H∞ performance γ > 0, a scalar β ∈ (0, 1), design desired distributed H∞ filters in the form of (4) such that
4
t−τl
l=1
(7)
Remark 5: Without multiplying the Lyapunov–Krasovskii functional (7) by an exponential factor eεt , where ε > 0 [20], the proposed Theorem 1 directly estimates the decay rate by taking full use of the matrices of the Lyapunov– Krasovskii functional and its derivative, which simplifies the proof. Moreover, these matrices may be viewed as some free parameters in the optimisation of the estimation [21]. 85 © The Institution of Engineering and Technology 2014
www.ietdl.org 4
Distributed H∞ filters design
Based on Theorem 1, the following theorem provides a sufficient condition on the existence of desired distributed H∞ filters of the form (4) to solve the proposed distributed H∞ filtering problem. Theorem 2: For given scalars γ > 0, τ ≥ τ > 0, μ(m) , β ∈ (0, 1), the proposed distributed H∞ filtering problem for the resultant filtering error system (6) is solvable if there exist real matrices Ql > 0, R > 0, P1,m > 0, Sl,m , diagonal real matrices P2,m > 0, Am , Bm and Cˆ m , m ∈ M; l ∈ L, of appropriate dimensions such that ⎡ ˆm
⎢ ⎢ ∗ ⎣ ∗
mT R −R δLτ ∗
mT
⎤
⎥ 0 ⎥ ⎦ < 0,
R ∗
Sl,m ≥ 0, R
−NI
P1,m − P2,m > 0,
m ∈ M; l ∈ L
(9)
ˆ m is derived from m by replacing its m(11) , where ˆ m(11) = P˜ m + sym(m ) + (1 +
m(12) and m(13) blocks by ˜ − δL IT RI, ˆ m(12) = veclL { ¯ (m) ˆ (13) = ˇ m with κδ)IT QI l } and m τ P1,m A¯ m = P2,m A¯ P1,m B¯ ˇm = P2,m B¯
Am , Am
Bm (Al ⊗ I )C¯ Bm (Al ⊗ I )C¯ ¯ veclL {Bm (Al ⊗ I )D} P1,m , Pm = ¯ P2,m veclL {Bm (Al ⊗ I )D} ¯ (m) = l
P2,m P2,m
Moreover, the filter parameters in (4) are given by Aˆ m = −1 −1 Am , Bˆ m = P2,m Bm and Cˆ m , m ∈ M. P2,m Proof: Applying Schur completement, Pm > 0 is ensured by P1,m − P2,m > 0 in (9). Denote Am = P2,m Aˆ m and Bm = P2,m Bˆ m . After some simple algebraic manipulations, one can see that (8) implies (9). This completes the proof.
5 5.1
Illustrative examples F-404 aircraft engine system
We use an F-404 aircraft engine system to illustrate the effectiveness of the obtained theoretical results. To estimate the state of the F-404 aircraft engine system, the measurement needs to be transmitted from the aircraft in air to the control flat on the ground via wireless communication channels. Coupling network-induced delays among the intercommunication between sensor nodes result from the sensor signal transmission collision over the wireless network, the computational load on the navigation computer, and the existence of the time delay in sensor signal processing [16]. In this example, to enhance the reliability of the filtering error system, we consider a sensor network consisting three intercommunicating sensor nodes to monitor the measurement. The network topology is characterised by a directed graph G = (V, E, A) with the nodes V = {1, 2, 3}, the set of edges E = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 1), (3, 3)} and the adjacency matrix A = [aij ]3×3 , where adjacency elements aij = 1 when (i, j) ∈ E; otherwise aij = 0. The linearised 86 © The Institution of Engineering and Technology 2014
deterministic model of the F-404 aircraft engine system is assumed to be represented by the system (1) and (2), where the nominal system matrix A and the measurement output matrices Ci (i ∈ V) on each sensor are taken from the decoupled linearised longitudinal dynamic equation of motion of the F-404 aircraft model given in [22] with some modifications −1.46 0 2.428 0.1643 −0.4 −0.3788 , C1 = 1 A= 0.3107 0 −2.23 C2 = 0 1 0 , C3 = 0.4 0.6 0
0
0
Other parameter matrices are given by B = [0.1 0.1 0.2]T , D1 = 0.7, D2 = 0.8, D3 = 0.5, E = [0 0 1] and F = 0. The objective is to design desired distributed H∞ filters over the sensor network to estimate the system’s third state x3 (t) by using the observed measurement of the system’s first state x1 (t) and second state x2 (t). Consider the following two cases of coupling intercommunication delays: Case I: L = 1, that is, ‘homogeneous’ intercommunication delays. In this case, delays induced through communication links starting from all neighbours j, j ∈ Ni , to node i are identical, that is, τij (r(t)) = τ1 (r(t)), (i, j) ∈ E; and Case II: L = Ni=1 Ni = 6, that is, ‘heterogeneous’ intercommunication delays. In this case, delays induced through communication links starting from all neighbours j to node i are non-identical. Therefore, it leads to six equivalent delays τl (r(t)), where l ∈ L = {1, 2, 3, 4, 5, 6}. Moreover, we assume that τ1 (r(t)) is involved in the intercommunication link (1, 1); τ2 (r(t)) is involved in the intercommunication link (1, 2); τ3 (r(t)) is involved in the intercommunication link (2, 2); τ4 (r(t)) is involved in the intercommunication link (2, 3); τ5 (r(t)) is involved in the intercommunication link (3, 1); and τ6 (r(t)) is involved in the intercommunication link (3, 3). Suppose that the network model has two working modes, that is, the low network load mode and the high network load mode. The transition probabilities modelling changes of the network modes are given as π11 = −0.3 and π22 = −0.5. Choose τ = 0.1, τ = 0.9, μ(1) = 0.4 and μ(2) = 0.35. Firstly, applying Theorem 2 in Case I, we obtain the relationship between the minimal value of the weighting average H∞ performance γmin and the weighting factor β, as shown in Fig. 3. Therefore the curve in Fig. 3 enables us to properly select the weighting factor β to obtain desirable weighting average H∞ performance, which means that the introduction of β increases the flexible dimension in the solution space for the formulated H∞ optimisation problem. Secondly, we set β = 0.9. Applying Theorem 2 in Case II, it is found that the proposed distributed H∞ filtering problem is solvable
Fig. 3
Relationship between γmin and β IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
www.ietdl.org with the minimal value of the weighting average H∞ disturbance attenuation performance level γmin = 0.1306. Finally, to illustrate the effectiveness of the designed distributed filsin(t) ters, we take the external disturbance as v(t) = 1+(t−1) 2 . The initial conditions are chosen as x0 = col3 {−0.1, 0, 0.1} and xˆ 0i = col3 {0, 0, 0}, i ∈ V. Applying the obtained distributed H∞ filters in Case II, Fig. 4 depicts the evolution of the objective output signal z(t) and its estimation signals zˆi (t), i ∈ V under the modes evolution generated by the Markov process {r(t), t ≥ 0}. It can be seen from Fig. 4 that the designed distributed H∞ filters well estimate the objective output signal z(t) = x3 (t). 5.2
Numerical example
This numerical example is to show the result using a large number of sensors. Consider the system (1) and (2) with the
following system parameter matrices
−1.2 0.9 A= , −0.4 −0.8 C1 = 0.9 0.2 , C2 C3 = 0.2 0.6 , C4 C5 = 0.6 0.7 , C6 D1 = 0.5,
D2 = 0.2,
D5 = 0.5,
D6 = 0.2
E = [0.5 0.5],
B= = 0.5 = 0.4 = 0.3
0.5 , −0.6 0.8 0.3 , 0.4
D3 = 0.3,
F = 0.1
over a sensor network consisting six intercommunicating sensor nodes. The network topology is characterised
Fig. 4
Objective output signal z(t) and its estimation signals zˆi (t), i ∈ V, on each node under modes evolution r(t)
Fig. 5
Objective output signal z(t) and its estimation signals zˆi (t), i ∈ V, on each node under modes evolution r(t)
IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
D4 = 0.4,
87 © The Institution of Engineering and Technology 2014
www.ietdl.org by a directed graph G = (V, E, A) with the nodes V = {1, 2, 3, 4, 5, 6}, the set of edges E = {(1, 1), (1,3), (1,5), (2,2), (2,3), (2,5), (3,1), (3,3), (3,4), (3,6), (4,2), (4,4), (4,6), (5,3), (5,5), (5,6), (6,1), (6,2), (6,4), (6,6)} and the adjacency matrix A = [aij ]6×6 , where adjacency elements aij = 1 when (i, j) ∈ E; otherwise aij = 0. The objective is to design desired distributed H∞ filters over the sensor network to estimate the system’s objective output z(t) by using the observed measurement. Choose L = 3, that is, there exist three equivalent delays τl (r(t)), where l ∈ L = {1, 2, 3}. Moreover, we assume that τ1 (r(t)) is involved in the intercommunication links (1, 1), (2, 3), (3, 1), (3, 3), (4, 2), (5, 3) and (6, 2); τ2 (r(t)) is involved in the intercommunication links (1, 3), (2, 2), (3, 4), (4, 6), (5, 5) and (6, 4); and τ3 (r(t)) is involved in the intercommunication links (1, 5), (2, 5), (3, 6), (4, 4), (5, 6), (6, 1) and (6, 6). In this example, the network model has two working modes with π11 = −0.3 and π22 = −0.5. Let β = 0.9, τ = 0.1, τ = 0.5, μ(1) = 0.2 and μ(2) = 0.15. Applying Theorem 2, it is found that the proposed distributed H∞ filtering problem is solvable with the minimal value of the weighting average H∞ disturbance attenuation performance level γmin = 0.2707. To illustrate the effectiveness of the designed distributed filters, we take the external disturbance as v(t) = 0.1 cos(t) exp(−0.2t). The initial conditions are chosen as x0 = col2 {0.3, −0.2} and xˆ 0i = col2 {0, 0}, i ∈ V. Performing the obtained distributed H∞ filters, Fig. 5 depicts the evolution of the objective output signal z(t) and its estimation signals zˆi (t), i ∈ V under the modes evolution generated by the Markov process {r(t), t ≥ 0}. It can be seen from Fig. 5 that the designed distributed H∞ filters perform well for estimating the objective output signal z(t), which verifies the effectiveness of the proposed filter design method.
8 1 2 3 4 5 6 7 8 9
10
11 12 13 14
6
Conclusion 15
The problem of distributed H∞ filtering for a continuoustime linear system over a sensor network subject to heterogeneous Markovian coupling intercommunication delays has been studied. The set of sensor nodes has formed a sensing and communication network whose topology has been modelled by a directed graph that describes measurement exchanged among neighbouring sensor nodes. Together with Kronecker product, a refined two-step decoupling technique has been proposed to deal with the complicated coupling of the exchanged measurement in the presence of heterogeneous Markovian coupling intercommunication delays. A sufficient condition on the existence of desired distributed H∞ filters has been provided such that the mean square exponential stability of the resultant filtering error system is preserved with the prescribed weighting average H∞ performance. The filter design problem has been posed in terms of LMIs. An F-404 aircraft engine system and a numerical example have been given to show the effectiveness of the obtained theoretical results.
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22
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7
Acknowledgments
This work was supported in part by the Australian Research Council Discovery Project under Grant DP1096780, the Research Advancement Awards Scheme Program (January 2010–December 2012) at Central Queensland University, Australia; Key project of Natural Science Foundation of Zhejiang Province of China (Grant no. Z13F030015). 88 © The Institution of Engineering and Technology 2014
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IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
www.ietdl.org 9
where φl (t) = col3 {I˜x(t − τl(m) (t)), I˜x(t − τ ), I˜x(t − τ )}, ˜ = col2 {˜x(t), I˜x(t − τ )} and φ(t)
Appendix
9.1
Appendix: Proof ofTheorem 1
Proof: Suppose that v(t) = 0 and let L be the weak infinite generator of {(˜x(t), r(t)), t ≥ 0}. Then from (7), one has LV1 (t, x˜ (t), m) = x˜ T (t)P˜ m x˜ (t) + 2˜xT (t)Pm x˙˜ (t)
LV2 (t, x˜ (t), m) ≤
M
πmn
L t
n=1
(n) t−τl (t)
l=1
(10)
x˜ T (s)IT Ql I˜x(s)ds
T
− (1 − μ(m) )
L
x˜ T (t − τl(m) (t))IT Ql
˜ m + δLτ ˜ mT R˜ m )ζ (t) LV (t, x˜ (t), m) ≤ ζ T (t)(
l=1
× I˜x(t − τl(m) (t)) ˜ x(t) − κ LV3 (t, x˜ (t), m) = κδ x˜ T (t)IT QI˜
t−τ
(11) ˜ x(s) ds x˜ T (s)IT QI˜
t−τ
(12)
LV4 (t, x˜ (t), m) = δLτ x˙˜ T (t)IT RIx˙˜ (t) t x˙˜ T (s)IT RIx˙˜ (s) ds − δL Note that πmn ≥ 0, ∀ m = n and πmm = − Then we have
(13)
πmn
n=1
=
L t
LV (t, x˜ (t), m) ≤ −ρ x˜ T (t)˜x(t)
0 (n)
+ πmm
(m)
t−τl
t−τ
E{V (t, x˜ (t), m)} ≥ E{˜xT (t)˜x(t)} x˜ (s)I Ql I˜x(s) ds T
(19)
T
t−τ
t
where = minm∈M (λmin (Pm ) > 0. Based on (18) and (19), it is clear that
x˜ T (s)IT Ql I˜x(s) ds t−τ
˜ x(s) ds x˜ (s)I QI˜ T
(18)
(t)
−πmm
+ πmm ≤κ
T
t
l=1
By
for each r(t) = m ∈ M and > 0. Moreover, understanding from (7) that
x˜ (s)I Ql I˜x(s) ds T
0
x˜ T (s)IT Ql I˜x(s) ds
t−τl (t)
n=m
˜ m − δLτ ˜ mT R˜ m )) > 0. where ρ = minm∈M (λmin (− Dynkin’s formula [26], we have
(17)
E{V ( , x˜ ( ), m)} − V (0, η(θ ), r(0)) = E{ LV (t, x˜ (t), m) dt} ≤ −ρ E{˜xT (t)˜x(t) dt}
t πmn
t
≤
πmn < 0.
x˜ T (s)IT Ql I˜x(s) ds
(n)
L
L
n=m
t−τl (t)
l=1
l=1
(16)
where ζ (t) = col4 {˜x(t), collL {I˜x(t − τl(m) (t))}, I˜x(t − τ ), I˜x ˜ m is derived from m in (8) by eliminat(t − τ )}, ing the third row and column of m , and ˜ m = vec4 ˜ veclL {IAˇ (m) {IA, l }, 0, 0}. ˜ m + δLτ ˜ mT R˜ m < 0 by virtue From (8), one can see that of Schur complement. Therefore it follows from (16) that
t−τ
M
⎤ T R + Sl,m T ⎦ −Sl,m , −R
Combining (10)–(15) and taking expectation on both side of the inequality, we obtain
˜ x(t) + x˜ (t)I QI˜ T
⎡ −sym(Sl,m ) − 2R R + Sl,m ∗ −R l,m = ⎣ ∗ ∗ ⎡ δL ⎤ δL T − IT RI I R ⎢ τ ⎥ τ ⎥ ˜ =⎢ ⎣ δL ⎦ ∗ − R τ
T
(14)
t−τ
E{˜xT ( )˜x( )} ≤ −1 V (0, η(θ ), r(0)) − −1 ρ E{˜xT (t)˜x(t) dt}
(20)
0
To handle the last term of the inequality (13), we apply the Jensen inequality [23] and the reciprocally convex approach [24, 25] to obtain t
x˙˜ T (s)IT RIx˙˜ (s) ds
−δL t−τ
t
= −δL t−τ
≤
L
x˙˜ T (s)IT RIx˙˜ (s) ds −
t−τ L x˙˜ T (s)IT RIx˙˜ (s) ds δ l=1
t−τ
˜ φ(t) ˜ φlT (t)l,m φl (t) + φ˜ T (t)
l=1
IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
(15)
Then apply Gronwall-Bellman lemma to obtain E{˜xT ( )˜x( )} ≤ −1 V (0, η(θ ), r(0)) exp(−−1 ρ ) (21) Note that there exists a positive scalar κ such that −1 V (0, η(θ ), r(0)) ≤ κsup−τ ≤θ ≤0 η(θ )2 . It follows immediately that the mean square exponential stability of the augmented filtering error system (6) can be guaranteed [26]. We now consider the weighting average H∞ perfor mance index J ( ) = E{ 0 [ N1 eT (t)e(t) − βγ 2 vT (t)v(t) − L (m) (m) 2 T l=1 (1 − β)γ v (t − τl (t))v(t − τl (t))]dt} for all nonzero v(t) ∈ L2 [0, ∞). Applying (7) to the filtering error 89 © The Institution of Engineering and Technology 2014
www.ietdl.org system (6), we have J ( ) + E LV (t, x˜ (t), m)] dt 0
≤ ζ˜ T (t)( m + δLτ mT Rm +
1 T m )ζ˜ (t) N m
90 © The Institution of Engineering and Technology 2014
(22)
where ζ˜ (t) = col5 {˜x(t), collL {I˜x(t − τl(m) (t))}, w(t), I˜x(t − τ ), I˜x(t − τ )}. Using Schur complement to the first inequality of (8), letting → ∞, and under zero initial conditions, we have J ( ) < 0, which means that N1 Ni=1 ei (t)2E < βγ 2 v(t)2 + Ll=1 (1 − β)γ 2 v(t − τl(m) (t))2 . This completes the proof.
IET Control Theory Appl., 2015, Vol. 9, Iss. 1, pp. 82–90 doi: 10.1049/iet-cta.2014.0006
