- Email: [email protected]
Finite-time asynchronous H∞ filtering for positive Markov jump systems
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
Available online at www.sciencedirect.com
Journal of the Franklin Institute xxx (xxxx) xxx www.elsevier.com/locate/jfranklin
Finite-time asynchronous H∞ filtering for positive Markov jump systems Hui Shang, Guangdeng Zong∗, Wenhai Qi School of Engineering, Qufu Normal University, Rizhao 276826, PR China Received 2 December 2018; received in revised form 19 July 2019; accepted 4 August 2019 Available online xxx
Abstract In this paper, the asynchronous positive H∞ filtering is addressed for positive Markov jump systems. The asynchrony between the filter’s and the system’s modes is described through a conditional probability matrix. Three distinct asynchronous positive filters named an upper-bounding filter, a lower-bounding filter and a weighted filter are built. Sufficient conditions are established such that the filtering error systems are positive and finite-time bounded with an H∞ disturbance attenuation level. It is proved that the weighted filter has more flexibilities than the bounding filters since it can adjust the parameter θ when estimating the given systems. Finally, the effectiveness of the theoretical results is validated through the fish’s population evolution example. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Positive systems are widely used to model many practical processes subject to positivity constraints, for which the state variables are usually flow, pressure, liquid level, concentration substances, densities, or quantity of virus [1,2]. Most of the existing results for general systems cannot be directly applied to positive systems due to their positivity constraints. Recently, positive Markov jump systems (PMJSs) have received increasing interests since they can be employed to represent systems subjected to positivity constraints, abrupt changes and the jump governed by a Markov chain. Many methodologies and approaches are reported for PMJSs [3–8]. ∗
Corresponding author. E-mail address: [email protected] (G. Zong).
https://doi.org/10.1016/j.jfranklin.2019.08.008 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
2
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
Control synthesis and filtering analysis are addressed in [9–14] under the assumption that the system’s modes are synchronous with the controller/filter’s at all times. This means that all the information of the plant’s mode can be obtained by controller or filter. It is possibly a strong restriction in practical application. On one hand, in networked control systems [15], the finite transmission channel often leads to loss of data and communication delays. On the other hand, imprecise information could cause the mismatch between the system and the filter. The asynchronous phenomena may reduce the system performance, even lead to unstability. Therefore, more and more researchers begin to study the asynchronous issues brought by different behaviors. For example, the asynchrony resulting from overlapped detection delay is discussed in [16–19]. Asynchronous filtering is considered as Markov chain over a finite-time span in [20–23]. Wu et al. [24] is firstly concerned with the asynchronous issue for hidden Markov model. Then some people extend the result to switched systems [3], fuzzy systems [25] and time-varying MJSs [26], etc. But for PMJSs, few works are found in the existing results, which motivates us to the present study. Filtering plays an important role in estimating the system output when some state variables are immeasurable and finds wide applications in target tracking and adaptive control [23,27,28] and many useful works are reported [11,29–33]. Remarkably, the positive filtering problem is discussed for positive T-S fuzzy systems in [11]. The authors in [30] have investigated l1 -gain filtering for PMJSs. A positive L1 -gain filter is designed in [33] for positive semi-Markovian switching systems. There are few results reported concerning filtering for PMJSs under asynchronous switching governed by a Markov chain. Due to the positivity of filtering error systems, the single filter only estimates the system output from one direction which proved to be less accurate. In addition, different from Lyapunov asymptotic stability, finite-time stability emphasizes the transient performance and has strong applications background [34]. A finite-time stable (FTS) system may be not asymptotically stable and vice versa. Therefore, the general stability results cannot be extended to cope with the finite time stability of PMJSs. It is imperative and interesting to study the asynchronous positive H∞ filtering problem of PMJSs in the meaning of finite-time stability. In this paper, we focus upon the asynchronous positive H∞ filtering for PMJSs subject to asynchronous switching. With the aid of hidden Markov model, the asynchrony between the filter’s and the system’s modes will be fully considered. Three different filtering design schemes are addressed under the finite-time bounded (FTB) frame during a finite time span. Sufficient convex optimization conditions are derived such that the filtering error (FE) systems are positive and FTB with an H∞ disturbance attenuation performance. The contributions of our work lie in four aspects: (i) Compared with the existing work [16–19], the hidden Markov model is attempted to deal with asynchronous H∞ filtering for PMJSs over a finite time span; (ii) The asynchrony between the the filter’s and the system’s modes is stochastic and obeys a conditional probability, which can reflect the real phenomenon better; (iii) Through introducing some slack matrices, two bounding positive asynchronous H∞ filters are constructed which make the FE systems be positive, FTB and have an H∞ performance index. The estimated output will stay between the upper-bounding filter’s state and lower-bounding filter’s state. A weighted filter is proposed which has more flexibilities than the bounding filters during the design process; (iv) The fish’s population evolution is modelled as PMJSs with disturbances and applied to verify the theoretical results. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
3
Notations: In this paper, Rn denotes the vectors of n-tuples of real number. For matrix Ai , Aci,g denote bth row, gth column of Ai and [Ai ]b,g denotes the element of bth row, gth column of Ai . For any symmetric matrix P, P > 0 (P < 0) implies that the matrix P is positive (negative) definite. For a real matrix or vector A, A 0(0) denotes all the elements of A are non-negative (positive). Ari,b ,
2. Problem formulation and preliminaries Consider the discrete-time PMJSs: x(k + 1) = Aα(k) x(k) + Bα(k) ν(k),
(1a)
y(k) = Cα(k) x(k) + Dα(k) ν(k),
(1b)
z(k) = Hα(k) x(k) + Lα(k) ν(k),
(1c)
where x(k) ∈ R is the system state vector, ν(k) ∈ R is the external disturbance input which satisfies ν T (k)ν(k) ≤ ϑ. ϑ is a known positive scalars. y(k) ∈ Rq is the measured output, z(k) ∈ Rm is the estimated output. {α(k), k ≥ 0} is a discrete-time homogeneous Markov chain which takes values in M = {1, 2, . . . , M}. The transition probability (TP) matrix {π ij } is defined as n
p
πi j = Pr {α(k + 1) = j|α(k) = i} ≥ 0, with Mj=1 πi j = 1, for all i, j ∈ M. When α(k) = i ∈ M, Aα(k) , Bα(k) , Cα(k) , Dα(k) , Hα(k) , Lα(k) are respectively denoted by Ai , Bi , Ci , Di , Hi , Li and assumed to be known with appropriate dimensions. Here, it is assumed that the state x(k) is not measurable. In this paper, we focus on asynchronous positive H∞ filtering for PMJSs (1). In order to get real-time information on the output, an upper-bounding estimation zˆ(k) and a lower-bounding estimation zˇ(k) will be constructed as follows: xˆ(k + 1) = Aˆ β(k) xˆ(k) + Bˆ β(k) y(k), zˆ(k) = Hˆ β(k) xˆ(k) + Lˆ β(k) y(k),
(2)
xˇ(k + 1) = Aˇ β(k) xˇ(k) + Bˇ β(k) y(k), zˇ(k) = Hˇ β(k) xˇ(k) + Lˇ β(k) y(k),
(3)
where xˆ(k) and xˇ(k) are the filter state vectors, zˆ(k) and zˇ(k) are the estimations of z(k). Aˆ β(k) , Bˆ β(k) , Hˆ β(k) , Lˆ β(k) , Aˇ β(k) , Bˇ β(k) , Hˇ β(k) , and Lˇ β(k) are unknown filter parameter matrices. The variable {β(k), k ≥ 0} is defined in another set N = {1, 2, . . . , N } and connected to {α(k), k ≥ 0} through a known conditional probability (CP) matrix = {ϕil } with Pr {β(k) = l|α(k) = i} = ϕil , where ϕ il ∈ [0, 1], and Nl=1 ϕil = 1 for ∀l ∈ N . Remark 1. Generally, it is impossible that all the information of system’s mode is available to the filter. In view of the above definition, we use the variable β(k) to describe the filter’s mode which is connected to system’s mode α(k) through a known conditional probability Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
4
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
matrix. The system’s mode is not observable directly and is hidden to filter’s mode, which constitutes the hidden Markov model (α(k), β(k), M, N ) [35]. Denote xˆF (k) [x T (k ) xˆeT (k )]T , zˆF (k ) zˆ(k) − z(k ), xˆe (k ) xˆ(k) − x(k). Let activated system mode α(k) = i and activated filtering mode β(k) = l for ∀i ∈ M, l ∈ N . From Eqs. (1) and (2), we get the following upper-bounding FE system: xˆ (k + 1) = Aˆ il xˆF (k) + Bˆ il ν(k),
eˆ : F (4) zˆF (k) = Hˆ il xˆF (k) + Lˆ il ν(k), where Aˆ il =
Ai Aˆ l + Bˆ l Ci − Ai
Hˆ il = Hˆ l + Lˆ l Ci − Hi
0 Bi ˆ il = , , B Aˆ l Bˆ l Di − Bi Hˆ l , Lˆ il = Lˆ l Di − Li .
Similarly, let xˇF (k) [x T (k ) xˇeT (k )]T , zˇF (k ) z(k) − zˇ(k ), xˇe (k ) x(k) − xˇ(k). The lowerbounding FE system reads xˇ (k + 1) = Aˇ il xˇF (k) + Bˇ il ν(k),
eˇ : F (5) zˇF (k) = Hˇ il xˇF (k) + Lˇ il ν(k), where Aˇ il =
Ai ˇ Ai − Bl Ci − Aˇ l
Hˇ il = Hi − Lˇ l Ci − Hˇ l
0 Bi ˇ il = , , B Aˇ l Bi − Bˇ l Di Hˇ l , Lˇ il = Li − Lˇ l Di .
Remark 2. In this paper, the hidden Markov model is attempted to describe asynchronous circumstances for PMJSs, where the filter mode cannot be determined directly by the system’s mode. In the existing results [16–18], asynchronous mode either depends on the modes at time k − 1 and time k or is entirely different from the system’s mode. These situations lead to some conservativeness when dealing with asynchronous problem. In this paper, the filter mode is not completely independent of system mode and only determined by observing the system mode at the current time k. The results in this paper are more appropriate to describe the practical issues. Remark 3. According to the above definition, the probability π ij represents the system transition probability from ith mode to jth mode for i, j ∈ M. Distinguished from the system mode α(k), a stochastic variable β(k) is introduced to denote the filter mode. These two stochastic variables are subject to a known CP matrix = {ϕil } which shows the asynchrony between the filters and the systems modes. ϕ il describes the possibility of the filter running in the lth mode, when the system lies in ith mode for i ∈ M, l ∈ N , which reflects the degree of asynchrony. In addition, if the MJSs include only one subsystem, i.e., M = 1, then the filter mode reduces to a mode-independent one. Moreover, the filter’s modes are synchronous with the system’s if the set M = N and the CP ϕil = 1 for i = l. Therefore, the asynchrony considered in this paper is more general. Lemma 1 [1]. System (1) is positive, if and only if Ai 0, Bi 0, Ci 0, Di 0, Hi 0 and Li 0 for all i ∈ M. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
5
Remark 4. The upper-bounding estimation system (2) and lower-bounding estimation system are designed to estimate the system output for PMJSs. The positivity for systems (2) and (3) are essential. The filter parameter matrices satisfy Aˆ β(k) 0, Bˆ β(k) 0, Hˆ β(k) 0, Lˆ β(k) 0, Aˇ β(k) 0, Bˇ β(k) 0, Hˇ β(k) 0, and Lˇ β(k) 0. Definition 1 [20]. Given scalars d2 > d1 > 0, integer K and a matrix R > 0. The system (1) is FTB with respect to (d1 , d2 , K, R, ϑ), if E {x T (0)Rx(0)} ≤ d1 ⇒ E {x T (k)Rx(k)} ≤ d2 , ν T (k)ν(k) ≤ ϑ for any k ∈ {1, 2, . . . , K }. Definition 2 [20]. For a scalar γ˜ > 0 and integer K, the system (1) is FTB with a required H∞ performance level γ˜ , if (i) system (1) is FTB with respect to (d1 , d2 , K, R, ϑ); (ii) under zero initial conditions, there holds K
E { z(k) 2 } < γ˜ 2
k=0
K
ν(k) 2 .
(6)
k=0
Lemma 2 [29]. For the functions g1 (x), g2 (x) and 0 ≤ g1 (x) ≤ g2 (x), if we define ψ (x) = δg1 (x) + (1 − δ)g2 (x), δ ∈ [0, 1], then we have g1 (x) ≤ ψ(x) ≤ g2 (x). Our main tasks are finding the filter parameters of the upper-bounding asynchronous positive filter in Eq. (2) and the lower-bounding asynchronous positive filter in Eq. (3) such that the corresponding FE systems (4) and (5) are respectively positive and FTB with required H∞ performance level. 3. Main results 3.1. Design of bounded filters In this section, the existence conditions concerning the asynchronous positive filters shall be first established. An upper-bounding asynchronous positive filter Eq. (2) and a lowerbounding asynchronous positive filter Eq. (3) will be designed such that the FE systems are positive, FTB and gain an H∞ performance level. Some feasible conditions will be provided to obtain the filters parameters. Theorem 1. Given positive scalars ϑ, ζ u > 1, c2 > c1 > 0, an integer K > 0, and a matrix R > 0, the upper-bounding FE system (4) is positive, FTB with an H∞ performance level γ˜u = γu ζuK , if there exist positive scalar γ u , matrices Pi > 0, Qil > 0 for any i ∈ M, l ∈ N such that Aˆ l + Bˆ l Ci − Ai 0, Bˆ l Di − Bi 0,
(7)
Hˆ l + Lˆ l Ci − Hi 0, Lˆ l Di − Li 0,
(8)
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
6
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
N
ϕil Qil < Pi ,
(9)
l=1
⎡
−P˜i−1 ⎢ ∗ ⎢ ⎣ ∗ ∗
0 −I ∗ ∗
Aˆ il Hˆ il −ζu Qil ∗
⎤ Bˆ il Lˆ il ⎥ ⎥ < 0, 0 ⎦ −γu2 I
(10)
1 1 ζuK c1 max{λmax (Pi ), i ∈ M} + ϑ ≤ c2 , min{λmin (Pi ), i ∈ M} ζu − 1 1 1 where P˜i = Mj=1 πi j Pj , Pi R− 2 Pi R− 2 .
(11)
Proof. By Lemma 1 and inequalities (7) and (8), there are matrices Aˆ il 0, Bˆ il 0, Hˆ il 0, Lˆ il 0 for all i ∈ M, l ∈ N . Hence, the upper-bounding FE system (4) is positive. Construct the following Lyapunov function candidate V (k, α(k), xˆF (k)) = xˆFT (k)Pα(k) xˆF (k).
(12)
When α(k) = i, α(k + 1) = j, β(k) = l, using Eq. (9), we get Jzν = E V (k + 1, j, xˆF (k + 1)) − ζuV (k, j, xˆF (k)) − γu2 ν T (k )ν(k ) + zˆFT (k )zˆF (k ) ⎧ ⎫ M ⎨ ⎬ πi j Pj xˆF (k + 1) − ζu xˆFT (k)Pi xˆF (k) − γu2 ν T (k)ν(k) + zˆFT (k)zˆF (k) = E xˆFT (k + 1) ⎩ ⎭ j=1 N T T =E ϕil ξ (k)il ξ (k) − ζu xˆF (k )Pi xˆF (k ) ≤E =E
l=1 N l=1 N
ϕil ξ T (k)il ξ (k) − ζu
N
ϕil xˆFT (k )Qil xˆF (k )
l=1
ϕil ξ T (k)il ξ (k) ,
(13)
l=1
T where ξ (k) = xˆFT (k) ν T (k) , P˜i = Mj=1 πi j Pj , Aˆ Til P˜i Aˆ il + Hˆ ilT Hˆ il Aˆ Til P˜i Bˆ il + Hˆ ilT Lˆ il il = , Bˆ ilT P˜i Aˆ il + Lˆ ilT Hˆ il Bˆ ilT P˜i Bˆ il + Lˆ ilT Lˆ il − γu2 I Aˆ Til P˜i Aˆ il + Hˆ ilT Hˆ il − ζu Qil Aˆ Til P˜i Bˆ il + Hˆ ilT Lˆ il il = . Bˆ ilT P˜i Aˆ il + Lˆ ilT Hˆ il Bˆ ilT P˜i Bˆ il + Lˆ ilT Lˆ il − γu2 I Schur complement formula and the condition (10) means il < 0 and Jzν < 0. Further, we can obtain E V (k + 1, α(k + 1), xˆF (k + 1)) ≤ E ζuV (k, α(k), xˆF (k)) + γu2 ν T (k)ν(k) − zˆFT (k)zˆF (k) . (14) Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
After some manipulations, we derive E V (k, α(k), xˆF (k)) ⎧ ⎫ k−1 ⎨ ⎬ ≤ ζuk V (0, α(0), xˆF (0)) + E ζuk−1−μ {γu2 ν T (μ)ν(μ) − zˆFT (μ)zˆF (μ)} , ⎩ ⎭ μ=0 ⎧ ⎫ k−1 ⎨ ⎬ k ≤ ζu V (0, α(0), xˆF (0)) + E ζuk−1−μ γu2 ν T (μ)ν(μ) . ⎩ ⎭
7
(15)
μ=0
In view of ν T (k)ν(k) ≤ ϑ and ζ u > 1, there holds k−1 E V (k, α(k), xˆF (k)) ) ≤ ζuk V (0, α(0), xˆF (0)) + ζuk−1−μ γu2 ϑ μ=0
ζk − 1 ≤ ζuk V (0, α(0), xˆF (0)) + γu2 ϑ u ζu − 1 1 k 2 ≤ ζu V (0, α(0), xˆF (0)) + γu ϑ ζu − 1 1 K T 2 ≤ ζu xˆF (0)P0 xˆF (0) + γu ϑ ζu − 1 1 K 2 . ≤ ζu c1 max{λmax (Pi ), i ∈ M} + γu ϑ ζu − 1
(16)
Observe E V (k, α(k), xˆF (k)) = E xˆFT (k)Pi xˆF (k)
≥ min{λmin (Pi ), i ∈ M}E xˆFT (k)RxˆF (k) .
(17)
Combining Eqs. (11), (16) with Eq. (17) yields 1 1 ≤ c2 . E xˆFT (k)RxˆF (k) ≤ ζuK c1 max{λmax (Pi ), i ∈ M} + γu2 ϑ min{λmin (Pi ), i ∈ M} ζu − 1 This together with Definition 1 implies that the upper-bounding FE system (4) is FTB. On the other hand, under the zero initial conditions, Eq. (15) implies ⎧ ⎫ k−1 ⎨ ⎬ 0 ≤ E V (k, α(k), xˆF (k)) ≤ E ζuk−1−μ {γu2 ν T (μ)ν(μ) − zˆFT (μ)zˆF (μ)} ⎩ ⎭ μ=0 ⎧ ⎫ ⎨K−1 ⎬ ≤E ζuK {γu2 ν T (μ)ν(μ) − zˆFT (μ)zˆF (μ)} . ⎩ ⎭ μ=0
Therefore, K−1 μ=0
K−1 E zˆFT (μ)zˆF (μ) ≤ ζuK γu2 ν T (μ)ν(μ), μ=0
which indicates the upper-bounding FE system (4) satisfies an H∞ performance level γ˜u = γu ζuK . This ends the proof. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
8
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
Remark 5. Due to the existence of coupling terms in Theorem 1, the conditions in Eqs. (7)–(11) cannot be easily solved. In the next, with the aid of the slack matrix, we shall cast them into convex optimization problems and provide filter parameters in a convenient way. Theorem 2. For positive scalars ϑ, ζ u > 1, c2 > c1 > 0, an integer K > 0, and a matrix R > 0; ˆ 1 > 0, λ ˆ 2 > 0, matrices Pi = diag{Pi1 , Pi2 }, Pi1 > 0, Pi2 > 0, Qil > 0, if there exist scalars γu , λ ˆ ˆ ˆ ˆ AF L 0, BF L 0, HF L 0, LF L 0 and positive definite diagonal matrix Vˆl2 , such that for any i ∈ M, l ∈ N c r [Aˆ F l ]b,g + Bˆ Fr l,bCi,g − Vˆl2,b Aci,g ≥ 0,
(18)
c r c Bˆ Fr l,b Di,g − Vˆl2,b Bi,g ≥ 0,
(19)
c [Hˆ F l ]b,g + Lˆ Fr l,bCi,g − [Hi ]b,g ≥ 0,
(20)
c Lˆ Fr l,b Di,g − [Li ]b,g ≥ 0,
(21)
⎡
√
−Pi ⎢ ∗ ⎢ ⎢ .. ⎣ .
ϕi1 Qi1 −Qi1
··· ..
∗
√
ϕiN QiN
.
⎤ ⎥ ⎥ ⎥ < 0, ⎦
(22)
−QiN ⎡
−I ˆ il = ⎣ ∗ ∗
Hˆ il −ζu Qil ∗
⎤ Lˆ il 0 ⎦ < 0, −γu2 I
(23)
ˆ il i⊥ < 0, Ti⊥
(24)
ˆ 1 R ≤ Pi ≤ λ ˆ 2 R, λ
(25)
1 ˆ 1, ≤ ζu−K c2 λ ζu − 1 where P˜i = Mj=1 πi j Pj , ⎡ ⎤ 0 0 Ai 0 Bi ⎢I ⎥ ˆ (12) ˜i − V˜ˆl P ⎢ ⎥ ˆ il I i⊥ = ⎣ , ⎦, il = I ˆ il ∗ I I ˆ ˆ ˆ ˆ ˆ ˆ ˆ (12) = 0 AF l + BF l Ci − Vl2 Ai AF l BF l Di − Vl2 Bi , Vˆ˜l = 0 il 0 Aˆ F l + Bˆ F l Ci − Vˆl2 Ai Aˆ F l Bˆ F l Di − Vˆl2 Bi ∗ ˆ 2 + γu2 ϑ c1 λ
(26)
Vˆl2 . Vˆl2T + Vˆl2
Then the upper-bounding FE system (4) is positive and FTB with an H∞ performance level γ˜u . Moreover, the filter parameters are given by Aˆ l = Vˆl2−1 Aˆ F l , Bˆ l = Vˆl2−1 Bˆ F l , Hˆ l = Hˆ F l , Lˆ l = Lˆ F l .
(27)
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
9
Proof. From Eqs. (18)–(21) and (27), we have Aˆ F l + Bˆ F l Ci − Vˆl2 Ai 0,
(28)
Bˆ F l Di − Vˆl2 Bi 0,
(29)
Hˆ F l + Lˆ F l Ci − Hi 0,
(30)
Lˆ F l Di − Li 0.
(31)
Then Aˆ l + Bˆ l Ci − Ai 0, Bˆ l Di − Bi 0, Hˆ l + Lˆ l Ci − Hi 0, Lˆ l Di − Li 0. By Lemma 1, one concludes that the upper-bounding FE system (4) is positive. In addition, Eq. (9) is equivalent to Eq. (22) using the Schur complement formula. Now, let’s introduce an invertible slack matrix Vˆ V Vil il1 ˆl2 , Vil3 Vl2 where Vil1 , Vˆl2 , Vil3 ∈ Rn×n and Vˆl2 is a positive definite diagonal matrix. From Eq. (10), we derive −1 ϒil −P˜i ˆ il < 0, ϒilT where ϒil = 0
Aˆ il
Bˆ il ,
⎡
−I ˆ il = ⎣ ∗ ∗
Hˆ il −ζu Qˆ il ∗
(32)
⎤ Lˆ il 0 ⎦. −γu2 I
Pre- and post-multiplying Eq. (32) by diag {Vil , I} and its transpose yields −Vil P˜i−1VilT Vil ϒil ˆ il < 0. ϒilT VilT
(33)
Since P˜i−1 > 0 and (P˜i − Vil )P˜i−1 (P˜i − Vil )T > 0, there holds −Vil P˜i−1VilT < P˜i − Vil − VilT . This together with Eq. (33) gives rise to P˜i − Vil − VilT Vil ϒil ˆ ˜ il = (34) ˆ il < 0, ϒilT VilT which implies Vil + VilT > 0 and Vil is invertible. ˆ˜ is rewritten as Denote Vil13 = [V T V T ]T . il il1
il3
ˆ˜ = T ˆ il + T Vil13 i + Ti Vil13 , il
(35)
where = [I2n×2n 02n×m 02n×n 02n×n 02n×p ], i = [−In×n 0n×n 0n×m Ai 0n×n Bi ]. Then, according to Eqs. (23) and (24), multiplying both sides of inequality (35) by T⊥ and its transpose leads to ˆ il ⊥ = ˆ i < 0, T⊥ (36) Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
10
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
where ⊥ are matrices whose column vectors constitute the bases of the null spaces of . Similarly, we obtain ˆ Ti⊥ ˆ il ˆ i⊥ < 0,
(37)
ˆ i⊥ whose column vectors constitute the bases of the null spaces of . ˆ Therefore, the where equivalence between Eqs. (34) and (36)–(37) is guaranteed through Finsler lemma. It means Eqs. (36) and (37) are the sufficient conditions for Eq. (10). Further, it follows from condition (25) that ˆ 1 I ≤ R− 21 Pi R− 21 = Pi ≤ λ ˆ 2I , λ which shows that ˆ 1 ≤ min{λmin (Pi ), i ∈ M} ≤ λ(Pi ) ≤ max{λmax (Pi ), i ∈ M} ≤ λ ˆ 2, λ which together with Eq. (26) indicates that Eq. (11) holds. This ends the proof. In the sequel, some results will be proposed such that the lower-bounding FE system (5) is positive and FTB with an H∞ performance level. A set of convex optimization conditions concerning the lower-bounding asynchronous positive filter will be provided. In order to avoid repetitions, here we provid the main proof process. Theorem 3. Given positive scalars ϑ, ζ > 1, d2 > d1 > 0, an integer K > 0, and a matrix R > 0, the lower-bounding FE system (5) is positive, FTB with an H∞ performance level γ˜ = γ ζK , if there exist scalar γ > 0, matrices Pi > 0, Qil > 0 for i ∈ M, l ∈ N such that Ai − Bˇ l Ci − Aˇ l 0, Bi − Bˇ l Di 0,
(38)
Hi − Lˇ l Ci − Hˇ l 0, Li − Lˇ l Di 0,
(39)
N
ϕil Qil < Pi ,
(40)
l=1
⎡
−P˜i−1 ⎢ ∗ ⎢ ⎣ ∗ ∗
0 −I ∗ ∗
Aˇ il Hˇ il −ζ Qil ∗
⎤ Bˇ il Lˇ il ⎥ ⎥ < 0, 0 ⎦ −γ2 I
1 1 ≤ d2 , ζK d1 max{λmax (Pi ), i ∈ M} + ϑ min{λmin (Pi ), i ∈ M} ζ − 1 1 1 where P˜i = Mj=1 πi j Pj , Pi R− 2 Pi R− 2 .
(41)
(42)
Proof. Considering Lemma 1 and inequalities (38)–(39), there are matrices Aˇ il 0, Bˇ il 0, Hˇ il 0, Lˇ il 0 for all i ∈ M, l ∈ N such that the lower-bounding FE system (5) is positive. Construct a Lyapunov function as follows: Vι (k, α(k), xˇF (k)) = xˇFT (k)Pα(k) xˇF (k).
(43)
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
11
Bearing in the mind Eqs. (40) and (41), one has Jzˇν =E Vι (k + 1, α(k + 1), xˇF (k + 1)) − ζ Vι (k, α(k), xˇF (k)) ≤ 0. Then, we can obtain E Vι (k + 1, α(k + 1), xˇF (k + 1)) ≤ E ζ Vι (k, α(k), xˇF (k)) + γ2 ν T (k)ν(k) − zˇFT (k)zˇF (k) . (44) Similar to Theorem 1, after some manipulations, we have E Vι (k, α(k), xˇF (k)) ⎧ ⎫ k−1 ⎨ ⎬ ≤ ζk Vι (0, α(0), xˇF (0)) + E ζk−1−μ γ2 ν T (μ)ν(μ) . ⎩ ⎭
(45)
μ=0
Noting that ν T (k)ν(k) ≤ ϑ and ζ > 1, it implies K E V ι(k, α(k), xˇF (k)) ) ≤ ζ d1 max{λmax (Pi ), i ∈ M} + γ2 ϑ
1 . ζ − 1
E xˇFT (k)Pi xˇF (k) ≥ min{λmin (Pi ), i ∈ M}E xˇFT (k)RxˇF (k) .
(46)
(47)
Combining (42), (46) with (47) implies that the lower-bounding FE system (5) is FTB. On the other hand, under the zero initial conditions, there holds ⎧ ⎫ k−1 ⎨ ⎬ 0 ≤ E V (k, α(k), xˇF (k)) ≤ E ζk−1−μ {γ2 ν T (μ)ν(μ) − zˇFT (μ)zˇF (μ)} ⎩ ⎭ μ=0 ⎧ ⎫ ⎨K−1 ⎬ ≤E ζK {γ2 ν T (μ)ν(μ) − zˇFT (μ)zˇF (μ)} , ⎩ ⎭ μ=0
which implies K−1 μ=0
K−1 E zˇFT (μ)zˇF (μ) ≤ ζK γ2 ν T (μ)ν(μ). μ=0
Then, the lower-bounding FE system (5) satisfies an H∞ performance level γ˜ = γ ζK . This ends the proof. Theorem 4. Given positive scalars ϑ, ζ > 1, d2 > d1 > 0, an integer K > 0, and a matrix ˇ 1 > 0, λ ˇ 2 > 0, matrices Pi = diag{Pi1 , Pi2 }, Pi1 > 0, Pi2 > 0, R > 0, if there exist scalars γ , λ Qil > 0, Aˇ F L 0, Bˇ F L 0, Hˇ F L 0, Lˇ F L 0 and positive definite diagonal matrix Vˇl2 , such that for any i ∈ M, l ∈ N r c Vˇl2,b Aci,g − Bˇ Fr l,bCi,g − [Aˇ F l ]b,g ≥ 0,
(48)
r c c Vˇl2,b Bi,g − Bˇ Fr l,bDi,g ≥ 0,
(49)
c [Hi ]b,g − Lˇ Fr l,bCi,g − [Hˇ F l ]b,g ≥ 0,
(50)
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
12
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
c [Li ]b,g − Lˇ Fr l,bDi,g ≥ 0,
⎡
√
−Pi ⎢ ∗ ⎢ ⎢ .. ⎣ .
ϕil Qi1 −Qi1
··· ..
(51) √
ϕiN QiN
.
∗
⎤ ⎥ ⎥ ⎥ < 0, ⎦
(52)
−QiN ⎡
Hˇ il −ζ Qil
−I ˇ il = ⎣ ∗ ∗
Lˇ il −γ2 I
⎤ ⎦ < 0,
(53)
ˇ il i⊥ < 0, Ti⊥
(54)
ˇ 1 R ≤ Pi ≤ λ ˇ 2 R, λ
(55)
1 ˇ 1, ≤ ζ−K d2 λ ζ − 1 where P˜i = Mj=1 πi j Pj , ⎡ ⎤ 0 0 Ai 0 Bi ˇ (12) ˜i − Vˇ˜l ⎢I ⎥ ˇ P il i⊥ = ⎣ , ⎦, il = I ˇ il ∗ I I I ˇ ˇ ˇ ˇ ˇ ˇ ˇ (12) = 0 Vl2 Ai − BF l Ci − AF l AF l Vl2 Bi − BF l Di , Vˇ˜l = 0 il 0 Vˇl2 Ai − Bˇ F l Ci − Aˇ F l Aˇ F l Vˇl2 Bi − Bˇ F l Di ∗ ˇ 2 + γ2 ϑ d1 λ
(56)
Vˇl2 . Vˇl2T + Vˇl2
Then the lower-bounding FE system (5) is positive, FTB with an H∞ performance level γ˜ . Moreover, the lower-bounding asynchronous positive filter parameters can be given by Aˇ l = Vˇl2−1 Aˇ F l , Bˇ l = Vˇl2−1 Bˇ F l , Hˇ l = Hˇ F l , Lˇ l = Lˇ F l . (57) Proof. According to Eqs. (49)–(51) and (57), the conditions (38), (39) can be proved, which implies the lower-bounding FE system (5) is positive. The equivalence between Eqs. (40) and (52) can be verified by Schur complement formula. Similar to Theorem 2, we will introduce an invertible slack matrix ι Vil1 Vˇl2 ι , Vil Vˇl2 Vι il3
ι ι where Vil1 , Vˇl2ι , Vil3 ∈ Rn×n and Vˇl2ι is a positive definite diagonal matrix. Pre- and post-multiplying the partitioned Eq. (41) by diag{Vilι , I }, we derive −Vilι P˜i−1VilιT Vilι ϒilι < 0, ˇ il ϒ ιT V ιT il
where ϒilι = 0
(58)
il
Aˇ il
Bˇ il ,
⎡
−I ˇ il = ⎣ ∗ ∗
Hˇ il −ζ Qˇ il ∗
⎤ Lˇ il 0 ⎦. −γ2 I
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
13
Considering P˜i−1 > 0 and (P˜i − Vilι )P˜i−1 (P˜i − Vilι )T > 0, we have −Vilι P˜i−1VilιT < P˜i − Vilι − VilιT which implies P˜i − Vilι − VilιT Vilι ϒilι ˇ ˜ il = (59) ˇ il < 0. ϒilιT VilιT ˇ˜ can be rewritten as Then, il ˇ˜ = ι ιT ˇ il + T Vil13 i + Ti Vil13 , il ι with = [I2n×2n 02n×m 02n×n 02n×n 02n×p ], i = [−In×n 0n×n 0n×m Ai 0n×n Bi ], Vil13 = ιT ιT T [Vil1 Vil3 ] . Similar to Theorem 2, condition (41) can be guaranteed. It is noted that Eq. (55) implies
ˇ 1 ≤ min{λmin (Pi ), i ∈ M} ≤ λ(Pi ) ≤ max{λmax (Pi ), i ∈ M} ≤ λ ˇ 2. λ
(60)
Combining (56) with (60) shows Eq. (42) holds. This completes the proof. Remark 6. The upper-bounding filter and the lower-bounding filter have been designed to make the FE systems be positive, FTB and have an H∞ performance level. The estimated output z(k) can be encapsulated between these two filters at all the time which implies the estimation can be better than the single filter. 3.2. Design of weighted filter In this section, on the basis of the developed filters in Eqs. (2) and (3), we shall establish another filter called weighted filter to acquire better filtering performance. Some feasible conditions will be derived such that the FE system is positive and FTB with an H∞ performance. Define the new controlled output z¯(k) as z¯(k) = θ zˆ(k) + (1 − θ )zˇ(k), θ ∈ [0, 1]. On account of the bounded filters in Theorems 2 and 4, we have 0 zˇ(k) z(k) zˆ(k). From Lemma 2, we derive zˆ(k) ≥ z¯(k) ≥ zˇ(k). Let’s construct the weighted filter as: x¯(k + 1) = A¯ l x¯(k) + B¯ l y(k), z¯(k) = H¯ l x¯(k) + L¯ l y(k), where x¯(k) = [θ xˆT (k) (1 − θ )xˇT (k)]T , A¯ l = diag{Aˆ l , Aˇ l }, B¯ l = [θ Bˆ lT [Hˆ l Hˇ l ], L¯ l = θ Lˆ l + (1 − θ )Lˇ l . Then the FE system reads x¯F (k + 1) = A¯ il x¯F (k) + B¯ il ν(k),
e¯ : z¯F (k) = H¯ il x¯F (k) + L¯ il ν(k),
(61) (1 − θ )Bˇ lT ]T , H¯ l =
(62)
where z¯F (k) z¯(k) − z(k) = θ zˆF (k) − (1 − θ )zˇF (k), Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
14
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
x¯F (k + 1) = [x T (k ) θ xˆeT (k ) (1 − θ )xˇeT (k)]T , ⎡ ⎤ Ai 0n×n 0n×n Aˆ l 0n×n ⎦, A¯ il = ⎣ θ (Aˆ l + Bˆ l Ci − Ai ) Aˇ l (1 − θ )(Ai − Bˇ l Ci − Aˇ l ) 0n×n ⎡ ⎤ Bi ˆ ⎣ ¯ θ ( B l D i − B i ) ⎦, Bil = (1 − θ )(Bi − Bˇ l Di ) H¯ il = θ (Hˆ l + Lˆ l Ci − Hi ) − (1 − θ )(Hi − Lˇ l Ci − Hˇ l )
Hˆ l
Hˇ l ,
L¯ il = θ (Lˆ l Di − Li ) + (1 − θ )(Li − Lˇ l Di ). Now we are in a position to illustrate the following results on the weighted asynchronous H∞ positive filter design. Theorem 5. For PMJSs (1), if all the conditions in Theorems 2 and 4 hold, then the weighted filter Eq. (61) is positive, and the weighted FE system (62) is FTB with an H∞ performance γ¯ = θ γ˜u + (1 − θ )γ˜ . Proof. According to the positivity of the bounding filters Eqs. (2), (3) and the FE systems (4), (5), we obtain xˆ(k) 0, xˇ(k) 0, zˆ(k) 0, zˇ(k) 0, which imply x¯(k) 0. From the definition of zˆ(k) and zˇ(k), we get 0 zˇ(k) zˆ(k). Bearing in mind Lemma 2 and z¯(k) = θ zˆ(k) + (1 − θ )zˇ(k), we get 0 zˇ(k) z¯(k) zˆ(k). So the weighted filter (61) is positive. Theorems 2 and 4 guarantee that the FE systems (4), (5) are FTB; which means that for given scalars c2 > c1 > 0, d2 > d1 > 0 and a matrix R > 0, there hold E {xˆFT (0)RxˆF (0)} ≤ c1 ⇒ E {xˆFT (k)RxˆF x(k)} ≤ c2 , ν T (k)ν(k) ≤ ϑ E {xˇFT (0)RxˇF (0)} ≤ d1 ⇒ E {xˇFT (k)RxˇF x(k)} ≤ d2 . ν T (k)ν(k) ≤ ϑ Therefore, one has E {x¯FT (0)Rx¯F (0)} ≤ θ c1 + (1 − θ )d1 ν T (k)ν(k) ≤ ϑ
⇒ E {x¯FT (k)Rx¯F x(k)} ≤ θ c1 + (1 − θ )d1 ,
(63)
which shows the weighted FE system (62) is FTB. Under the zero initial conditions, it follows from Theorems 2 and 4 K
E { zˆF (k) 2 } < γ˜u2
k=0 K
K
ν(k) 2 ,
k=0
E { zˇF (k) 2 } < γ˜2
k=0
K
ν(k) 2 .
k=0
Due to z¯F (k) = θ zˆF (k) − (1 − θ )zˇF (k), we get z¯F (k) = θ zˆF (k) − (1 − θ )zˇF (k) ≤ θ zˆF (k) + (1 − θ )zˇF (k) , K k=0
E { z¯F (k) 2 } < γ¯ 2
K
ν(k) 2 ,
k=0
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
15
where γ¯ = θ γ˜u + (1 − θ )γ˜ . Hence, the weighted FE system (62) is FTB and has an H∞ performance. This ends the proof. The weighted filter in Theorem 5 is designed based on Theorems 2 and 4. All the conditions have been converted to linear matrix inequalities (LMIs). Hence, the conditions can be solved with the aid of Matlab LMI Toolbox. An effective algorithm is provided as follows: Step 1. Take initial value x0 , the TP matrix and the CP matrix . Step 2. Select the parameters ϑ > 0, ζ u > 1, ζ > 1, c1 > c2 > 0, d1 > d2 > 0, θ , the positive integer K and positive definite matrix R. ˆ 1, λ ˆ 2 , positive Step 3. Solving the LMI conditions (18)–(26), if there exists positive scalars λ ˆ definite matrices Pi , Qil and positive definite diagonal matrices Vl2 such that conditions (18)–(26) are feasible, go to Step 4; otherwise go back to Step 2, and reselect the parameter values. ˇ 1, λ ˇ 2 , positive Step 4. Solving the LMI conditions (48)–(56), if there exists positive scalars λ definite matrices Pi , Qil and positive definite diagonal matrices Vˇl2 such that conditions (48)–(56) are feasible, go to Step 5; otherwise go back to Step 2, and reselect the parameter values. Step 5. According to the filter gain matrices Aˆ l , Aˇ l , Bˆ l , Bˇ l , Hˆ l , Hˇ l , Lˆ l , Lˇ l , compute the weighted filter gain matrices. Here, the optimal values γ u , γ can be solved by solver mincx. Remark 7. The weighted filter has been designed to acquire better filtering performance and provides more flexibilities through adjusting the parameter θ in estimating the given systems than the above bounding filters. Meanwhile, when θ = 1 and θ = 0, the weighted filter (61) will respectively reduce to the upper-bounding filter (2) and the lower-bounding filter (3). 4. Simulation example In this section, we borrow the fish’s structured population model from [8,36] and describe it as PMJSs to validate the theoretical results. Let x(k) denote the fish’s number in a pond. Due to the influence of various factors, such as temperature, feed intake or predation, and the non-stop movement of fish, it is difficult to obtain the accurate population description. Here, we sample the fish’s number twice a day, once in the day and once at night. The fish’s population is modelled as system (1) with the parameters given below: 0.51 0.31 0.25 , B1 = , A1 = C1 = 0.72 0.53 , D1 = 0.75, 0.28 0.15 0.15 0.42 0.60 0.16 , B2 = , A2 = C2 = 0.59 0.67 , D2 = 0.49, 0.30 0.36 0.14 H1 = 0.43 0.18 , H2 = 0.43 0.21 , L1 = 0.33, L2 = 0.11. Take K = 20, ϑ = 0.2, ζu = ζ = 1.1. The TP matrix is given as 0.2 0.8 . = 0.6 0.4 Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
16
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
Table 1 The different CP matrices.
Case I Case II Case III
π 11
π 12
π 21
π 22
0.3 1 1
0.7 0 0
0.5 0.5 0
0.5 0.5 1
Table 2 The optimal H∞ performance indices for different CP matrices. γ˜u γ˜
Case I
Case II
Case III
1.7569 1.3595
1.7034 1.3328
1.6515 1.3168
In order to show the asynchrony effects on optimal H∞ performance index, we provide Tables 1 and 2. Table 2 shows different H∞ performance indices can be obtained under different CP matrices. For completely asynchronous case III, solving Theorem 2, we obtain the upper-bounding filter parameters 0.1184 0.1751 ˆ 0.9369 ˆ Aˆ 1 = , B1 = , H1 = 0.2834 0.1293 , Lˆ 1 = 0.4668, 0.1664 0.1659 0.5084 0.0789 0.1452 ˆ 0.9095 ˆ , B2 = , H2 = 0.2313 0.1094 , Lˆ 2 = 0.4657. Aˆ 2 = 0.1207 0.1435 0.4815 Similarly, solving Theorem 4, we obtain the lower-bounding filter parameters 0.0336 0.0353 ˇ 0.0872 ˇ ˇ , B1 = , H1 = 0.2449 0.0682 , Lˇ 1 = 0.1307, A1 = 0.0274 0.0214 0.0470 0.0134 0.0139 ˇ 0.0960 ˇ ˇ , B2 = , H2 = 0.2373 0.0741 , Lˇ 2 = 0.1786. A2 = 0.0143 0.0104 0.0553
Fig. 1. The modes of the system and the bounding filters. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
17
Fig. 2. Response of system output z(k) and its estimations.
In this example, the initial states of the upper-bounding FE system and the lower-bounding FE system are respectively taken as [0.2 0.4 0.3 0.5]T and [0.2 0.4 0.15 0.3]T . The external disturbance is ν(k) = 0.4e−0.4k . Fig. 1 presents the switching signal of the system and the filter, which shows the modes of the system and the filter are switching asynchronously. It’s worth noting that α(k) and β(k) have two modes respectively. There are four combinations between system’s modes and filter’s modes. Denote α(k) = 1, β(k) = 1 as case 1, α(k) = 1, β(k) = 2 as case 2, α(k) = 2, β(k) = 1 as case 3 and the rest as case 4. Fig. 1 presents the switching among the four combinations clearly. The responses of the system output z(k), the upper-bounding estimation zˆ(k) and the lower-bounding estimation zˇ(k) are depicted in Fig. 2. One can see that the system output z(k) always runs between the estimations zˆ(k) and zˇ(k) from Fig. 2; which coincides with
Fig. 3. Responses of xˆT (k)Ru xˆ(k) and xˇT (k)R xˇ(k). Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
18
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
Fig. 4. Response of system output z(k) and its estimations.
the theoretical results. Fig. 3 is the trajectories of xˆT (k)Ru xˆ(k) and xˇT (k)R xˇ(k); which also show that the FE systems are FTB and gain an H∞ performance level. Fig. 4 presents the responses of the upper-bounding FE systems, the lower-bounding FE systems, and the weighted FE system for parameter θ = 0.3. We can see they are all FTB and have an H∞ performance level. 5. Conclusion In this paper, we have investigated the asynchronous positive H∞ filtering problem for a class of PMJSs. Based on the hidden Markov model, sufficient conditions have been proposed such that the FE systems are positive and FTB with an H∞ performance. In terms of the stochastic system theory and linear matrix inequality techniques, we have built three different asynchronous filters and also discussed their relations to gain more flexibilities during the design. The fish’s population evolution has been modelled as a PMJS which validates the effectiveness of our theoretical results. Acknowledgment This work was supported in part by the National Natural Science Foundation of China under grant (61773225, 61773226, 61873331, 61803225), in part by the Taishan Scholar Project of Shandong Province (TSQN20161033) and in part by Interdisciplinary Scientific Research Projects of Qufu Normal University (xkjjc201905). References [1] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley, New York, 2000. [2] Y. Yin, G. Zong, X. Zhao, Improved stability criteria for switched positive linear systems with average dwell time switching, J. Frankl. Inst. 354 (8) (2017) 3472–3484. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
19
[3] H. Ren, G. Zong, T. Li, Event-triggered finite-time control for networked switched linear systems with asynchronous switching, IEEE Trans. Syst. Man Cybern. Syst. 48 (11) (2018) 1874–1884. [4] W. Qi, G. Zong, H.R. Karimi, L∞ control for positive delay systems with semi-Markov process and application to a communication network model, IEEE Trans. Ind. Electron. 66 (3) (2019) 2081–2091. [5] Y. Guo, Stabilization of positive Markov jump systems, J. Frankl. Inst. 353 (14) (2016) 3428–3440. [6] W. Qi, H.P. Ju, J. Cheng, Y. Kao, X. Gao, Exponential stability and L1 -gain analysis for positive time-delay Markovian jump systems with switching transition rates subject to average dwell time, Inf. Sci. 424 (2018) 224–234. [7] S. Li, Z. Xiang, Stochastic stability analysis and L∞ -gain controller design for positive Markov jump systems with time-varying delays, Nonlinear Anal. Hybrid Syst. 22 (2016) 31–42. [8] V.H. Le, An LP approach to full-order and reduced-order state estimations of positive Markov jump systems with delay, Int. J. Syst. Sci. 48 (12) (2017) 2534–2543. [9] S. Zhu, B. Wang, C. Zhang, Delay-dependent stochastic finite-time l1 -gain filtering for discrete-time positive Markov jump linear systems with time-delay, J. Frankl. Inst. 354 (15) (2017) 6894–6913. [10] Y. Wang, X. Liu, J. Xiao, Y. Shen, Output formation-containment of interacted heterogeneous linear systems by distributed hybrid active control, Automatica 93 (2018) 26–32. [11] X. Chen, J. Lam, H.K. Lam, Positive filtering for positive Takagi-Sugeno fuzzy systems under l1 performance, Inf. Sci. 299 (2015) 32–41. [12] D. Yang, G. Zong, H.R. Karimi, H∞ refined anti-disturbance control of switched LPV systems with application to aero-engine, IEEE Trans. Ind. Electron. (2019), doi:10.1109/TIE.2019.2912780. [13] H. Sun, Y. Li, G. Zong, L. Hou, Disturbance attenuation and rejection for stochastic Markovian jump system with partially known transition probabilities, Automatica 89 (2018) 349–357. [14] D. Yang, G. Zong, S.K. Nguang, H∞ bumpless transfer reliable control of Markovian switching LPV systems subject to actuator failures, Inf. Sci. (2019), doi:10.1016/j.ins.2019.07.059. [15] Y. Wang, W. Yang, J. Xiao, Z. Zeng, Impulsive multisynchronization of coupled multistable neural networks with time-varying delay, IEEE Trans. Neural Netw. Learn. Syst. 28 (7) (2017) 1560–1571. [16] Y. Ren, J.E. Meng, G. Sun, Asynchronous L1 positive filter design for switched positive systems with overlapped detection delay, IET Control Theory Appl. 11 (3) (2016) 319–328. [17] S. Li, Z. Xiang, Stabilisation of a class of positive switched nonlinear systems under asynchrononus switching, Int. J. Syst. Sci. 48 (7) (2017) 1537–1547. [18] Y. Li, H. Zhang, Asynchronous L1 -gain control of uncertain switched positive linear systems with dwell time, ISA Trans. 75 (2018) 25–37. [19] H. Ren, G. Zong, K. Hamid Reza, Asynchronous finite-time filtering of networked switched systems and its application: an event-driven method, IEEE Trans. Circuits Syst. I Regul. Pap. 66 (1) (2019) 391–402. [20] H. Shen, F. Li, Z. Wu, H.P. Ju, Finite-time asynchronous H∞ filtering for discrete-time Markov jump systems over a lossy network, Int. J. Robust Nonlinear Control 26 (17) (2016) 3831–3848. [21] M. Chen, X. Yang, H. Shen, F. Yao, Finite-time asynchronous H∞ control for Markov jump repeated scalar non-linear systems with input constraints, Appl. Math. Comput. 275 (2016) 172–180. [22] M. Zhang, P. Shi, Z. Liu, H. Su, L. Ma, Fuzzy model-based asynchronous H∞ filter design of discrete-time Markov jump systems, J. Frankl. Inst. 354 (18) (2017) 8444–8460. [23] Z. Wu, P. Shi, H. Su, J. Chu, Asynchronous L2 − L∞ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities, Automatica 50 (1) (2014) 180–186. [24] Z. Wu, P. Shi, Z. Shu, H. Su, R. Lu, Passivity-based asynchronous control for Markov jump systems, IEEE Trans. Autom. Control 62 (4) (2017) 2020–2025. [25] Z. Wu, S. Dong, H. Su, C. Li, Asynchronous dissipative control for fuzzy Markov jump systems., IEEE Trans. Cybern. 48 (8) (2018) 2426–2436. [26] J. Song, Y. Niu, Y. Zou, Asynchronous output feedback control of time-varying Markovian jump systems within a finite-time interval, J. Frankl. Inst. 354 (15) (2017) 6747–6765. [27] H. Dong, Z. Wang, D.W.C. Ho, H. Gao, Robust filtering for Markovian jump systems with randomly occurring nonlinearities and sensor saturation: the finite-horizon case, IEEE Trans. Signal Process. 59 (7) (2011) 3048–3057. [28] S. Xu, J. Lam, Positive Linear Systems: Theory and Applications, Springer-Verlag, Berlin, 2006. [29] Y. Ren, G. Sun, Z. Feng, J.E. Meng, S. Wu, Weighted-average l1 filtering for switched positive systems, Int. J. Robust Nonlinear Control 27 (18) (2017) 5097–5112. [30] S. Zhu, Q. Han, C. Zhang, L1 -gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: a linear programming approach, Automatica 50 (8) (2014) 2098–2107. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
20
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
[31] Y. Zheng, S.S. Ge, Y. Wang, Non-weighted L1 filtering for positive switched delay systems, IET Control Theory Appl. 12 (8) (2018) 1046–1054. [32] H. Wang, D. Zhang, R. Lu, Event-triggered H∞ filter design for Markovian jump systems with quantization, Nonlinear Anal. Hybrid Syst. 28 (2018) 23–41. [33] W. Qi, J.H.P. Park, G. Zong, J. Cheng, L1 finite-time stabilization for positive semi-Markovian switching systems, Inf. Sci. 47 (2019) 321–333. [34] G. Zong, H. Ren, Guaranteed cost finite-time control for semi-Markov jump systems with event-triggered scheme and quantization input, Int. J. Robust Nonlinear Control, 29 (15) (2019) 5251–5273. [35] L. Rabiner, A tutorial on hidden Markov models and selected applications in speech recognition, Proc. IEEE, 77 (1989) 257–286. [36] S. Li, Z. Xiang, H. Lin, H.R. Karimi, State estimation on positive Markovian jump systems with time-varying delay and uncertain transition probabilities, Inf. Sci. 369 (2016) 251–266.
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
[m1+;October 15, 2019;10:21]
Available online at www.sciencedirect.com
Journal of the Franklin Institute xxx (xxxx) xxx www.elsevier.com/locate/jfranklin
Finite-time asynchronous H∞ filtering for positive Markov jump systems Hui Shang, Guangdeng Zong∗, Wenhai Qi School of Engineering, Qufu Normal University, Rizhao 276826, PR China Received 2 December 2018; received in revised form 19 July 2019; accepted 4 August 2019 Available online xxx
Abstract In this paper, the asynchronous positive H∞ filtering is addressed for positive Markov jump systems. The asynchrony between the filter’s and the system’s modes is described through a conditional probability matrix. Three distinct asynchronous positive filters named an upper-bounding filter, a lower-bounding filter and a weighted filter are built. Sufficient conditions are established such that the filtering error systems are positive and finite-time bounded with an H∞ disturbance attenuation level. It is proved that the weighted filter has more flexibilities than the bounding filters since it can adjust the parameter θ when estimating the given systems. Finally, the effectiveness of the theoretical results is validated through the fish’s population evolution example. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Positive systems are widely used to model many practical processes subject to positivity constraints, for which the state variables are usually flow, pressure, liquid level, concentration substances, densities, or quantity of virus [1,2]. Most of the existing results for general systems cannot be directly applied to positive systems due to their positivity constraints. Recently, positive Markov jump systems (PMJSs) have received increasing interests since they can be employed to represent systems subjected to positivity constraints, abrupt changes and the jump governed by a Markov chain. Many methodologies and approaches are reported for PMJSs [3–8]. ∗
Corresponding author. E-mail address: [email protected] (G. Zong).
https://doi.org/10.1016/j.jfranklin.2019.08.008 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
2
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
Control synthesis and filtering analysis are addressed in [9–14] under the assumption that the system’s modes are synchronous with the controller/filter’s at all times. This means that all the information of the plant’s mode can be obtained by controller or filter. It is possibly a strong restriction in practical application. On one hand, in networked control systems [15], the finite transmission channel often leads to loss of data and communication delays. On the other hand, imprecise information could cause the mismatch between the system and the filter. The asynchronous phenomena may reduce the system performance, even lead to unstability. Therefore, more and more researchers begin to study the asynchronous issues brought by different behaviors. For example, the asynchrony resulting from overlapped detection delay is discussed in [16–19]. Asynchronous filtering is considered as Markov chain over a finite-time span in [20–23]. Wu et al. [24] is firstly concerned with the asynchronous issue for hidden Markov model. Then some people extend the result to switched systems [3], fuzzy systems [25] and time-varying MJSs [26], etc. But for PMJSs, few works are found in the existing results, which motivates us to the present study. Filtering plays an important role in estimating the system output when some state variables are immeasurable and finds wide applications in target tracking and adaptive control [23,27,28] and many useful works are reported [11,29–33]. Remarkably, the positive filtering problem is discussed for positive T-S fuzzy systems in [11]. The authors in [30] have investigated l1 -gain filtering for PMJSs. A positive L1 -gain filter is designed in [33] for positive semi-Markovian switching systems. There are few results reported concerning filtering for PMJSs under asynchronous switching governed by a Markov chain. Due to the positivity of filtering error systems, the single filter only estimates the system output from one direction which proved to be less accurate. In addition, different from Lyapunov asymptotic stability, finite-time stability emphasizes the transient performance and has strong applications background [34]. A finite-time stable (FTS) system may be not asymptotically stable and vice versa. Therefore, the general stability results cannot be extended to cope with the finite time stability of PMJSs. It is imperative and interesting to study the asynchronous positive H∞ filtering problem of PMJSs in the meaning of finite-time stability. In this paper, we focus upon the asynchronous positive H∞ filtering for PMJSs subject to asynchronous switching. With the aid of hidden Markov model, the asynchrony between the filter’s and the system’s modes will be fully considered. Three different filtering design schemes are addressed under the finite-time bounded (FTB) frame during a finite time span. Sufficient convex optimization conditions are derived such that the filtering error (FE) systems are positive and FTB with an H∞ disturbance attenuation performance. The contributions of our work lie in four aspects: (i) Compared with the existing work [16–19], the hidden Markov model is attempted to deal with asynchronous H∞ filtering for PMJSs over a finite time span; (ii) The asynchrony between the the filter’s and the system’s modes is stochastic and obeys a conditional probability, which can reflect the real phenomenon better; (iii) Through introducing some slack matrices, two bounding positive asynchronous H∞ filters are constructed which make the FE systems be positive, FTB and have an H∞ performance index. The estimated output will stay between the upper-bounding filter’s state and lower-bounding filter’s state. A weighted filter is proposed which has more flexibilities than the bounding filters during the design process; (iv) The fish’s population evolution is modelled as PMJSs with disturbances and applied to verify the theoretical results. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
3
Notations: In this paper, Rn denotes the vectors of n-tuples of real number. For matrix Ai , Aci,g denote bth row, gth column of Ai and [Ai ]b,g denotes the element of bth row, gth column of Ai . For any symmetric matrix P, P > 0 (P < 0) implies that the matrix P is positive (negative) definite. For a real matrix or vector A, A 0(0) denotes all the elements of A are non-negative (positive). Ari,b ,
2. Problem formulation and preliminaries Consider the discrete-time PMJSs: x(k + 1) = Aα(k) x(k) + Bα(k) ν(k),
(1a)
y(k) = Cα(k) x(k) + Dα(k) ν(k),
(1b)
z(k) = Hα(k) x(k) + Lα(k) ν(k),
(1c)
where x(k) ∈ R is the system state vector, ν(k) ∈ R is the external disturbance input which satisfies ν T (k)ν(k) ≤ ϑ. ϑ is a known positive scalars. y(k) ∈ Rq is the measured output, z(k) ∈ Rm is the estimated output. {α(k), k ≥ 0} is a discrete-time homogeneous Markov chain which takes values in M = {1, 2, . . . , M}. The transition probability (TP) matrix {π ij } is defined as n
p
πi j = Pr {α(k + 1) = j|α(k) = i} ≥ 0, with Mj=1 πi j = 1, for all i, j ∈ M. When α(k) = i ∈ M, Aα(k) , Bα(k) , Cα(k) , Dα(k) , Hα(k) , Lα(k) are respectively denoted by Ai , Bi , Ci , Di , Hi , Li and assumed to be known with appropriate dimensions. Here, it is assumed that the state x(k) is not measurable. In this paper, we focus on asynchronous positive H∞ filtering for PMJSs (1). In order to get real-time information on the output, an upper-bounding estimation zˆ(k) and a lower-bounding estimation zˇ(k) will be constructed as follows: xˆ(k + 1) = Aˆ β(k) xˆ(k) + Bˆ β(k) y(k), zˆ(k) = Hˆ β(k) xˆ(k) + Lˆ β(k) y(k),
(2)
xˇ(k + 1) = Aˇ β(k) xˇ(k) + Bˇ β(k) y(k), zˇ(k) = Hˇ β(k) xˇ(k) + Lˇ β(k) y(k),
(3)
where xˆ(k) and xˇ(k) are the filter state vectors, zˆ(k) and zˇ(k) are the estimations of z(k). Aˆ β(k) , Bˆ β(k) , Hˆ β(k) , Lˆ β(k) , Aˇ β(k) , Bˇ β(k) , Hˇ β(k) , and Lˇ β(k) are unknown filter parameter matrices. The variable {β(k), k ≥ 0} is defined in another set N = {1, 2, . . . , N } and connected to {α(k), k ≥ 0} through a known conditional probability (CP) matrix = {ϕil } with Pr {β(k) = l|α(k) = i} = ϕil , where ϕ il ∈ [0, 1], and Nl=1 ϕil = 1 for ∀l ∈ N . Remark 1. Generally, it is impossible that all the information of system’s mode is available to the filter. In view of the above definition, we use the variable β(k) to describe the filter’s mode which is connected to system’s mode α(k) through a known conditional probability Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
4
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
matrix. The system’s mode is not observable directly and is hidden to filter’s mode, which constitutes the hidden Markov model (α(k), β(k), M, N ) [35]. Denote xˆF (k) [x T (k ) xˆeT (k )]T , zˆF (k ) zˆ(k) − z(k ), xˆe (k ) xˆ(k) − x(k). Let activated system mode α(k) = i and activated filtering mode β(k) = l for ∀i ∈ M, l ∈ N . From Eqs. (1) and (2), we get the following upper-bounding FE system: xˆ (k + 1) = Aˆ il xˆF (k) + Bˆ il ν(k),
eˆ : F (4) zˆF (k) = Hˆ il xˆF (k) + Lˆ il ν(k), where Aˆ il =
Ai Aˆ l + Bˆ l Ci − Ai
Hˆ il = Hˆ l + Lˆ l Ci − Hi
0 Bi ˆ il = , , B Aˆ l Bˆ l Di − Bi Hˆ l , Lˆ il = Lˆ l Di − Li .
Similarly, let xˇF (k) [x T (k ) xˇeT (k )]T , zˇF (k ) z(k) − zˇ(k ), xˇe (k ) x(k) − xˇ(k). The lowerbounding FE system reads xˇ (k + 1) = Aˇ il xˇF (k) + Bˇ il ν(k),
eˇ : F (5) zˇF (k) = Hˇ il xˇF (k) + Lˇ il ν(k), where Aˇ il =
Ai ˇ Ai − Bl Ci − Aˇ l
Hˇ il = Hi − Lˇ l Ci − Hˇ l
0 Bi ˇ il = , , B Aˇ l Bi − Bˇ l Di Hˇ l , Lˇ il = Li − Lˇ l Di .
Remark 2. In this paper, the hidden Markov model is attempted to describe asynchronous circumstances for PMJSs, where the filter mode cannot be determined directly by the system’s mode. In the existing results [16–18], asynchronous mode either depends on the modes at time k − 1 and time k or is entirely different from the system’s mode. These situations lead to some conservativeness when dealing with asynchronous problem. In this paper, the filter mode is not completely independent of system mode and only determined by observing the system mode at the current time k. The results in this paper are more appropriate to describe the practical issues. Remark 3. According to the above definition, the probability π ij represents the system transition probability from ith mode to jth mode for i, j ∈ M. Distinguished from the system mode α(k), a stochastic variable β(k) is introduced to denote the filter mode. These two stochastic variables are subject to a known CP matrix = {ϕil } which shows the asynchrony between the filters and the systems modes. ϕ il describes the possibility of the filter running in the lth mode, when the system lies in ith mode for i ∈ M, l ∈ N , which reflects the degree of asynchrony. In addition, if the MJSs include only one subsystem, i.e., M = 1, then the filter mode reduces to a mode-independent one. Moreover, the filter’s modes are synchronous with the system’s if the set M = N and the CP ϕil = 1 for i = l. Therefore, the asynchrony considered in this paper is more general. Lemma 1 [1]. System (1) is positive, if and only if Ai 0, Bi 0, Ci 0, Di 0, Hi 0 and Li 0 for all i ∈ M. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
5
Remark 4. The upper-bounding estimation system (2) and lower-bounding estimation system are designed to estimate the system output for PMJSs. The positivity for systems (2) and (3) are essential. The filter parameter matrices satisfy Aˆ β(k) 0, Bˆ β(k) 0, Hˆ β(k) 0, Lˆ β(k) 0, Aˇ β(k) 0, Bˇ β(k) 0, Hˇ β(k) 0, and Lˇ β(k) 0. Definition 1 [20]. Given scalars d2 > d1 > 0, integer K and a matrix R > 0. The system (1) is FTB with respect to (d1 , d2 , K, R, ϑ), if E {x T (0)Rx(0)} ≤ d1 ⇒ E {x T (k)Rx(k)} ≤ d2 , ν T (k)ν(k) ≤ ϑ for any k ∈ {1, 2, . . . , K }. Definition 2 [20]. For a scalar γ˜ > 0 and integer K, the system (1) is FTB with a required H∞ performance level γ˜ , if (i) system (1) is FTB with respect to (d1 , d2 , K, R, ϑ); (ii) under zero initial conditions, there holds K
E { z(k) 2 } < γ˜ 2
k=0
K
ν(k) 2 .
(6)
k=0
Lemma 2 [29]. For the functions g1 (x), g2 (x) and 0 ≤ g1 (x) ≤ g2 (x), if we define ψ (x) = δg1 (x) + (1 − δ)g2 (x), δ ∈ [0, 1], then we have g1 (x) ≤ ψ(x) ≤ g2 (x). Our main tasks are finding the filter parameters of the upper-bounding asynchronous positive filter in Eq. (2) and the lower-bounding asynchronous positive filter in Eq. (3) such that the corresponding FE systems (4) and (5) are respectively positive and FTB with required H∞ performance level. 3. Main results 3.1. Design of bounded filters In this section, the existence conditions concerning the asynchronous positive filters shall be first established. An upper-bounding asynchronous positive filter Eq. (2) and a lowerbounding asynchronous positive filter Eq. (3) will be designed such that the FE systems are positive, FTB and gain an H∞ performance level. Some feasible conditions will be provided to obtain the filters parameters. Theorem 1. Given positive scalars ϑ, ζ u > 1, c2 > c1 > 0, an integer K > 0, and a matrix R > 0, the upper-bounding FE system (4) is positive, FTB with an H∞ performance level γ˜u = γu ζuK , if there exist positive scalar γ u , matrices Pi > 0, Qil > 0 for any i ∈ M, l ∈ N such that Aˆ l + Bˆ l Ci − Ai 0, Bˆ l Di − Bi 0,
(7)
Hˆ l + Lˆ l Ci − Hi 0, Lˆ l Di − Li 0,
(8)
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
6
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
N
ϕil Qil < Pi ,
(9)
l=1
⎡
−P˜i−1 ⎢ ∗ ⎢ ⎣ ∗ ∗
0 −I ∗ ∗
Aˆ il Hˆ il −ζu Qil ∗
⎤ Bˆ il Lˆ il ⎥ ⎥ < 0, 0 ⎦ −γu2 I
(10)
1 1 ζuK c1 max{λmax (Pi ), i ∈ M} + ϑ ≤ c2 , min{λmin (Pi ), i ∈ M} ζu − 1 1 1 where P˜i = Mj=1 πi j Pj , Pi R− 2 Pi R− 2 .
(11)
Proof. By Lemma 1 and inequalities (7) and (8), there are matrices Aˆ il 0, Bˆ il 0, Hˆ il 0, Lˆ il 0 for all i ∈ M, l ∈ N . Hence, the upper-bounding FE system (4) is positive. Construct the following Lyapunov function candidate V (k, α(k), xˆF (k)) = xˆFT (k)Pα(k) xˆF (k).
(12)
When α(k) = i, α(k + 1) = j, β(k) = l, using Eq. (9), we get Jzν = E V (k + 1, j, xˆF (k + 1)) − ζuV (k, j, xˆF (k)) − γu2 ν T (k )ν(k ) + zˆFT (k )zˆF (k ) ⎧ ⎫ M ⎨ ⎬ πi j Pj xˆF (k + 1) − ζu xˆFT (k)Pi xˆF (k) − γu2 ν T (k)ν(k) + zˆFT (k)zˆF (k) = E xˆFT (k + 1) ⎩ ⎭ j=1 N T T =E ϕil ξ (k)il ξ (k) − ζu xˆF (k )Pi xˆF (k ) ≤E =E
l=1 N l=1 N
ϕil ξ T (k)il ξ (k) − ζu
N
ϕil xˆFT (k )Qil xˆF (k )
l=1
ϕil ξ T (k)il ξ (k) ,
(13)
l=1
T where ξ (k) = xˆFT (k) ν T (k) , P˜i = Mj=1 πi j Pj , Aˆ Til P˜i Aˆ il + Hˆ ilT Hˆ il Aˆ Til P˜i Bˆ il + Hˆ ilT Lˆ il il = , Bˆ ilT P˜i Aˆ il + Lˆ ilT Hˆ il Bˆ ilT P˜i Bˆ il + Lˆ ilT Lˆ il − γu2 I Aˆ Til P˜i Aˆ il + Hˆ ilT Hˆ il − ζu Qil Aˆ Til P˜i Bˆ il + Hˆ ilT Lˆ il il = . Bˆ ilT P˜i Aˆ il + Lˆ ilT Hˆ il Bˆ ilT P˜i Bˆ il + Lˆ ilT Lˆ il − γu2 I Schur complement formula and the condition (10) means il < 0 and Jzν < 0. Further, we can obtain E V (k + 1, α(k + 1), xˆF (k + 1)) ≤ E ζuV (k, α(k), xˆF (k)) + γu2 ν T (k)ν(k) − zˆFT (k)zˆF (k) . (14) Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
After some manipulations, we derive E V (k, α(k), xˆF (k)) ⎧ ⎫ k−1 ⎨ ⎬ ≤ ζuk V (0, α(0), xˆF (0)) + E ζuk−1−μ {γu2 ν T (μ)ν(μ) − zˆFT (μ)zˆF (μ)} , ⎩ ⎭ μ=0 ⎧ ⎫ k−1 ⎨ ⎬ k ≤ ζu V (0, α(0), xˆF (0)) + E ζuk−1−μ γu2 ν T (μ)ν(μ) . ⎩ ⎭
7
(15)
μ=0
In view of ν T (k)ν(k) ≤ ϑ and ζ u > 1, there holds k−1 E V (k, α(k), xˆF (k)) ) ≤ ζuk V (0, α(0), xˆF (0)) + ζuk−1−μ γu2 ϑ μ=0
ζk − 1 ≤ ζuk V (0, α(0), xˆF (0)) + γu2 ϑ u ζu − 1 1 k 2 ≤ ζu V (0, α(0), xˆF (0)) + γu ϑ ζu − 1 1 K T 2 ≤ ζu xˆF (0)P0 xˆF (0) + γu ϑ ζu − 1 1 K 2 . ≤ ζu c1 max{λmax (Pi ), i ∈ M} + γu ϑ ζu − 1
(16)
Observe E V (k, α(k), xˆF (k)) = E xˆFT (k)Pi xˆF (k)
≥ min{λmin (Pi ), i ∈ M}E xˆFT (k)RxˆF (k) .
(17)
Combining Eqs. (11), (16) with Eq. (17) yields 1 1 ≤ c2 . E xˆFT (k)RxˆF (k) ≤ ζuK c1 max{λmax (Pi ), i ∈ M} + γu2 ϑ min{λmin (Pi ), i ∈ M} ζu − 1 This together with Definition 1 implies that the upper-bounding FE system (4) is FTB. On the other hand, under the zero initial conditions, Eq. (15) implies ⎧ ⎫ k−1 ⎨ ⎬ 0 ≤ E V (k, α(k), xˆF (k)) ≤ E ζuk−1−μ {γu2 ν T (μ)ν(μ) − zˆFT (μ)zˆF (μ)} ⎩ ⎭ μ=0 ⎧ ⎫ ⎨K−1 ⎬ ≤E ζuK {γu2 ν T (μ)ν(μ) − zˆFT (μ)zˆF (μ)} . ⎩ ⎭ μ=0
Therefore, K−1 μ=0
K−1 E zˆFT (μ)zˆF (μ) ≤ ζuK γu2 ν T (μ)ν(μ), μ=0
which indicates the upper-bounding FE system (4) satisfies an H∞ performance level γ˜u = γu ζuK . This ends the proof. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
8
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
Remark 5. Due to the existence of coupling terms in Theorem 1, the conditions in Eqs. (7)–(11) cannot be easily solved. In the next, with the aid of the slack matrix, we shall cast them into convex optimization problems and provide filter parameters in a convenient way. Theorem 2. For positive scalars ϑ, ζ u > 1, c2 > c1 > 0, an integer K > 0, and a matrix R > 0; ˆ 1 > 0, λ ˆ 2 > 0, matrices Pi = diag{Pi1 , Pi2 }, Pi1 > 0, Pi2 > 0, Qil > 0, if there exist scalars γu , λ ˆ ˆ ˆ ˆ AF L 0, BF L 0, HF L 0, LF L 0 and positive definite diagonal matrix Vˆl2 , such that for any i ∈ M, l ∈ N c r [Aˆ F l ]b,g + Bˆ Fr l,bCi,g − Vˆl2,b Aci,g ≥ 0,
(18)
c r c Bˆ Fr l,b Di,g − Vˆl2,b Bi,g ≥ 0,
(19)
c [Hˆ F l ]b,g + Lˆ Fr l,bCi,g − [Hi ]b,g ≥ 0,
(20)
c Lˆ Fr l,b Di,g − [Li ]b,g ≥ 0,
(21)
⎡
√
−Pi ⎢ ∗ ⎢ ⎢ .. ⎣ .
ϕi1 Qi1 −Qi1
··· ..
∗
√
ϕiN QiN
.
⎤ ⎥ ⎥ ⎥ < 0, ⎦
(22)
−QiN ⎡
−I ˆ il = ⎣ ∗ ∗
Hˆ il −ζu Qil ∗
⎤ Lˆ il 0 ⎦ < 0, −γu2 I
(23)
ˆ il i⊥ < 0, Ti⊥
(24)
ˆ 1 R ≤ Pi ≤ λ ˆ 2 R, λ
(25)
1 ˆ 1, ≤ ζu−K c2 λ ζu − 1 where P˜i = Mj=1 πi j Pj , ⎡ ⎤ 0 0 Ai 0 Bi ⎢I ⎥ ˆ (12) ˜i − V˜ˆl P ⎢ ⎥ ˆ il I i⊥ = ⎣ , ⎦, il = I ˆ il ∗ I I ˆ ˆ ˆ ˆ ˆ ˆ ˆ (12) = 0 AF l + BF l Ci − Vl2 Ai AF l BF l Di − Vl2 Bi , Vˆ˜l = 0 il 0 Aˆ F l + Bˆ F l Ci − Vˆl2 Ai Aˆ F l Bˆ F l Di − Vˆl2 Bi ∗ ˆ 2 + γu2 ϑ c1 λ
(26)
Vˆl2 . Vˆl2T + Vˆl2
Then the upper-bounding FE system (4) is positive and FTB with an H∞ performance level γ˜u . Moreover, the filter parameters are given by Aˆ l = Vˆl2−1 Aˆ F l , Bˆ l = Vˆl2−1 Bˆ F l , Hˆ l = Hˆ F l , Lˆ l = Lˆ F l .
(27)
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
9
Proof. From Eqs. (18)–(21) and (27), we have Aˆ F l + Bˆ F l Ci − Vˆl2 Ai 0,
(28)
Bˆ F l Di − Vˆl2 Bi 0,
(29)
Hˆ F l + Lˆ F l Ci − Hi 0,
(30)
Lˆ F l Di − Li 0.
(31)
Then Aˆ l + Bˆ l Ci − Ai 0, Bˆ l Di − Bi 0, Hˆ l + Lˆ l Ci − Hi 0, Lˆ l Di − Li 0. By Lemma 1, one concludes that the upper-bounding FE system (4) is positive. In addition, Eq. (9) is equivalent to Eq. (22) using the Schur complement formula. Now, let’s introduce an invertible slack matrix Vˆ V Vil il1 ˆl2 , Vil3 Vl2 where Vil1 , Vˆl2 , Vil3 ∈ Rn×n and Vˆl2 is a positive definite diagonal matrix. From Eq. (10), we derive −1 ϒil −P˜i ˆ il < 0, ϒilT where ϒil = 0
Aˆ il
Bˆ il ,
⎡
−I ˆ il = ⎣ ∗ ∗
Hˆ il −ζu Qˆ il ∗
(32)
⎤ Lˆ il 0 ⎦. −γu2 I
Pre- and post-multiplying Eq. (32) by diag {Vil , I} and its transpose yields −Vil P˜i−1VilT Vil ϒil ˆ il < 0. ϒilT VilT
(33)
Since P˜i−1 > 0 and (P˜i − Vil )P˜i−1 (P˜i − Vil )T > 0, there holds −Vil P˜i−1VilT < P˜i − Vil − VilT . This together with Eq. (33) gives rise to P˜i − Vil − VilT Vil ϒil ˆ ˜ il = (34) ˆ il < 0, ϒilT VilT which implies Vil + VilT > 0 and Vil is invertible. ˆ˜ is rewritten as Denote Vil13 = [V T V T ]T . il il1
il3
ˆ˜ = T ˆ il + T Vil13 i + Ti Vil13 , il
(35)
where = [I2n×2n 02n×m 02n×n 02n×n 02n×p ], i = [−In×n 0n×n 0n×m Ai 0n×n Bi ]. Then, according to Eqs. (23) and (24), multiplying both sides of inequality (35) by T⊥ and its transpose leads to ˆ il ⊥ = ˆ i < 0, T⊥ (36) Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
10
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
where ⊥ are matrices whose column vectors constitute the bases of the null spaces of . Similarly, we obtain ˆ Ti⊥ ˆ il ˆ i⊥ < 0,
(37)
ˆ i⊥ whose column vectors constitute the bases of the null spaces of . ˆ Therefore, the where equivalence between Eqs. (34) and (36)–(37) is guaranteed through Finsler lemma. It means Eqs. (36) and (37) are the sufficient conditions for Eq. (10). Further, it follows from condition (25) that ˆ 1 I ≤ R− 21 Pi R− 21 = Pi ≤ λ ˆ 2I , λ which shows that ˆ 1 ≤ min{λmin (Pi ), i ∈ M} ≤ λ(Pi ) ≤ max{λmax (Pi ), i ∈ M} ≤ λ ˆ 2, λ which together with Eq. (26) indicates that Eq. (11) holds. This ends the proof. In the sequel, some results will be proposed such that the lower-bounding FE system (5) is positive and FTB with an H∞ performance level. A set of convex optimization conditions concerning the lower-bounding asynchronous positive filter will be provided. In order to avoid repetitions, here we provid the main proof process. Theorem 3. Given positive scalars ϑ, ζ > 1, d2 > d1 > 0, an integer K > 0, and a matrix R > 0, the lower-bounding FE system (5) is positive, FTB with an H∞ performance level γ˜ = γ ζK , if there exist scalar γ > 0, matrices Pi > 0, Qil > 0 for i ∈ M, l ∈ N such that Ai − Bˇ l Ci − Aˇ l 0, Bi − Bˇ l Di 0,
(38)
Hi − Lˇ l Ci − Hˇ l 0, Li − Lˇ l Di 0,
(39)
N
ϕil Qil < Pi ,
(40)
l=1
⎡
−P˜i−1 ⎢ ∗ ⎢ ⎣ ∗ ∗
0 −I ∗ ∗
Aˇ il Hˇ il −ζ Qil ∗
⎤ Bˇ il Lˇ il ⎥ ⎥ < 0, 0 ⎦ −γ2 I
1 1 ≤ d2 , ζK d1 max{λmax (Pi ), i ∈ M} + ϑ min{λmin (Pi ), i ∈ M} ζ − 1 1 1 where P˜i = Mj=1 πi j Pj , Pi R− 2 Pi R− 2 .
(41)
(42)
Proof. Considering Lemma 1 and inequalities (38)–(39), there are matrices Aˇ il 0, Bˇ il 0, Hˇ il 0, Lˇ il 0 for all i ∈ M, l ∈ N such that the lower-bounding FE system (5) is positive. Construct a Lyapunov function as follows: Vι (k, α(k), xˇF (k)) = xˇFT (k)Pα(k) xˇF (k).
(43)
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
11
Bearing in the mind Eqs. (40) and (41), one has Jzˇν =E Vι (k + 1, α(k + 1), xˇF (k + 1)) − ζ Vι (k, α(k), xˇF (k)) ≤ 0. Then, we can obtain E Vι (k + 1, α(k + 1), xˇF (k + 1)) ≤ E ζ Vι (k, α(k), xˇF (k)) + γ2 ν T (k)ν(k) − zˇFT (k)zˇF (k) . (44) Similar to Theorem 1, after some manipulations, we have E Vι (k, α(k), xˇF (k)) ⎧ ⎫ k−1 ⎨ ⎬ ≤ ζk Vι (0, α(0), xˇF (0)) + E ζk−1−μ γ2 ν T (μ)ν(μ) . ⎩ ⎭
(45)
μ=0
Noting that ν T (k)ν(k) ≤ ϑ and ζ > 1, it implies K E V ι(k, α(k), xˇF (k)) ) ≤ ζ d1 max{λmax (Pi ), i ∈ M} + γ2 ϑ
1 . ζ − 1
E xˇFT (k)Pi xˇF (k) ≥ min{λmin (Pi ), i ∈ M}E xˇFT (k)RxˇF (k) .
(46)
(47)
Combining (42), (46) with (47) implies that the lower-bounding FE system (5) is FTB. On the other hand, under the zero initial conditions, there holds ⎧ ⎫ k−1 ⎨ ⎬ 0 ≤ E V (k, α(k), xˇF (k)) ≤ E ζk−1−μ {γ2 ν T (μ)ν(μ) − zˇFT (μ)zˇF (μ)} ⎩ ⎭ μ=0 ⎧ ⎫ ⎨K−1 ⎬ ≤E ζK {γ2 ν T (μ)ν(μ) − zˇFT (μ)zˇF (μ)} , ⎩ ⎭ μ=0
which implies K−1 μ=0
K−1 E zˇFT (μ)zˇF (μ) ≤ ζK γ2 ν T (μ)ν(μ). μ=0
Then, the lower-bounding FE system (5) satisfies an H∞ performance level γ˜ = γ ζK . This ends the proof. Theorem 4. Given positive scalars ϑ, ζ > 1, d2 > d1 > 0, an integer K > 0, and a matrix ˇ 1 > 0, λ ˇ 2 > 0, matrices Pi = diag{Pi1 , Pi2 }, Pi1 > 0, Pi2 > 0, R > 0, if there exist scalars γ , λ Qil > 0, Aˇ F L 0, Bˇ F L 0, Hˇ F L 0, Lˇ F L 0 and positive definite diagonal matrix Vˇl2 , such that for any i ∈ M, l ∈ N r c Vˇl2,b Aci,g − Bˇ Fr l,bCi,g − [Aˇ F l ]b,g ≥ 0,
(48)
r c c Vˇl2,b Bi,g − Bˇ Fr l,bDi,g ≥ 0,
(49)
c [Hi ]b,g − Lˇ Fr l,bCi,g − [Hˇ F l ]b,g ≥ 0,
(50)
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
12
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
c [Li ]b,g − Lˇ Fr l,bDi,g ≥ 0,
⎡
√
−Pi ⎢ ∗ ⎢ ⎢ .. ⎣ .
ϕil Qi1 −Qi1
··· ..
(51) √
ϕiN QiN
.
∗
⎤ ⎥ ⎥ ⎥ < 0, ⎦
(52)
−QiN ⎡
Hˇ il −ζ Qil
−I ˇ il = ⎣ ∗ ∗
Lˇ il −γ2 I
⎤ ⎦ < 0,
(53)
ˇ il i⊥ < 0, Ti⊥
(54)
ˇ 1 R ≤ Pi ≤ λ ˇ 2 R, λ
(55)
1 ˇ 1, ≤ ζ−K d2 λ ζ − 1 where P˜i = Mj=1 πi j Pj , ⎡ ⎤ 0 0 Ai 0 Bi ˇ (12) ˜i − Vˇ˜l ⎢I ⎥ ˇ P il i⊥ = ⎣ , ⎦, il = I ˇ il ∗ I I I ˇ ˇ ˇ ˇ ˇ ˇ ˇ (12) = 0 Vl2 Ai − BF l Ci − AF l AF l Vl2 Bi − BF l Di , Vˇ˜l = 0 il 0 Vˇl2 Ai − Bˇ F l Ci − Aˇ F l Aˇ F l Vˇl2 Bi − Bˇ F l Di ∗ ˇ 2 + γ2 ϑ d1 λ
(56)
Vˇl2 . Vˇl2T + Vˇl2
Then the lower-bounding FE system (5) is positive, FTB with an H∞ performance level γ˜ . Moreover, the lower-bounding asynchronous positive filter parameters can be given by Aˇ l = Vˇl2−1 Aˇ F l , Bˇ l = Vˇl2−1 Bˇ F l , Hˇ l = Hˇ F l , Lˇ l = Lˇ F l . (57) Proof. According to Eqs. (49)–(51) and (57), the conditions (38), (39) can be proved, which implies the lower-bounding FE system (5) is positive. The equivalence between Eqs. (40) and (52) can be verified by Schur complement formula. Similar to Theorem 2, we will introduce an invertible slack matrix ι Vil1 Vˇl2 ι , Vil Vˇl2 Vι il3
ι ι where Vil1 , Vˇl2ι , Vil3 ∈ Rn×n and Vˇl2ι is a positive definite diagonal matrix. Pre- and post-multiplying the partitioned Eq. (41) by diag{Vilι , I }, we derive −Vilι P˜i−1VilιT Vilι ϒilι < 0, ˇ il ϒ ιT V ιT il
where ϒilι = 0
(58)
il
Aˇ il
Bˇ il ,
⎡
−I ˇ il = ⎣ ∗ ∗
Hˇ il −ζ Qˇ il ∗
⎤ Lˇ il 0 ⎦. −γ2 I
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
13
Considering P˜i−1 > 0 and (P˜i − Vilι )P˜i−1 (P˜i − Vilι )T > 0, we have −Vilι P˜i−1VilιT < P˜i − Vilι − VilιT which implies P˜i − Vilι − VilιT Vilι ϒilι ˇ ˜ il = (59) ˇ il < 0. ϒilιT VilιT ˇ˜ can be rewritten as Then, il ˇ˜ = ι ιT ˇ il + T Vil13 i + Ti Vil13 , il ι with = [I2n×2n 02n×m 02n×n 02n×n 02n×p ], i = [−In×n 0n×n 0n×m Ai 0n×n Bi ], Vil13 = ιT ιT T [Vil1 Vil3 ] . Similar to Theorem 2, condition (41) can be guaranteed. It is noted that Eq. (55) implies
ˇ 1 ≤ min{λmin (Pi ), i ∈ M} ≤ λ(Pi ) ≤ max{λmax (Pi ), i ∈ M} ≤ λ ˇ 2. λ
(60)
Combining (56) with (60) shows Eq. (42) holds. This completes the proof. Remark 6. The upper-bounding filter and the lower-bounding filter have been designed to make the FE systems be positive, FTB and have an H∞ performance level. The estimated output z(k) can be encapsulated between these two filters at all the time which implies the estimation can be better than the single filter. 3.2. Design of weighted filter In this section, on the basis of the developed filters in Eqs. (2) and (3), we shall establish another filter called weighted filter to acquire better filtering performance. Some feasible conditions will be derived such that the FE system is positive and FTB with an H∞ performance. Define the new controlled output z¯(k) as z¯(k) = θ zˆ(k) + (1 − θ )zˇ(k), θ ∈ [0, 1]. On account of the bounded filters in Theorems 2 and 4, we have 0 zˇ(k) z(k) zˆ(k). From Lemma 2, we derive zˆ(k) ≥ z¯(k) ≥ zˇ(k). Let’s construct the weighted filter as: x¯(k + 1) = A¯ l x¯(k) + B¯ l y(k), z¯(k) = H¯ l x¯(k) + L¯ l y(k), where x¯(k) = [θ xˆT (k) (1 − θ )xˇT (k)]T , A¯ l = diag{Aˆ l , Aˇ l }, B¯ l = [θ Bˆ lT [Hˆ l Hˇ l ], L¯ l = θ Lˆ l + (1 − θ )Lˇ l . Then the FE system reads x¯F (k + 1) = A¯ il x¯F (k) + B¯ il ν(k),
e¯ : z¯F (k) = H¯ il x¯F (k) + L¯ il ν(k),
(61) (1 − θ )Bˇ lT ]T , H¯ l =
(62)
where z¯F (k) z¯(k) − z(k) = θ zˆF (k) − (1 − θ )zˇF (k), Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
14
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
x¯F (k + 1) = [x T (k ) θ xˆeT (k ) (1 − θ )xˇeT (k)]T , ⎡ ⎤ Ai 0n×n 0n×n Aˆ l 0n×n ⎦, A¯ il = ⎣ θ (Aˆ l + Bˆ l Ci − Ai ) Aˇ l (1 − θ )(Ai − Bˇ l Ci − Aˇ l ) 0n×n ⎡ ⎤ Bi ˆ ⎣ ¯ θ ( B l D i − B i ) ⎦, Bil = (1 − θ )(Bi − Bˇ l Di ) H¯ il = θ (Hˆ l + Lˆ l Ci − Hi ) − (1 − θ )(Hi − Lˇ l Ci − Hˇ l )
Hˆ l
Hˇ l ,
L¯ il = θ (Lˆ l Di − Li ) + (1 − θ )(Li − Lˇ l Di ). Now we are in a position to illustrate the following results on the weighted asynchronous H∞ positive filter design. Theorem 5. For PMJSs (1), if all the conditions in Theorems 2 and 4 hold, then the weighted filter Eq. (61) is positive, and the weighted FE system (62) is FTB with an H∞ performance γ¯ = θ γ˜u + (1 − θ )γ˜ . Proof. According to the positivity of the bounding filters Eqs. (2), (3) and the FE systems (4), (5), we obtain xˆ(k) 0, xˇ(k) 0, zˆ(k) 0, zˇ(k) 0, which imply x¯(k) 0. From the definition of zˆ(k) and zˇ(k), we get 0 zˇ(k) zˆ(k). Bearing in mind Lemma 2 and z¯(k) = θ zˆ(k) + (1 − θ )zˇ(k), we get 0 zˇ(k) z¯(k) zˆ(k). So the weighted filter (61) is positive. Theorems 2 and 4 guarantee that the FE systems (4), (5) are FTB; which means that for given scalars c2 > c1 > 0, d2 > d1 > 0 and a matrix R > 0, there hold E {xˆFT (0)RxˆF (0)} ≤ c1 ⇒ E {xˆFT (k)RxˆF x(k)} ≤ c2 , ν T (k)ν(k) ≤ ϑ E {xˇFT (0)RxˇF (0)} ≤ d1 ⇒ E {xˇFT (k)RxˇF x(k)} ≤ d2 . ν T (k)ν(k) ≤ ϑ Therefore, one has E {x¯FT (0)Rx¯F (0)} ≤ θ c1 + (1 − θ )d1 ν T (k)ν(k) ≤ ϑ
⇒ E {x¯FT (k)Rx¯F x(k)} ≤ θ c1 + (1 − θ )d1 ,
(63)
which shows the weighted FE system (62) is FTB. Under the zero initial conditions, it follows from Theorems 2 and 4 K
E { zˆF (k) 2 } < γ˜u2
k=0 K
K
ν(k) 2 ,
k=0
E { zˇF (k) 2 } < γ˜2
k=0
K
ν(k) 2 .
k=0
Due to z¯F (k) = θ zˆF (k) − (1 − θ )zˇF (k), we get z¯F (k) = θ zˆF (k) − (1 − θ )zˇF (k) ≤ θ zˆF (k) + (1 − θ )zˇF (k) , K k=0
E { z¯F (k) 2 } < γ¯ 2
K
ν(k) 2 ,
k=0
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
15
where γ¯ = θ γ˜u + (1 − θ )γ˜ . Hence, the weighted FE system (62) is FTB and has an H∞ performance. This ends the proof. The weighted filter in Theorem 5 is designed based on Theorems 2 and 4. All the conditions have been converted to linear matrix inequalities (LMIs). Hence, the conditions can be solved with the aid of Matlab LMI Toolbox. An effective algorithm is provided as follows: Step 1. Take initial value x0 , the TP matrix and the CP matrix . Step 2. Select the parameters ϑ > 0, ζ u > 1, ζ > 1, c1 > c2 > 0, d1 > d2 > 0, θ , the positive integer K and positive definite matrix R. ˆ 1, λ ˆ 2 , positive Step 3. Solving the LMI conditions (18)–(26), if there exists positive scalars λ ˆ definite matrices Pi , Qil and positive definite diagonal matrices Vl2 such that conditions (18)–(26) are feasible, go to Step 4; otherwise go back to Step 2, and reselect the parameter values. ˇ 1, λ ˇ 2 , positive Step 4. Solving the LMI conditions (48)–(56), if there exists positive scalars λ definite matrices Pi , Qil and positive definite diagonal matrices Vˇl2 such that conditions (48)–(56) are feasible, go to Step 5; otherwise go back to Step 2, and reselect the parameter values. Step 5. According to the filter gain matrices Aˆ l , Aˇ l , Bˆ l , Bˇ l , Hˆ l , Hˇ l , Lˆ l , Lˇ l , compute the weighted filter gain matrices. Here, the optimal values γ u , γ can be solved by solver mincx. Remark 7. The weighted filter has been designed to acquire better filtering performance and provides more flexibilities through adjusting the parameter θ in estimating the given systems than the above bounding filters. Meanwhile, when θ = 1 and θ = 0, the weighted filter (61) will respectively reduce to the upper-bounding filter (2) and the lower-bounding filter (3). 4. Simulation example In this section, we borrow the fish’s structured population model from [8,36] and describe it as PMJSs to validate the theoretical results. Let x(k) denote the fish’s number in a pond. Due to the influence of various factors, such as temperature, feed intake or predation, and the non-stop movement of fish, it is difficult to obtain the accurate population description. Here, we sample the fish’s number twice a day, once in the day and once at night. The fish’s population is modelled as system (1) with the parameters given below: 0.51 0.31 0.25 , B1 = , A1 = C1 = 0.72 0.53 , D1 = 0.75, 0.28 0.15 0.15 0.42 0.60 0.16 , B2 = , A2 = C2 = 0.59 0.67 , D2 = 0.49, 0.30 0.36 0.14 H1 = 0.43 0.18 , H2 = 0.43 0.21 , L1 = 0.33, L2 = 0.11. Take K = 20, ϑ = 0.2, ζu = ζ = 1.1. The TP matrix is given as 0.2 0.8 . = 0.6 0.4 Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
16
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
Table 1 The different CP matrices.
Case I Case II Case III
π 11
π 12
π 21
π 22
0.3 1 1
0.7 0 0
0.5 0.5 0
0.5 0.5 1
Table 2 The optimal H∞ performance indices for different CP matrices. γ˜u γ˜
Case I
Case II
Case III
1.7569 1.3595
1.7034 1.3328
1.6515 1.3168
In order to show the asynchrony effects on optimal H∞ performance index, we provide Tables 1 and 2. Table 2 shows different H∞ performance indices can be obtained under different CP matrices. For completely asynchronous case III, solving Theorem 2, we obtain the upper-bounding filter parameters 0.1184 0.1751 ˆ 0.9369 ˆ Aˆ 1 = , B1 = , H1 = 0.2834 0.1293 , Lˆ 1 = 0.4668, 0.1664 0.1659 0.5084 0.0789 0.1452 ˆ 0.9095 ˆ , B2 = , H2 = 0.2313 0.1094 , Lˆ 2 = 0.4657. Aˆ 2 = 0.1207 0.1435 0.4815 Similarly, solving Theorem 4, we obtain the lower-bounding filter parameters 0.0336 0.0353 ˇ 0.0872 ˇ ˇ , B1 = , H1 = 0.2449 0.0682 , Lˇ 1 = 0.1307, A1 = 0.0274 0.0214 0.0470 0.0134 0.0139 ˇ 0.0960 ˇ ˇ , B2 = , H2 = 0.2373 0.0741 , Lˇ 2 = 0.1786. A2 = 0.0143 0.0104 0.0553
Fig. 1. The modes of the system and the bounding filters. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
17
Fig. 2. Response of system output z(k) and its estimations.
In this example, the initial states of the upper-bounding FE system and the lower-bounding FE system are respectively taken as [0.2 0.4 0.3 0.5]T and [0.2 0.4 0.15 0.3]T . The external disturbance is ν(k) = 0.4e−0.4k . Fig. 1 presents the switching signal of the system and the filter, which shows the modes of the system and the filter are switching asynchronously. It’s worth noting that α(k) and β(k) have two modes respectively. There are four combinations between system’s modes and filter’s modes. Denote α(k) = 1, β(k) = 1 as case 1, α(k) = 1, β(k) = 2 as case 2, α(k) = 2, β(k) = 1 as case 3 and the rest as case 4. Fig. 1 presents the switching among the four combinations clearly. The responses of the system output z(k), the upper-bounding estimation zˆ(k) and the lower-bounding estimation zˇ(k) are depicted in Fig. 2. One can see that the system output z(k) always runs between the estimations zˆ(k) and zˇ(k) from Fig. 2; which coincides with
Fig. 3. Responses of xˆT (k)Ru xˆ(k) and xˇT (k)R xˇ(k). Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
ARTICLE IN PRESS
JID: FI
18
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
Fig. 4. Response of system output z(k) and its estimations.
the theoretical results. Fig. 3 is the trajectories of xˆT (k)Ru xˆ(k) and xˇT (k)R xˇ(k); which also show that the FE systems are FTB and gain an H∞ performance level. Fig. 4 presents the responses of the upper-bounding FE systems, the lower-bounding FE systems, and the weighted FE system for parameter θ = 0.3. We can see they are all FTB and have an H∞ performance level. 5. Conclusion In this paper, we have investigated the asynchronous positive H∞ filtering problem for a class of PMJSs. Based on the hidden Markov model, sufficient conditions have been proposed such that the FE systems are positive and FTB with an H∞ performance. In terms of the stochastic system theory and linear matrix inequality techniques, we have built three different asynchronous filters and also discussed their relations to gain more flexibilities during the design. The fish’s population evolution has been modelled as a PMJS which validates the effectiveness of our theoretical results. Acknowledgment This work was supported in part by the National Natural Science Foundation of China under grant (61773225, 61773226, 61873331, 61803225), in part by the Taishan Scholar Project of Shandong Province (TSQN20161033) and in part by Interdisciplinary Scientific Research Projects of Qufu Normal University (xkjjc201905). References [1] L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley, New York, 2000. [2] Y. Yin, G. Zong, X. Zhao, Improved stability criteria for switched positive linear systems with average dwell time switching, J. Frankl. Inst. 354 (8) (2017) 3472–3484. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
19
[3] H. Ren, G. Zong, T. Li, Event-triggered finite-time control for networked switched linear systems with asynchronous switching, IEEE Trans. Syst. Man Cybern. Syst. 48 (11) (2018) 1874–1884. [4] W. Qi, G. Zong, H.R. Karimi, L∞ control for positive delay systems with semi-Markov process and application to a communication network model, IEEE Trans. Ind. Electron. 66 (3) (2019) 2081–2091. [5] Y. Guo, Stabilization of positive Markov jump systems, J. Frankl. Inst. 353 (14) (2016) 3428–3440. [6] W. Qi, H.P. Ju, J. Cheng, Y. Kao, X. Gao, Exponential stability and L1 -gain analysis for positive time-delay Markovian jump systems with switching transition rates subject to average dwell time, Inf. Sci. 424 (2018) 224–234. [7] S. Li, Z. Xiang, Stochastic stability analysis and L∞ -gain controller design for positive Markov jump systems with time-varying delays, Nonlinear Anal. Hybrid Syst. 22 (2016) 31–42. [8] V.H. Le, An LP approach to full-order and reduced-order state estimations of positive Markov jump systems with delay, Int. J. Syst. Sci. 48 (12) (2017) 2534–2543. [9] S. Zhu, B. Wang, C. Zhang, Delay-dependent stochastic finite-time l1 -gain filtering for discrete-time positive Markov jump linear systems with time-delay, J. Frankl. Inst. 354 (15) (2017) 6894–6913. [10] Y. Wang, X. Liu, J. Xiao, Y. Shen, Output formation-containment of interacted heterogeneous linear systems by distributed hybrid active control, Automatica 93 (2018) 26–32. [11] X. Chen, J. Lam, H.K. Lam, Positive filtering for positive Takagi-Sugeno fuzzy systems under l1 performance, Inf. Sci. 299 (2015) 32–41. [12] D. Yang, G. Zong, H.R. Karimi, H∞ refined anti-disturbance control of switched LPV systems with application to aero-engine, IEEE Trans. Ind. Electron. (2019), doi:10.1109/TIE.2019.2912780. [13] H. Sun, Y. Li, G. Zong, L. Hou, Disturbance attenuation and rejection for stochastic Markovian jump system with partially known transition probabilities, Automatica 89 (2018) 349–357. [14] D. Yang, G. Zong, S.K. Nguang, H∞ bumpless transfer reliable control of Markovian switching LPV systems subject to actuator failures, Inf. Sci. (2019), doi:10.1016/j.ins.2019.07.059. [15] Y. Wang, W. Yang, J. Xiao, Z. Zeng, Impulsive multisynchronization of coupled multistable neural networks with time-varying delay, IEEE Trans. Neural Netw. Learn. Syst. 28 (7) (2017) 1560–1571. [16] Y. Ren, J.E. Meng, G. Sun, Asynchronous L1 positive filter design for switched positive systems with overlapped detection delay, IET Control Theory Appl. 11 (3) (2016) 319–328. [17] S. Li, Z. Xiang, Stabilisation of a class of positive switched nonlinear systems under asynchrononus switching, Int. J. Syst. Sci. 48 (7) (2017) 1537–1547. [18] Y. Li, H. Zhang, Asynchronous L1 -gain control of uncertain switched positive linear systems with dwell time, ISA Trans. 75 (2018) 25–37. [19] H. Ren, G. Zong, K. Hamid Reza, Asynchronous finite-time filtering of networked switched systems and its application: an event-driven method, IEEE Trans. Circuits Syst. I Regul. Pap. 66 (1) (2019) 391–402. [20] H. Shen, F. Li, Z. Wu, H.P. Ju, Finite-time asynchronous H∞ filtering for discrete-time Markov jump systems over a lossy network, Int. J. Robust Nonlinear Control 26 (17) (2016) 3831–3848. [21] M. Chen, X. Yang, H. Shen, F. Yao, Finite-time asynchronous H∞ control for Markov jump repeated scalar non-linear systems with input constraints, Appl. Math. Comput. 275 (2016) 172–180. [22] M. Zhang, P. Shi, Z. Liu, H. Su, L. Ma, Fuzzy model-based asynchronous H∞ filter design of discrete-time Markov jump systems, J. Frankl. Inst. 354 (18) (2017) 8444–8460. [23] Z. Wu, P. Shi, H. Su, J. Chu, Asynchronous L2 − L∞ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities, Automatica 50 (1) (2014) 180–186. [24] Z. Wu, P. Shi, Z. Shu, H. Su, R. Lu, Passivity-based asynchronous control for Markov jump systems, IEEE Trans. Autom. Control 62 (4) (2017) 2020–2025. [25] Z. Wu, S. Dong, H. Su, C. Li, Asynchronous dissipative control for fuzzy Markov jump systems., IEEE Trans. Cybern. 48 (8) (2018) 2426–2436. [26] J. Song, Y. Niu, Y. Zou, Asynchronous output feedback control of time-varying Markovian jump systems within a finite-time interval, J. Frankl. Inst. 354 (15) (2017) 6747–6765. [27] H. Dong, Z. Wang, D.W.C. Ho, H. Gao, Robust filtering for Markovian jump systems with randomly occurring nonlinearities and sensor saturation: the finite-horizon case, IEEE Trans. Signal Process. 59 (7) (2011) 3048–3057. [28] S. Xu, J. Lam, Positive Linear Systems: Theory and Applications, Springer-Verlag, Berlin, 2006. [29] Y. Ren, G. Sun, Z. Feng, J.E. Meng, S. Wu, Weighted-average l1 filtering for switched positive systems, Int. J. Robust Nonlinear Control 27 (18) (2017) 5097–5112. [30] S. Zhu, Q. Han, C. Zhang, L1 -gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: a linear programming approach, Automatica 50 (8) (2014) 2098–2107. Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008
JID: FI
20
ARTICLE IN PRESS
[m1+;October 15, 2019;10:21]
H. Shang, G. Zong and W. Qi / Journal of the Franklin Institute xxx (xxxx) xxx
[31] Y. Zheng, S.S. Ge, Y. Wang, Non-weighted L1 filtering for positive switched delay systems, IET Control Theory Appl. 12 (8) (2018) 1046–1054. [32] H. Wang, D. Zhang, R. Lu, Event-triggered H∞ filter design for Markovian jump systems with quantization, Nonlinear Anal. Hybrid Syst. 28 (2018) 23–41. [33] W. Qi, J.H.P. Park, G. Zong, J. Cheng, L1 finite-time stabilization for positive semi-Markovian switching systems, Inf. Sci. 47 (2019) 321–333. [34] G. Zong, H. Ren, Guaranteed cost finite-time control for semi-Markov jump systems with event-triggered scheme and quantization input, Int. J. Robust Nonlinear Control, 29 (15) (2019) 5251–5273. [35] L. Rabiner, A tutorial on hidden Markov models and selected applications in speech recognition, Proc. IEEE, 77 (1989) 257–286. [36] S. Li, Z. Xiang, H. Lin, H.R. Karimi, State estimation on positive Markovian jump systems with time-varying delay and uncertain transition probabilities, Inf. Sci. 369 (2016) 251–266.
Please cite this article as: H. Shang, G. Zong and W. Qi, Finite-time asynchronous H∞ filtering for positive Markov jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.08.008