Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 825143, 8 pages http://dx.doi.org/10.1155/2013/825143
Research Article Finite-Frequency Filter Design for Networked Control Systems with Missing Measurements Dan Ye,1,2 Yue Long,3 and Guang-Hong Yang1 1
College of Information Science and Engineering, Northeastern University, Liaoning, Shenyang 110004, China State Key Laboratory of Robotics, Shenyang Institute of Automation, (CAS), Liaoning, Shenyang 110016, China 3 College of Information, Shenyang Institute of Engineering, Liaoning, Shenyang 110136, China 2
Correspondence should be addressed to Dan Ye;
[email protected] Received 26 March 2013; Revised 22 May 2013; Accepted 2 June 2013 Academic Editor: Bo Shen Copyright © 2013 Dan Ye et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with the problem of robust filter design for networked control systems (NCSs) with random missing measurements. Different from existing robust filters, the proposed one is designed in finite-frequency domain. With consideration of possible missing data, the NCSs are first modeled to Markov jump systems (MJSs). A finite-frequency stochastic 𝐻∞ performance is subsequently given that extends the standard 𝐻∞ performance, and then a sufficient condition guaranteeing the system to be with such a performance is derived in terms of linear matrix inequality (LMI). With the aid of this condition, a procedure of filter synthesis is proposed to deal with noises in the low-, middle-, and high-frequency domains, respectively. Finally, an example about the lateral-directional dynamic model of the NASA High Alpha Research Vehicle (HARV) is carried out to illustrate the effectiveness of the proposed method.
1. Introduction Due to the rapid development in communication network and computer technology, networked control systems (NCSs) have received much more research attentions. Networked control systems (NCSs) are control systems in which controller and plant are connected via a communication channel. The defining feature of an NCS is that information (reference input, plant output, control input, etc.) is exchanged using a network among control system components (sensors, controller, actuators, etc.). NCSs have many advantages such as easy diagnosis, low cost, and high mobility. And thus NCSs have been applied in many industrial systems such as automobiles, manufacturing plants, and aircrafts; see, for example, [1, 2]. Motivated by this wide spectrum of applications, the new problems arising from the limit resources of the communication channel are gradually taken into account when designing the NCSs. For example, the issues of network-induced delay, packet dropout, and quantization are considered in [3–12], respectively. On the other hand, state estimation has been widely studied and has found many practical applications over the
past decades [13]. When a priori statistical information on the external noise signals is unknown, the celebrated Kalman filtering cannot be employed. To address this issue, 𝐻∞ filtering is introduced, which aims to make the worst case 𝐻∞ norm from the process noise to the estimation error minimized. More recently, there have appeared a few results on 𝐻∞ filter design [14–18]. However, it should be noticed that all the aforementioned methods are proposed in fullfrequency domain. Nevertheless, practical industry systems often employ large, complex, or lightweight structures, which include finite-frequency fundamental vibration modes [19]. In these situations, it is more reasonable and precise to design filters in finite-frequency domain. Fortunately, the KalmanYakubovich-Popov (KYP) lemma is generalized in [20] to characterize frequency domain inequalities with (semi)finitefrequency ranges in terms of linear matrix inequalities (LMIs). The generalized KYP lemma is an effective tool to deal with the finite-frequency problem of linear timeinvariant systems [20–26]. There are some results concerned with this meaningful problem, for example, [27] proposed an effective filter for fuzzy nonlinear systems. However, to the best of the author’s knowledge, for system engaging random
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Mathematical Problems in Engineering
packet dropout, few results have been published for such class of NCSs. This instance motivates our present investigation. This paper studies the robust filter design problem with finite-frequency specifications for networked control systems (NCSs) subject to random missing measurements. First, the considered systems are modeled in the framework of Markov jump systems (MJSs). Motivated by Iwasaki et al. [22], a definition of finite-frequency stochastic 𝐻∞ norm is subsequently given to measure the robustness, which extends the standard 𝐻∞ norm and contains the frequency information of noises. Then based on Projection Lemma, an analysis condition is presented to guarantee the MJS being with such a performance in the framework of linear matrix inequalities (LMIs). Further, a procedure of filter synthesis is designed to deal with noises in low-, middle-, and highfrequency domains, respectively. Finally, an example about the lateral-directional dynamic model of the NASA High Alpha Research Vehicle (HARV) is given to illustrate the effectiveness of the proposed method. The rest of the paper is organized as follows. The problem statement for NCSs with random packet dropout is formulated in Section 2. Section 3 provides sufficient condition to meet the performance request and a design procedure of the robust filter. In Section 4, an example is given to illustrate the effectiveness of the proposed method. Finally, some conclusions end the paper in Section 5. Notations. Throughout the paper, the superscripts 𝑇 and −1 stand for, respectively, the transposition and the inverse of a matrix; 𝑀 > 0 means that 𝑀 is real symmetric and positive definite; ‖ ⋅ ‖ denotes the Euclidean norm; 𝑙2 denotes the Hilbert space of square integrable functions. In block symmetric matrices or long matrix expressions, we use ∗ to represent a term that is induced by symmetry. The sum of a square matrix 𝐴 and its transposition 𝐴𝑇 is denoted by 𝐻𝑒(𝐴) := 𝐴 + 𝐴𝑇 .
w(t)
Plant
y(k)
Sensor
Missing measurements
yr (k)
z(k) e(k)
+
−
ẑ(k)
Filter
Figure 1: The structure of the NCSs.
by a time-homogeneous Markov chain 𝛼𝑘 with the range set S = {1, 2, . . . , 𝑁} and the transition probability matrix Λ = (𝜆 𝑖𝑗 )𝑖,𝑗∈S = (P (𝛼𝑘+1 = 𝑗 | 𝛼𝑘 = 𝑖))𝑖,𝑗∈S 𝜆 11 𝜆 12 [ 𝜆 21 𝜆 22 [ = [ .. .. [ . . 𝜆 𝜆 𝑁1 𝑁2 [
. . . 𝜆 1𝑁 . . . 𝜆 2𝑁 ] ] . ]. d .. ]
(2)
. . . 𝜆 𝑁𝑁]
In this situation, the dynamics of plant (1) together with the missing measurements at time instant 𝑡𝑘 can be approximated by 𝑥 (𝑘 + 1) = 𝐴 𝛼𝑘 𝑥 (𝑘) + 𝐵𝛼𝑘 𝑤 (𝑘) , 𝑦𝑟 (𝑘) = 𝐶𝑥 (𝑘) + 𝐷𝑤 (𝑘) ,
(3)
𝑧 (𝑘) = 𝐸𝑥 (𝑘) , where 𝑡0 = 0, 𝑡𝑘 = ∑𝑘−1 𝑖=0 𝛼𝑖 ℎ (𝑘 ≥ 1), 𝑥(𝑘) = 𝑥(𝑡𝑘 ), 𝐴 𝛼𝑘 = 𝛼 ℎ
2. Problem Formulation Considering the NCS depicted in Figure 1, the continuoustime plant model is 𝑥̇ (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑤 (𝑡) , 𝑦 (𝑡) = 𝐶𝑥 (𝑡) + 𝐷𝑤 (𝑡) ,
(1)
𝑧 (𝑡) = 𝐸𝑥 (𝑡) , where 𝑥(𝑡) ∈ R𝑛 is the state, 𝑦(𝑡) ∈ R𝑚 is the measured output, 𝑧(𝑡) ∈ R𝑝 is the controlled output, and 𝑤(𝑡) ∈ R𝑑 is the exogenous disturbance which belongs to 𝑙2 [0, ∞). 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸 are known real constant matrices with appropriate dimensions. It is assumed that, as shown in Figure 1, the measurement signals will be transmitted via the networks wherein missing data may occur. Further, assume the interval between the 𝑘th and the (𝑘 + 1)th successfully received measurements at the filter is 𝛼𝑘 ℎ, where ℎ is the sampling period of the sensor. It is obvious that the number of the missed packets at time instant (𝑘 + 1)ℎ is 𝛼𝑘 − 1, which can be modeled
𝑒𝐴 𝑐 (𝛼𝑘 ℎ) , and 𝐵𝛼𝑘 = ∫0 𝑘 𝑒𝐴 𝑐 𝑡 𝐵𝑐 𝑑𝑡. It can be seen that, after the above treatment, the possible missing measurements can be converted to the jumping parameter of the MJS (3) with the transition probability Λ. In this paper, the filter is chosen as the following form: 𝑥̂ (𝑘 + 1) = 𝐴 𝑓𝛼𝑘 𝑥̂ (𝑘) + 𝐵𝑓𝛼𝑘 𝑦𝑟 (𝑘) , 𝑧̂ (𝑘) = 𝐶𝑓𝛼𝑘 𝑥 (𝑘) + 𝐷𝑓𝛼𝑘 𝑦𝑟 (𝑘) ,
(4)
̂ where 𝑥(𝑘) ∈ R𝑛 is the filter’s state, 𝑧̂(𝑘) is the estimated output, and 𝐴 𝑓𝛼𝑘 , 𝐵𝑓𝛼𝑘 , 𝐶𝑓𝛼𝑘 , and 𝐷𝑓𝛼𝑘 are filter gains to be designed. To ensure the achievement of filter design objective, a basic assumption, that is, 𝐴 𝑐 is stable, is also assumed to be valid. Remark 1. This assumption is required to get a stable filtering error dynamics. If this assumption is not satisfied, a stabilizing output feedback controller is required. For convenience, 𝐴 𝛼𝑘 , 𝐵𝛼𝑘 , 𝐴 𝑓𝛼𝑘 , 𝐵𝑓𝛼𝑘 , 𝐶𝑓𝛼𝑘 , and 𝐷𝑓𝛼𝑘 are notated as 𝐴 𝑖 , 𝐵𝑖 , 𝐴 𝑓𝑖 , 𝐵𝑓𝑖 , 𝐶𝑓𝑖 , and 𝐷𝑓𝑖 when 𝛼𝑘 = 𝑖,
Mathematical Problems in Engineering
3
respectively. Denoting 𝜂(𝑘) = [𝑥𝑇 (𝑘) 𝑥̂𝑇 (𝑘)]𝑇 and 𝑒(𝑘) = 𝑧(𝑘) − 𝑧̂(𝑘), the filtering error system can be described by the following system: 𝜂 (𝑘 + 1) = A𝑖 𝜂 (𝑘) + B𝑖 𝑤 (𝑘) ,
(5)
𝑒𝑟 (𝑘) = C𝑖 𝜂 (𝑘) + D𝑖 𝑤 (𝑘) , where [
𝐴𝑖 0 𝐵𝑖 A𝑖 B𝑖 𝐴 𝑓𝑖 𝐵𝑓𝑖 𝐷 ] . ] = [ 𝐵𝑓𝑖 𝐶 C𝑖 D𝑖 [ 𝐸 − 𝐷𝑓𝑖 𝐶 −𝐶𝑓𝑖 −𝐷𝑓𝑖 𝐷 ]
(6)
In order to present the objective of this paper clearly, the following definition is first given. Definition 2 (MSS). The filter error system (5) is said to be mean-square stable (MSS) with 𝑤(𝑘) = 0, if 2 lim E {𝜂 (𝑘) } = 0 𝑘→∞
(7)
holds for all 𝛼𝑘 = 𝑖 ∈ S. Definition 3 (finite-frequency stochastic 𝐻∞ norm). For all the solutions of (5) which satisfied the following inequalities under zero initial condition for nonzero disturbance: (i) for the low-frequency range |𝜃| ≤ 𝜗𝑙 ∞
𝑇
(8)
𝜗 2∞ ≤ (2 sin 𝑙 ) ∑ 𝜂 (𝑘) 𝜂(𝑘)𝑇 , 2 𝑘=0 (ii) for the middle-frequency range 𝜗1 ≤ 𝜃 ≤ 𝜗2 𝑗𝜗𝑤
𝑒
∞
𝑗𝜗1
∑ (𝜂 (𝑘 + 1) − 𝑒
Now, the problem to be addressed in this paper can be formulated as follows: design a stable filter (4) such that the filter error system (5) is mean-square stable, and with prescribed finite-frequency stochastic 𝐻∞ norm 𝛾 for the external disturbance 𝑤(𝑘).
3. Main Results The filter design problem proposed in the above section will be discussed in this section. 3.1. Conditions for Robustness. Before proceeding further, the following lemma will be recalled to help us derive our main results. Lemma 5 (Projection Lemma [27]). For arbitrary Γ, Λ, Θ, there exists matrix F satisfying ΓFΛ + (ΓFΛ)𝑇 + Θ < 0 if and only if the following two conditions hold: 𝑇
< 0.
(12)
𝜂 (𝑘))
𝑇
A B𝑖 A𝑖 B𝑖 ] Ξ𝑖 [ 𝑖 ] 𝐼 0 𝐼 0
(9)
𝑇
(13)
C D𝑖 C D𝑖 +[ 𝑖 ] Π[ 𝑖 ] < 0, 0 𝐼 0 𝐼
𝑇
× (𝜂 (𝑘 + 1) − 𝑒−𝑗𝜗2 𝜂 (𝑘)) ≤ 0, 𝐼
0
where Π = [ 0 −𝛾2 𝐼 ] and
where 𝜗𝑤 = (𝜗2 − 𝜗1 )/2, (iii) for the high-frequency range |𝜃| ≥ 𝜗ℎ ∞
∑ (𝜂 (𝑘 + 1) − 𝜂 (𝑘)) (𝜂 (𝑘 + 1) − 𝜂 (𝑘))
(i) for the low-frequency range |𝜃| ≤ 𝜗𝑙 𝑇
𝑘=0
𝜗 2∞ ≥ (2 sin ℎ ) ∑ 𝜂 (𝑘) 𝜂(𝑘)𝑇 . 2 𝑘=0
Ξ𝑖 = [ (10)
∞
∞
𝑘=0
𝑘=0
2 E ( ∑ 𝑒𝑟 (𝑘) ) ≤ 𝛾2 E ( ∑ ‖𝑤 (𝑘)‖2 )
(11)
𝑄𝑖 −𝑃𝑖 ], 𝑄𝑖 𝑃𝑖 − 2 cos 𝜗𝑙 𝑄𝑖
(14)
(ii) for the middle-frequency range 𝜗1 ≤ 𝜃 ≤ 𝜗2 Ξ𝑖 = [
the given constant 𝛾 > 0 is said to be the finite-frequency stochastic 𝐻∞ norm of (5) if the following inequality
holds.
⊥𝑇
Λ𝑇 ΘΛ𝑇
Lemma 6. Assume the MJLS (5) is mean-square stable; let 𝛾 > 0 be a given constant, then system (5) has a finite-frequency stochastic 𝐻∞ norm 𝛾 if there exist mode-dependent matrices 𝑃𝑖 = 𝑃𝑖𝑇 , 𝑄𝑖 = 𝑄𝑖𝑇 > 0, 𝑖 ∈ S such that the following inequalities hold: [
𝑘=0
⊥
Γ⊥ ΘΓ⊥ < 0,
The following lemma will be given which provides a sufficient condition for the desired performance (11) of system (5).
∑ (𝜂 (𝑘 + 1) − 𝜂 (𝑘)) (𝜂 (𝑘 + 1) − 𝜂 (𝑘)) 𝑘=0
Remark 4. Definition 3 is motivated by the work of [20, 27], which can be regarded as an extension in finite-frequency domain of standard 𝐻∞ norm. Noticeably, it expresses the robustness from 𝑤(𝑘) to 𝑒(𝑘) in finite-frequency, that is, the smaller it is, the more robust to 𝑤(𝑘) the error 𝑒(𝑘) becomes.
𝑒𝑗𝜗𝑐 𝑄𝑖 −𝑃𝑖 ], 𝑒 𝑄𝑖 𝑃𝑖 − 2 cos 𝜗𝑤 𝑄𝑖 −𝑗𝜗𝑐
(15)
where 𝜗𝑐 = (𝜗2 + 𝜗1 )/2, 𝜗𝑤 = (𝜗2 − 𝜗1 )/2, (iii) for the high-frequency range |𝜃| ≥ 𝜗ℎ −𝑄𝑖 −𝑃 Ξ𝑖 = [ 𝑖 ], −𝑄𝑖 𝑃𝑖 + 2 cos 𝜗ℎ 𝑄𝑖 where 𝑃𝑖 = ∑𝑗∈S 𝜆 𝑖𝑗 𝑃𝑗 .
(16)
4
Mathematical Problems in Engineering
Proof. We first consider the middle-frequency case for the system (5). Assume (13) holds, before and after multiplying it by [𝜂𝑇 (𝑘) 𝑤𝑇 (𝑘)] and its transpose, then we can derive
where 𝑃𝑖 = ∑𝑗∈S 𝜆 𝑖𝑗 𝑃𝑗 and Ψ33 = 𝑃𝑖1 − 2 cos 𝜗𝑙 𝑄𝑖1 + He (𝑈𝑖 𝐴 𝑖 + B𝑓𝑖 𝐶)
𝜂𝑇 (𝑘) 𝑃𝑖 𝜂 (𝑘) − 𝜂𝑇 (𝑘 + 1) 𝑃𝑖 𝜂 (𝑘 + 1) + 𝑒𝑟𝑇 (𝑘) 𝑒𝑟 (𝑘)
𝑇 𝐸) , + 𝐸𝑇 𝐸 − He (𝐶𝑇 𝐷𝑓𝑖
− 𝛾2 𝑤𝑇 (𝑘) 𝑤 (𝑘) + tr {𝑄 (𝑒𝑗𝜗𝑐 𝜂 (𝑘) 𝜂𝑇 (𝑘 + 1)
𝑇 𝐸, Ψ34 = 𝑃𝑖2 − 2 cos 𝜗𝑙 𝑄𝑖2 + 𝑊𝑖 𝐴 𝑖 + B𝑓𝑖 𝐶 + A𝑇𝑓𝑖 − 𝐶𝑓𝑖
+ 𝑒−𝑗𝜗𝑐 𝜂 (𝑘 + 1) 𝜂𝑇 (𝑘)
Ψ44 = 𝑃𝑖3 − 2 cos 𝜗𝑙 𝑄𝑖3 + He (A𝑓𝑖 ) ,
−2 cos 𝜗𝑤 𝜂 (𝑘) 𝜂𝑇 (𝑘))} ≤ 0. (17)
Ψ35 = 𝐵𝑖𝑇 𝑈𝑖𝑇 + 𝐷𝑇 B𝑇𝑓𝑖 − 𝐷𝑇 𝐷𝑓𝑇 𝐸,
Since 𝜂(0) = 0 and system (5) is mean-square stable, summing up (17) from 0 to ∞ with respect to 𝑘, it is straightforward to see that (17) is equal to ∞
E ( ∑ (𝑒𝑟𝑇 (𝑘) 𝑒𝑟 (𝑘) − 𝛾2 𝑤𝑇 (𝑘) 𝑤 (𝑘)) + tr {𝑄𝑖 𝑆}) ≤ 0,
Ψ45 = 𝐵𝑖𝑇 𝑊𝑖𝑇 + 𝐷𝑇 B𝑇𝑓𝑖 . Proof. It can be concluded from Lemma 6 that if inequality (13) holds for all 𝛼𝑘 = 𝑖 ∈ S, the performance of (11) can be reached. Further, (13) is equivalent to 𝑇
𝑘=0
(18) where ∞
(19) 𝑇
−2 cos 𝜗𝑤 𝜂 (𝑘) 𝜂 (𝑘)) . It is easy to prove that −𝑆 is equal to the left-hand side of (9), so the 𝑆 is semipositive definite. Also, since 𝑄𝑖 > 0, the term tr{𝑄𝑖 𝑆} is nonnegative when (9) is satisfied. Hence, we have 𝑇 2 𝑇 E(∑∞ 𝑘=0 (𝑒𝑟 (𝑘)𝑒𝑟 (𝑘) − 𝛾 𝑤 (𝑘)𝑤(𝑘)) ≤ 0, which is equivalent to condition (11) for middle frequency in Definition 3. Similarly, the results follow by choosing 𝜗1 := −𝜗𝑙 and 𝜗2 := 𝜗𝑙 for low-frequency case and 𝜗1 := 𝜗ℎ and 𝜗2 := 2𝜋 − 𝜗ℎ for high-frequency case, respectively. The proof is completed. Remark 7. If all the matrices in Lemma 6 are independent on 𝑖, the MJS will be reduced to a determinate linear system. In this case, Lemma 6 is equivalent to the GKYP in [20], which has been proved to be an effective tool to deal with the finitefrequency problem of linear time-invariant systems. Theorem 8. Consider system (5) for all 𝛼𝑘 = 𝑖 ∈ S; assume it is mean-square stable; for a given scalar 𝛾 > 0, the performance of (11) is guaranteed if there exist matrices 𝑈𝑖 , 𝑊𝑖 , 𝑉𝑖 , A𝑓𝑖 , B𝑓𝑖 , 𝐶𝑓𝑖 , 𝐷𝑓𝑖 , and 𝑃𝑖𝑇
𝑃 ∗ = 𝑃𝑖 = [ 𝑖1 ], 𝑃𝑖2 𝑃𝑖3
𝑄𝑖𝑇
𝑄 ∗ = 𝑄𝑖 = [ 𝑖1 ], 𝑄𝑖2 𝑄𝑖3
(20)
∗ ∗ ∗ Ψ44 Ψ45 𝐶𝑓
∗ ∗] ] ∗] ] ] < 0, ∗ ∗] ] ] ] −𝛾2 𝐼 ∗ ] 𝐷𝑓 𝐷 −𝐼]
(23)
Θ𝑖 = 𝐽Ξ𝑖 𝐽𝑇 + 𝐻𝑖 Π𝐻𝑖𝑇 ,
(24)
𝐼 0
0 0
00
D𝑖 𝐼
with 𝐽 = [ 0 𝐼 ] and 𝐻𝑖 = [ C𝑖 0 ]. On the other hand, (13) implies that [
𝑇
0 −𝑃 𝐼 0 0 𝐼 0 0 ] < 0. ] Θ𝑖 [ ]=[ 𝑖 0 0 𝐼 0 0 𝐼 0 D𝑇𝑖 D𝑖 − 𝛾2 𝐼
∗ ∗ ∗
(21)
(25)
Combining (23) and (25), from Lemma 5, one can easily derive that (13) holds if and only if −𝐼 [ 𝑇] 𝑇 ] Θ𝑖 + 𝐻𝑒 ([ [ A𝑖 ] 𝑋𝑖 [0 𝐼 0]) < 0, 𝑇 [B𝑖 ]
(26)
where matrix 𝑋𝑖𝑇 is the slack variable with appropriate dimensions which is introduced by Lemma 5. Rewrite 𝑋𝑖 as the form of 𝑈 𝑉 𝑋𝑖 = [ 𝑖 𝑖 ] . 𝑊𝑖 𝑉𝑖
(27)
One can conclude that the following inequality provides a sufficient condition for (26): ∗ ∗ −𝑃𝑖 ] < 0, [𝑄𝑖 − 𝑋𝑖 𝜓 ∗ 𝑇 𝑇 𝑇 𝑇 2 𝑋 + D C D D − 𝛾 𝐼 0 B ] [ 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
such that the following LMIs hold: ∗ ∗ −𝑃𝑖1 [ −𝑃 −𝑃 ∗ [ 𝑖2 𝑖3 [ 𝑄 − 𝑈 𝑄𝑇 − 𝑉 Ψ3 [ 1𝑖 𝑖 𝑖 2𝑖 3 [ [𝑄 − 𝑊 𝑄 − 𝑉 Ψ 4 [ 2𝑖 𝑖 3𝑖 𝑖 3 [ [ [ 0 0 Ψ35 0 𝐷𝑓 𝐶 [ 0
A𝑖 B𝑖 A𝑖 B𝑖 [ 𝐼 0 ] Θ𝑖 [ 𝐼 0 ] < 0, [0 𝐼] [0 𝐼] where
𝑆 := ∑ (𝑒𝑗𝜗𝑐 𝜂 (𝑘) 𝜂𝑇 (𝑘 + 1) + 𝑒−𝑗𝜗𝑐 𝜂 (𝑘 + 1) 𝜂𝑇 (𝑘) 𝑘=0
(22)
(28)
where 𝜓 = 𝑃𝑖 − 2 cos 𝜗𝑙 𝑄𝑖 + 𝐻𝑒(𝑋𝑖 A𝑖 ) + C𝑇𝑖 C𝑖 . After partitioning the matrices 𝑃𝑖 and 𝑄𝑖 as the following form 𝑃 ∗ 𝑃𝑖 = [ 𝑖1 ], 𝑃𝑖2 𝑃𝑖3
𝑄𝑖 = [
𝑄𝑖1 ∗ ] 𝑄𝑖2 𝑄𝑖3
(29)
Mathematical Problems in Engineering
5
4. An Illustrative Example
and defining the following new variables A𝑓𝑖 = 𝑉𝑖 𝐴 𝑓𝑖 ,
B𝑓𝑖 = 𝑉𝑖 𝐵𝑓𝑖 ,
(30)
inequality (21) can be derived. The proof is completed. Remark 9. In Theorem 8, by introducing a variable 𝑋𝑖 , the coupling between the variable 𝑃𝑖 and the filter gains will be eliminated. Such a matrix does not present any structure constraint; on the contrary, it may lead to potentially less conservative results. 3.2. Conditions for Stability. Theorem 8 can guarantee the filtering error system to be with a specific robust performance in a certain frequency range of relevance. However, the stability has not been captured, and hence, one may wish to include a stability constraint as an additional design specification. The following theorem will give a result for stability. Theorem 10. The system (5) is mean-square stable with 𝑤(𝑘) = 0 if there exist matrices 𝑈𝑖 , 𝑊𝑖 , 𝑉𝑖 , A𝑓𝑖 , B𝑓𝑖 , 𝐶𝑓𝑖 , 𝐷𝑓𝑖 , and 𝑃𝑠𝑖𝑇
∗ 𝑃 = 𝑃𝑠𝑖 = [ 𝑠𝑖1 ] > 0, 𝑃𝑠𝑖2 𝑃𝑠𝑖3
(31)
In this section, an example is given to illustrate the effectiveness of the proposed method. According to some previous researches of the aircraft dynamic model such as [28, 29], it is easily concluded that the nonlinear aircraft dynamic model can be established according to Newton’s Second Law of motion. Furthermore, in order to decouple the nonlinear dynamics, the model is decomposed into two models along with longitudinal- and lateral-directional motions, respectively. The model used in this example is the lateral-directional dynamic model of the NASA High Alpha Research Vehicle (HARV) utilized in [30], that is, −0.166 0.629 −0.9971 𝑥̇ (𝑡) = [−12.97 −1.761 0.5083 ] 𝑥 (𝑡) [ 3.191 −0.1417 −0.1529] 1.8 + [ 0.7 ] 𝑤 (𝑡) , [−1.46] 𝑦 (𝑡) = [
0 1 0 0.96 ] 𝑥 (𝑡) + [ ] 𝑤 (𝑡) , 0 0 1 −0.4
𝑧 (𝑡) = [
2.8 −0.52 1.3 ] 𝑥 (𝑡) , 1.7 0.9 −0.4
such that the following inequality holds: 𝑃𝑠𝑖1 − 𝐻𝑒 (𝑈𝑖 )
∗
∗
∗
[ ] [ ] [ 𝑃𝑠𝑖2 − 𝑉𝑇 − 𝑊𝑖 𝑃𝑠𝑖3 − 𝐻𝑒 (𝑉𝑖 ) ] ∗ ∗ 𝑖 [ ] [ ] < 0, [ 𝑇 𝑇 ] [𝐴 𝑖 𝑈𝑖 + 𝐶𝑇 B𝑇𝑓𝑖 𝐴𝑇𝑖 𝑊𝑖𝑇 + 𝐶𝑇 B𝑇𝑓𝑖 −𝑃𝑠𝑖1 ∗ ] [ ] [ ] 𝑇 𝑇 A A −𝑃 −𝑃 𝑠𝑖2 𝑠𝑖3 ] 𝑓𝑖 𝑓𝑖 [ (32) where 𝑃𝑠𝑖 = ∑𝑗∈S 𝜆 𝑖𝑗 𝑃𝑠𝑗 . Proof. Combing the knowledge of the existing stability criteria for MJSs and the lemma, following the line of the proof for Theorem 8, the conclusion can be derived easily. We do not explain it specifically here. Remark 11. It should be pointed out that inequalities (28)– (33) are all linear matrix inequalities which can be solved through LMI toolbox of MATLAB. Based on the above analysis, a set of optimal solutions A𝑓𝑖 , B𝑓𝑖 , 𝐶𝑓𝑖 , and 𝐷𝑓𝑖 can be obtained by solving the following optimization problem: 𝛾
min s.t.
(21) , (32) .
𝐶𝑓𝑖 = 𝐶𝑓𝑖 ,
𝐵𝑓𝑖 = 𝑉−1 B𝑓𝑖 , 𝐷𝑓𝑖 = 𝐷𝑓𝑖 .
where the system state and output are, respectively, sideslip angle (∘ ) ] [ ] [ state = [ roll rate (∘ /𝑠) ] , ] [ ∘ [ yaw rate ( /𝑠) ]
(36)
roll rate (∘ /𝑠) [ ]. output = ∘ [yaw rate ( /𝑠)] Assume that the sampling period is ℎ = 1 s, and the sampled data are transmitted through a network, where the data packets may be lost. Further, the quantity of the lost packet (𝛼𝑘 − 1), 𝛼𝑘 ∈ S = {1, 2, 3} at each sampling period is shown in Figure 2, which is subject to the following transition probability matrix:
(33)
Then the filter gains can be computed by the following equalities: 𝐴 𝑓𝑖 = 𝑉−1 A𝑓𝑖 ,
(35)
(34)
0.9 0.1 0 Λ = [0.8 0.1 0.1] . [0.8 0.2 0 ]
(37)
For prescribed 𝜗𝑙 = 0.8, solving the optimization problem (33), we can obtain the optimal value for finite-frequency
6
Mathematical Problems in Engineering 1
0.5
1
w(k)
Quantity of lost packets
2
0
−0.5 0 0
20
40
60
80
−1
100
0
20
40 60 Time steps
Time steps
Figure 2: The quantity of the dropped packet.
𝐴 𝑓1
−0.5552 0.2072 1.8402 = [ 1.0360 −1.5484 −10.5804] , [ 0.4397 0.4041 2.0462 ] 𝐵𝑓1
𝐶𝑓1 = [
0.2367 1.8933 = [−1.3950 −11.7654] , [ 0.2054 1.8380 ] −2.8000 −1.1960 −5.4184 ], −1.7000 0.0944 2.7866
𝐷𝑓1 = [
−1.7160 −4.1184 ], 0.9944 2.3866
0.2543 0.0869 −0.3916 𝐴 𝑓2 = [−0.2131 −0.2798 3.0482 ] , [ 0.0533 0.0319 −0.4729] 0.0765 −0.3750 𝐵𝑓2 = [−0.5082 3.0383 ] , [ 0.0340 −0.7283] 𝐶𝑓2 = [
−2.8000 −0.2844 −3.2306 ], −1.7000 −0.3677 1.6775
𝐷𝑓2 = [
−0.8044 −1.9306 ], 0.5323 1.2775
−0.1377 −0.0739 0.0753 𝐴 𝑓3 = [ 0.3593 0.4907 −0.6657] , [ 0.0806 −0.1414 0.0216 ] −0.0615 0.0800 𝐵𝑓3 = [ 0.5174 −0.9498] , [−0.1909 −0.0263]
100
Figure 3: The disturbance input 𝑤(𝑡). ×10−3 1 The estimation of e1 (k)
stochastic 𝐻∞ norm, that is, the robust performance is 𝛾 = 0.001 with the corresponding filter gains as follows:
80
0.5 0 −0.5 −1
0
10
20
30
40 50 60 Time steps
70
80
90
100
By our method Standard H∞ filtering
Figure 4: The estimation error 𝑒1 (𝑘).
−2.8000 0.7070 −0.8511 𝐶𝑓3 = [ ], −1.7000 −0.7467 0.7680 𝐷𝑓3 = [
0.1870 0.4489 ]. 0.1533 0.3680 (38)
In order to show the advantage of the proposed method, we compare it with standard 𝐻∞ filtering method for MJSs which can be found in many researches. In the following, the system will be simulated under zero initial condition, and the disturbance input 𝑤(𝑡) is 𝑤 (𝑡) = {
sin (0.5𝑡) , 0,
0 s ≤ 𝑡 ≤ 40 s, otherwise,
(39)
which is shown in Figure 3. It is easy to see that the disturbance considered in the this paper is an instantaneous sinusoidal disturbance and its frequency is considered as zero, which belongs to the low frequency. The simulation results are shown in Figures 4 and 5, which confirm that all the expected system performance requirement are well achieved. Compared with the standard
Mathematical Problems in Engineering
7
×10−3 1 The estimation of e2 (k)
[4] 0.5 0
[5]
−0.5 −1
0
10
20
30
40 50 60 Time steps
70
80
90
100
By our method Standard H∞ filtering
Figure 5: The estimation error 𝑒2 (𝑘).
[6]
[7]
[8]
𝐻∞ filtering method, our method performs better even in the case of possible missing measurements.
[9]
5. Conclusions [10]
In this paper, we have studied the robust filtering with finite-frequency specifications for NCSs subject to random missing measurements. Here, the NCSs are first modeled into MJLs. Then, a new robust filtering method has been proposed that makes full use of the frequency information of noises to reduce design conservatism by the introduction of finite-frequency stochastic 𝐻∞ index for MJLSs. The design problem is formulated into solving a set of linear matrix inequalities, which can be computed by the LMI Control Toolbox. An example is included to show the effectiveness of the obtained theoretical results.
Acknowledgments This work is supported by National Natural Science Foundation of China (nos. 61273155, 61273148), New Century Excellent Talents in University (no. NCET-11-0083), a Foundation for the Author of National Excellent Doctoral Dissertation of P.R. China (no. 201157), the Fundamental Research Funds for the Central Universities (Grant no. N120504003), and the Foundation of State Key Laboratory of Robotics (no. 2012001).
[11]
[12]
[13]
[14]
[15]
[16]
[17]
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