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Stability analysis of networked control systems with round-robin scheduling and packet dropouts
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Journal of the Franklin Institute 350 (2013) 2013–2027 www.elsevier.com/locate/jfranklin
Stability analysis of networked control systems with round-robin scheduling and packet dropouts$ Yong Xua, Hongye Sua,n, Ya-Jun Panb, Zheng-Guang Wua, Weihua Xua a
State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Yuquan Campus, Hangzhou, Zhejiang 310027, PR China b Department of Mechanical Engineering, Dalhousie University, P.O. Box 15000, Halifax, Nova Scotia, Canada B3H 4R2 Received 25 November 2012; received in revised form 17 March 2013; accepted 20 May 2013 Available online 1 June 2013
Abstract In this paper, the stability of networked control systems (NCSs) with communication constraints at both channels is investigated. A Conventional Round-Robin Scheduling (CRRS) is applied to deal with the communication constraints issue for its simple structure. Furthermore, a Dynamic Round-Robin Scheduling (DRRS), which can preserve the controllability and the detectability of the systems, is considered. For the unreliable communication channels, two independent homogeneous Markov chains are selected to model the packet dropouts phenomenon in the sensor-to-controller (S/C) channel and the controller-to-actuator (C/A) channel. According to the periodic property of the Round-Robin Scheduling (RRS), an auxiliary system with augmented Markov chain is established by the lifting technique to facilitate the stability analysis of the closed-loop system. A necessary and sufficient condition of the exponential mean-square stability for the NCSs is derived. Two illustrative examples are shown to demonstrate the effectiveness of the proposed stability analysis method. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Networked control systems, which connect the systems and controllers via a shared channel, have received numerous research attention on the system modeling, stability analysis, controller ☆ This work is supported by National Natural Science Foundation of PR China (NSFC: 61134007), National Natural Science Foundation of PR China (NSFC: 61104102), National Natural Science Foundation of PR China (NSFC: 61174029). n Corresponding author. Tel.: +86 0571 87951075. E-mail addresses: [email protected] (Y. Xu), [email protected] (H. Su), [email protected] (Y.-J. Pan).
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.05.024
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or estimator design in the past decade [1]. NCSs are provided with more advantages, including shortening the installation period, reducing wire cost and simplifying system maintenance, than the classical wired point-to-point feedback control systems [2]. NCSs have been applied in many fields of unmanned aerial vehicles [3], vehicular networks and traffic control [4], and teleoperation systems [5] due to its aforementioned merits. However, the communication channel brings a lot of challenges to the users such as quantization errors [6], packet dropouts and communication delays [7–10] and random sampling [11,12], which may degrade the system performance and even lead the system to instability. Due to the channel capacity constraints, not all of the system outputs and controller outputs of the Multiple-Input-Multiple-Output (MIMO) system can be transmitted through the channels at the same time. Therefore, some scheduling methods have been proposed in the past years such as TryOnce-Discard (TOD) protocol and Round-Robin (RR) protocol. Both protocols allow only one output to be transmitted via the channel at a time. The RR protocol is a static one, which transmits the packets in a fixed order. The TOD protocol is a dynamic protocol, which chooses the greatest weighted error node to be transmitted. In [13] the TOD protocol was introduced for the first time to deal with the packets scheduling and the maximum allowable transfer interval was then obtained. Based on the small gain theorem, in [14] an input–output LP stability condition was derived, which improves the maximum allowable transfer interval and the result was suitable for both the TOD and the RR protocols. In [15], the error of the output brought by the transmission delay was modeled as system uncertainty and the switched linear system approach was used to analyze the stability condition both for the TOD and the RR protocols. In [16] the RR protocol was applied to the sampled data control systems and the direct Lyapunov Krasovskii approach was selected to analyze the stability condition. However, the works mentioned above just take into account the system in the continuous time domain. Due to the nature of the NCSs, discrete time systems have been widely addressed. In [17], the necessary and sufficient stability condition was obtained for random time delay systems with output feedback controller. In [18], the sector bounded approach was introduced to deal with the quantized controller signal and the stability condition of the system was achieved by the quantizer density and the unstable eigenvalue of the systems. In [19], the synchronization stability of the chaotic Lur'e systems with random sampling was analyzed. Therefore, this paper wants to extend the RRS method to the discrete time system. Packet dropouts, caused by node failure or data collisions, are mainly modeled as Markov jumping systems due to two factors. On one hand, Markov jumping systems have been studied by many researches in the past two decades and some technical problems have been solved, which facilitate our analysis processes [20]. On the other hand, compared with Bernoulli process, Markov jumping systems are able to capture the possible temporal correlation of the channel variation. According to analyzing the random Riccati recursion and introducing the notion of peak covariance, the sufficient bounded condition of the estimation error covariance for the general systems with Markov based packet dropouts was obtained, which was also a necessary condition for the scalar case [21]. In [22], by exploiting the system structure, the necessary and sufficient condition for the stability of the mean estimation error covariance matrices of secondorder systems was derived, and the result could be extended to certain classes of higher-order systems. In [23], the separation principle was obtained for the NCSs with Markov-modeled packet dropouts which appeared simultaneously in the multiple S/C and multiple C/A channels. The H ∞ performance of the NCSs with Markov-based packet dropouts, which appears in the multiple S/C and the multiple C/A channels, was studied and the sufficient condition was proposed [24]. In the proposed paper, we mainly consider NCSs with independent packet dropouts in both communication channels and model it by Markov chains.
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Motivated by the above discussion, this paper studies the stability analysis of the discrete time NCSs with RRS and packet dropouts at both channels. Two scheduling protocols (CRRS and DRRS) are considered in the paper to deal with the communication constraint problem. For the imperfect channels, two independent homogenous Markov chains are introduced to describe the random packet dropouts. Then, a necessary and sufficient condition for the exponential mean-square stability of the closed-loop systems is derived. Finally, two illustrative examples are shown to demonstrate the effectiveness of the proposed stability analysis approach. The main contributions of the paper can be summarized as follows: (1) in order to establish a more general model, we consider the RRS and the packet dropouts together in this paper, rather than just address one problem in some existing papers [13,16,21]; (2) in order to overcome the periodic parameter varying caused by RRS, an auxiliary system is established via the lifting technique to facilitate the analysis. And an augmented Markov chain is obtained to describe the packet dropouts of the auxiliary system; (3) the equivalence of the exponential mean-square stability between the original system and the auxiliary system is established. This paper is organized as follows. The model of the NCSs is described in Section 2, where CRRS, DRRS, and Markov packet dropouts are analyzed and a closed-loop system is obtained. In Section 3, an auxiliary system is derived, some properties of the auxiliary system are considered, and the necessary and sufficient conditions for the exponentially mean-square stability of the NCSs are proposed. Two illustrative examples are presented in Section 4. Section 5 draws the conclusions. Notations: Throughout this paper, R stands for real numbers and N for nonnegative integers. Rn and Rnm denote, respectively, the n dimensional Euclidean space and the set of all n m real matrices. The superscript MT denotes the transposition of M. r s ðMÞ means the spectral radius of matrix M. For the vector x∈Rn , ∥x∥2 ≜xT x. PrðxðtÞÞ denotes the probability of xðtÞ. EðxðtÞÞ defines the expectation of xðtÞ. The notation X≥Y (X4Y), where X and Y are symmetric matrices, means that X−Y is positive semidefinite (positive definite).
2. NCS modeling As shown in Fig. 1, in this paper, the following discrete-time linear invariant system is considered: ( xðt þ 1Þ ¼ AxðtÞ þ BuðtÞ; ð1Þ yðtÞ ¼ CxðtÞ; t∈N; where xðtÞ∈Rn , yðtÞ ¼ ½y1 ðtÞ⋯yq ðtÞT ∈Rq , and uðtÞ ¼ ½u1 ðtÞ⋯um ðtÞT ∈Rm are the state, output
Fig. 1. Schematic diagram of NCS.
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and input vectors of the system, respectively. A∈Rnn , C ¼ ½CT1 ; …; C Tq T ∈Rqn are known real constant matrices.
B ¼ ½B1 ; …; Bm ∈Rnm ,
2.1. Communication scheduling The RRS is selected to deal with the communication constraints issue. At each time stamp, only one output of the plant can access the S/C channel to communicate with the observer, and the C/A channel works similarly. According to the transmission order, the RRS can be divided into the following CRRS and DRRS:
The CRRS transmits the packets in a fixed order as follows, i.e. y1 ð1Þ; y2 ð2Þ; …; yq ðqÞ; y1 ðq þ 1Þ; y2 ðq þ 2Þ…;
ð2Þ
u1 ð1Þ; u2 ð2Þ; …; um ðmÞ; u1 ðm þ 1Þ; u2 ðm þ 2Þ…:
ð3Þ
and
where yi ðtÞ; i∈f1; 2; …; qg means that at time t, yi can access to the S/C channel to communicate with the observer, uj ðtÞ; j∈f1; 2; …; mg works similarly. The DRRS transmits the packets based on the detectability and the controllability indices as in [25]: y1 ð1Þ; y1 ð2Þ; …; y1 ðn1 Þ; y2 ðn1 þ 1Þ; y2 ðn1 þ 2Þ; …; y2 ðn1 þ n2 Þ; …; yq ðn1 þ ⋯ þ nq−1 þ 1Þ; yq ðn1 þ ⋯ þ nq−1 þ 2Þ; …; y1 ðnÞ; y1 ðn þ 1Þ; y1 ðn þ 2Þ; …; y1 ðn þ n1 Þ; y2 ðn þ n1 þ 1Þ; y2 ðn þ n1 þ 2Þ; …
ð4Þ
u1 ð1Þ; u1 ð2Þ; …; u1 ðn 1 Þ; u2 ðn 1 þ 1Þ; u2 ðn 1 þ 2Þ; …; u2 ðn 1 þ n 2 Þ; …; uq ðn 1 þ ⋯ þ n q−1 þ 1Þ; uq ðn 1 þ ⋯ þ n q−1 þ 2Þ; …; u1 ðnÞ; u1 ðn þ 1Þ; u1 ðn þ 2Þ; …; u1 ðn þ n 1 Þ; u2 ðn þ n 1 þ 1Þ; u2 ðn þ n 1 þ 2Þ; …
ð5Þ
and
where ni and n i are the transmit times of yi and ui with respectively.
∑qi ¼ 1
ni ¼ n and ∑m i ¼ 1 n i ¼ n,
Remark 1. In the communication fields, the CRRS is always considered to overcome the communication constraints issue for its simple structure. However, when the CRRS is applied to the control fields, we have to analyze the controllability and the detectability of the dynamic system. Recently, the periodic connection methods are proposed to deal with the aforementioned issue under the assumption that the matrix A is invertible [25,26]. We define this method as DRRS, because this technique has to design the transmission order for different systems. 2.2. Unreliable communication channel For the communication channels, packet dropouts are an unavoidable phenomenon. In this paper two random variables θs ðtÞ∈f0; 1g and θa ðtÞ∈f0; 1g are employed to describe the packet transmission conditions for the S/C and the C/A channels, respectively. 1 denotes that the packet transmits successfully, while 0 means that the packet is lost.
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Assumption 1 (Seiler and Sengupta [27]). The independent transition probability matrices Ts and Ta of the stochastic variables θs ðtÞ∈f0; 1g and θa ðtÞ∈f0; 1g are defined as " # " # qs qa 1−qs 1−qa T s≜ ; T a≜ ; ð6Þ ps 1−ps pa 1−pa where 0ops o1 and 0opa o1 are the failure rates, 0oqs o1 and 0oqa o1 are named as the recovery rates. In each step, θðtÞ≜ðθs ðtÞ; θa ðtÞÞ is applied to describe the possible communication cases of the S/C and the C/A channels, where θðtÞ∈Θ≜fð0; 0Þ; ð0; 1Þ; ð1; 0Þ; ð1; 1Þg. Remark 2. For the wireless networked control systems, the main result, which leads to packet dropouts, is that the wireless channel (like the Zigbee model) is buss, then parts of the packets are unable to access the channel. And the channel buss condition of the sensor input side and the controller input side is independent. Therefore, we assume that the packet dropouts in the S/C and the C/A channels are independent. 2.3. Observer-based controller with imperfect information According to the RRS and the packet dropouts model introduced aforementioned, an observerbased controller is given as follows: 8 ^ ^ ^ ^ ^ > < xðt þ 1Þ ¼ AxðtÞ þ θa ðtÞBj uðtÞ þ θs ðtÞLðyi ðtÞ−y i ðtÞÞ; xð0Þ ¼ 0; ^ ¼ K xðtÞ; ^ uðtÞ ð7Þ > : yðtÞ ^ ¼ C xðtÞ; ^ where i∈f1; …; qg, and j∈f1; …; mg for the CRRS case and i; j∈f1; …; ng for the DRRS case, n ^ ^ , yðtÞ ∈Rq are the state and the output of the observer, respectively. K∈R1n and xðtÞ∈R n1 L∈R are the assigned controller and the observer gains, respectively. The state of the system ^ þ 1Þ ¼ AxðtÞ ^ þ θa ðtÞBj uðtÞ ^ when the packet is lost. If the packet is received, is estimated by xðt the estimate state is improved through the term Lðyi ðtÞ−y^ i ðtÞÞ. As shown in Fig. 1, in order to construct the observer in Eq. (7), the TCP protocol is introduced in the C/A channel that is the packet dropouts acknowledgment is available to the observer. 2.4. Closed-loop system Define the state of the closed-loop system as " # xðtÞ xðtÞ≜ ; eðtÞ
ð8Þ
^ where eðtÞ ¼ xðtÞ−xðtÞ. Combining the system (1) and the observer-based controller (7), the closed-loop system can be written as xðt þ 1Þ ¼ Aði; j; θs ðtÞ; θa ðtÞÞxðtÞ; where
"
A þ θa ðtÞBj K Aði; j; θs ðtÞ; θa ðtÞÞ ¼ 0
ð9Þ # −θa ðtÞBj K : A−θs ðtÞLC i
ð10Þ
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Definition 1 (Costa et al. [28]). The closed-loop system (9) is said to be exponentially meansquare stable if there exist constants 0oαo1 and β41 such that ∀ xð0Þ∈R2n ; ∀θð0Þ∈Θ, and ∀t∈N satisfy Eð∥xðtÞ∥2 jxð0Þ; θð0Þ Þoβαt ∥xð0Þ∥2 :
ð11Þ
Remark 3. The RRS successfully overcomes the communication constraints issue, but it causes two additional problems: (1) The parameters of the system are periodic time varying, which make it difficult to directly study the stability of the closed-loop system (9) in sampling times; (2) Once the RRS is introduced, the controllability and the detectability of certain MIMO system (1) cannot be preserved [25]. 3. Main results In this section, first, two auxiliary systems are established based on the lifting technique, which transforms the periodic time varying systems to time invariant systems for the CRRS and the DRRS. Then, the augmented Markov chains and the stability of the auxiliary systems are obtained. At last, the necessary and sufficient exponential mean-square stability conditions of the closed-loop systems (9) are derived by proving the stability equivalence between the closed-loop systems (9) and the auxiliary systems. 3.1. Auxiliary system Define a new time scale t k ≜kΠ; k∈N and Π is the least common multiple period of the RRS in the S/C and the C/A channels. It is obvious that Π takes different numbers for different RRS protocols. Now define the following auxiliary system: ~ s ðt k Þ; θ a ðt k ÞÞzðt k Þ; zðt kþ1 Þ ¼ Aðθ
ð12Þ
where zðt k Þ is the state of the auxiliary system. In the next part, we will construct the ~ s ðt k Þ and θ a ðt k ÞÞ for the CRRS and the DRRS cases, respectively. Aðθ 3.1.1. Auxiliary system of CRRS For the CRRS case, the Π can be expressed as Π ¼ mc m ¼ qc q;
mc ∈f1; qg;
qc ∈f1; mg:
ð13Þ
From Eq. (13), Π can be known as the minimum period including all possible closed-loop systems combined by ui ; i∈f1; …; mg, and yh ; h∈f1; …; qg in the order (2) and (3), respectively. For example, when m ¼ 2 and q¼ 3, the minimum period is 6 and all the possible closed-loop systems are PK ðy1 ; u1 Þ; PK ðy2 ; u2 Þ; PK ðy3 ; u1 Þ; PK ðy1 ; u2 Þ; PK ðy2 ; u1 Þ; PK ðy3 ; u2 Þ:
ð14Þ
Define the index j∈fjj0oj≤Π; j∈Ng, and parameters Π jm ∈N; Π qm ∈N; ojm ∈N, and oqm ∈N satisfy Π−j ¼ Π jm m þ ojm ; 0≤ojm om; j−1 ¼ Π jq q þ ojq ; 0≤ojq oq:
ð15Þ
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~ s ðt k Þ; θ a ðt k ÞÞ in Eqs. (12) and (13) is described as follows: For the CRRS case, Aðθ " # Ξ K ð1; mc Þc A~ 12 ðθ s ðt k Þ; θ a ðt k ÞÞc ~Aðθ s ðt k Þ; θ a ðt k ÞÞ ¼ ; 0 Ξ L ð1; qc Þc
ð16Þ
with A~ 12 ðθ s ðt k Þ; θ a ðt k ÞÞc Π
¼ ∑ ½Ξ K ðmc þ 1−Π jm ; mc Þc Ψ K ðm−ojm j¼1
þ1; mÞc θa ðm−ojm ÞBm−ojm KΨ L ð1; ojm ; Π jq Þc Ξ L ð1; Π jq Þc ; y
Ξ K ðx; yÞc ¼ ∏ ½ðA þ θa ðimÞBm KÞðA þ θa ðim−1ÞBm−1 KÞ⋯ðA þ θa ðði−1Þm þ 1ÞB1 KÞ; i¼x y
Ξ L ðx; yÞc ¼ ∏ ½ðA−θs ðiqÞLC q ÞðA−θs ðiq−1ÞLCq−1 Þ⋯ðA−θs ðði−1Þq þ 1ÞLC 1 Þ; i¼x
( Ψ K ðx; yÞc ¼
ðA þ θa ðyÞBy KÞðA þ θa ðy−1ÞBy−1 KÞ⋯ðA þ θa ðxÞBx KÞ; 0≤x≤y≤m; 1;
y≤x;
Ψ L ðx; y; Π jq Þc ¼ ðA−θs ðΠ jq q þ yÞLy CÞðA−θs ðΠ jq q þ y−1ÞLy−1 CÞ⋯ðA−θs ðΠ jq q þ xÞLx CÞ; 0≤x≤y≤q: In this paper,
∏li ¼ 1 JðiÞ
is a multiplication of the matrices J(i) in the inverse order, i.e.
y
y
∏ JðiÞ≜J y J y−1 ⋯ J x ;
∏ JðiÞ ¼ 1
i¼x
i¼x
ð17Þ
if yox:
3.1.2. Auxiliary system of DRRS For the DRRS case, Π ¼ n because of the transmission sequences (4) and (5). The matrix θ s ðt k Þ; θ a ðt k Þ in Eq. (12) can be expressed as " # Ξ K ð1; nÞd A~ 12 ðθ s ðt k Þ; θ a ðt k ÞÞd ~ s ðt k Þ; θ a ðt k ÞÞ ¼ Aðθ ; ð18Þ 0 Ξ L ð1; nÞd where Π
Ξ K ð1; nÞd ¼ ∏ ððA þ θa ðiÞBi KÞ; i¼1 Π
Ξ L ð1; nÞd ¼ ∏ ððA−θs ðiÞLC i Þ; i¼1
Π
A~ 12 ðθ s ðt k Þ; θ a ðt k ÞÞd ¼ ∑
j¼1
Bi ¼ Brb ;
rb −1
rb
s¼0
s¼0
m
j−1
i ¼ jþ1
i¼1
!
∏ ðA þ θa ðiÞBi KÞθa ðjÞBj K ∏ ðA−θs ðiÞLC i Þ ;
∑ ns oi≤ ∑ ns ;
0or b ≤m;
n0 ¼ 0;
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C i ¼ C rc ;
r c −1
rc
s¼0
s¼0
∑ n s oi≤ ∑ n s ;
0or c ≤q;
n 0 ¼ 0:
ð19Þ
3.2. Properties of the auxiliary system The auxiliary system (12) is obtained by lumping Π steps of the original system (1). Therefore, the packet dropouts result in κ ¼ 22Π cases of the auxiliary system. Define a set Ω≜ð1; 2; 3; …; 22Π Þ:
ð20Þ
All of the packet dropouts cases can be one-to-one mapped to the set Ω as the following lemma. Lemma 1. The packet dropouts cases ðθ s ðt k Þ; θ a ðt k ÞÞ of the auxiliary system (12) can be mapped to the sequence r k ; ðr k ∈ΩÞ by the following one-to-one mapping ΓðÞ: Π
2Π
r k ¼ Γððθ s ðt k Þ; θ a ðt k ÞÞÞ≜1 þ ∑ θa ðt k−1 þ iÞ2i−1 þ ∑ θs ðt k−1 þ i−ΠÞ2i−1 ; i¼1 i ¼ Πþ1 where
ð21Þ
θ s ðt k Þ ¼ ðθs ðt k−1 þ 1Þ; θs ðt k−1 þ 2Þ; …; θs ðt k−1 þ ΠÞÞ; θ a ðt k Þ ¼ ðθa ðt k−1 þ 1Þ; θa ðt k−1 þ 2Þ; …; θa ðt k−1 þ ΠÞÞ: At the same time, if rk is given, the packet dropouts condition ðθ~ s ðr k Þ; θ~ a ðr k ÞÞ can be derived as follows:
Step1: Step2: Step3: Step4:
set i ¼ 2Π and ρ ¼ r k −1; θ~ rk ðiÞ ¼ ff0; 1gjρ ¼ 2n þ θ~ rk ðiÞ; n∈Ng; if i40, then ρ ¼ n and i ¼ i þ 1, return to step 2; we can get ðθ~ s ðr k Þ; θ~ a ðr k ÞÞ.
Then the augmented transition probability matrix of rk can be derived as follows. Lemma 2. The stochastic process rk, which takes values in the set Ω ¼ ð1; …22Π Þ, is still a Markov chain, and the transition probability matrix Δ≜½δi;j ; i; j∈Ω can be derived as follows: δi;j ¼ Prðr kþ1 ¼ jjr k ¼ iÞ ¼ Prðθs ðΠÞ ¼ θ~ j ð2ΠÞjθs ðΠ−1Þ ¼ θ~ j ð2Π−1ÞÞ Prðθs ðΠ−1Þ ¼ θ~ j ð2Π−1Þjθs ðΠ−2Þ ¼ θ~ j ð2Π−2ÞÞ ⋯ Prðθs ð1Þ ¼ θ~ j ðΠ þ 1Þjθs ðΠÞ ¼ θ~ i ð2ΠÞÞ Prðθa ðΠÞ ¼ θ~ j ðΠÞjθa ðΠ−1Þ ¼ θ~ j ðΠ−1ÞÞ ⋯ Prðθa ð1Þ ¼ θ~ j ð1Þjθa ðΠÞ ¼ θ~ i ðΠÞÞ
ð22Þ
Proof. δi;j can be expressed as δi;j ¼ Prðr kþ1 ¼ jjr k ¼ iÞ
ð23Þ
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According to Lemma 1, we can obtain that δi;j ¼ Prððθ s ðr kþ1 Þ; θ a ðr kþ1 ÞÞ ¼ ðθ~ s ðjÞ; θ~ a ðjÞÞjðθ s ðr k Þ; θ a ðr k ÞÞ ¼ ðθ~ s ðiÞ; θ~ a ðiÞÞÞ Because θs ðtÞ and θa ðtÞ are independent, Eq. (24) is equivalent to Eq. (22).
ð24Þ
□
According to the above lemmas and the results in the literature [28], the exponential meansquare stability of the auxiliary system (12) can be summarized in Lemma 3. Lemma 3 (Costa et al. [28]). Under Assumption 1, the following assertions are equivalent: (1) The auxiliary system (12) is exponentially mean-square stable. ~ ~ (2) r s ðAÞo1, where A ¼ ðΔT ⊗I ð2nÞ2 Þdiag½AðiÞ⊗ AðiÞ. (3) There exists a set of solutions fPðrÞ40; r∈Ωg such that Π
T ~ A~ ðrÞ ∑ δrι PðιÞAðrÞ−PðrÞo0; ι¼1
∀r∈Ω:
ð25Þ
3.3. Stability of the networked control systems Based on the properties of the auxiliary system (12), the stability conditions of the closed-loop system (9) are expressed as follows. Theorem 1. Under Assumption 1, the closed-loop system (9) is exponentially mean-square stable if and only if the auxiliary system (12) is exponentially mean-square stable. Proof. Sufficiency: Given that the auxiliary system (12) is exponentially mean-square stable, there exist constants 0oα1 o1 and β1 41, which are independent of zð0Þ satisfying Eð∥zðkÞ∥2 jzð0Þ; r0 Þoβ1 αk1 ∥zð0Þ∥2 :
ð26Þ
Since Π is finite, a constant ε41 can be found such that for Δk ∈ð1; 2; …; ΠÞ, Eð∥xðkΠ þ Δk Þ∥2 jxð0Þ; θ0 ÞoεEð∥xðΠkÞ∥2 jxð0Þ; θ0 Þ:
ð27Þ
From Eq. (27), for all t∈ðkΠ þ 1; kΠ þ 2; …; ðk þ 1ÞΠÞ, Eð∥xðtÞ∥2 jxð0Þ; θ0 ÞoεEð∥zðkÞ∥2 jzð0Þ r0 Þoεβ1 αk1 ∥zð0Þ∥2 :
ð28Þ
A constant β^ 1 41 exists because Π is finite, which leads to ∥zð0Þ∥ oβ^1 ∥xð0Þ∥ . And due to toðk þ 1ÞΠ, we have β β^ 1=Π ð29Þ Eð∥xðtÞ∥2 jxð0Þ; θ0 Þoε 1 1 ðα1 Þt ∥xð0Þ∥2 : α1 2
2
The sufficient condition is proven. Necessity: If the closed-loop system (9) is exponentially mean-square stable, then there also exist constants 0oα2 o1 and β2 41, which are independent of xð0Þ, satisfying Eð∥xðtÞ∥2 jxð0Þ; θ0 Þoβ2 αt2 ∥xð0Þ∥2 :
ð30Þ
Hence the following relationship is true: 2 k 2 Eð∥zðkÞ∥2 jzð0Þ; r0 Þ ¼ Eð∥xðkΠÞ∥2 jxð0Þ; θ0 Þoβ2 αkΠ 2 ∥xð0Þ∥ ≤β 2 α2 ∥zð0Þ∥ :
ð31Þ
2022
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The necessary condition is proven. And according to Lemma 3, the proof of Theorem 1 is completed. □ 4. Simulations In this section, two examples are employed to illustrate the effectiveness of the proposed method. Example 1. (CRRS) Consider the following unstable system with parameter perturbation over CRRS channels: 1 0 1 0 ! 0 1 0 0 1 c1 1 1 0 C B 0 C B 0 1 A; B ¼ ðb1 b2 Þ ¼ @ 1 0 A; C ¼ ; ¼ A¼@ c2 1 0 1 0:5 þ a 1 þ b −0:4 1 1 where the two uncertain parameters satisfy a40 and b40. In each period, the transmission sequence of the plant and the controller outputs are ðy1 ; y2 Þ and ðu1 ; u2 Þ, respectively. We consider two cases of the communication channels, they are ideal channel and packet dropouts channels modeled by Markov chain. For the packet dropouts modeled by Markov chains in the S/C and the C/A channels, the transition probability matrices are 0:2 0:8 0:15 0:85 Ts ¼ ; Ta ¼ : 0:1 0:9 0:2 0:8 Based on Lemma 3(2), we choose the controller and observer gains as follows: K ¼ ð−1:1 −0:51 1:15Þ;
L ¼ ð0:68 0:9 0:81Þ;
respectively, for the closed-loop system (9) with CRRS. Since Theorem 1 is the necessary and sufficient condition, when the feedback and the observer gains are provided, we are able to distinguish the stable and unstable regions in the uncertain plant parameter space for the ideal channel as well as the Markov modeled packet dropouts channels, respectively. The stable region in the plant parameter space for the ideal channel is shown as the area consisting of the curve, x-axis and y-axis in Fig. 2. For the unreliable channel modeled by Markov chains, the exponentially mean-square stable region in the plant parameter space is depicted as the area composed of the dashed curve, x-axis and y-axis in Fig. 2. When we choose a¼ 0.225 and b¼ 0, the trajectory of the system under the packets dropouts in Fig. 3 is shown in Fig. 4, where the system is stable. Example 2. (DRRS) Consider the following unstable system with parameter perturbation over DRRS channels: ! ! c1 0 2:52 þ a 1:2 0 0:5 0 A¼ ; B ¼ ðb1 b2 Þ ¼ ; C¼ ¼ ; c2 0:4 þ b 0 0 0:4 0 1 where the two uncertain parameters satisfy a40 and b40. It is necessary to point out that the controllability and detectability of the system cannot be preserved, if we still select the CRRS. Therefore, we consider the DRRS, which transmits the packets by taking into account the index of the controllability and the detectability. We choose the transmission sequence as ðy1 ; y1 Þ and ðu2 ; u2 Þ in a period. Here we also analyze the two similar communication channel cases as
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0.35 Markov case Ideal case
0.3 0.25
b
0.2 0.15 0.1 0.05 0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
a Fig. 2. Stable regions of the closed-loop system under CRRS.
θa θs
1 0.9 the states of θa and θs
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
10
20
30 k
40
50
60
Fig. 3. The packet dropouts of channels under CRRS.
expressed in Example 1. The transition probability matrices for the Markov chains are 0:1 0:9 0:1 0:9 ; Ta ¼ : Ts ¼ 0:1 0:9 0:05 0:95 According to Lemma 3(2), select the following controller and the observer gains: K ¼ ð−0:2 −1:9Þ;
L ¼ ð1:3 0:1Þ;
respectively, for the closed-loop system (9) with DRRS. According to the necessary and sufficient condition in Theorem 1, we can depict the stable and unstable regions in the uncertain plant parameter space for the ideal channel and the unreliable channel in Fig. 5, when the controller and the observer gains are given. The stable region in the plant parameter space for the ideal channel is the area made up of the curve, x-axis and y-axis in Fig. 5. For the Markov
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1
x1 x2 x3
the trajectory of the system
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
10
20
30
40
50
k Fig. 4. The trajectory of the system under CRRS.
1.4 Ideal case Markov case
1.2 1
b
0.8 0.6 0.4 0.2 0
0
0.02
0.04
0.06
0.08 a
0.1
0.12
0.14
0.16
Fig. 5. Stable regions of the closed-loop system under DRRS.
chain modeled unreliable channel, the exponentially mean-square stable region in the plant parameter space is the area composed of the dashed-dotted curve, x-axis and y-axis in Fig. 5. If we choose a¼ 0.04 and b¼ 0, then the trajectory of the system under the packets dropouts in Fig. 6 is in Fig. 7, where the system is stable. Based on the above examples, we can conclude that the stability of the closed-loop system (9) depends on the model uncertainty and the reliability of the communication channels. In addition, according to analyzing the stable region of the closed-loop system (9), we can determine the request of the accuracy of the model and the quality of the communication channels before constructing the NCSs. 5. Conclusions The stability analysis of a discrete-time networked control system with communication constrains is studied. CRRS and DRRS are presented to deal with the communication constrains
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θa θs
1 0.9 the states of θa and θs
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
10
20
30 k
40
50
60
Fig. 6. The packet dropouts of channels under DRRS.
1.5
x1 x2
the trajectory of the system
1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −3
0
10
20
30
40
50
k Fig. 7. The trajectory of the system under DRRS.
issue at both channels. Two homogeneous Markov chains are introduced to describe the packet dropouts in the S/C and the C/A channels. Based on the lifting technique, an auxiliary system is proposed with the augmented Markov chain and certain properties of the auxiliary system are analyzed. A necessary and sufficient condition for the exponential mean-square stability of the closed-loop system (9) is derived based on the stability equivalence between the closed-loop system (9) and the auxiliary system (12). Two illustrative examples are shown to demonstrate the effectiveness of the proposed stability analysis method. There are still some interesting problems needed to be studied in the future work. On one hand, how to extend our method to the system with stochastic switching dynamics with
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perturbations [29–31]. On the other hand, it is worthwhile to investigate about the NCSs under RRS condition with the quantization, transmission induced delay and sensor nonlinearity [32,33].
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[24] D. Wu, J. Wu, S. Chen, Robust H ∞ control for networked control systems with uncertainties and multiple-packet transmission, IET Control Theory and Applications 4 (5) (2010) 701–709. [25] L. Zhang, D. Hristu-Varsakelis, Communication and control co-design for networked control systems, Automatica 42 (6) (2006) 953–958. [26] D. Hristu-Varsakelis, L. Zhang, LQG control of networked control systems with access constraints and delays, International Journal of Control 81 (8) (2008) 1266–1280. [27] P. Seiler, R. Sengupta, An H ∞ approach to networked control, IEEE Transactions on Automatic Control 50 (3) (2005) 356–364. [28] O. Costa, M. Fragoso, R. Marques, Discrete-time Markovian Jump Linear Systems, Springer-Verlag, London, 2005. [29] M. Liu, D. Ho, Y. Niu, Stabilization of Markovian jump linear system over networks with random communication delay, Automatica 45 (2) (2009) 416–421. [30] M. Liu, P. Shi, L. Zhang, X. Zhao, Fault tolerant control for nonlinear Markovian jump systems via proportional and derivative sliding mode observer technique, IEEE Transactions on Circuits and Systems—Part I 58 (11) (2011) 2755–2764. [31] D. Zhang, L. Yu, H ∞ filtering for linear neutral systems with mixed time-varying delays and nonlinear perturbations, Journal of the Franklin Institute 347 (7) (2010) 1374–1390. [32] M. Liu, D. Ho, Y. Niu, Robust filtering design for stochastic system with mode-dependent output quantization, IEEE Transactions on Signal Processing 58 (12) (2010) 6410–6416. [33] Y. Xu, H. Su, Y. Pan, Z. Wu, Robust H ∞ filtering for networked stochastic systems with randomly occurring sensor nonlinearities and packet dropouts, Signal Processing 93 (7) (2013) 1794–1803.
Journal of the Franklin Institute 350 (2013) 2013–2027 www.elsevier.com/locate/jfranklin
Stability analysis of networked control systems with round-robin scheduling and packet dropouts$ Yong Xua, Hongye Sua,n, Ya-Jun Panb, Zheng-Guang Wua, Weihua Xua a
State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Yuquan Campus, Hangzhou, Zhejiang 310027, PR China b Department of Mechanical Engineering, Dalhousie University, P.O. Box 15000, Halifax, Nova Scotia, Canada B3H 4R2 Received 25 November 2012; received in revised form 17 March 2013; accepted 20 May 2013 Available online 1 June 2013
Abstract In this paper, the stability of networked control systems (NCSs) with communication constraints at both channels is investigated. A Conventional Round-Robin Scheduling (CRRS) is applied to deal with the communication constraints issue for its simple structure. Furthermore, a Dynamic Round-Robin Scheduling (DRRS), which can preserve the controllability and the detectability of the systems, is considered. For the unreliable communication channels, two independent homogeneous Markov chains are selected to model the packet dropouts phenomenon in the sensor-to-controller (S/C) channel and the controller-to-actuator (C/A) channel. According to the periodic property of the Round-Robin Scheduling (RRS), an auxiliary system with augmented Markov chain is established by the lifting technique to facilitate the stability analysis of the closed-loop system. A necessary and sufficient condition of the exponential mean-square stability for the NCSs is derived. Two illustrative examples are shown to demonstrate the effectiveness of the proposed stability analysis method. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Networked control systems, which connect the systems and controllers via a shared channel, have received numerous research attention on the system modeling, stability analysis, controller ☆ This work is supported by National Natural Science Foundation of PR China (NSFC: 61134007), National Natural Science Foundation of PR China (NSFC: 61104102), National Natural Science Foundation of PR China (NSFC: 61174029). n Corresponding author. Tel.: +86 0571 87951075. E-mail addresses: [email protected] (Y. Xu), [email protected] (H. Su), [email protected] (Y.-J. Pan).
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.05.024
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or estimator design in the past decade [1]. NCSs are provided with more advantages, including shortening the installation period, reducing wire cost and simplifying system maintenance, than the classical wired point-to-point feedback control systems [2]. NCSs have been applied in many fields of unmanned aerial vehicles [3], vehicular networks and traffic control [4], and teleoperation systems [5] due to its aforementioned merits. However, the communication channel brings a lot of challenges to the users such as quantization errors [6], packet dropouts and communication delays [7–10] and random sampling [11,12], which may degrade the system performance and even lead the system to instability. Due to the channel capacity constraints, not all of the system outputs and controller outputs of the Multiple-Input-Multiple-Output (MIMO) system can be transmitted through the channels at the same time. Therefore, some scheduling methods have been proposed in the past years such as TryOnce-Discard (TOD) protocol and Round-Robin (RR) protocol. Both protocols allow only one output to be transmitted via the channel at a time. The RR protocol is a static one, which transmits the packets in a fixed order. The TOD protocol is a dynamic protocol, which chooses the greatest weighted error node to be transmitted. In [13] the TOD protocol was introduced for the first time to deal with the packets scheduling and the maximum allowable transfer interval was then obtained. Based on the small gain theorem, in [14] an input–output LP stability condition was derived, which improves the maximum allowable transfer interval and the result was suitable for both the TOD and the RR protocols. In [15], the error of the output brought by the transmission delay was modeled as system uncertainty and the switched linear system approach was used to analyze the stability condition both for the TOD and the RR protocols. In [16] the RR protocol was applied to the sampled data control systems and the direct Lyapunov Krasovskii approach was selected to analyze the stability condition. However, the works mentioned above just take into account the system in the continuous time domain. Due to the nature of the NCSs, discrete time systems have been widely addressed. In [17], the necessary and sufficient stability condition was obtained for random time delay systems with output feedback controller. In [18], the sector bounded approach was introduced to deal with the quantized controller signal and the stability condition of the system was achieved by the quantizer density and the unstable eigenvalue of the systems. In [19], the synchronization stability of the chaotic Lur'e systems with random sampling was analyzed. Therefore, this paper wants to extend the RRS method to the discrete time system. Packet dropouts, caused by node failure or data collisions, are mainly modeled as Markov jumping systems due to two factors. On one hand, Markov jumping systems have been studied by many researches in the past two decades and some technical problems have been solved, which facilitate our analysis processes [20]. On the other hand, compared with Bernoulli process, Markov jumping systems are able to capture the possible temporal correlation of the channel variation. According to analyzing the random Riccati recursion and introducing the notion of peak covariance, the sufficient bounded condition of the estimation error covariance for the general systems with Markov based packet dropouts was obtained, which was also a necessary condition for the scalar case [21]. In [22], by exploiting the system structure, the necessary and sufficient condition for the stability of the mean estimation error covariance matrices of secondorder systems was derived, and the result could be extended to certain classes of higher-order systems. In [23], the separation principle was obtained for the NCSs with Markov-modeled packet dropouts which appeared simultaneously in the multiple S/C and multiple C/A channels. The H ∞ performance of the NCSs with Markov-based packet dropouts, which appears in the multiple S/C and the multiple C/A channels, was studied and the sufficient condition was proposed [24]. In the proposed paper, we mainly consider NCSs with independent packet dropouts in both communication channels and model it by Markov chains.
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2015
Motivated by the above discussion, this paper studies the stability analysis of the discrete time NCSs with RRS and packet dropouts at both channels. Two scheduling protocols (CRRS and DRRS) are considered in the paper to deal with the communication constraint problem. For the imperfect channels, two independent homogenous Markov chains are introduced to describe the random packet dropouts. Then, a necessary and sufficient condition for the exponential mean-square stability of the closed-loop systems is derived. Finally, two illustrative examples are shown to demonstrate the effectiveness of the proposed stability analysis approach. The main contributions of the paper can be summarized as follows: (1) in order to establish a more general model, we consider the RRS and the packet dropouts together in this paper, rather than just address one problem in some existing papers [13,16,21]; (2) in order to overcome the periodic parameter varying caused by RRS, an auxiliary system is established via the lifting technique to facilitate the analysis. And an augmented Markov chain is obtained to describe the packet dropouts of the auxiliary system; (3) the equivalence of the exponential mean-square stability between the original system and the auxiliary system is established. This paper is organized as follows. The model of the NCSs is described in Section 2, where CRRS, DRRS, and Markov packet dropouts are analyzed and a closed-loop system is obtained. In Section 3, an auxiliary system is derived, some properties of the auxiliary system are considered, and the necessary and sufficient conditions for the exponentially mean-square stability of the NCSs are proposed. Two illustrative examples are presented in Section 4. Section 5 draws the conclusions. Notations: Throughout this paper, R stands for real numbers and N for nonnegative integers. Rn and Rnm denote, respectively, the n dimensional Euclidean space and the set of all n m real matrices. The superscript MT denotes the transposition of M. r s ðMÞ means the spectral radius of matrix M. For the vector x∈Rn , ∥x∥2 ≜xT x. PrðxðtÞÞ denotes the probability of xðtÞ. EðxðtÞÞ defines the expectation of xðtÞ. The notation X≥Y (X4Y), where X and Y are symmetric matrices, means that X−Y is positive semidefinite (positive definite).
2. NCS modeling As shown in Fig. 1, in this paper, the following discrete-time linear invariant system is considered: ( xðt þ 1Þ ¼ AxðtÞ þ BuðtÞ; ð1Þ yðtÞ ¼ CxðtÞ; t∈N; where xðtÞ∈Rn , yðtÞ ¼ ½y1 ðtÞ⋯yq ðtÞT ∈Rq , and uðtÞ ¼ ½u1 ðtÞ⋯um ðtÞT ∈Rm are the state, output
Fig. 1. Schematic diagram of NCS.
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and input vectors of the system, respectively. A∈Rnn , C ¼ ½CT1 ; …; C Tq T ∈Rqn are known real constant matrices.
B ¼ ½B1 ; …; Bm ∈Rnm ,
2.1. Communication scheduling The RRS is selected to deal with the communication constraints issue. At each time stamp, only one output of the plant can access the S/C channel to communicate with the observer, and the C/A channel works similarly. According to the transmission order, the RRS can be divided into the following CRRS and DRRS:
The CRRS transmits the packets in a fixed order as follows, i.e. y1 ð1Þ; y2 ð2Þ; …; yq ðqÞ; y1 ðq þ 1Þ; y2 ðq þ 2Þ…;
ð2Þ
u1 ð1Þ; u2 ð2Þ; …; um ðmÞ; u1 ðm þ 1Þ; u2 ðm þ 2Þ…:
ð3Þ
and
where yi ðtÞ; i∈f1; 2; …; qg means that at time t, yi can access to the S/C channel to communicate with the observer, uj ðtÞ; j∈f1; 2; …; mg works similarly. The DRRS transmits the packets based on the detectability and the controllability indices as in [25]: y1 ð1Þ; y1 ð2Þ; …; y1 ðn1 Þ; y2 ðn1 þ 1Þ; y2 ðn1 þ 2Þ; …; y2 ðn1 þ n2 Þ; …; yq ðn1 þ ⋯ þ nq−1 þ 1Þ; yq ðn1 þ ⋯ þ nq−1 þ 2Þ; …; y1 ðnÞ; y1 ðn þ 1Þ; y1 ðn þ 2Þ; …; y1 ðn þ n1 Þ; y2 ðn þ n1 þ 1Þ; y2 ðn þ n1 þ 2Þ; …
ð4Þ
u1 ð1Þ; u1 ð2Þ; …; u1 ðn 1 Þ; u2 ðn 1 þ 1Þ; u2 ðn 1 þ 2Þ; …; u2 ðn 1 þ n 2 Þ; …; uq ðn 1 þ ⋯ þ n q−1 þ 1Þ; uq ðn 1 þ ⋯ þ n q−1 þ 2Þ; …; u1 ðnÞ; u1 ðn þ 1Þ; u1 ðn þ 2Þ; …; u1 ðn þ n 1 Þ; u2 ðn þ n 1 þ 1Þ; u2 ðn þ n 1 þ 2Þ; …
ð5Þ
and
where ni and n i are the transmit times of yi and ui with respectively.
∑qi ¼ 1
ni ¼ n and ∑m i ¼ 1 n i ¼ n,
Remark 1. In the communication fields, the CRRS is always considered to overcome the communication constraints issue for its simple structure. However, when the CRRS is applied to the control fields, we have to analyze the controllability and the detectability of the dynamic system. Recently, the periodic connection methods are proposed to deal with the aforementioned issue under the assumption that the matrix A is invertible [25,26]. We define this method as DRRS, because this technique has to design the transmission order for different systems. 2.2. Unreliable communication channel For the communication channels, packet dropouts are an unavoidable phenomenon. In this paper two random variables θs ðtÞ∈f0; 1g and θa ðtÞ∈f0; 1g are employed to describe the packet transmission conditions for the S/C and the C/A channels, respectively. 1 denotes that the packet transmits successfully, while 0 means that the packet is lost.
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Assumption 1 (Seiler and Sengupta [27]). The independent transition probability matrices Ts and Ta of the stochastic variables θs ðtÞ∈f0; 1g and θa ðtÞ∈f0; 1g are defined as " # " # qs qa 1−qs 1−qa T s≜ ; T a≜ ; ð6Þ ps 1−ps pa 1−pa where 0ops o1 and 0opa o1 are the failure rates, 0oqs o1 and 0oqa o1 are named as the recovery rates. In each step, θðtÞ≜ðθs ðtÞ; θa ðtÞÞ is applied to describe the possible communication cases of the S/C and the C/A channels, where θðtÞ∈Θ≜fð0; 0Þ; ð0; 1Þ; ð1; 0Þ; ð1; 1Þg. Remark 2. For the wireless networked control systems, the main result, which leads to packet dropouts, is that the wireless channel (like the Zigbee model) is buss, then parts of the packets are unable to access the channel. And the channel buss condition of the sensor input side and the controller input side is independent. Therefore, we assume that the packet dropouts in the S/C and the C/A channels are independent. 2.3. Observer-based controller with imperfect information According to the RRS and the packet dropouts model introduced aforementioned, an observerbased controller is given as follows: 8 ^ ^ ^ ^ ^ > < xðt þ 1Þ ¼ AxðtÞ þ θa ðtÞBj uðtÞ þ θs ðtÞLðyi ðtÞ−y i ðtÞÞ; xð0Þ ¼ 0; ^ ¼ K xðtÞ; ^ uðtÞ ð7Þ > : yðtÞ ^ ¼ C xðtÞ; ^ where i∈f1; …; qg, and j∈f1; …; mg for the CRRS case and i; j∈f1; …; ng for the DRRS case, n ^ ^ , yðtÞ ∈Rq are the state and the output of the observer, respectively. K∈R1n and xðtÞ∈R n1 L∈R are the assigned controller and the observer gains, respectively. The state of the system ^ þ 1Þ ¼ AxðtÞ ^ þ θa ðtÞBj uðtÞ ^ when the packet is lost. If the packet is received, is estimated by xðt the estimate state is improved through the term Lðyi ðtÞ−y^ i ðtÞÞ. As shown in Fig. 1, in order to construct the observer in Eq. (7), the TCP protocol is introduced in the C/A channel that is the packet dropouts acknowledgment is available to the observer. 2.4. Closed-loop system Define the state of the closed-loop system as " # xðtÞ xðtÞ≜ ; eðtÞ
ð8Þ
^ where eðtÞ ¼ xðtÞ−xðtÞ. Combining the system (1) and the observer-based controller (7), the closed-loop system can be written as xðt þ 1Þ ¼ Aði; j; θs ðtÞ; θa ðtÞÞxðtÞ; where
"
A þ θa ðtÞBj K Aði; j; θs ðtÞ; θa ðtÞÞ ¼ 0
ð9Þ # −θa ðtÞBj K : A−θs ðtÞLC i
ð10Þ
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Definition 1 (Costa et al. [28]). The closed-loop system (9) is said to be exponentially meansquare stable if there exist constants 0oαo1 and β41 such that ∀ xð0Þ∈R2n ; ∀θð0Þ∈Θ, and ∀t∈N satisfy Eð∥xðtÞ∥2 jxð0Þ; θð0Þ Þoβαt ∥xð0Þ∥2 :
ð11Þ
Remark 3. The RRS successfully overcomes the communication constraints issue, but it causes two additional problems: (1) The parameters of the system are periodic time varying, which make it difficult to directly study the stability of the closed-loop system (9) in sampling times; (2) Once the RRS is introduced, the controllability and the detectability of certain MIMO system (1) cannot be preserved [25]. 3. Main results In this section, first, two auxiliary systems are established based on the lifting technique, which transforms the periodic time varying systems to time invariant systems for the CRRS and the DRRS. Then, the augmented Markov chains and the stability of the auxiliary systems are obtained. At last, the necessary and sufficient exponential mean-square stability conditions of the closed-loop systems (9) are derived by proving the stability equivalence between the closed-loop systems (9) and the auxiliary systems. 3.1. Auxiliary system Define a new time scale t k ≜kΠ; k∈N and Π is the least common multiple period of the RRS in the S/C and the C/A channels. It is obvious that Π takes different numbers for different RRS protocols. Now define the following auxiliary system: ~ s ðt k Þ; θ a ðt k ÞÞzðt k Þ; zðt kþ1 Þ ¼ Aðθ
ð12Þ
where zðt k Þ is the state of the auxiliary system. In the next part, we will construct the ~ s ðt k Þ and θ a ðt k ÞÞ for the CRRS and the DRRS cases, respectively. Aðθ 3.1.1. Auxiliary system of CRRS For the CRRS case, the Π can be expressed as Π ¼ mc m ¼ qc q;
mc ∈f1; qg;
qc ∈f1; mg:
ð13Þ
From Eq. (13), Π can be known as the minimum period including all possible closed-loop systems combined by ui ; i∈f1; …; mg, and yh ; h∈f1; …; qg in the order (2) and (3), respectively. For example, when m ¼ 2 and q¼ 3, the minimum period is 6 and all the possible closed-loop systems are PK ðy1 ; u1 Þ; PK ðy2 ; u2 Þ; PK ðy3 ; u1 Þ; PK ðy1 ; u2 Þ; PK ðy2 ; u1 Þ; PK ðy3 ; u2 Þ:
ð14Þ
Define the index j∈fjj0oj≤Π; j∈Ng, and parameters Π jm ∈N; Π qm ∈N; ojm ∈N, and oqm ∈N satisfy Π−j ¼ Π jm m þ ojm ; 0≤ojm om; j−1 ¼ Π jq q þ ojq ; 0≤ojq oq:
ð15Þ
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~ s ðt k Þ; θ a ðt k ÞÞ in Eqs. (12) and (13) is described as follows: For the CRRS case, Aðθ " # Ξ K ð1; mc Þc A~ 12 ðθ s ðt k Þ; θ a ðt k ÞÞc ~Aðθ s ðt k Þ; θ a ðt k ÞÞ ¼ ; 0 Ξ L ð1; qc Þc
ð16Þ
with A~ 12 ðθ s ðt k Þ; θ a ðt k ÞÞc Π
¼ ∑ ½Ξ K ðmc þ 1−Π jm ; mc Þc Ψ K ðm−ojm j¼1
þ1; mÞc θa ðm−ojm ÞBm−ojm KΨ L ð1; ojm ; Π jq Þc Ξ L ð1; Π jq Þc ; y
Ξ K ðx; yÞc ¼ ∏ ½ðA þ θa ðimÞBm KÞðA þ θa ðim−1ÞBm−1 KÞ⋯ðA þ θa ðði−1Þm þ 1ÞB1 KÞ; i¼x y
Ξ L ðx; yÞc ¼ ∏ ½ðA−θs ðiqÞLC q ÞðA−θs ðiq−1ÞLCq−1 Þ⋯ðA−θs ðði−1Þq þ 1ÞLC 1 Þ; i¼x
( Ψ K ðx; yÞc ¼
ðA þ θa ðyÞBy KÞðA þ θa ðy−1ÞBy−1 KÞ⋯ðA þ θa ðxÞBx KÞ; 0≤x≤y≤m; 1;
y≤x;
Ψ L ðx; y; Π jq Þc ¼ ðA−θs ðΠ jq q þ yÞLy CÞðA−θs ðΠ jq q þ y−1ÞLy−1 CÞ⋯ðA−θs ðΠ jq q þ xÞLx CÞ; 0≤x≤y≤q: In this paper,
∏li ¼ 1 JðiÞ
is a multiplication of the matrices J(i) in the inverse order, i.e.
y
y
∏ JðiÞ≜J y J y−1 ⋯ J x ;
∏ JðiÞ ¼ 1
i¼x
i¼x
ð17Þ
if yox:
3.1.2. Auxiliary system of DRRS For the DRRS case, Π ¼ n because of the transmission sequences (4) and (5). The matrix θ s ðt k Þ; θ a ðt k Þ in Eq. (12) can be expressed as " # Ξ K ð1; nÞd A~ 12 ðθ s ðt k Þ; θ a ðt k ÞÞd ~ s ðt k Þ; θ a ðt k ÞÞ ¼ Aðθ ; ð18Þ 0 Ξ L ð1; nÞd where Π
Ξ K ð1; nÞd ¼ ∏ ððA þ θa ðiÞBi KÞ; i¼1 Π
Ξ L ð1; nÞd ¼ ∏ ððA−θs ðiÞLC i Þ; i¼1
Π
A~ 12 ðθ s ðt k Þ; θ a ðt k ÞÞd ¼ ∑
j¼1
Bi ¼ Brb ;
rb −1
rb
s¼0
s¼0
m
j−1
i ¼ jþ1
i¼1
!
∏ ðA þ θa ðiÞBi KÞθa ðjÞBj K ∏ ðA−θs ðiÞLC i Þ ;
∑ ns oi≤ ∑ ns ;
0or b ≤m;
n0 ¼ 0;
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Y. Xu et al. / Journal of the Franklin Institute 350 (2013) 2013–2027
C i ¼ C rc ;
r c −1
rc
s¼0
s¼0
∑ n s oi≤ ∑ n s ;
0or c ≤q;
n 0 ¼ 0:
ð19Þ
3.2. Properties of the auxiliary system The auxiliary system (12) is obtained by lumping Π steps of the original system (1). Therefore, the packet dropouts result in κ ¼ 22Π cases of the auxiliary system. Define a set Ω≜ð1; 2; 3; …; 22Π Þ:
ð20Þ
All of the packet dropouts cases can be one-to-one mapped to the set Ω as the following lemma. Lemma 1. The packet dropouts cases ðθ s ðt k Þ; θ a ðt k ÞÞ of the auxiliary system (12) can be mapped to the sequence r k ; ðr k ∈ΩÞ by the following one-to-one mapping ΓðÞ: Π
2Π
r k ¼ Γððθ s ðt k Þ; θ a ðt k ÞÞÞ≜1 þ ∑ θa ðt k−1 þ iÞ2i−1 þ ∑ θs ðt k−1 þ i−ΠÞ2i−1 ; i¼1 i ¼ Πþ1 where
ð21Þ
θ s ðt k Þ ¼ ðθs ðt k−1 þ 1Þ; θs ðt k−1 þ 2Þ; …; θs ðt k−1 þ ΠÞÞ; θ a ðt k Þ ¼ ðθa ðt k−1 þ 1Þ; θa ðt k−1 þ 2Þ; …; θa ðt k−1 þ ΠÞÞ: At the same time, if rk is given, the packet dropouts condition ðθ~ s ðr k Þ; θ~ a ðr k ÞÞ can be derived as follows:
Step1: Step2: Step3: Step4:
set i ¼ 2Π and ρ ¼ r k −1; θ~ rk ðiÞ ¼ ff0; 1gjρ ¼ 2n þ θ~ rk ðiÞ; n∈Ng; if i40, then ρ ¼ n and i ¼ i þ 1, return to step 2; we can get ðθ~ s ðr k Þ; θ~ a ðr k ÞÞ.
Then the augmented transition probability matrix of rk can be derived as follows. Lemma 2. The stochastic process rk, which takes values in the set Ω ¼ ð1; …22Π Þ, is still a Markov chain, and the transition probability matrix Δ≜½δi;j ; i; j∈Ω can be derived as follows: δi;j ¼ Prðr kþ1 ¼ jjr k ¼ iÞ ¼ Prðθs ðΠÞ ¼ θ~ j ð2ΠÞjθs ðΠ−1Þ ¼ θ~ j ð2Π−1ÞÞ Prðθs ðΠ−1Þ ¼ θ~ j ð2Π−1Þjθs ðΠ−2Þ ¼ θ~ j ð2Π−2ÞÞ ⋯ Prðθs ð1Þ ¼ θ~ j ðΠ þ 1Þjθs ðΠÞ ¼ θ~ i ð2ΠÞÞ Prðθa ðΠÞ ¼ θ~ j ðΠÞjθa ðΠ−1Þ ¼ θ~ j ðΠ−1ÞÞ ⋯ Prðθa ð1Þ ¼ θ~ j ð1Þjθa ðΠÞ ¼ θ~ i ðΠÞÞ
ð22Þ
Proof. δi;j can be expressed as δi;j ¼ Prðr kþ1 ¼ jjr k ¼ iÞ
ð23Þ
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According to Lemma 1, we can obtain that δi;j ¼ Prððθ s ðr kþ1 Þ; θ a ðr kþ1 ÞÞ ¼ ðθ~ s ðjÞ; θ~ a ðjÞÞjðθ s ðr k Þ; θ a ðr k ÞÞ ¼ ðθ~ s ðiÞ; θ~ a ðiÞÞÞ Because θs ðtÞ and θa ðtÞ are independent, Eq. (24) is equivalent to Eq. (22).
ð24Þ
□
According to the above lemmas and the results in the literature [28], the exponential meansquare stability of the auxiliary system (12) can be summarized in Lemma 3. Lemma 3 (Costa et al. [28]). Under Assumption 1, the following assertions are equivalent: (1) The auxiliary system (12) is exponentially mean-square stable. ~ ~ (2) r s ðAÞo1, where A ¼ ðΔT ⊗I ð2nÞ2 Þdiag½AðiÞ⊗ AðiÞ. (3) There exists a set of solutions fPðrÞ40; r∈Ωg such that Π
T ~ A~ ðrÞ ∑ δrι PðιÞAðrÞ−PðrÞo0; ι¼1
∀r∈Ω:
ð25Þ
3.3. Stability of the networked control systems Based on the properties of the auxiliary system (12), the stability conditions of the closed-loop system (9) are expressed as follows. Theorem 1. Under Assumption 1, the closed-loop system (9) is exponentially mean-square stable if and only if the auxiliary system (12) is exponentially mean-square stable. Proof. Sufficiency: Given that the auxiliary system (12) is exponentially mean-square stable, there exist constants 0oα1 o1 and β1 41, which are independent of zð0Þ satisfying Eð∥zðkÞ∥2 jzð0Þ; r0 Þoβ1 αk1 ∥zð0Þ∥2 :
ð26Þ
Since Π is finite, a constant ε41 can be found such that for Δk ∈ð1; 2; …; ΠÞ, Eð∥xðkΠ þ Δk Þ∥2 jxð0Þ; θ0 ÞoεEð∥xðΠkÞ∥2 jxð0Þ; θ0 Þ:
ð27Þ
From Eq. (27), for all t∈ðkΠ þ 1; kΠ þ 2; …; ðk þ 1ÞΠÞ, Eð∥xðtÞ∥2 jxð0Þ; θ0 ÞoεEð∥zðkÞ∥2 jzð0Þ r0 Þoεβ1 αk1 ∥zð0Þ∥2 :
ð28Þ
A constant β^ 1 41 exists because Π is finite, which leads to ∥zð0Þ∥ oβ^1 ∥xð0Þ∥ . And due to toðk þ 1ÞΠ, we have β β^ 1=Π ð29Þ Eð∥xðtÞ∥2 jxð0Þ; θ0 Þoε 1 1 ðα1 Þt ∥xð0Þ∥2 : α1 2
2
The sufficient condition is proven. Necessity: If the closed-loop system (9) is exponentially mean-square stable, then there also exist constants 0oα2 o1 and β2 41, which are independent of xð0Þ, satisfying Eð∥xðtÞ∥2 jxð0Þ; θ0 Þoβ2 αt2 ∥xð0Þ∥2 :
ð30Þ
Hence the following relationship is true: 2 k 2 Eð∥zðkÞ∥2 jzð0Þ; r0 Þ ¼ Eð∥xðkΠÞ∥2 jxð0Þ; θ0 Þoβ2 αkΠ 2 ∥xð0Þ∥ ≤β 2 α2 ∥zð0Þ∥ :
ð31Þ
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The necessary condition is proven. And according to Lemma 3, the proof of Theorem 1 is completed. □ 4. Simulations In this section, two examples are employed to illustrate the effectiveness of the proposed method. Example 1. (CRRS) Consider the following unstable system with parameter perturbation over CRRS channels: 1 0 1 0 ! 0 1 0 0 1 c1 1 1 0 C B 0 C B 0 1 A; B ¼ ðb1 b2 Þ ¼ @ 1 0 A; C ¼ ; ¼ A¼@ c2 1 0 1 0:5 þ a 1 þ b −0:4 1 1 where the two uncertain parameters satisfy a40 and b40. In each period, the transmission sequence of the plant and the controller outputs are ðy1 ; y2 Þ and ðu1 ; u2 Þ, respectively. We consider two cases of the communication channels, they are ideal channel and packet dropouts channels modeled by Markov chain. For the packet dropouts modeled by Markov chains in the S/C and the C/A channels, the transition probability matrices are 0:2 0:8 0:15 0:85 Ts ¼ ; Ta ¼ : 0:1 0:9 0:2 0:8 Based on Lemma 3(2), we choose the controller and observer gains as follows: K ¼ ð−1:1 −0:51 1:15Þ;
L ¼ ð0:68 0:9 0:81Þ;
respectively, for the closed-loop system (9) with CRRS. Since Theorem 1 is the necessary and sufficient condition, when the feedback and the observer gains are provided, we are able to distinguish the stable and unstable regions in the uncertain plant parameter space for the ideal channel as well as the Markov modeled packet dropouts channels, respectively. The stable region in the plant parameter space for the ideal channel is shown as the area consisting of the curve, x-axis and y-axis in Fig. 2. For the unreliable channel modeled by Markov chains, the exponentially mean-square stable region in the plant parameter space is depicted as the area composed of the dashed curve, x-axis and y-axis in Fig. 2. When we choose a¼ 0.225 and b¼ 0, the trajectory of the system under the packets dropouts in Fig. 3 is shown in Fig. 4, where the system is stable. Example 2. (DRRS) Consider the following unstable system with parameter perturbation over DRRS channels: ! ! c1 0 2:52 þ a 1:2 0 0:5 0 A¼ ; B ¼ ðb1 b2 Þ ¼ ; C¼ ¼ ; c2 0:4 þ b 0 0 0:4 0 1 where the two uncertain parameters satisfy a40 and b40. It is necessary to point out that the controllability and detectability of the system cannot be preserved, if we still select the CRRS. Therefore, we consider the DRRS, which transmits the packets by taking into account the index of the controllability and the detectability. We choose the transmission sequence as ðy1 ; y1 Þ and ðu2 ; u2 Þ in a period. Here we also analyze the two similar communication channel cases as
Y. Xu et al. / Journal of the Franklin Institute 350 (2013) 2013–2027
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0.35 Markov case Ideal case
0.3 0.25
b
0.2 0.15 0.1 0.05 0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
a Fig. 2. Stable regions of the closed-loop system under CRRS.
θa θs
1 0.9 the states of θa and θs
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
10
20
30 k
40
50
60
Fig. 3. The packet dropouts of channels under CRRS.
expressed in Example 1. The transition probability matrices for the Markov chains are 0:1 0:9 0:1 0:9 ; Ta ¼ : Ts ¼ 0:1 0:9 0:05 0:95 According to Lemma 3(2), select the following controller and the observer gains: K ¼ ð−0:2 −1:9Þ;
L ¼ ð1:3 0:1Þ;
respectively, for the closed-loop system (9) with DRRS. According to the necessary and sufficient condition in Theorem 1, we can depict the stable and unstable regions in the uncertain plant parameter space for the ideal channel and the unreliable channel in Fig. 5, when the controller and the observer gains are given. The stable region in the plant parameter space for the ideal channel is the area made up of the curve, x-axis and y-axis in Fig. 5. For the Markov
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1
x1 x2 x3
the trajectory of the system
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
10
20
30
40
50
k Fig. 4. The trajectory of the system under CRRS.
1.4 Ideal case Markov case
1.2 1
b
0.8 0.6 0.4 0.2 0
0
0.02
0.04
0.06
0.08 a
0.1
0.12
0.14
0.16
Fig. 5. Stable regions of the closed-loop system under DRRS.
chain modeled unreliable channel, the exponentially mean-square stable region in the plant parameter space is the area composed of the dashed-dotted curve, x-axis and y-axis in Fig. 5. If we choose a¼ 0.04 and b¼ 0, then the trajectory of the system under the packets dropouts in Fig. 6 is in Fig. 7, where the system is stable. Based on the above examples, we can conclude that the stability of the closed-loop system (9) depends on the model uncertainty and the reliability of the communication channels. In addition, according to analyzing the stable region of the closed-loop system (9), we can determine the request of the accuracy of the model and the quality of the communication channels before constructing the NCSs. 5. Conclusions The stability analysis of a discrete-time networked control system with communication constrains is studied. CRRS and DRRS are presented to deal with the communication constrains
Y. Xu et al. / Journal of the Franklin Institute 350 (2013) 2013–2027
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θa θs
1 0.9 the states of θa and θs
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
10
20
30 k
40
50
60
Fig. 6. The packet dropouts of channels under DRRS.
1.5
x1 x2
the trajectory of the system
1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −3
0
10
20
30
40
50
k Fig. 7. The trajectory of the system under DRRS.
issue at both channels. Two homogeneous Markov chains are introduced to describe the packet dropouts in the S/C and the C/A channels. Based on the lifting technique, an auxiliary system is proposed with the augmented Markov chain and certain properties of the auxiliary system are analyzed. A necessary and sufficient condition for the exponential mean-square stability of the closed-loop system (9) is derived based on the stability equivalence between the closed-loop system (9) and the auxiliary system (12). Two illustrative examples are shown to demonstrate the effectiveness of the proposed stability analysis method. There are still some interesting problems needed to be studied in the future work. On one hand, how to extend our method to the system with stochastic switching dynamics with
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Y. Xu et al. / Journal of the Franklin Institute 350 (2013) 2013–2027
perturbations [29–31]. On the other hand, it is worthwhile to investigate about the NCSs under RRS condition with the quantization, transmission induced delay and sensor nonlinearity [32,33].
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