Research Article  Open Access
Yafeng Guo, Tianhong Pan, "Robust Stability of Uncertain Systems over Network with Bounded Packet Loss", Journal of Applied Mathematics, vol. 2012, Article ID 945240, 11 pages, 2012. https://doi.org/10.1155/2012/945240
Robust Stability of Uncertain Systems over Network with Bounded Packet Loss
Abstract
This paper investigates the problem of robust stability of uncertain linear discretetime system over network with bounded packet loss. A new Lyapunov functional is constructed. It can more fully utilize the characteristics of the packet loss; hence the established stability criterion is more effective to deal with the effect of packet loss on the stability. Numerical examples are given to illustrate the effectiveness and advantage of the proposed methods.
1. Introduction
A networked control system (NCS) is a system whose feedback loop or (and) control loop is (are) connected via a communication network, which may be shared with other devices. The main advantages of NCS are low cost, reduced weight, high reliability, simple installation, and maintenance. As a result, the NCSs have been applied in many fields, such as mobile sensor networks, manufacturing systems, teleoperation of robots, and aircraft systems [1].
However, the insertion of the communication networks in control loops will bring some new problems. One of the most common problems in NCSs, especially in wireless sensor networks, is packet dropout, that is, packets can be lost due to communication noise, interference, or congestion [2]. Some results on this issue have been available. Generally, in these results there are two types of packetloss model. One is stochastic packet loss ([2–5]; etc), another is arbitrary but bounded packet loss ([6–9]; etc.).
Here, we are concerned about the arbitrary but bounded packet loss. For this case, there are two approaches available. One approach is based on switched system theory; another one is based on the theory of timevaryingdelay system. Yu et al. [7] modeled the packetloss process as an arbitrary but finite switching signal. This enables them to apply the theory from switched systems to stabilize the NCS. However, Yu et al. [7] adopted a common Lyapunov function and the results are quadratic. Xiong and Lam [9] utilized a packetlossdependent Lyapunov function to establish the stabilization condition, which is less conservative than that of Yu et al. [7]. Unfortunately, however, in the stability condition of their approaches the system matrices appear in the forms of power and crossmultiplication among them. Therefore, it is difficult to deal with the systems with parametric uncertainty by using these approaches. In contrast, if utilizing the delay system approach, the system matrices are affine in the stability condition. Hence, this approach suits the uncertain systems. However, it may be very conservative if directly using the existing delay system approaches (e.g., [10–14]) to deal with the bounded packet loss. The main reason is that the existing approaches can not fully utilize the characteristic of packet loss. Therefore, for the systems in the simultaneous presence of parameter uncertainties and bounded packet loss, the problem of robust stability has not been fully investigated and remains to be challenging, which motivates the present study.
In this paper, we study the robust stability problem for uncertain discretetime systems with bounded networked packet loss. First, we transform the packet loss into a timevarying input delay. Second, we note that the considered timevarying delay has a new characteristic. It is different with the general timevarying delay, that is, the considered time delay will change with some laws in the interval of two consecutive successful transmissions of the network, which is not possessed by general timevarying delay. In order to utilize this characteristic, we define a new Lyapunov functional. It does not only depend on the bound of the delay, but also on the rate of its change. Due to more fully utilizing the properties of the packet loss (that is the timevarying delay induced by the packet loss), the established stability criterion shows its less conservativeness. The construction of Lyapunov functional is inspired by Fridman [15], where the stability of sampleddata control systems is considered. It does not mean that the method developed in this paper is trivial. In fact, as it is shown in the Section 3 of this paper that the properties of induceddelay are more complicate than that in Fridman [15], such that the method of Fridman [15] cannot directly be applied to the problem considered in this paper. Finally, three examples are provided to illustrate the effectiveness of the developed results.
2. Problem Formulation
The framework of NCSs considered in the paper is depicted in Figure 1. The plant to be controlled is modeled by linear discretetime system: where is the time step, and are the system state and control input, respectively. is the initial state. and are known real constant matrices with appropriate dimension. and are unknown matrices describing parameter uncertainties.
In this paper, the parameter uncertainties are assumed to be of the form , where , , and are known real constant matrices of appropriate dimensions, and is an unknown realvalued timevarying matrix satisfying .
Networks exist between sensor and controller and between controller and zeroorder holder (ZOH). The sensor is clock driven, the controller and ZOH are event driven and the data are transmitted in a single packet at each time step. As have been mentioned in Section 1, this paper only considers the network packet loss. Then it is assumed that there is not any networkinduced delay.
Let denote the sequence of time points of successful data transmissions from the sensor to the zeroorder hold at the actuator side and for any .
Assumption 2.1. The number of consecutive packet loss in the network is less than , that is
Remark 2.2. Assumption 2.1 is similar to that in Liu et al. [8]. From the physical point of view, it is natural to assume that only a finite number of consecutive packet losses can be tolerated in order to avoid the NCS becoming open loop. Thus, the number of consecutive packet loss in the networks should be less than the finite number .
The networked controller is a statefeedback controller: From the viewpoint of the ZOH, the control input is The initial inputs are set to zero: , . Hence the closedloop system becomes for . The objective of this paper is to analyze the robust stability of NCS (2.5).
Remark 2.3. The packet loss process can take place in the sensorcontroller link and the controlleractuator link. Since the considered controller is static in this paper, it is equivalent to incorporate the doublesided packet loss as a singlepacket loss process. This is just the reason that this paper only considers the singlepacket loss process. However, if the controller is online implemented, then one should clearly consider the doublesided packet loss rather than incorporate them as a singlepacket loss process. For such case, readers can be referred to Ding [16], which systematically addressed the modeling and analysis methods for doublesided packet loss process.
3. Stability of Networked Control Systems
In this section, we analyze the stability property of NCSs. Here we firstly investigate the stability of NCS (2.5) when the plant (2.1) without any uncertainty, that is, and . we have the following result.
Theorem 3.1. Assuming and , NCS (2.5) with arbitrary packetloss process is asymptotically stable if there exist matrices , , , , , , and such that the following LMI holds where and
Proof. Define
then the NCS (2.5) can be represented as a delay system:
Inspired by Fridman [15], we construct the following new functional candidate as:
with and
where and , , , are to be determined.
From (2.2) and (3.6), we know that . Therefore, similar with the discussion of Fridman [15], it can be seen that (3.4) guarantees (3.8) to be a Lyapunov functional. For , we, respectively, calculate the forward difference of the functional (3.8) along the solution of system (3.7) by two cases.
Case 1 . In this case, we have . Then,
where .
In addition, for any appropriately dimensioned matrices and the following relationships always hold:
Then, from (3.10)–(3.13), we have
where .
Now we prove (3.1) and (3.2) guaranteeing that . By Schur complement, (3.2) is equivalent to
Then from (3.1) and (3.15), we know that and . Hence for any scalar , the following inequality holds:
Noting that in this case , then we have . Therefore, . By setting , from (3.16) we obtain that . Due to , the inequality above implies holds. Therefore, in this case holds.
Case 2 . In this case, we have and . The
By the Jensen's inequality [17], we have
In addition, for any appropriately dimensioned matrix the following relationship always holds:
Then, from (3.17)–(3.20), we have
In this case, we can see that (3.3) guarantees .
From both Cases 1 and 2, we can conclude for , for all . Then, from the Lyapunov stability theory, the NCS (2.5) with arbitrary packetloss process satisfying (2.2) is asymptotically stable.
Remark 3.2. The proposed stability criterion in Theorem 3.1 is dependent on the bound of the packet loss. Furthermore, from the proof of Theorem 3.1, we can see that the varying rate of packetlossinduced delays is fully utilized to obtain the stability condition. According to the difference of induced delays' varying rates, we separate into two parts, that is and . For the two cases, we, respectively, calculate the forward difference of the functional and guarantee it less than zero, such that the NCS is asymptotically stable. Theorem 3.1 is more effective to deal with packet loss than the existing timevarying delay system approaches in the sense that Theorem 3.1 can allow a larger upper bound of the packet loss, which will be demonstrated in an example in next section.
Remark 3.3. In Fridman [15], the continuoustime sampled control system is considered. The varying rate of samplinginduced delays is constant when the derivative of the Lyapunov functional is calculated. However, in our paper, the varying rate of packetlossinduced delays will be changing when the difference of the Lyapunov functional is calculated. Therefore, the method of Fridman [15] for continuoustime domain cannot directly be applied to the problem of discretetime domain considered in this paper.
Note that in LMIs (3.1)–(3.3) the system matrices and appear in affine form, thus the stability condition presented in Theorem 3.1 can be readily extended to cope with uncertain systems (2.1). By using Theorem 3.4 and the wellknown procedure, we can easily obtain the following theorem, and hence its proof is omitted.
Theorem 3.4. NCS (2.5) is robustly asymptotically if there exist matrices , , , , , , and scalar , , satisfying (3.4) and the following LMIs: where , , are given in (3.5) and .
Remark 3.5. Because the LMIs of Theorem 3.1 are affine in the system matrices and , it is readily extended to deal with the systems with normbounded parameter uncertainty (i.e., Theorem 3.4). With similar reason it can also be easily extended to deal with the systems with polytopictype uncertainty. The reason why we only consider one of the cases is to avoid the paper being too miscellaneous.
Remark 3.6. It is worth to reiterate that if there is only packet loss, the method of this paper is more suitable than the general timedelay method. However, if there simultaneously exist networkinduced delay and packet loss, the method of this paper is not applicable, but the general timedelay method is still valid. For example, Yue et al. [18], Gao and Chen [19], and Huang and Nguang [20] considered the networked control systems with both networkinduced delay and packet loss, where Yue et al. [18] and Gao and Chen [19] are the methods of continuoustime domain, Huang and Nguang [20] is the method of discretetime domain. Yue et al. [18] investigated the regulating control for networkbased uncertain systems. Gao and Chen [19] studied the output tracking control for networkbased uncertain systems. For the uncertain networked control system with random time delays, Huang and Nguang [20] analyzed robust disturbance attenuation performance and proposed the corresponding design method for the controllers.
4. Numerical Examples
In this section, three examples are provided to illustrate the effectiveness and advantage of the proposed stability results.
Example 4.1. Borrow the system considered by Gao and Chen [10], where , and
Here we are interested in the allowable maximum bound of dropout loss that guarantees the asymptotic stability of the closedloop system. For extensive comparison purpose, we let the controller gain matrices take two different values: and . By using different methods, the calculated results are presented in Table 1. From the table, it is easy to see that the method proposed in this paper is more effective than the others. But it is never to say that the proposed method in this paper is more suitable to deal with the timedelay; it is only to show that the proposed method is more suitable to deal with the packetloss than the general timedelay methods.
Example 4.2. Borrow the system considered by Wang et al. [14], where , and
When lower bound of the equivalent delay is 0, the allowable maximum upper bound of the equivalent delay is 13 as reported in Wang et al. [14]. Therefore, if there is only bounded packet loss, by using the method of Wang et al. [14], the allowable maximum bound of dropout loss is 13. However, by using Theorem 3.1 of this paper, one can obtain that the allowable maximum bound of dropout loss is 190. This example shows again that the proposed method is more suitable to deal with the packetloss than the general timedelay methods.
Example 4.3. Consider the following uncertain system:
where . The system matrices can be written in the form of (2.1) with matrices given by
Now assume that the controller gain matrix is , and our purpose is to determine the upper value of such that the closedloop system is robustly stable. By using Theorem 3.4, the detail calculated result is shown in Table 2.
In the following, we will present some simulation results. Assume the initial condition to be for . Let changes randomly between and , which is shown in Figure 2(a). In addition, let the upper of dropout loss is 13, which is shown in Figure 2(b). Then, the state response of the closeloop system is given in Figure 2(c). It can be seen from this figure that the system is robustly asymptotically stable, which shows the validity of the method proposed in this paper.

(a)
(b)
(c)
5. Conclusions
The problem of robust stability analysis for uncertain systems over network with bounded packet loss has been considered in this paper. A new Lyapunov functional is constructed. This Lyapunov functional not only utilizes the bound of the packet loss but also utilizes the varying rate of the packetlossinduced delays, which aims at reducing the conservatism of the results. Numerical examples are also presented to demonstrate the effectiveness and advantages of the proposed approach.
Acknowledgments
This work was supported by National Natural Science Foundation of China (61104115 and 60904053), Research Fund for the Doctoral Program of Higher Education of China (20110072120018), and the Fundamental Research Funds for the Central Universities.
References
 G. C. Walsh, H. Ye, and L. G. Bushnell, “Stability analysis of networked control systems,” IEEE Transactions on Control System Technology, vol. 10, no. 3, pp. 438–446, 2002. View at: Google Scholar
 L. Schenato, “To zero or to hold control inputs with lossy links?” IEEE Transactions on Automatic Control, vol. 54, no. 5, pp. 1093–1099, 2009. View at: Publisher Site  Google Scholar
 O. C. Imer, S. Yüksel, and T. Başar, “Optimal control of LTI systems over unreliable communication links,” Automatica, vol. 42, no. 9, pp. 1429–1439, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. I. Jordan, and S. S. Sastry, “Kalman filtering with intermittent observations,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1453–1464, 2004. View at: Publisher Site  Google Scholar
 S. Hu and W.Y. Yan, “Stability robustness of networked control systems with respect to packet loss,” Automatica, vol. 43, no. 7, pp. 1243–1248, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 G.P. Liu, J. X. Mu, D. Rees, and S. C. Chai, “Design and stability analysis of networked control systems with random communication time delay using the modified MPC,” International Journal of Control, vol. 79, no. 4, pp. 288–297, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Yu, L. Wang, T. Chu, and G. Xie, “Stabilization of networked control systems with data packet dropout and network delays via switching system approach,” in Proceedings of the 43rd IEEE Conference on Decision and Control, pp. 3539–3544, Atlantis, Bahamas, 2004. View at: Google Scholar
 G.P. Liu, Y. Xia, J. Chen, D. Rees, and W. Hu, “Networked predictive control of systems with random network delays in both forward and feedback channels,” IEEE Transactions on Industrial Electronics, vol. 54, no. 3, pp. 1282–1297, 2007. View at: Google Scholar
 J. Xiong and J. Lam, “Stabilization of linear systems over networks with bounded packet loss,” Automatica, vol. 43, no. 1, pp. 80–87, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. Gao and T. Chen, “New results on stability of discretetime systems with timevarying state delay,” IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 328–334, 2007. View at: Publisher Site  Google Scholar
 B. Zhang, S. Xu, and Y. Zou, “Improved stability criterion and its applications in delayed controller design for discretetime systems,” Automatica, vol. 44, no. 11, pp. 2963–2967, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. Xiong and J. Lam, “Stabilization of networked control systems with a logic ZOH,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 358–363, 2009. View at: Publisher Site  Google Scholar
 X. Meng, J. Lam, B. Du, and H. Gao, “A delaypartitioning approach to the stability analysis of discretetime systems,” Automatica, vol. 46, pp. 610–614, 2010. View at: Google Scholar
 P. Wang, C. Han, and B. Ding, “Stability of discretetime networked control systems and its extension for robust ${H}_{\infty}$ control,” International Journal of Systems Science. In press. View at: Publisher Site  Google Scholar
 E. Fridman, “A refined input delay approach to sampleddata control,” Automatica, vol. 46, no. 2, pp. 421–427, 2010. View at: Google Scholar
 B. Ding, “Stabilization of linear systems over networks with bounded packet loss and its use in model predictive control,” Automatica, vol. 47, no. 11, pp. 2526–2533, 2011. View at: Google Scholar
 K. Gu, “An integral inequality in the stability problem of timedelay systems,” in Proceedings of the 39th IEEE Conference on Decision and Control, pp. 2805–2810, Sydney, Australia, 2000. View at: Google Scholar
 D. Yue, Q.L. Han, and J. Lam, “Networkbased robust ${H}_{\infty}$ control of systems with uncertainty,” Automatica, vol. 41, no. 6, pp. 999–1007, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 H. Gao and T. Chen, “Networkbased ${H}_{\infty}$ output tracking control,” IEEE Transactions on Automatic Control. Transactions on Automatic Control, vol. 53, no. 3, pp. 655–667, 2008. View at: Publisher Site  Google Scholar
 D. Huang and S. K. Nguang, “Robust disturbance attenuation for uncertain networked control systems with random time delays,” IET Control Theory & Applications, vol. 2, no. 11, pp. 1008–1023, 2008. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2012 Yafeng Guo and Tianhong Pan. 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.