Mathematical Problems in Engineering

Volume 2013, Article ID 825143, 8 pages

http://dx.doi.org/10.1155/2013/825143

## Finite-Frequency Filter Design for Networked Control Systems with Missing Measurements

^{1}College of Information Science and Engineering, Northeastern University, Liaoning, Shenyang 110004, China^{2}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

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.

#### Abstract

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 performance is subsequently given that extends the standard 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, filtering is introduced, which aims to make the worst case norm from the process noise to the estimation error minimized. More recently, there have appeared a few results on filter design [14–18]. However, it should be noticed that all the aforementioned methods are proposed in full-frequency 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 Kalman-Yakubovich-Popov (KYP) lemma is generalized in [20] to characterize frequency domain inequalities with (semi)finite-frequency 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 time-invariant 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 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 norm is subsequently given to measure the robustness, which extends the standard 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 high-frequency 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 and stand for, respectively, the transposition and the inverse of a matrix; means that is real symmetric and positive definite; denotes the Euclidean norm; 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 and its transposition is denoted by .

#### 2. Problem Formulation

Considering the NCS depicted in Figure 1, the continuous-time plant model is where is the state, is the measured output, is the controlled output, and is the exogenous disturbance which belongs to . , , , , and 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 th and the th successfully received measurements at the filter is , where is the sampling period of the sensor. It is obvious that the number of the missed packets at time instant is , which can be modeled by a time-homogeneous Markov chain with the range set and the transition probability matrix In this situation, the dynamics of plant (1) together with the missing measurements at time instant can be approximated by where , , , , and . 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 .

In this paper, the filter is chosen as the following form: where is the filter's state, is the estimated output, and , , , and are filter gains to be designed.

To ensure the achievement of filter design objective, a basic assumption, that is, 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, , , , , , and are notated as , , , , , and when , respectively. Denoting and , the filtering error system can be described by the following system: where

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 , if
holds for all .

*Definition 3 (finite-frequency stochastic 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 (ii)for the middle-frequency range
where ,(iii)for the high-frequency range the given constant is said to be the finite-frequency stochastic norm of (5) if the following inequality
holds.

*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 norm. Noticeably, it expresses the robustness from to in finite-frequency, that is, the smaller it is, the more robust to the error becomes.

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 norm for the external disturbance .

#### 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 , there exists matrix satisfying if and only if the following two conditions hold:
*

The following lemma will be given which provides a sufficient condition for the desired performance (11) of system (5).

Lemma 6. *Assume the MJLS (5) is mean-square stable; let be a given constant, then system (5) has a finite-frequency stochastic norm if there exist mode-dependent matrices , , such that the following inequalities hold:
**
where and*(i)*for the low-frequency range *(ii)*for the middle-frequency range
where , ,*(iii)*for the high-frequency range
where .*

*Proof. *We first consider the middle-frequency case for the system (5). Assume (13) holds, before and after multiplying it by and its transpose, then we can derive
Since and system (5) is mean-square stable, summing up (17) from to with respect to , it is straightforward to see that (17) is equal to
where
It is easy to prove that is equal to the left-hand side of (9), so the is semipositive definite. Also, since , the term is nonnegative when (9) is satisfied. Hence, we have , which is equivalent to condition (11) for middle frequency in Definition 3.

Similarly, the results follow by choosing and for low-frequency case and and for high-frequency case, respectively. The proof is completed.

*Remark 7. *If all the matrices in Lemma 6 are independent on , 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 finite-frequency problem of linear time-invariant systems.

Theorem 8. *Consider system (5) for all ; assume it is mean-square stable; for a given scalar , the performance of (11) is guaranteed if there exist matrices , , , , , , , and
**
such that the following LMIs hold:
**
where and
*

*Proof. *It can be concluded from Lemma 6 that if inequality (13) holds for all , the performance of (11) can be reached. Further, (13) is equivalent to
where
with and .

On the other hand, (13) implies that

Combining (23) and (25), from Lemma 5, one can easily derive that (13) holds if and only if
where matrix is the slack variable with appropriate dimensions which is introduced by Lemma 5.

Rewrite as the form of
One can conclude that the following inequality provides a sufficient condition for (26):
where .

After partitioning the matrices and as the following form
and defining the following new variables
inequality (21) can be derived. The proof is completed.

*Remark 9. *In Theorem 8, by introducing a variable , the coupling between the variable 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 if there exist matrices , , , , , , , and
**
such that the following inequality holds:
**
where .*

*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 , , , and can be obtained by solving the following optimization problem:

Then the filter gains can be computed by the following equalities:

#### 4. An Illustrative Example

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, where the system state and output are, respectively,

Assume that the sampling period is , and the sampled data are transmitted through a network, where the data packets may be lost. Further, the quantity of the lost packet , at each sampling period is shown in Figure 2, which is subject to the following transition probability matrix:

For prescribed , solving the optimization problem (33), we can obtain the optimal value for finite-frequency stochastic norm, that is, the robust performance is with the corresponding filter gains as follows:

In order to show the advantage of the proposed method, we compare it with standard 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 is 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 filtering method, our method performs better even in the case of possible missing measurements.

#### 5. Conclusions

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 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. 2012-001).

#### References

- P. Seiler and R. Sengupta, “An ${H}_{\infty}$ approach to networked control,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 3, pp. 356–364, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Shi and H. Fang, “Kalman filter-based identification for systems with randomly missing measurements in a network environment,”
*International Journal of Control*, vol. 83, no. 3, pp. 538–551, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. R. Karimi and H. Gao, “New delay-dependent exponential H∞ synchronization for uncertain neural networks with mixed time delays,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 40, no. 1, pp. 173–185, 2010. View at Publisher · View at Google Scholar · View at Scopus - H. L. Dong, Z. D. Wang, and H. J. Gao, “Robust ${H}_{\infty}$ filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts,”
*IEEE Transactions on Signal Processing*, vol. 58, no. 4, pp. 1957–1966, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - B. Shen, Z. D. Wang, and X. H. Liu, “Sampled-data synchronization control of dynamical networks with stochastic sampling,”
*IEEE Transactions on Automatic Control*, vol. 57, no. 10, pp. 2644–2650, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - D. R. Ding, Z. D. Wang, B. Shen, and H. S. Shu, “${H}_{\infty}$ state estimation for discrete-time complex networks with randomly occurring sensor saturations and randomly varying sensor delays,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 23, no. 5, pp. 725–736, 2012. View at Publisher · View at Google Scholar - D. Yue, Q.-L. Han, and J. Lam, “Network-based robust ${H}_{\infty}$ control of systems with uncertainty,”
*Automatica*, vol. 41, no. 6, pp. 999–1007, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. P. Hespanha, P. Naghshtabrizi, and Y. Xu, “A survey of recent results in networked control systems,”
*Proceedings of the IEEE*, vol. 95, no. 1, pp. 138–162, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. Wu and T. W. Chen, “Design of networked control systems with packet dropouts,”
*IEEE Transactions on Automatic Control*, vol. 52, no. 7, pp. 1314–1319, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. N. Nair, F. Fagnani, S. Zampieri, and R. J. Evans, “Feedback control under data rate constraints: an overview,”
*Proceedings of the IEEE*, vol. 95, no. 1, pp. 108–137, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. Y. Fu and L. H. Xie, “Finite-level quantized feedback control for linear systems,”
*IEEE Transactions on Automatic Control*, vol. 54, no. 5, pp. 1165–1170, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - J. Hu, Z. D. Wang, B. Shen, and H. J. Gao, “Gain-constrained recursive filtering with stochastic nonlinearities and probabilistic sensor delays,”
*IEEE Transactions on Signal Processing*, vol. 61, no. 5, pp. 1230–1238, 2013. View at Publisher · View at Google Scholar - Z. D. Wang, F. W. Yang, D. W. C. Ho, and X. H. Liu, “Robust ${H}_{\infty}$ filtering for stochastic time-delay systems with missing measurements,”
*IEEE Transactions on Signal Processing*, vol. 54, no. 7, pp. 2579–2587, 2006. View at Publisher · View at Google Scholar · View at Scopus - G.-H. Yang and D. Ye, “Adaptive reliable ${H}_{\infty}$ filtering against sensor failures,”
*IEEE Transactions on Signal Processing*, vol. 55, no. 7, pp. 3161–3171, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - B. Shen, Z. D. Wang, H. S. Shu, and G. L. Wei, “On nonlinear ${H}_{\infty}$ filtering for discrete-time stochastic systems with missing measurements,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 9, pp. 2170–2180, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J. Liu, J. L. Wang, and G.-H. Yang, “Reliable guaranteed variance filtering against sensor failures,”
*IEEE Transactions on Signal Processing*, vol. 51, no. 5, pp. 1403–1411, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - X. He, Z. D. Wang, and D. H. Zhou, “Robust ${H}_{\infty}$ filtering for time-delay systems with probabilistic sensor faults,”
*IEEE Ignal Processing Letters*, vol. 16, no. 5, pp. 442–445, 2009. View at Publisher · View at Google Scholar - L. J. Karam and J. H. McClellan, “Chebyshev digital FIR filter design,”
*Signal Processing*, vol. 76, no. 1, pp. 17–36, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - T. Iwasaki and S. Hara, “Generalized KYP lemma: unified frequency domain inequalities with design applications,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 1, pp. 41–59, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - Y.-S. Chou, C.-C. Lin, and Y.-L. Chang, “Robust $H$-infinity filtering in finite frequency domain for polytopic systems with application to in-band quantisation noise reduction in uncertain cascaded sigma-delta modulators,”
*IET Control Theory & Applications*, vol. 6, no. 9, pp. 1155–1171, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - T. Iwasaki, S. Hara, and A. L. Fradkov, “Time domain interpretations of frequency domain inequalities on (semi)finite ranges,”
*Systems & Control Letters*, vol. 54, no. 7, pp. 681–691, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Iwasaki and S. Hara, “Feedback control synthesis of multiple frequency domain specifications via generalized KYP lemma,”
*International Journal of Robust and Nonlinear Control*, vol. 17, no. 5-6, pp. 415–434, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. W. Li and H. J. Gao, “Robust finite frequency ${H}_{\infty}$ filtering for uncertain 2-D Roesser systems,”
*Automatica*, vol. 48, no. 6, pp. 1163–1170, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Wang and G.-H. Yang, “A finite frequency approach to filter design for uncertain discrete-time systems,”
*International Journal of Adaptive Control and Signal Processing*, vol. 22, no. 6, pp. 533–550, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - H. J. Gao and X. W. Li, “${H}_{\infty}$ filtering for discrete-time state-delayed systems with finite frequency specifications,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 12, pp. 2935–2941, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - D.-W. Ding and G.-H. Yang, “Fuzzy filter design for nonlinear systems in finite-frequency domain,”
*IEEE Transactions on Fuzzy Systems*, vol. 18, no. 5, pp. 935–945, 2010. View at Publisher · View at Google Scholar · View at Scopus - D. Mcituer, I. Ashkenas, and D. Graham,
*Aircraft Dynamics and Automatic Control*, Princeton University Press, Princeton, NJ, USA, 1973. - R. J. Adams, J. M. Buffngton, A. G. Sparks, and S. S. Banda, “An introduction to multivariable flight control system design,”
*Wright Lab. Report*WL-TR-92-3110, Wright Lab Wright-Patterson AFB, Wright-Patterson AFB, Ohio, USA, 1992. View at Google Scholar - W. Siwakosit, S. A. Snell, and R. A. Hess, “Robust flight control design with handling qualities constraints using scheduled linear dynamic inversion and loop-shaping,”
*IEEE Transactions on Control Systems Technology*, vol. 8, no. 3, pp. 483–494, 2000. View at Publisher · View at Google Scholar · View at Scopus