Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 930927, 9 pages

http://dx.doi.org/10.1155/2013/930927

Research Article

Delay-Probability-Distribution-Dependent FIR Filtering Design with Envelope Constraints

Cheng Peng,¹ Zhandong Yu,² Hamid Reza Karimi,³ Weizhi Wang,¹ and Nan Wang¹

¹The Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China

²The Bohai University, Jinzhou 121013, China

³The Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

Received 23 August 2013; Accepted 11 September 2013

Academic Editor: Zhiguang Feng

Copyright © 2013 Cheng Peng 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 studies the problem of finite-impulse response (FIR) filtering design of time-delay system. The time-delay considered here is time-varying meanwhile with a certain stochastic characteristic, and the probability of delay distribution is assumed to be known. Furthermore, the requirement of pulse-shape is also considered in filter design. Employing the information about the size and probability distribution of delay, a delay-probability-distribution-dependent criterion is proposed for the filtering error system. Based on a Lyapunov-Krasovskii functional, a set of linear matrix inequalities (LMIs) are formulated to solve the problem. At last, a numerical example is used to demonstrate the effectiveness of the filter design approach proposed in the paper.

1. Introduction

In the studies about filtering problem, one most significant approach frequently applied in the past decades is Kalman filtering, the main idea of which is to minimize the variance of the estimation error assuming considered system dynamics to be exactly known and the external disturbances to be stationary Gaussian noises with known statistical properties [1, 2]. However, in many practical engineering applications, the statistical details about external noise are not available [3–8]. In these cases, many approaches are introduced to improve systems’ robustness, such as , , and mixed filtering [2, 9–13]. In this paper, the filtering approach is utilized.

On the other hand, time-delays are frequently encountered in practical engineering systems, such as manufacturing systems, power systems, and networked control systems [14–17]. Existence of delay makes the analysis and synthesis of systems a much more difficult task; meanwhile it is also the source of instability and poor performance in many cases [13, 18–20]. The main approaches to solve delay problems can be classified into delay-dependent approach and delay-independent approach. It has been shown in [21, 22] that the results obtained using delay-dependent approaches are generally less conservative than the delay-independent approaches ones [23]. Acknowledging this fact, the delay-dependent approach is applied in this paper.

In fact, the variation of delay may often stick to some probability distribution in spite of its varying and underivable property [24, 25]. Furthermore, in many real systems such as networked control systems, the time-varying delay may have some abrupt burst, leading to very large delay with a very small probability [26]. In this sense, the discussion about time-delay should not only depend on its size but also on its probability distribution. In this paper, a new filter design approach and new stability criteria for the filtering error system taking the stochastic characteristic of time-varying delay into account is proposed.

While an optimal filter can catch the frequency-domain property, the time-domain constraints such as envelope constraints or bounds on signals cannot be handled by this frequency-domain approach [27]. Among various time-domain specifications, envelope constraints, which make requirement on the pulse-shape, have significant applications in many practical engineering systems, such as communication systems, radar, sonar systems, and signal processing systems [28–31]. For instance, in deconvolution filtering and data channel equalization problems, it is extremely important to achieve a desired pulse-shape through designing an appropriate filter [27].

Therefore, aiming at incorporating both frequency-domain and time-domain constraints into the problem, we intend to design a filter satisfying the performance and subject to envelope constraints in outputs. Meanwhile, time-varying delays with certain stochastic characteristics in the transmission channel are also taken into account. With the proposed filter design approach, a more general condition of time-varying delay problem can be solved. As in most situations, although detailed and exact information about delay cannot be achieved, the delay’s probability distribution characteristics can be predicted or observed relatively easily. Once the probability information is gotten, the filter design approach can be developed.

In this paper, based on a Lyapunov-Krasovskii functional, we first present an optimal solution to the design of a finite-impulse response (FIR) filter using information about the range of time-varying delay and its probability distribution. Then, the envelope constraints are taken into consideration. The resultant filter is called an optimal Envelope-Constrained FIR (ECFIR) filter. We obtain the solution via solving an LMI optimization problem. At last, a numerical example is presented to illustrate the effectiveness of the proposed filtering design approach.

2. Problem Formulation and Preliminaries

Consider a filtering system shown in Figure 1, where represents a linear dynamic system with state-space realization given by

where is the model state vector, is the input signal, is the source signal generated by the model, and , , are known constant matrices with appropriate dimensions. Then the output is transmitted through a channel with time-varying delay modeled by

where is the channel state vector, is the time-varying delay with an upper bound of , is the output of the channel, and is the disturbance input; , , , , , are all known constant system matrices with appropriate dimensions. As is shown in (2), the source signal suffers from influence of time-varying delay and disturbance from the environment represented by . The output of transmission channel is , which is also the input signal of the filter. We are going to use the corrupted signal to reconstruct original source signal.

Figure 1: Filtering system.

Assumption 1. changes randomly and for a constant , and the probability of and can be known. The following sets and functions are defined:

Obviously, it can be seen from the definition that is equal to the occurrence of event and means that the event occurs. Therefore, a stochastic variable can be defined as

Assumption 2. is a Bernoulli distributed sequence with

where is a constant.

Remark 3. From Assumption 2, it is easy to see that and . As and , and also denote the probability of taking values in and , respectively.

According to Assumptions 1 and 2, the system model described by (2) can be rewritten as

At the receiving end, we are interested in designing a linear filter with state-realization as follows:

where is the filter state vector, , is the estimated signal of source signal and , , , have the following form:

The transfer function of the filter is given by

where , , and are parameters to be determined. Define the filtering error as . Then, via augmenting the models and , the filtering error system is given as follows:

where

Before giving the main results, we need following definitions at first.

Definition 4. For a given function , its stochastic difference operator is defined as

Definition 5 (see [32]). The filtering error system in (10) is said to be stochastically stable if for any initial condition and zero exogenous noise , there exists a positive definite independent of , such that the following condition is satisfied:

Definition 6. System (10) is said to be stochastically stable with an norm bound , if the following conditions hold.

(1)The filtering error system with is stochastically stable.(2)For all nonzero and under zero initial conditions, the following inequality holds:

Now, with the definitions above, we present the objective of this paper.

Given the filtering system shown in Figure 1, we are interested in designing a filter in the form of (7)-(8) such that

(a)the filtering error system (10) is asymptotically stable in the stochastic sense;(b)the filtering error system (10) possesses a minimized performance level ;(c)a time-domain envelope constraint is imposed on the output signal as follows:

where and are the lower and upper bounds of the time-domain mask, respectively.

3. Main Results

In this section, based on the Lyapunov-Krasovskii stability theorem, a delay-probability-distribution-dependent approach is proposed to solve the FIR filter design problem subject to envelope constraints described in (15). First, a stability criterion for the filtering error system described in (10) is proposed. Then the envelope constraints are taken into consideration. An optimal ECFIR filter design approach is given at last.

Theorem 7. Given the system in Figure 1, for some given constants , , and , the filtering error system (10) is stochastically stable with performance if there exist matrices , , , , of appropriate dimensions such that the following optimization problem has solutions,

subject to the following LMI constraint:

where

and , , , , , and are defined in (11).

Proof. First, define a Lyapunov-Krasovskii functional as follows:

where

and , , , , and are Lyapunov matrices to be determined.

Then using the stochastic difference operator defined in (12), we obtain

Using the Jensen inequality [33], the following expressions are obtained:

Thus, we have

Thus, we obtain

where

By Schur complement, it can be concluded from (17) that . By similar lines as in [32], the stochastic stability can be guaranteed if condition (17) holds.

Then, define the performance index as follows:

Considering the fact that , under the zero initial condition, we have

Thus, is equal to

where

Through applying Schur complement, it is shown that can be guaranteed by condition (17). That is to say, once (17) is satisfied, the performance can be guaranteed to be less than . Thus, the proof is completed.

At this point, the second desired property of the system will be considered, which is the envelope constraints demand. First, some notations are introduced [34]:

where is an matrix, ,

is a given signal, and

Are, respectively, the upper and lower bounds. Therefore, the constraint of (15) is equal to

where denotes a conversion from a vertical vector to a diagonal matrix.

Based on Theorem 7 and (33), we can establish another theorem to determine the filter that satisfies the envelope constraint meanwhile possessing optimal performance.

Theorem 8. An optimal filter of the form (7)-(8) satisfying envelope constraint in (15) can be obtained by solving the following LMI optimization problem:

subject to

where is defined in (17).

4. An Illustrative Example

In this section, an example is given to support the filter design method proposed in the paper. Consider a filtering system as shown in Figure 1. The parameters for are given by

The parameters for the delay channel are given by

Using Theorem 8, the optimal filter is obtained via using the LMI toolbox of MATLAB with chosen to be 5. The resultant optimal is 8.5057 and filter gains are given as follows:

The expected envelope constraints and (the output of ) corresponding to a particular case where input signal is chosen to be unit impulse signal are shown in Figure 2. The transmitted signal through which is generated with no noise added is also given in the figure. The filter output and filtering error are given in Figures 3 and 4, respectively.

Figure 2: Source signal , filter input signal , and envelope bounds.

Figure 3: Output of the filter without disturbance.

Figure 4: Filtering error without disturbance.

Furthermore, to illustrate the performance of the designed filter, we add the disturbance signal chosen as white noise with mean of zero and variance of into the system. The resultant filter output and filtering error are shown in Figures 5 and 6, respectively. It is shown that the designed filter is effective.

Figure 5: Output of the filter with disturbance.

Figure 6: Filtering error with disturbance.

5. Conclusions

In this paper, we have solved the filtering design problem of time-delay system. The time-delay considered here is time-varying meanwhile with a certain stochastic characteristic, and the probability of delay distribution is assumed to be known. Furthermore, the envelope constraints are also considered in the process of filtering design. The delay-distribution-dependent criterion is formed for the filtering error system, employing the information about not only the size of delay but also its probability distribution. A set of linear matrix inequalities (LMIs) are formulated to solve the problem. Through solving the LMI optimization problem, the performance is minimized and pulse-shape demand imposed by envelope constraints is satisfied. Finally, an illustrative example is presented to demonstrate the effectiveness of the filtering design approach. For future research directions, extending the filter design approach proposed in this paper to networked control systems and distributed systems is an interesting issue. Besides, more general filter design approaches considering delays in different forms with different characteristics also deserve further investigation.

Acknowledgments

The authors are grateful to the associate editor and anonymous reviewers for their constructive comments based on which the presentation of this paper has been greatly improved.

References

R. E. Kalman, “A new approach to linear filtering and prediction problems,” Transanctions of the ASME: Journal of Basic Engineering, vol. 82, no. 1, pp. 35–45, 1960.

B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ, USA, 1979.

S. Yin, S. Ding, and H. Luo, “Real-time implementation of fault tolerant control system with performance optimization,” IEEE Transactions on Industrial Electronics, 2013.

S. Yin, S. Ding, A. Haghani, H. Hao, and P. Zhang, “A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process,” Journal of Process Control, vol. 22, no. 9, pp. 1567–1581, 2012.

S. Yin, S. X. Ding, A. H. A. Sari, and H. Hao, “Data-driven monitoring for stochastic systems and its application on batch process,” International Journal of Systems Science, vol. 44, no. 7, pp. 1366–1376, 2013.

S. Tong, Y. Li, Y. Li, and Y. Liu, “Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 41, no. 6, pp. 1693–1704, 2011.

S. Tong and Y. Li, “Observer-based fuzzy adaptive control for strict-feedback nonlinear systems,” Fuzzy Sets and Systems, vol. 160, no. 12, pp. 1749–1764, 2009.

Y. Li, S. Tong, and T. Li, “Adaptive fuzzy output feedback control of uncertain nonlinear systems with unknown backlash-like hysteresis,” Information Sciences, vol. 198, pp. 130–146, 2012.

L. Xie, C. E. de Souza, and M. Fu, “ estimation for discrete-time linear uncertain systems,” International Journal of Robust and Nonlinear Control, vol. 1, no. 2, pp. 111–123, 1991.

R. M. Palhares and P. L. D. Peres, “Robust filtering with guaranteed energy-to-peak performance—an approach,” Automatica, vol. 36, no. 6, pp. 851–858, 2000.

H. D. Tuan, P. Apkarian, and T. Q. Nguyen, “Robust and reduced-order filtering: new LMI-based characterizations and methods,” IEEE Transactions on Signal Processing, vol. 49, no. 12, pp. 2975–2984, 2001.

H. Gao, J. Lam, L. Xie, and C. Wang, “New approach to mixed filtering for polytopic discrete-time systems,” IEEE Transactions on Signal Processing, vol. 53, no. 8, part 2, pp. 3183–3192, 2005.

J. Qiu, G. Feng, and J. Yang, “Delay-dependent non-synchronized robust state estimation for discrete-time piecewise linear delay systems,” International Journal of Adaptive Control and Signal Processing, vol. 23, no. 12, pp. 1082–1096, 2009.

R. Zhang, T. Li, and L. Guo, “ control for flexible spacecraft with time-varying input delay,” Mathematical Problems in Engineering, vol. 2013, Article ID 839108, 6 pages, 2013.

T. Li, K. Zhang, and B. Zheng, “Fault detection for non-Gaussian stochastic systems with timevarying delay,” Mathematical Problems in Engineering, vol. 2013, Article ID 958954, 8 pages, 2013.

B. Zhang and W. X. Zheng, “ filter design for nonlinear networked control systems with uncertain packet-loss probability,” Signal Processing, vol. 92, no. 6, pp. 1499–1507, 2012.

J. Qiu, G. Feng, and J. Yang, “A new design of delay-dependent robust filtering for discrete-time T-S fuzzy systems with time-varying delay,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 5, pp. 1044–1058, 2009.

Z. Feng and J. Lam, “Robust reliable dissipative filtering for discrete delay singular systems,” Signal Processing, vol. 92, no. 12, pp. 3010–3025, 2012.

Z. Feng, J. Lam, and H. Gao, “Delay-dependent robust controller synthesis for discrete singular delay systems,” International Journal of Robust and Nonlinear Control, vol. 21, no. 16, pp. 1880–1902, 2011.

S. Tong, W. Wang, and L. Qu, “Decentralized robust control for uncertain T-S fuzzy large-scale systems with time-delay,” International Journal of Innovative Computing, Information and Control, vol. 3, no. 3, pp. 657–672, 2007.

H. Gao and C. Wang, “Delay-dependent robust and filtering for a class of uncertain nonlinear time-delay systems,” IEEE Transactions on Automatic Control, vol. 48, no. 9, pp. 1661–1666, 2003.

J. Qiu, G. Feng, and J. Yang, “Improved delay-dependent filtering design for discrete-time polytopic linear delay systems,” IEEE Transactions on Circuits and Systems II, vol. 55, no. 2, pp. 178–182, 2008.

C. E. de Souza, R. M. Palhares, and P. L. D. Peres, “Robust filter design for uncertain linear systems with multiple time-varying state delays,” IEEE Transactions on Signal Processing, vol. 49, no. 3, pp. 569–576, 2001.

S. Wang, R. Nathuji, R. Bettati, and W. Zhao, “Providing statistical delay guarantees in wireless networks,” in Proceedings of the 24th International Conference on Distributed Computing Systems, pp. 48–55, Tokyo, Japan, March 2004.

Y. Wei, M. Wang, and J. Qiu, “A new approach to delay-dependent filtering for discretetime Markovian jump systems with time-varying delay and incomplete transition descriptions,” IET Control Theory and Applications, vol. 7, no. 5, pp. 684–696, 2013.

J. Liu, D. Yue, Z. Gu, and E. Tian, “ filtering for systems with time-varying delay satisfying a certain stochastic characteristic,” IET Signal Processing, vol. 5, no. 8, pp. 757–766, 2011.

Z. Tan, Y. C. Soh, and L. Xie, “Envelope-constrained FIR filter design,” IEEE Transactions on Circuits and Systems II, vol. 47, no. 1, pp. 79–82, 2000.

K. L. Teo, A. Cantoni, and X. G. Lin, “New approach to the optimization of envelope-constrained filters with uncertain input,” IEEE Transactions on Signal Processing, vol. 42, no. 2, pp. 426–429, 1994.

R. J. Evans, T. E. Fortmann, and A. Cantoni, “Envelope-constrained filters. I. Theory and applications,” IEEE Transactions on Information Theory, vol. IT-23, no. 4, pp. 421–434, 1977.

R. J. Evans, A. Cantoni, and K. M. Ahmed, “Envelope-constrained filters with uncertain input,” Circuits, Systems, and Signal Processing, vol. 2, no. 2, pp. 131–154, 1983.

B. N. Vo, A. Cantoni, and K. L. Teo, “Envelope constrained filter with linear interpolator,” IEEE Transactions on Signal Processing, vol. 45, no. 6, pp. 1405–1414, 1997.

H. Gao, T. Chen, and L. Wang, “Robust fault detection with missing measurements,” International Journal of Control, vol. 81, no. 5, pp. 804–819, 2008.

K. Gu, “An integral inequality in the stability problem of time-delay systems,” in Proceedings of the 39th IEEE Confernce on Decision and Control, pp. 2805–2810, Sydney, Australia, December 2000.

Z. Tan, Y. Soh, and L. Xie, “ optimal envelope-constrained FIR filter design: an LMI approach,” in Proceedings of the 5th International Symposium on Signal Processing and Its Applications, pp. 951–954, Brisbane, Australia, 1999.