- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 310915, 7 pages
Filtering of Time-Varying Nonlinear Systems with Randomly Occurring Output Degradation
The 28th Research Institute of China Electronics Technology Group Corporation, Nanjing 210007, China
Received 13 September 2013; Accepted 22 October 2013
Academic Editor: Lifeng Ma
Copyright © 2013 Zhengwang Jia and Yongcheng Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper deals with the filtering problem for nonlinear systems with randomly occurring output degradation phenomenon. Such a phenomenon is described by a stochastic variable which obeys the Bernoulli distribution with probability known priorly. A sufficient condition is derived for the nonlinear system to reach the required performance. An iterative algorithm is then proposed to obtain the filter parameters recursively by solving the corresponding linear matrix inequality. A numerical example is presented to show the effectiveness of the proposed method.
Many practical engineering systems, like the radar systems which are used for tracking the hostile weapon systems, are always encountering failures. For example, the radar systems will fail from time to time due to the electromagnetic interference from the enemy. Moreover, other reasons that lead the sensors to failures mainly include the external disturbance and changes of working conditions, to name just a few; see [1–3], for example. It is worth pointing out that, in the systems mentioned above, the sensor failure is not persistent all the time but is intermittent stochastically. In other words, the sensor failure occurs at random time points in a probabilistic way. Such phenomena are called the randomly occurring phenomena which would drastically degrade the system performance. Therefore, in recent years, the randomly occurring phenomena have stirred quite a lot of research interests due to its clear engineering insights and many results have been reported in the literature; see [4–11] for some latest publications. However, in spite of its clear physical insight and importance in engineering application, the filtering problem for nonlinear time-varying systems under the circumstance of randomly occurring output degradation has not yet been studied sufficiently.
On another research frontier, it is well known that the nonlinearities are inevitable in practical engineering systems, and the analysis and synthesis of nonlinear systems have been attracting more and more research attention, among which the sector bound nonlinearity which could cover several class of well-studied nonlinearities has drawn particular research focus since many sensor failures like missing measurements, signal saturations can be easily converted into the nonlinearity belonging to a known sector; see [11–15] and the references therein. On the other hand, in recent years, time-varying systems have started to receive attention due to the fact that there are virtually no strictly time-invariant systems since the working circumstances, operating points, or equipment deterioration levels are inherently time-varying in nature; see [7, 8, 16, 17] for some latest results. For time-varying Markov jumping system, in , the fault detection problem has been solved in the presence of sensor saturation and randomly occurring nonlinearities. The distributed estimation algorithm has been proposed in  for a class of nonlinear time-varying stochastic systems. A distributed filter has been designed by  for a type of time-varying systems while quantization errors and packet dropouts are happening. Recently, in [10, 17], mixed controllers have been designed for a special type of nonlinear stochastic systems over a finite horizon. Unfortunately, despite the importance of the time-varying nature in system modeling, the filtering problem for nonlinear time-varying systems with randomly occurring output degradation has not yet received enough research.
Motivated by the above discussion and in order to meet the ever-increasing practical engineering requirements toward the system reliability, in this paper, we aim to design an filter for nonlinear time-varying stochastic systems in the presence of randomly occurring output degradation. The main contributions of this paper lie in the following two aspects: for the first time, the phenomena of randomly occurring output degradation are considered to better illustrate the system's complex working environment and its impact on the performance. Moreover, such a model can stand for a wide range of output degradations, including the signal quantization phenomenon and saturation as special cases; by virtue of the iterative linear matrix inequality method, the filtering problem is solved for a class of nonlinear time-varying systems which are much more general than those investigated in the existing literature. The rest of the paper is arranged as follows. Section 2 formulates the filtering problem for the nonlinear time-varying systems with randomly occurring output degradation. In Section 3, the performance is analyzed in terms of certain matrix inequality. Then, it gives the methodology to solve the addressed filtering problem and outlines the computational algorithm to recursively obtain the required filter parameters. A numerical example is presented in Section 4 to show the effectiveness and applicability of the proposed algorithm. Section 5 draws the conclusion.
2. Problem Formulation
Let us consider the discrete-time nonlinear stochastic system and measurement equations defined on : where , , , and represent the state, measured output, estimated output, and disturbance belonging to , respectively. , , , , and are known real time-varying matrices with appropriate dimensions. and are nonlinear functions.
It is assumed that the nonlinear function has the following constraint: where is known positive-definite matrix sequence with appropriate dimensions describing the shape of the ellipsoids with being the center of the ellipsoids.
Remark 1. The nonlinear function stands for the nonlinearity that is unknown, bounded, and deterministic but reside within an ellipsoidal set. Such a type of nonlinearity is usually occurring in practical engineering practice and always being a main origin for the degradation of the system performance.
Definition 2. A nonlinear function is said to satisfy the sector-bounded condition if for some real matrices and , where is a symmetric positive-definite matrix. In this case, we say that belongs to .
The nonlinear function in (1) belongs to sector where and are real matrices of appropriate dimensions; that is
Remark 3. The nonlinear function is used to describe the nonlinearity which belongs to a specified sector. Such a kind of nonlinearity is frequently seen in both theoretical research and engineering practice and could cover quite a lot of nonlinearities as special cases, for example, the Lipschitz type nonlinearity.
The nonlinear function has the following form: where is a Bernoulli distributed random sequence taking values between and with the following probability:
Remark 4. Equation (5) is introduced to illustrate the constraints on the output sampling when the output signal is confronted with degradation. The nonlinear function which will be interpreted in detail later stands for the sector bound type output degradation phenomenon.
Remark 5. The Bernoulli distributed random variable is employed to describe the phenomenon of the randomly occurring output degradations. To be specific, when , the output signal will not be confronted with degradation; otherwise there might be possible signal missing during the output sampling. Compared to the traditional models where the output signals are assumed to be always available, such a model proposed in this paper could obviously better reflect the real-world engineering practice especially when the working condition is changing.
As discussed in , there exist matrices and such that ; the sensor fault function can be written as where is a nonlinear vector-valued function satisfying the sector-bounded condition with and . In this case, can be represented as follows: where .
The objective of this paper is to design a filter with the following form for system (1): where is the state estimate, is the output to be estimated, and and are the filter parameters to be determined.
Let us define the estimation error by and then obtain the following filtering error system: where
Given a performance parameter , the objective of this paper is to obtain and in (9) such that are satisfied when the sensor faults are randomly happening.
3. Main Results
Before giving the main results, some useful lemmas are firstly introduced.
Lemma 6. Let be quadratic functions of the variable : with . If there exist scalars such that then the following is true:
Lemma 7 (Schur complement equivalence). Given constant matrices , , and where and , then if and only if or equivalently
3.1. Performance Analysis
The following theorem gives a sufficient condition under which the required performance can be satisfied.
Theorem 8. Given a performance and the initial matrix , let the filter parameters and be given. If there exist three sequences of positive scalars , , and and a sequence of positive definite matrices satisfying such that the following matrix inequality is feasible, then performance can be satisfied.
one can get from (10) that
Adding the zero term to both sides of (20), one can get where
Next, by summing up (23) on both sides from 0 to with respect to , one can have which is equivalent to
Taking the property of which is contained in random matrices and in into consideration, by taking the mathematical expectation of both sides of (26), one can have where From (27), since and the initial condition , one can see that the performance can be satisfied if is satisfied.
Next, one can see from the sector-bounded condition of nonlinear function (4) that where
Thirdly, one can get from the ellipsoidal nonlinear function (2) that where
Moreover, it also can be obtained from the sensor fault constraint (8) that where
By Schur complement equivalence lemma, the matrix inequality (18) can be rewritten as follows:
By Lemma 6, can be satisfied, which means that the required performance can be satisfied. The proof is complete.
3.2. Filter Design
Theorem 9. Let the performance and the initial condition matrix be given. If there exist a sequence of matrices , a sequence of matrices , three sequences of positive scalars , , and , a sequence of positive-definite matrices , and a sequence of positive-definite matrices satisfying such that the following matrix inequality where is satisfied, then the required filter parameters at each time can be obtained by solving the corresponding matrix inequality.
3.3. Computational Algorithm
Algorithm 10 ( filter design algorithm). Step 1.Set the required , the initial condition , the initial , and . Select the initial values for and satisfying the initial condition constraint.Step 2.Set . Solve (36) for , , , and . Then and can also be obtained.Step 3.Set . Using the obtained and , solve (36) for , , , and .Step 4.If , then stop. Else go to Step 3.
Remark 11. The filter design algorithm gives a recursive way to obtain the numerical values of the desired filter parameters at each time point . It should be pointed out that the existence of the filter is expressed as the feasibility of certain linear matrix inequalities that can be solved forward in time. The possible research topic in the future is to consider more performance indices and give the filtering schemes that are able to satisfy multiple requirements simultaneously.
4. An Illustrative Example
In this section, an illustrative example is presented to show the effectiveness of the proposed filtering design algorithm.
Let us consider the nonlinear system (1) with the parameters given below: Let and set . Let .
Assume that the nonlinear function which is chosen as belongs to sector with
The problem of filtering for nonlinear stochastic systems when the measurement output is confronted with possible randomly occurring degradation is considered. Such a phenomenon is described by a Bernoulli sequence with known probability. A sufficient condition is proposed under which the required performance can be satisfied even if the randomly occurring output degradation happens. A recursive algorithm is given in order to obtain the required filter parameters iteratively. Finally, an illustrative example is presented to show the effectiveness of the proposed method.
- S. Varma and K. Kumar, “Fault tolerant satellite attitude control using solar radiation pressure based on nonlinear adaptive sliding mode,” Acta Astronautica, vol. 66, pp. 486–500, 2010.
- F. Yang and Y. Li, “Set-membership filtering for systems with sensor saturation,” Automatica, vol. 45, no. 8, pp. 1896–1902, 2009.
- H. Dong, Z. Wang, and H. Gao, “Distributed filtering for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks,” IEEE Transactions on Industrial Electronics, vol. 60, no. 10, pp. 4665–4672, 2013.
- G. Wei, Z. Wang, and B. Shen, “Probability-dependent gain-scheduled control for discrete stochastic delayed systems with randomly occurring nonlinearities,” International Journal of Robust and Nonlinear Control, vol. 23, no. 7, pp. 815–826, 2013.
- D. Ding, Z. Wang, J. Hu, and H. Shu, “Dissipative control for state-saturated discrete time-varying systems with randomly occurring nonlinearities and missing measurements,” International Journal of Control, vol. 86, no. 4, pp. 674–688, 2013.
- J. Hu, Z. Wang, B. Shen, and H. 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.
- H. Dong, Z. Wang, and H. Gao, “Fault detection for Markovian jump systems with sensor saturations and randomly varying nonlinearities,” IEEE Transactions on Circuits and Systems, vol. 59, no. 10, pp. 2354–2362, 2012.
- D. Ding, Z. Wang, B. Shen, and H. Shu, “ 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.
- J. Hu, Z. Wang, H. Gao, and L. K. Stergioulas, “Robust sliding mode control for discrete stochastic systems with mixed time delays, randomly occurring uncertainties, and randomly occurring nonlinearities,” IEEE Transactions on Industrial Electronics, vol. 59, no. 7, pp. 3008–3015, 2012.
- L. Ma, Z. Wang, Y. Bo, and Z. Guo, “A game theory approach to mixed / control for a class of stochastic time-varying systems with randomly occurring nonlinearities,” Systems & Control Letters, vol. 60, no. 12, pp. 1009–1015, 2011.
- J. Hu, Z. Wang, H. Gao, and L. K. Stergioulas, “Probability-guaranteed finite-horizon filtering for a class of nonlinear time-varying systems with sensor saturations,” Systems & Control Letters, vol. 61, no. 4, pp. 477–484, 2012.
- H. K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, USA, 1996.
- Q. L. Han, “Absolute stability of time-delay systems with sector-bounded non-linearity,” Automatica, vol. 41, no. 12, pp. 2171–2176, 2005.
- S. J. Kim and I. J. Ha, “A state-space approach to analysis of almost periodic nonlinear systems with sector nonlinearities,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 66–70, 1999.
- J. Lam, H. Gao, S. Xu, and C. Wang, “ and / model reduction for system input with sector nonlinearities,” Journal of Optimization Theory and Applications, vol. 125, no. 1, pp. 137–155, 2005.
- H. Dong, Z. Wang, and H. Gao, “Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts,” IEEE Transactions on Signal Processing, vol. 60, no. 6, pp. 3164–3173, 2012.
- L. Ma, Z. Wang, Y. Bo, and Z. Guo, “Finite-horizon / control for a class of nonlinear Markovian jump systems with probabilistic sensor failures,” International Journal of Control, vol. 84, no. 11, pp. 1847–1857, 2011.