Robust finite-time filtering for uncertain systems subject to missing measurements
© Deng; licensee Springer. 2013
Received: 26 October 2012
Accepted: 24 April 2013
Published: 9 May 2013
In this paper, the robust finite-time filter design problem for uncertain systems subject to missing measurements is investigated. It is assumed that the system is subject to the norm-bounded uncertainties and the measurements of the output are intermittent. For the model of the missing measurements, the Bernoulli process is adopted. A full-order filter is proposed to estimate the signal which can track the signal to be estimated. By augmenting the system vector, a stochastic augmented system is obtained. Based on the analysis of the robust stochastic finite-time stability and the performance, the filter design method is obtained. The filter parameters can be calculated by solving a sequence of linear matrix inequalities. Finally, a numerical example is used to show the design procedure and the effectiveness of the proposed design approach.
Keywordsfinite-time stability robust filtering filtering linear matrix inequalities
In the modern control, a filter plays an important role since the filter can be used to estimate the unavailable state and filter the external noise. Therefore, the filter design has been a hot research topic since the original development of the modern control. It is well known that the Kalman filter is an effective way to estimate state. However, the Kalman filter requires the preliminary knowledge of the spectrum of the noise and the precise system model. However, in many practical cases, these requirements cannot be satisfied. In these cases, the filter is a great alternative. The filter, which was originally proposed in the late 1980s , has attracted a lot of attention due to the fact that the filter can be easily utilized to deal with the uncertainties and the attenuation effect from the external input to the estimated signal [2–5].
In the state-space model, it is always assumed that system matrices are precise. However, in the real world, these matrices are unavoidable to contain uncertainties which can result from the modeling error or variations of the system parameters. During the past 20 years, the norm-bounded uncertainties have been widely adopted in the system modeling for practical plants, such as the works in [6–10]. In , the norm-bounded uncertainties were used in the time-delay linear systems. While in , the norm-bounded uncertainties were used in the neutral systems.
In the literature, most of the works on the filtering were based on the Lyapunov asymptotic stability. However, in many practical applications, the asymptotic stability is not enough if large values of the state are not acceptable, see [4, 12–27] and the references therein. Although the finite-time stability was early proposed in 1960s , it was not a hot research topic in the following 40 years. Recently, as the development and the application of the linear matrix inequalities [28, 29], the finite-time stability has been devoted considerable efforts.
The missing measurements have been attracting a great number of attention due to the fact that the measurements are missing when sensors temporally fail [30–34]. If the phenomenon of missing measurements is not considered during the filter design, the actual missing measurements may deteriorate the designed filters. Although, there are many results on the filtering, uncertain systems, and finite-time stability, there are few results on the filtering for uncertain systems subject to missing measurements. This fact motivates me to do the research. In this paper, the contributions can be summarized as follows. The missing measurements are considered the finite-time framework. Due to the existence of the stochastic variable in the augmented system, the robust stochastic finite-time boundedness is studied for the uncertain stochastic system. Moreover, the filtering with the robust stochastic finite-time stability is investigated.
2 Problem formulation
where is a given scalar.
where the stochastic variable is a Bernoulli distributed white sequence taking values in the set .
where is the state of the filter, is an estimation of , and , and are filter parameters to be designed later.
Note that there is a stochastic variable and some norm-bounded uncertainties in the augmented system in (8). Therefore, the challenge now is how to design the filter such that the augmented system in (8) is robustly stochastically finite-time bounded and the effect of the disturbance input to the signal to be estimated is constrained to a prescribed level.
Before proceeding, the following definitions are introduced.
Definition 1 (Finite-time stable (FTS) )
is said to be FTS with respect to , where R is a positive definite matrix, and , if , then for all .
Definition 2 (Robustly stochastically finite-time stable (RSFTS))
is said to be RSFTS with respect to , where the system matrix has the uncertainty and the stochastic variable, R is a positive definite matrix, and , if for all admissible uncertainties ΔA, stochastic variable , , then for all .
Definition 3 (Robustly stochastically finite-time bounded (RSFTB))
is said to be RSFTB with respect to , where the system matrix has the uncertainty and the stochastic variable, the input matrix contains the norm-bounded uncertainty, R is a positive definite matrix, and , if for all admissible uncertainties ΔA and ΔB, stochastic variable , , then for all .
With the above definitions, the main objectives in this paper can be summarized as follows. For the uncertainty in 1, design the full-order filter (7) such that for all the admissible uncertainties and the missing measurements,
the augmented system (8) is RSFTS;
under the zero-initial condition, the signal to be estimated satisfies(12)
for all -bounded , where the prescribed value γ is the attenuation level.
In addition, some useful lemmas are also needed.
Lemma 1 (Schur complement )
Given a symmetric matrix , the following three conditions are equivalent to each other:
3 Main results
3.1 Finite-time stability and performance analysis
In this section, the finite-time stability, robust finite-time stability, and robust stochastic finite-time stability will be analyzed by assuming the parameters of the filter to be designed are given.
Therefore, if the conditions (15) and (16) hold, the augmented system (8) is RSFTB. The proof is completed. □
It is noticed that there is a positive-definite matrix in Theorem 2. The matrix can be randomly chosen. For considering the performance, other sufficient conditions are provided in the following theorem.
where and .
Proof To prove the theorem, and in Theorem 1 can be replaced with P and , respectively. The proof is completed. □
The robust stochastic finite-time stability and the robust stochastic finite-time boundedness of the augmented system (8) have been offered. Now, we are going to consider the performance.
where and .
The cost function can be revaluated with similar lines in Theorem 1. □
3.2 Filter design
The robust stochastic finite-time stability and the performance have been investigated in the above subsection. In this subsection, the filter design method will be proposed.
Moreover, the filter parameters can be calculated as and .
the conditions (34) and (35) can guarantee that the condition (31) is satisfied. □
4 Numerical example
In this paper, the robust finite-time filter design problem of discrete-time systems subject to missing measurements has been investigated. The uncertainties in the system matrices are assumed to be norm-bounded. The measurements of the system output are intermittent and a Bernoulli process is used to model the intermittent measurements. Based on the results of the robust stochastic finite-time stability and the performance, the filter design approach was proposed. Finally, an illustrative example was used to show the design procedure and the effectiveness of the proposed design approach.
- Elsayed A, Grimble MJ:A new approach to the design of optimal digital linear filters. IMA J. Math. Control Inf. 1989, 6(2):233–251. 10.1093/imamci/6.2.233MathSciNetView ArticleGoogle Scholar
- Grigoriadis KM, Watson JT:Reduced-order and filtering via linear matrix inequalities. IEEE Trans. Aerosp. Electron. Syst. 1997, 33(4):1326–1338.View ArticleGoogle Scholar
- Wang Q, Lam J, Xu S, Gao H: Delay-dependent and delay-independent energy-to-peak model approximation for systems with time-varying delay. Int. J. Syst. Sci. 2005, 36(8):445–460. 10.1080/00207720500139773MathSciNetView ArticleGoogle Scholar
- Meng Q, Shen Y: Finite-time stabilization via dynamic output feedback. Commun. Nonlinear Sci. Numer. Simul. 2009, 14(4):1043–1049. 10.1016/j.cnsns.2008.03.010MathSciNetView ArticleGoogle Scholar
- Wu L, Wang Z, Gao H, Wang C: and filtering for two-dimensional linear parameter-varying systems. Int. J. Robust Nonlinear Control 2007, 17(12):1129–1154. 10.1002/rnc.1169MathSciNetView ArticleGoogle Scholar
- Han Q-L, Gu K: On robust stability of time-delay systems with norm-bounded uncertainty. IEEE Trans. Autom. Control 2001, 46(9):1426–1431. 10.1109/9.948471MathSciNetView ArticleGoogle Scholar
- Garcia G, Bernussou J, Arzelier D: Robust stabilization of discrete-time linear systems with norm-bounded time-varying uncertainty. Syst. Control Lett. 1994, 22(5):327–339. 10.1016/0167-6911(94)90030-2MathSciNetView ArticleGoogle Scholar
- Zhou K, Khargonekar PP: Robust stabilization of linear systems with norm-bounded time-varying uncertainty. Syst. Control Lett. 1988, 10(1):17–20. 10.1016/0167-6911(88)90034-5MathSciNetView ArticleGoogle Scholar
- Han Q-L: On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty. Automatica 2004, 40(6):1087–1092. 10.1016/j.automatica.2004.01.007MathSciNetView ArticleGoogle Scholar
- Yu L, Chu J, Su H:Robust memoryless controller design for linear time-delay systems with norm-bounded time-varying uncertainty. Automatica 1996, 32(12):1759–1762. 10.1016/S0005-1098(96)80016-1MathSciNetView ArticleGoogle Scholar
- Shi P, Agarwal RK, Boukas E-K, Shue S-P:Robust state feedback control of discrete time-delay linear systems with norm-bounded uncertainty. Int. J. Syst. Sci. 2000, 31(4):409–415. 10.1080/002077200290984View ArticleGoogle Scholar
- Weiss L, Infante EF: Finite time stability under perturbing forces and on product spaces. IEEE Trans. Autom. Control 1967, AC-12(1):54–59.MathSciNetView ArticleGoogle Scholar
- Amato F, Ariola M: Finite-time control of discrete-time linear systems. IEEE Trans. Autom. Control 2005, 50(5):724–729.MathSciNetView ArticleGoogle Scholar
- Liu H, Shen Y: finite-time control for switched linear systems with time-varying delay. Intell. Control Autom. 2011, 2(2):203–213.View ArticleGoogle Scholar
- Shen Y: Finite-time control of linear parameter-varying systems with norm-bounded exogenous disturbance. J. Control Theory Appl. 2008, 6(2):184–188. 10.1007/s11768-008-6176-1MathSciNetView ArticleGoogle Scholar
- Yin Y, Liu F, Shi P: Finite-time gain-scheduled control on stochastic bioreactor systems with partially known transition jump rates. Circuits Syst. Signal Process. 2011, 30(3):609–627. 10.1007/s00034-010-9236-yMathSciNetView ArticleGoogle Scholar
- Luan X, Liu F, Shi P: Robust finite-time control for a class of extended stochastic switching systems. Int. J. Syst. Sci. 2011, 42(7):1197–1205. 10.1080/00207720903428898MathSciNetView ArticleGoogle Scholar
- Zhao S, Sun J, Liu L: Finite-time stability of linear time-varying singular systems with impulsive effects. Int. J. Control 2008, 81(11):1824–1829. 10.1080/00207170801898893MathSciNetView ArticleGoogle Scholar
- Amato F, Ariola M, Dorato P: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 2001, 37(9):1459–1463. 10.1016/S0005-1098(01)00087-5View ArticleGoogle Scholar
- Karafyllis I: Finite-time global stabilization using time-varying distributed delay feedback. SIAM J. Control Optim. 2006, 45(1):320–342. 10.1137/040616383MathSciNetView ArticleGoogle Scholar
- Bhat SP, Bernstein DS: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 2000, 38(3):751–766. 10.1137/S0363012997321358MathSciNetView ArticleGoogle Scholar
- Amatoa F, Ariolab M, Cosentino C: Finite-time stabilization via dynamic output feedback. Automatica 2006, 42(2):337–342. 10.1016/j.automatica.2005.09.007MathSciNetView ArticleGoogle Scholar
- Hong Y, Xu Y, Huang J: Finite-time control for robot manipulators. Syst. Control Lett. 2002, 46(4):243–263. 10.1016/S0167-6911(02)00130-5MathSciNetView ArticleGoogle Scholar
- Moulay E, Perruquetti W: Finite-time stability and stabilization of a class of continuous systems. J. Math. Anal. Appl. 2006, 323(2):1430–1443. 10.1016/j.jmaa.2005.11.046MathSciNetView ArticleGoogle Scholar
- Bhat SP, Bernstein DS: Geometric homogeneity with applications to finite-time stability. Math. Control Signals Syst. 2005, 17(1):101–127.MathSciNetView ArticleGoogle Scholar
- Bhat SP, Bernstein DS: Continuous finite-time stabilization of the translational and rotational double integrator. IEEE Trans. Autom. Control 1998, 43(5):678–682. 10.1109/9.668834MathSciNetView ArticleGoogle Scholar
- Moulay E, Dambrine M, Perruquetti NYW: Finite-time stability and stabilization of time-delay systems. Syst. Control Lett. 2008, 57(7):561–566. 10.1016/j.sysconle.2007.12.002MathSciNetView ArticleGoogle Scholar
- Rajchakit M, Rajchakit G: Mean square robust stability of stochastic switched discrete-time systems with convex polytopic uncertainties. J. Inequal. Appl. 2012., 2012: Article ID 135Google Scholar
- Zhu E, Xu Y: Pathwise estimation of stochastic functional Kolmogorov-type systems with infinite delay. J. Inequal. Appl. 2012., 2012: Article ID 171Google Scholar
- Hounkpevi FO, Yaz EE: Robust minimum variance linear state estimators for multiple sensors with different failure rates. Automatica 2007, 43(7):1274–1280. 10.1016/j.automatica.2006.12.025MathSciNetView ArticleGoogle Scholar
- Wang Z, Yang F, Ho DWC, Liu X:Robust control for networked systems with random packet losses. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 2007, 37(4):916–924.View ArticleGoogle Scholar
- Huang M, Dey S: Stability of Kalman filtering with Markovian packet losses. Automatica 2007, 43(4):598–607. 10.1016/j.automatica.2006.10.023MathSciNetView ArticleGoogle Scholar
- Wei G, Wang Z, Shu H: Robust filtering with stochastic nonlinearities and multiple missing measurements. Automatica 2009, 45(3):836–841. 10.1016/j.automatica.2008.10.028MathSciNetView ArticleGoogle Scholar
- Wang Z, Ho DWC, Liu X: Variance-constrained filtering for uncertain stochastic systems with missing measurements. IEEE Trans. Autom. Control 2003, 48(7):1254–1258. 10.1109/TAC.2003.814272MathSciNetView ArticleGoogle Scholar
- Nahi NE: Optimal recursive estimation with uncertain observation. IEEE Trans. Inf. Theory 1969, 15(4):457–462. 10.1109/TIT.1969.1054329MathSciNetView ArticleGoogle Scholar
- Wang Z, Yang F, Ho DWC, Liu X: Robust finite-horizon filtering for stochastic systems with missing measurements. IEEE Signal Process. Lett. 2005, 12(6):437–440.View ArticleGoogle Scholar
- Shi P, Boukas E-K, Agarwal RK: Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay. IEEE Trans. Autom. Control 2005, 44(11):2139–2144.MathSciNetGoogle Scholar
- Song S-H, Kim J-K: control of discrete-time linear systems with norm-bounded uncertainties and time delay in state. Automatica 1998, 34(1):137–139. 10.1016/S0005-1098(97)00182-9MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.