Mathematical Problems in Engineering

Volume 2013, Article ID 975968, 8 pages

http://dx.doi.org/10.1155/2013/975968

## Robust and Nonfragile Kalman-Type Filter Design for Parameter-Uncertain Time-Delay Systems: PLMI Approach

School of Electronics Engineering, Kyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 702-701, Republic of Korea

Received 24 July 2013; Revised 30 September 2013; Accepted 2 October 2013

Academic Editor: Jyh-Horng Chou

Copyright © 2013 Seung Hyeop Yang and Hong Bae Park. 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 describes the synthesis of a robust and nonfragile Kalman-type filter design for a class of time-delay systems with polytopic uncertainties, filter-gain variations, and disturbances. We present the sufficient condition for filter existence and the method for designing a robust nonfragile filter by using LMIs (Linear Matrix Inequalities) technique. Because the obtained sufficient condition can be represented as PLMIs (Parameterized Linear Matrix Inequalities), which can generate infinite LMIs, we use a relaxation technique to find finite solutions for a robust nonfragile filter. We show that the proposed filter can minimize estimation error in terms of parameter uncertainties, filter-fragility, and disturbances.

#### 1. Introduction

The Kalman filtering approach is a popular state estimation method for a class of systems presented in linear models and thus has been widely applied in previous research on control engineering and target tracking, among others [1–3]. However, this approach does not provide reliable performance if there exist system uncertainties due to no full knowledge of the system model. So, the robust Kalman filtering for uncertain system has been studied on various issues [4–9]. Another serious issue in filter design is fragility. In practice, a filter designed by robust theory can be fragile because of the uncertainty associated with the implemented filter itself (e.g., limited word lengths and round-off error, etc.) [10]. To solve this problem in [11–13], nonfragile design methods have been investigated using LMIs and Riccati equation. Also, in practical applications, the noise effects must be considered to reduce estimation error and improve filter performance. To deal with this problem, filtering approach was proposed [13–22].

The aforementioned research has motivated our research to construct the filter for practical application with better performance against uncertainties, various disturbances, and time-delay. This paper examines additional methods to design a robust and nonfragile Kalman-type filter for a class of uncertain systems with disturbances and time-delay by taking the PLMIs (parameterized linear matrix inequalities) approach. Here, we represent the uncertain linear system by polytopic uncertainties, filter-gain variation, time-delay, and disturbances. Next, we rewrite the sufficient condition for filter existence as PLMIs and use a relaxation technique to find finite feasible solutions. Finally, a numerical example is provided to verify the performance of the proposed filter design method. The main contributions of this article are summarized as follows.(1)Since uncertainties in the system and filter are designed as structured uncertainties, it is possible to find filter-gain region which guarantees filter performance using PLMI and relaxation technique with less computational efforts.(2)The design method included all of factor increasing estimation error such as system uncertainties, filter-gain variations, disturbances, and time-delay. Therefore, it is more suitable for practical filter applications.

#### 2. Problem Statement

Consider the following linear system with parameter uncertainties and time-varying delay: where is the state, is the measured output, is the time-varying delay, is the disturbance, and is constant matrix with appropriate dimension. Here, we define the parameter uncertainties as follows:

Here, the value of represents the number of parameter uncertainties for system matrices, and the time-varying delay satisfies

We can express a full-order filter for system (1) as where is the filter state, is the estimated output, and and are filter matrices. In addition, we can define the estimation error of filter as and calculate next state of estimation error from (5) as where

Here, we consider a robust and nonfragile filter for the system (4) as where is the region of filter-gain variations, is the vertex of the polytope, and is the measure of nonfragility against filter-gain variation.

#### 3. Robust and Nonfragile Filter Design

In this section, we consider system performance and obtain the filter-gain . We can solve the parameters by using PLMI method because the system (1) and the filter (4) are transformed into matrix inequalities through the Lyapunov method.

Theorem 1. *The estimation error (5) is asymptotically minimizable with norm bound with the filter-gain (8) if there are positive definite matrices and such that *

*Proof. *We define a Lyapunov candidate functional as

When assuming the zero input, we have

In the next place, assume the zero initial condition and introduce

Noting
and further substituting (11) to (13).

Let , then
where is defined as

Therefore, when , , the estimation error is minimized with norm bound . The inequality (15) is transformed into (9) using Schur complements [23].

We present the proposed condition for a robust and nonfragile filter by PLMIs, which involve an infinite number of LMIs, and thus transform PLMIs into a finite number of LMIs by using, relaxation technique.

Theorem 2. *The estimation error from linear parameter uncertain system (1) and filter (4) with time-delay is asymptotically minimizable with disturbance attenuation and nonfragility whenever there exist matrices , , positive definite matrices , , and positive constant such that
**
where is the decision variable, , , , are affine symmetric matrix-valued functions of . Here, (16) holds for , , , , and defined below. **
where
*

*Proof. *The inequality (9) in Theorem 1 is same as follows:The inequality (20) is converted to the (16) using the relaxation technique [24], because the fourth term in (16) is a tight upper bound of the fourth term in (20).

Since the filter implementation accompanied by imprecision inherent in analog-digital and digital-analog conversion, finite word length, and finite resolution measuring instruments and round-off errors in numerical computation, we have to consider a design procedure which has sufficient space for coefficients readjustment. Inequality (16) provides a sufficient condition for the existence of the robust filter under additive filter-gain perturbations.

#### 4. Numerical Example

Consider nominal system matrices for parameter uncertain linear system with and assume a system uncertainty range as ±10%. Then, we can obtain the parameter uncertain matrices and as and the filter system matrices are

To solve this problem, we use the LMI toolbox with a parameter relaxation technique and obtain all feasible solutions as follows:

In this way, we obtain robust and nonfragile filter-gain with time-delay and vertices of the perturbation satisfying the nonfragility condition as follows:

For the simulation, we define the time-varying parameter as follows:

If the initial values of all states are zero and the value of is defined as

We simulate that the proposed nominal filter-gain for the system (1) with uncertainties and disturbances in Figure 1~Figure 4. It is demonstrated that the proposed nominal filter-gain guarantees the asymptotic minimization and disturbance attenuation with from the simulation results Figures 2 and 3 because the trajectories of filter (estimated) output converged to system output through computer simulation. Figure 4 shows the minimized estimation error between nominal system and filter.

Next, it is demonstrated that the practical filter with filter-gain perturbations has the nonfragility as well as the robustness in Figures 5, 6, and 7. Through the simulation results, the proposed robust and nonfragile filter guarantees the asymptotic minimization of estimation error and disturbance attenuation with from the results in Figures 5 and 7. Figure 8 demonstrates that the proposed filter shows good performance even though there exist uncertainties, disturbances, and time-delay in system.

#### 5. Conclusion

This paper proposes a method for designing a robust and nonfragile Kalman-type filter for linear time-delay systems with parameter uncertainties. We obtain filter regions satisfying the nonfragility condition by taking the relaxation technique based on PLMI approach, and the proposed filter is robust against parameter uncertainties in the system. Finally, even though there exist filter-gain variations in the polytopic region and disturbances, the obtained Kalman-type filter ensures the satisfactory filter performance and disturbance attenuation for the system.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research was financially supported by the Ministry of Education (MOE) and National Research Foundation of Korea (NRF) through the Human Resource Training Project for Regional Innovation (No. 2013H1B8A2032081).

#### References

- B. D. O. Anderson and J. B. Moore,
*Optimal Control: Linear Quadratic Methods*, Prentice Hall, Englewood Cliffs, NJ, USA, 1990. - H. Yang, Y. Xia, P. Shi, and B. Liu, “Guaranteed cost control of networked control systems based on delta operator Kalman filter,”
*International Journal of Adaptive Control and Signal Processing*, vol. 27, no. 8, pp. 701–717, 2013. View at Publisher · View at Google Scholar - P. Shi, E.-K. Boukas, and R. K. Agarwal, “Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 8, pp. 1592–1597, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Bolzern, P. Colaneri, and G. de Nicolao, “Optimal robust filtering with time-varying parameter uncertainty,”
*International Journal of Control*, vol. 63, no. 3, pp. 557–576, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. M. Haddad and D. S. Bernstein, “Robust reduced-order, nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds,”
*IEEE Transactions on Automatic Control*, vol. 33, no. 6, pp. 591–595, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. Zhu, J. Park, K.-S. Lee, and M. Spiryagin, “Guaranteed performance robust kalman filter for continuous-time markovian jump nonlinear system with uncertain noise,”
*Mathematical Problems in Engineering*, vol. 2008, Article ID 583947, 12 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - I. R. Petersen and D. C. McFarlane, “Optimal guaranteed cost control and filtering for uncertain linear systems,”
*IEEE Transactions on Automatic Control*, vol. 39, no. 9, pp. 1971–1977, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - U. Shaked and C. E. de Souza, “Robust minimum variance filtering,”
*IEEE Transactions on Signal Processing*, vol. 43, no. 11, pp. 2474–2483, 1995. View at Publisher · View at Google Scholar · View at Scopus - L. Xie and Y. C. Soh, “Robust Kalman filtering for uncertain systems,”
*Systems & Control Letters*, vol. 22, no. 2, pp. 123–129, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. H. Keel and S. P. Bhattacharyya, “Robust, fragile, or optimal?”
*IEEE Transactions on Automatic Control*, vol. 42, no. 8, pp. 1098–1105, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Famularo, C. T. Abdallah, A. Jadbabais, P. Dorato, and W. M. Haddad, “Robust nonfragile LQ controllers: the static state feedback case,” in
*Proceedings of the American Control Conference*, pp. 109–113, Philadelphia, Pa, USA, 1998. - G.-H. Yang and J. L. Wang, “Robust nonfragile Kalman filtering for uncertain linear systems with estimator gain uncertainty,”
*IEEE Transactions on Automatic Control*, vol. 46, no. 2, pp. 343–348, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G.-H. Yang and W.-W. Che, “Non-fragile ${H}_{\infty}$ filter design for linear continuous-time systems,”
*Automatica*, vol. 44, no. 11, pp. 2849–2856, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. S. Bernstein and W. M. Haddad, “Steady-state Kalman filtering with an ${H}_{\infty}$ error bound,”
*Systems & Control Letters*, vol. 12, no. 1, pp. 9–16, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fu, C. E. de Souza, and L. Xie, “${H}_{\infty}$ estimation for uncertain systems,”
*International Journal of Robust and Nonlinear Control*, vol. 2, pp. 87–105, 1992. View at Google Scholar - M. J. Grimble and A. Elsayed, “Solution of the ${H}_{\infty}$ optimal linear filtering problem for discrete-time systems,”
*IEEE Transactions on Acoustics, Speech, and Signal Processing*, vol. 38, no. 7, pp. 1092–1104, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Li and M. Fu, “Linear matrix inequality approach to robust ${H}_{\infty}$ filtering,”
*IEEE Transactions on Signal Processing*, vol. 45, no. 9, pp. 2338–2350, 1997. View at Publisher · View at Google Scholar · View at Scopus - K. M. Nagpal and P. P. Khargonekar, “Filtering and smoothing in an ${H}_{\infty}$ setting,”
*IEEE Transactions on Automatic Control*, vol. 36, no. 2, pp. 152–166, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. O. S. Teixeira, J. Chandrasekar, H. J. Palanthandalam-Madapusi, L. A. B. Tôrres, L. A. Aguirre, and D. S. Bernstein, “Gain-constrained Kalman filtering for linear and nonlinear systems,”
*IEEE Transactions on Signal Processing*, vol. 56, no. 9, pp. 4113–4123, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - L. Xie, C. E. de Souza, and M. Fu, “${H}_{\infty}$ estimation for discrete-time linear uncertain systems,”
*International Journal of Robust and Nonlinear Control*, vol. 1, pp. 111–123, 1991. View at Publisher · View at Google Scholar - X. Wang, S. Zhang, and M. Liu, “Comparison of Kalman filter, ${H}_{\infty}$ filter and robust mixed Kalman/${H}_{\infty}$ filter,” in
*Proceedings of the 30th Chinese Control Conference (CCC '11)*, pp. 3277–3281, July 2011. View at Scopus - X.-G. Guo and G.-H. Yang, “Non-fragile ${H}_{\infty}$ filter design for delta operator formulated systems with circular region pole constraints: an LMI optimization approach,”
*Acta Automatica Sinica*, vol. 35, no. 9, pp. 1209–1215, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - H. D. Tuan and P. Apkarian, “Relaxations of parameterized LMIs with control applications,”
*International Journal of Robust and Nonlinear Control*, vol. 9, no. 2, pp. 59–84, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet