Research Article | Open Access
Seung Hyeop Yang, Hong Bae Park, "Robust and Nonfragile Kalman-Type Filter Design for Parameter-Uncertain Time-Delay Systems: PLMI Approach", Mathematical Problems in Engineering, vol. 2013, Article ID 975968, 8 pages, 2013. https://doi.org/10.1155/2013/975968
Robust and Nonfragile Kalman-Type Filter Design for Parameter-Uncertain Time-Delay Systems: PLMI Approach
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.
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.) . 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.
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 .
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 , 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.
(a) Trajectories of the
(b) Trajectories of the
(a) System state and filter state
(b) System state and filter state
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.
(a) System state and filter state
(b) System state and filter state
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.
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).
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- U. Shaked and C. E. de Souza, “Robust minimum variance filtering,” IEEE Transactions on Signal Processing, vol. 43, no. 11, pp. 2474–2483, 1995.
- L. Xie and Y. C. Soh, “Robust Kalman filtering for uncertain systems,” Systems & Control Letters, vol. 22, no. 2, pp. 123–129, 1994.
- L. H. Keel and S. P. Bhattacharyya, “Robust, fragile, or optimal?” IEEE Transactions on Automatic Control, vol. 42, no. 8, pp. 1098–1105, 1997.
- 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.
- G.-H. Yang and W.-W. Che, “Non-fragile filter design for linear continuous-time systems,” Automatica, vol. 44, no. 11, pp. 2849–2856, 2008.
- D. S. Bernstein and W. M. Haddad, “Steady-state Kalman filtering with an error bound,” Systems & Control Letters, vol. 12, no. 1, pp. 9–16, 1989.
- M. Fu, C. E. de Souza, and L. Xie, “ estimation for uncertain systems,” International Journal of Robust and Nonlinear Control, vol. 2, pp. 87–105, 1992.
- M. J. Grimble and A. Elsayed, “Solution of the optimal linear filtering problem for discrete-time systems,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 38, no. 7, pp. 1092–1104, 1990.
- H. Li and M. Fu, “Linear matrix inequality approach to robust filtering,” IEEE Transactions on Signal Processing, vol. 45, no. 9, pp. 2338–2350, 1997.
- K. M. Nagpal and P. P. Khargonekar, “Filtering and smoothing in an setting,” IEEE Transactions on Automatic Control, vol. 36, no. 2, pp. 152–166, 1991.
- 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.
- 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, pp. 111–123, 1991.
- X. Wang, S. Zhang, and M. Liu, “Comparison of Kalman filter, filter and robust mixed Kalman/ filter,” in Proceedings of the 30th Chinese Control Conference (CCC '11), pp. 3277–3281, July 2011.
- X.-G. Guo and G.-H. Yang, “Non-fragile 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.
- 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.
- 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.
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