Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 635709, 13 pages

http://dx.doi.org/10.1155/2012/635709

## Robust Observer Design for Takagi-Sugeno Fuzzy Systems with Mixed Neutral and Discrete Delays and Unknown Inputs

^{1}Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway^{2}Laboratory of Modeling, Information and Systems, University of Picardie Jules Verne, Amiens, France

Received 30 September 2012; Accepted 3 November 2012

Academic Editor: Peng Shi

Copyright © 2012 Hamid Reza Karimi and Mohammed Chadli. 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

A robust observer design is proposed for Takagi-Sugeno fuzzy neutral models with unknown inputs. The model consists of a mixed neutral and discrete delay, and the disturbances are imposed on both state and output signals. Delay-dependent sufficient conditions for the design of an unknown input T-S observer with time delays are given in terms of linear matrix inequalities. Some relaxations are introduced by using intermediate variables. A numerical example is given to illustrate the effectiveness of the given results.

#### 1. Introduction

In recent years, there have been rapidly growing interests in stability analysis and synthesis of fuzzy control systems; several works concerning stability and state estimation for a class of systems described by Takagi-Sugeno (T-S) fuzzy models [1–7] have been carried out. Furthermore, some recent applications of fuzzy theory in engineering are reported in [8–10]. Based on the Lyapunov method, design conditions for controllers and observers are given in linear matrix inequalities (LMIs) formulation (see among others [11–19]). In the literature of the study, all of the existing results concern T-S fuzzy models with known inputs (see, e.g., [15, 17]). However, it is well known that state estimation for dynamic systems with time delays and unknown inputs or disturbances is an interesting research topic in the fields of robust control, system supervision, and fault-tolerant control [20–23]. Recently, the problem of model reduction for Takagi-Sugeno (TS) fuzzy stochastic systems in [24], the problem of model approximation for discrete-time Takagi-Sugeno (T-S) fuzzy time-delay systems in [25], and the problem of filtering for a class of discrete-time T-S fuzzy time-varying delay systems in [26] have been fully investigated.

On the other hand, stability of neutral systems proves to be a more complex issue because the system involves the derivative of the delayed state. Especially, in the past few decades, increased attention has been devoted to the problem of robust delay-dependent stability and stabilization via different approaches for linear neutral systems with delayed state and/or input and parameter uncertainties [27–30]. Recently, the problem of network-based feedback control for systems with mixed delays based on quantization and dropout compensation has been studied in [31].

However, to the best of our knowledge, the class of unknown input T-S neutral models has not yet been fully investigated in the past and remains to be important and challenging. This motivates the present study. Thus, the contributions of this paper are two-fold: (i) design of observers for T-S fuzzy models with mixed neutral and discrete time delays and unknown inputs which influence states and outputs simultaneously and (ii) for the addressed problem, the observer gains are computed by solving a convex optimization technique.

This paper is organized as follows. First, the considered observer structure for T-S fuzzy model with mixed time delays and unknown inputs is given. In Section 3, the main results are given for T-S models. Section 4 gives a numerical example to show the validity of the given results. At last, we conclude the paper in Section 5.

*Notation*. Throughout this paper, the notation , where and are symmetric matrices, means that is positive definite, and and denote, the -dimensional Euclidean space and the set of all real matrices respectively. Superscript “” denotes matrix transposition, is the identity matrix with compatible dimensions, the symbol denotes the transpose elements in the symmetric positions, , and denotes any generalized inverse of matrix with . The operator represents a block diagonal matrix, and the operator represents .

#### 2. Problem Formulation

Now, consider the following T-S fuzzy models with unknown inputs and different neutral and discrete time delays:
with
where is the number of submodels, is the state vector, is the input vector, is the unknown input, is the external disturbance vector, and is the measured output. , , , and define the th local model. Matrices and represent the influence matrices of the unknown inputs, and and represent the influence matrices of the disturbances. The *activation functions * depend on the decision vector assumed to depend on measurable variables. It can depend on the measurable state variables and be a function of the measurable outputs of the system and possibly of the known inputs [1, 32]. The time-varying function is continuous vector-valued initial function, and and are constant time delays with .

In this paper, we are concerned with the reconstruction of state variable of unknown inputs T-S model (2.1) using measurable signals, that is, known input and measured output . In order to estimate the state of the unknown input T-S fuzzy model (2.1), the considered unknown input observer structure has the following form: where the observer considers the same activation functions as used for the T-S model (2.1). The variables , , , , and are the observer gains to be determined in order to estimate the state of the unknown input T-S model (2.1). The time-varying function is continuous vector-valued initial function. Now let us define the state estimation error

From estimation error (2.4) with the expression of given by the observer (2.3) and T-S model (2.1), we get

The dynamic of state estimation error is then given by with

Then, we get if the following constraints hold

For description brevity, (2.8) can be written as where , , and .

*Remark 2.1. *It is noting that if is determined, we get from (2.7) and deduce directly , , and from (2.9)–(2.15). Then, it suffices to guarantee the stability of the dynamic system (2.8) under the constraint (2.9)–(2.15). Furthermore, (2.5) is simplified to .

In the following, LMIs design conditions satisfying when are given for continuous-time systems.

#### 3. Synthesis Conditions

This section deals with the continuous-time T-S models. Sufficient LMIs conditions guaranteeing the global asymptotic convergence of state estimation error (2.4) are given by using slack variable to introduce relaxation.

Theorem 3.1. *The observer (2.3) converges asymptotically to the state of the continuous-time T-S model (2.1), if there exist matrices , , , , and such that the conditions (2.9)–(2.15) and the following hold for all :
**
where , , , , , and .*

*Proof. *To investigate the delay-dependent asymptotically stable analysis of the error system (2.8), we define a class of Lyapunov-Krasovskii functions as follows:
where
Time derivative of along the system trajectory (2.16) becomes
Then by taking the time derivative of and , one can read
Moreover, from (2.16), the following equation holds for any matrices and with appropriate dimensions:
Furthermore, from the Leibniz-Newton formula, the following equations hold for any matrices , with appropriate dimensions:
From the obtained derivative terms in (3.4)–(3.6) and adding the left-hand side of (3.7)–(24) into , we obtain the following result for :
or equivalently,
with , , , , , , and
where , , , , , , and .

If a constant scalar satisfies the following condition:
where
then
From the previous inequality, we can easily obtain, for all ,
Based on Lyapunov stability theory, the error system (2.16) is asymptotically stable.

However, the condition (3.12) is not strict LMI due to the nonconvex constraints in the matrix indices, and, thus, one always has difficulties to get solutions satisfying the constraint. In order to find the solutions of (3.12), the obtained sufficient condition is now changed by some manipulations. The inequality yields (by Schur complements)
with
where , , , and . It is clear that result in LMI stabilization conditions.

It can be easily seen that the matrices and are nonsingular. Let . By premultiplying , postmultiplying to , and using the definitions , , , , , , , and , one can obtain the LMI (3.1). This completes the proof.

*Remark 3.2. *The equality constraints (2.13)–(2.15) can be rewritten in the following equivalent form:
with
where , , , and . A necessary and sufficient condition for the existence of a solution to (3.18) is [33]
Note that under this condition, a solution is obtained by
where is an arbitrary matrix [33].

*Remark 3.3. *According to Theorem 3.1, the observer parameters can be calculated in the following: (1)calculate , (2)compute by solving (2.10) and calculate from , (3)compute by solving (2.11), (4)compute by solving (2.12), (5)compute , (6)compute by solving (2.9), (7)compute from .

*Remark 3.4. *It is worth noting that the number of the variables to be determined in the LMI (3.1) is .

*Remark 3.5. *The reduced conservatism of Theorem 3.1 benefits from the construction of the Lyapunov-Krasovskii functional in (3.2), introducing some free weighting matrices to express the relationship among the system matrices, and neither the model transformation approach nor any bounding technique is needed to estimate the inner product of the involved crossing terms. It can be easily seen that results of this paper are quite different from existing results in the literature in the following perspective. The structures at most of references, for instance [20], consider a delay-free T-S fuzzy system and in comparison to our case do not center on time delays, that is, the results in the previous reference cannot be directly applied to the T-S fuzzy models with unknown inputs and different neutral and discrete-time delays.

#### 4. Numerical Example

To show the validness of the proposed results, a numerical example is proposed for the discrete-time T-S model (2.1) with the following data:

In the light of Theorem 3.1 and Remark 3.2, we solved LMIs and obtained the following observer gains by using Matlab LMI Control Toolbox [34] For simulation purpose, we simply choose as the disturbance, as the input signal, and and as activation functions. The error signals for an input are depicted in Figure 1. It is seen that the state estimation for the systems is performed as well.

#### 5. Conclusion

In this paper, a robust observer design was proposed for Takagi-Sugeno (T-S) fuzzy neutral models with unknown inputs. The model consists of a mixed neutral and discrete delay, and the disturbances are imposed on both state and output signals. Delay-dependent sufficient conditions for the design of an unknown input T-S observer with time delays were given in terms of linear matrix inequalities (LMIs). Some relaxations were introduced by using intermediate variables. A numerical example was given to illustrate the effectiveness of the given results. Extension to the case of unmeasured decision variables is considered as a challenging problem. A numerical example has shown the effectiveness of the proposed results. Future work will investigate fault detection and Markovian jump systems for fuzzy systems with unknown inputs and time delays (see for instance [25–27, 35–37]).

#### Acknowledgments

The authors would like to thank the editor and anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper.

#### References

- T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to modelling and control,”
*IEEE Transactions on Systems, Man, and Cybernetics*, vol. 15, no. 1, pp. 116–132, 1985. View at Google Scholar - Y. Zhao, H. Gao, J. Lam, and B. Du, “Stability and stabilization of delayed T-S fuzzy systems: a delay partitioning approach,”
*IEEE Transactions on Fuzzy Systems*, vol. 17, no. 4, pp. 750–762, 2009. View at Google Scholar - B. Jiang, K. Zhang, and P. Shi, “Integrated fault estimation and accommodation design for discrete-time Takagi-Sugeno fuzzy systems with actuator faults,”
*IEEE Trans on Fuzzy Systems*, vol. 19, no. 2, pp. 291–304, 2011. View at Google Scholar - S. K. Nguang and P. Shi, “Robust ${H}_{\infty}$ output feedback control design for Takagi- Sugeno systems with Markovian jumps: a linear matrix inequality approach,”
*Journal of Dynamic Systems, Measurement and Control, ASME Transaction*, vol. 128, no. 3, pp. 617–625, 2006. View at Google Scholar - M. Nachidi, F. Tadeo, and A. Benzouia, “Controller design for Takagi-Sugeno systems in continuous-time,”
*International Journal of Innovative Computing, Information and Control*, vol. 8, no. 9, pp. 6389–6400, 2012. View at Google Scholar - T.-E. Lee, J.-P. Su, K.-W. Yu, and K.-H. Hsia, “Multi-objective fuzzy optimal design of alpha-beta estimators for nonlinear variable structure control,”
*International Journal of Innovative Computing, Information and Control A*, vol. 7, no. 5, pp. 2123–2140, 2011. View at Google Scholar - V. Kumar, K. P. S. Rana, and A. K. Sinha, “Performance, and stability analysis of a formula-based fuzzy PI controller,”
*International Journal of Innovative Computing, Information and ControlB*, vol. 7, no. 7, pp. 4291–4308, 2011. View at Google Scholar - K. Zrar Ghafoor, K. Abu Bakar, S. Salleh et al., “Fuzzy logic-assisted geographical routing over vehicular Ad Hoc networks,”
*International Journal of Innovative Computing Information and Control B*, vol. 8, no. 7, pp. 5095–5120, 2012. View at Google Scholar - J. Yang and J. Watada, “Fuzzy clustering analysis of data mining: application to an accident mining system,”
*International Journal of Innovative Computing Information and Control*, vol. 8, no. .8, pp. 5715–5724, 2012. View at Google Scholar - J. Botzheim and P. Foldesi, “Parametric approximation of fuzzy exponent for computationally intensive problems,”
*International Journal of Innovative Computing Information and Control*, vol. 8, no. 8, pp. 5725–5744, 2012. View at Google Scholar - N. Yagiz and Y. Hacioglu, “Robust control of a spatial robot using fuzzy sliding modes,”
*Mathematical and Computer Modelling*, vol. 49, no. 1-2, pp. 114–127, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Gao, Y. Zhao, and T. Chen, “${H}_{\infty}$ Fuzzy control of nonlinear systems under unreliable communication links,”
*IEEE Transactions on Fuzzy Systems*, vol. 17, no. 2, pp. 265–278, 2009. View at Google Scholar - H. Gao, X. Liu, and J. Lam, “Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay,”
*IEEE Transactions on Systems, Man and Cybernetics B*, vol. 39, no. 2, pp. 306–317, 2009. View at Google Scholar - L. Wu and D. W. C. Ho, “Fuzzy filter design for Ito stochastic systems with application to sensor detection,”
*IEEE Trans Fuzzy Systems*, vol. 17, no. 1, pp. 233–242, 2009. View at Google Scholar - L. Xiaodiong and Z. Qingling, “New approaches to ${H}_{\infty}$ controller designs based on fuzzy observers for T-S fuzzy systems via LMIs,”
*Automatica*, vol. 39, no. 9, pp. 1571–1582, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Tanaka, H. Ohtake, and H. O. Wang, “A descriptor system approach to fuzzy control system design via fuzzy lyapunov functions,”
*IEEE Transactions on Fuzzy Systems*, vol. 15, no. 3, pp. 333–341, 2007. View at Google Scholar - W. Assawinchaichote, S. K. Nguang, and P. Shi,
*Fuzzy Control and Filter Design for Uncertain Fuzzy Systems*, vol. 347, Springer, Berlin, Germany, 2006. - A. Kruszewski, R. Wang, and T. M. Guerra, “Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time TS fuzzy models: a new approach,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 2, pp. 606–611, 2008. View at Publisher · View at Google Scholar - A. Sala and C. Ariño, “Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya's theorem,”
*Fuzzy Sets and Systems*, vol. 158, no. 24, pp. 2671–2686, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Chadli and H. R. Karimi, “Robust observer design for unknown inputs Takagi-Sugeno models,”
*IEEE Transactions on Fuzzy Systems*. In press. View at Publisher · View at Google Scholar - M. Blanke, M. Kinnaert, and J. M. Lunze,
*Staroswiecki, Diagnosis and Fault-Tolerant Control*, Hardcover, 2nd edition, 2006. - R. J. Patton, “Fault-tolerant control systems: the 1997 situation,” in
*IFAC Symposium on Fault Detection Supervision and Safety for Technical Processes*, vol. 3, pp. 1033–1054, 1997. - M. Oudghiri, M. Chadli, and A. El Hajjaji, “Robust observer-based fault tolerant control for vehicle lateral dynamics,”
*International Journal of Vehicle Design*, vol. 48, no. 3-4, pp. 173–189, 2008. View at Google Scholar - X. Su, L. Wu, P. Shi, and Y. D. Song, “${H}_{\infty}$ model reduction of T-S fuzzy stochastic systems,”
*IEEE Transactions on Systems, Man, and Cybernetics Part B*, vol. 42, no. 6, pp. 1574–1585, 2012. View at Publisher · View at Google Scholar - L. Wu, X. Su, P. Shi, and J. Qiu, “Model approximation for discrete-time state-delay systems in the T-S fuzzy framework,”
*IEEE Transactions on Fuzzy Systems*, vol. 19, no. 2, Article ID 366378, 2011. View at Google Scholar - X. Su, P. Shi, L. Wu, and Y. Song, “A novel approach to filter design for T-S fuzzy discrete-time systems with time-varying delay,”
*IEEE Transactions on Fuzzy Systems*. In press. View at Publisher · View at Google Scholar - Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,”
*Systems & Control Letters*, vol. 51, no. 1, pp. 57–65, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Yue and Q. L. Han, “A Delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model,”
*IEEE Transactions on Circuits and Systems Part II*, vol. 51, pp. 685–680, 2004. View at Google Scholar - H. R. Karimi and P. Maass, “Delay-range-dependent exponential ${H}_{\infty}$ synchronization of a class of delayed neural networks,”
*Chaos, Solitons and Fractals*, vol. 41, no. 3, pp. 1125–1135, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. R. Karimi and H. Gao, “New delay-dependent exponential ${H}_{\infty}$
synchronization for uncertain neural networks with mixed time-delays?”
*IEEE Transactions on Systems, Man and Cybernetics B*, vol. 40, no. 1, pp. 173–185, 2010. View at Google Scholar - R. Yang, P. Shi, G.-P. Liu, and H. Gao, “Network-based feedback control for systems with mixed delays based on quantization and dropout compensation,”
*Automatica*, vol. 47, no. 12, pp. 2805–2809, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Murray-Smith and T. Johansen,
*Multiple Model Approaches to Modelling and Control*, Taylor & Francis, 1999. - C. R. Rao and S. K. Mitra,
*Generalized Inverse of Matrices and Its Applications*, John Wiley, 1971. - P. Gahinet, A. Nemirovsky, A. J. Laub, and M. Chilali,
*LMI Control Toolbox: For Use with Matlab*, The MATH Works, Natik, Mass, USA, 1995. - L. Zhang, E.-K. Boukas, and J. Lam, “Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 10, pp. 2458–2464, 2008. View at Publisher · View at Google Scholar - L. Zhang, E.-K. Boukas, L. Baron, and H. R. Karimi, “Fault detection for discrete-time Markov jump linear systems with partially known transition probabilities,”
*International Journal of Control*, vol. 83, no. 8, pp. 1564–1572, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. R. Karimi, “Robust delay-dependent ${H}_{\infty}$ control of uncertain time-delay systems with mixed neutral, discrete, and distributed time-delays and Markovian switching parameters,”
*IEEE Transactions on Circuits and Systems. I*, vol. 58, no. 8, pp. 1910–1923, 2011. View at Publisher · View at Google Scholar