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

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1 Mathematical Problems in Engineering Volume 2012, Article ID , 13 pages doi:101155/2012/ Research Article Robust Observer Design for Takagi-Sugeno Fuzzy Systems with Mixed Neutral and Discrete Delays and Unknown Inputs Hamid Reza Karimi 1 and Mohammed Chadli 2 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 Correspondence should be addressed to Hamid Reza Karimi, hamidrkarimi@uiano Received 30 September 2012; Accepted 3 November 2012 Academic Editor: Peng Shi Copyright q 2012 H R Karimi and M 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 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 In the literature of the study, all of the existing results concern T-S fuzzy models with known inputs see, eg, 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 Recently, the problem of H model reduction for Takagi-Sugeno TS fuzzy stochastic systems in 24, the problem of H model approximation for discrete-time Takagi-Sugeno T-S fuzzy time-delay systems

2 2 Mathematical Problems in Engineering 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 Recently, the problem of networkbased 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 X>Y, where X and Y are symmetric matrices, means that X Y is positive definite, and R n and R n m denote, the n-dimensional Euclidean space and the set of all n m real matrices respectively Superscript T denotes matrix transposition, I is the identity matrix with compatible dimensions, the symbol denotes the transpose elements in the symmetric positions, I M {1, 2,,M}, andσ denotes any generalized inverse of matrix Σ with ΣΣ Σ Σ The operator diag{ } represents a block diagonal matrix, and the operator sym A represents A A T 2 Problem Formulation Now, consider the following T-S fuzzy models with unknown inputs and different neutral and discrete time delays: ẋ t M μ i ξ t A i x t A τi ẋ t τ A hi x t h B i u t R i d t H i w t i 1 x t φ t, t κ, 0, y t Cx t Fd t Jw t, 21 with μ i ξ t 0, M μ i ξ t 1, i 1 22 where M is the number of submodels, x t R n is the state vector, u t R m is the input vector, d t R q d is the unknown input, w t R q w is the external disturbance vector, and

3 Mathematical Problems in Engineering 3 y R p is the measured output A i R n n, A τi R n n, B i R n m,andc R p n define the ith local model Matrices R i R n q d and F R p q d represent the influence matrices of the unknown inputs, and H i R n q w and J R p q w represent the influence matrices of the disturbances The activation functions μ i depend on the decision vector ξ t 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 φ t is continuous vector-valued initial function, and τ and h are constant time delays with κ : max{τ, h} In this paper, we are concerned with the reconstruction of state variable x t of unknown inputs T-S model 21 using measurable signals, that is, known input u t and measured output y t In order to estimate the state of the unknown input T-S fuzzy model 21, the considered unknown input observer structure has the following form: ż t M μ i ξ t ( N i z t N τi ż t τ N hi z t h G i u t L i y t ) i 1 z t ϕ t, t κ, 0, x t z t Ey t, 23 where the observer considers the same activation functions μ i as used for the T-S model 21 The variables N i R n n, N τi R n n, G i R n m, L i R n p,ande R n p are the observer gains to be determined in order to estimate the state of the unknown input T-S model 21 The time-varying function ϕ t is continuous vector-valued initial function Now let us define the state estimation error e t x t x t 24 From estimation error 24 with the expression of x t given by the observer 23 and T-S model 21, weget e t I EC x t z t EFd t EJw t 25 The dynamic of state estimation error is then given by ė t M μ i ξ N i e t TA i K i C N i x t TA τi ẋ t τ N τi ż t τ i 1 TA hi x t h N hi z t h TB i G i u t TR i K i F d t TH i K i J w t EF d t EJẇ t, 26 with T I EC, K i N i E L i 27

4 4 Mathematical Problems in Engineering Then, we get ė t M μ i ξ t N i e t N τi ė t τ N hi e t h, i 1 28 if the following constraints hold N i TA i K i C, TA τi N τi T 0, TA hi N hi T 0, TB i G i 0, TR i K i F 0, E F J 0, TH i K i J For description brevity, 28 can be written as ė t N μ e t N μτ ė t τ N μh e t h, 216 where N μ M i 1 μ i ξ t N i, N μh M i 1 μ i ξ t N hi,andn μτ M i 1 μ i ξ t N τi Remark 21 It is noting that if E is determined, we get T from 27 and deduce directly G i, N hi,andn τi from Then, it suffices to guarantee the stability of the dynamic system 28 under the constraint Furthermore, 25 is simplified to e t I EC x t z t In the following, LMIs design conditions satisfying e t 0 when t 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 24 are given by using slack variable to introduce relaxation Theorem 31 The observer 23 converges asymptotically to the state of the continuous-time T-S model 21, if there exist matrices Q j > 0, j 1,,4, S 1, U l > 0, l 1,,3,

5 Mathematical Problems in Engineering 5 and P 2, P 3, N i, N hi, N di i I M : such that the conditions and the following hold for all Σ 11i Σ 12i Σ 13i Q 3 Q T 4 N τi Σ 22 N hi 0 N τi Σ < 0, 31 Σ 44 0 U 2 where Σ 11i sym{n i Q 1 Q 3 } U 1 S 1, Σ 12i P T 2 N T i, Σ 13i N hi Q 1 Q T 2, Σ 22 sym{ P 3 } U 2 du 3 hs 1, Σ 33 S 1 sym{q 2 }, and Σ 44 U 1 sym{q 4 } Proof To investigate the delay-dependent asymptotically stable analysis of the error system 28, we define a class of Lyapunov-Krasovskii functions as follows: V t 3 V i t, i 1 32 where V 1 t e T t P 1 e t, V 2 t V 3 t t h t τ 0 e T θ S 1 e θ dθ ė T θ S 2 ė τ dθ dτ, h t τ 0 e T θ U 1 e θ dθ ė T s U 2 ė s ds t τ τ t τ ė T θ U 3 ė τ dθ dτ 33 Time derivative of V 1 t along the system trajectory 216 becomes V 1 t 2e T t P 1 ( Nμ e t N μτ ė t τ N μh e t h ), 34 Then by taking the time derivative of V 2 t and V 3 t, one can read V 2 t &e T t S 1 e t e T t h S 1 e t h hė T t S 1 ė t ė T θ S 1 ė θ dθ, t h 35 V 3 t e T t U 1 e t e T t τ U 1 e t τ ė T t U 2 ė t ė T t τ U 2 ė t τ τė T t U 3 ė t ė T τ U 3 ė τ dτ t τ 36 Moreover, from 216, the following equation holds for any matrices P 2 and P 3 with appropriate dimensions: ( ) 2 e T t P T 2 ėt t P T 3 ( ė t N μ e t N μτ ė t τ N μh e t h ) 0 37

6 6 Mathematical Problems in Engineering Furthermore, from the Leibniz-Newton formula, the following equations hold for any matrices Q j,j 1,,4, with appropriate dimensions: ( ) ( ) 2 e T t Q 1 e T t h Q 2 e t e t h ė s ds 0, t h ( ) ( ) 2 e T t Q 3 e T t τ Q 4 e t e t τ ė s ds 0 t τ 38 From the obtained derivative terms in and adding the left-hand side of into V t, we obtain the following result for V t : V t χ T t Πχ t ( ) Tds (χ T t ν 1 ė s S 1 )S 1 χ T t ν 1 ė s S 1 t h ( ) Tds, (χ T t ν 2 ė s U 3 )U 3 χ T t ν 2 ė s U 3 t τ 39 or equivalently, V t χ T t Πχ t, 310 with Π : Ξ hν 1 S 1 ν T 1 τν 2U 3 ν T 2, S 1 : S 1 1, U 3 : U 1 3, χ t col{e t, ė t,e t h,e t τ, ė t τ }, ν 1 col{q 1, 0,Q 2, 0, 0}, ν 2 col{q 3, 0, 0,Q 4, 0}, and Σ 11 Σ 12 Σ 13 Q 3 Q T 4 Σ 15 Σ 22 P T 3 N μ h 0 P T 3 N μ τ Ξ Σ , 311 Σ 44 0 U 2 where Σ 11 sym{ P 1 P T 2 N μ Q 1 Q 3 } U 1 S 1, Σ 12 P T 2 NT μ P 3, Σ 13 P 1 P T 2 N μ h Q 1 Q T 2, Σ 15 P 1 P T 2 N μ τ, Σ 22 sym{ P 3 } U 2 τu 3 hs 1, Σ 33 S 1 sym{q 2 },and Σ 44 U 1 sym{q 4 } If a constant scalar κ>0 satisfies the following condition: Π Π Π < 0, 312

7 Mathematical Problems in Engineering 7 where κi Π :, then ( ) V t χ T t Πχ t χ T t Π Π χ t < χ T t Πχ t 314 From the previous inequality, we can easily obtain, for all ê t / 0, V t < κ ê t 2 < Based on Lyapunov stability theory, the error system 216 is asymptotically stable However, the condition 312 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 312, the obtained sufficient condition is now changed by some manipulations The inequality Π < 0 yields by Schur complements M Ξ i Π hν 1 τν 2 μ i ξ t hs 1 0 < 0, i 1 τu with Σ 11i Σ 12i Σ 13i Q 3 Q T 4 Σ 15i Σ 22 P T 3 N h i 0 P T 3 N τ i Ξ i Σ , 317 Σ 44 0 U 2 where Σ 11i sym{ P 1 P T 2 N i Q 1 Q 3 } U 1 S 1, Σ 12i P T 2 NT i P 3, Σ 13i P 1 P T 2 N h i Q 1 Q T 2,andΣ 15 i P 1 P T 2 N τ i It is clear that Ξ i < 0 result in LMI stabilization conditions It can be easily seen that the matrices P 1 P 2 T and P 3 are nonsingular Let ς : diag{ P 1 P 2 T 1,P 1 3, P 1 P 2 T 1, P 1 P 2 T 1,P 1 3 } By premultiplying ς, postmultiplying ς T to Ξ i < 0, and using the definitions N i : N i P 1 P 2 1, N hi : N hi P 1 P 2 1, N τi : N τi P 1 3, Q i : P 1 P T 2 1 Q i P 1 P 2 1, U 1 : P 1 P T 2 1 U 1 P 1 P 2 1, U 2 : P 1 3 U 1 P T 3 1, P 2 : P 1 3 P 2 P 1 P 2 1,andP 3 : P 1 3, one can obtain the LMI 31 This completes the proof

8 8 Mathematical Problems in Engineering Remark 32 The equality constraints can be rewritten in the following equivalent form: [ ] [ ] CR CH F J E K [ R H 0 ], 318 F J 0 with H [ H 1 H 2 H M ], K [ K1 K 2 K M ], R [ ] R 1 R 2 R M, F 0 0 J F 0 J F, J M times, 0 F 0 J 319 where F R M p M q d, H R n M q w, R R n M q d,andk R n M p A necessary and sufficient condition for the existence of a solution E, K to 318 is 33 CR CH F J [ ] rank F J 0 CR CH F J rank 320 F J 0 R H 0 Note that under this condition, a solution is obtained by [ ] [ ] [ CR CH F J E K R H 0 F J 0 ( [ ][ ] ) CR CH F J CR CH F J Z I, F J 0 F J 0 ] 321 where Z is an arbitrary matrix 33 Remark 33 According to Theorem 31, the observer parameters can be calculated in the following: 1 calculate N τi N τi P 3, 2 compute T by solving 210 and calculate E from T I EC, 3 compute N hi by solving 211, 4 compute G i by solving 212, 5 compute N i N i N 1 h i N hi, 6 compute K i by solving 29, 7 compute L i from L i K i N i E

9 Mathematical Problems in Engineering 9 Remark 34 It is worth noting that the number of the variables to be determined in the LMI 31 is 3n 2 M 2 4n Remark 35 The reduced conservatism of Theorem 31 benefits from the construction of the Lyapunov-Krasovskii functional in 32, 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 21 with the following data: A , A , A h , A h , A τ , A τ , [ ] B 1 05, B , F, [ ] R 1 04, R , C, J 0, H 0, τ 005, h 01, A τ1 01I 3, A τ2 07I 3 41

10 10 Mathematical Problems in Engineering x 1 (t) ꉱx 1 (t) x 2 (t) ꉱx 2 (t) t (s) t (s) e3 (t) e2(t) e1(t) t (s) t (s) t (s) x 3 (t) t (s) Figure 1: Time behaviour of states and corresponding error signals In the light of Theorem 31 and Remark 32, we solved LMIs and obtained the following observer gains by using Matlab LMI Control Toolbox E , G , G ,

11 Mathematical Problems in Engineering N , N , L , L For simulation purpose, we simply choose w t t/ 1 t 2 as the disturbance, u t e t sin t as the input signal, and μ 1 ξ 1 sin x 2 t /2 andμ 2 ξ 1 sin x 2 t /2 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, 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 1 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 , 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 , B Jiang, K Zhang, and P Shi, Integrated fault estimation and accommodation design for discretetime Takagi-Sugeno fuzzy systems with actuator faults, IEEE Trans on Fuzzy Systems, vol 19, no 2, pp , S K Nguang and P Shi, Robust H output feedback control design for Takagi- Sugeno systems with Markovian jumps: a linear matrix inequality approach, Dynamic Systems, Measurement and Control, ASME Transaction, vol 128, no 3, pp , 2006

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