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

Authors: Hamid Reza Karimi; Mohammed Chadli
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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 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 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 Section3, 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 Rn and Rn×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, IM={1,2,…,M}, and Σ+ 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+AT.


## 2. Problem Formulation

Now, consider the following T-S fuzzy models with unknown inputs and different neutral and discrete time delays:(2.1)x˙(t)=∑i=1M‍μi(ξ(t))(Aix(t)+Aτix˙(t-τ)+Ahix(t-h)+Biu(t)+Rid(t)+Hiw(t)Aix(t)+Aτix˙(t-τ)+Ahix(t-h))x(t)=ϕ(t),t∈[-κ,0],y(t)=Cx(t)+Fd(t)+Jw(t),
with
(2.2)μi(ξ(t))≥0,∑i=1M‍μi(ξ(t))=1,
where M is the number of submodels, x(t)∈Rn is the state vector, u(t)∈Rm is the input vector, d(t)∈Rqd is the unknown input, w(t)∈Rqw is the external disturbance vector, and y∈Rp is the measured output. Ai∈Rn×n, Aτi∈Rn×n, Bi∈Rn×m, and C∈Rp×n define the ith local model. Matrices Ri∈Rn×qd and F∈Rp×qd represent the influence matrices of the unknown inputs, and Hi∈Rn×qw and J∈Rp×qw represent the influence matrices of the disturbances. The activation functionsμi(·) 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 variablex(t) of unknown inputs T-S model (2.1) 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 (2.1), the considered unknown input observer structure has the following form:
(2.3)z˙(t)=∑i=1M‍μi(ξ(t))(Niz(t)+Nτiz˙(t-τ)+Nhiz(t-h)+Giu(t)+Liy(t))z(t)=φ(t),t∈[-κ,0],x^(t)=z(t)-Ey(t),
where the observer considers the same activation functions μi(·) as used for the T-S model (2.1). The variables Ni∈Rn×n, Nτi∈Rn×n, Gi∈Rn×m, Li∈Rn×p, and E∈Rn×p 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 φ(t) is continuous vector-valued initial function. Now let us define the state estimation error
(2.4)e(t)=x(t)-x^(t).From estimation error (2.4) with the expression of x^(t) given by the observer (2.3) and T-S model (2.1), we get
(2.5)e(t)=(I+EC)x(t)-z(t)+EFd(t)+EJw(t).The dynamic of state estimation error is then given by(2.6)e˙(t)=∑i=1M‍μi(ξ)(Nie(t)+(TAi-KiC-Ni)x(t)+TAτix˙(t-τ)-Nτiz˙(t-τ)+TAhix(t-h)-Nhiz(t-h)+(TBi-Gi)u(t)+(TRi-KiF)d(t)+(THi-KiJ)w(t)Aτi)+EFd˙(t)+EJw˙(t),
with
(2.7)T=I+EC,Ki=NiE+Li.Then, we get(2.8)e˙(t)=∑i=1M‍μi(ξ(t))(Nie(t)+Nτie˙(t-τ)+Nhie(t-h)),
if the following constraints hold
(2.9)Ni=TAi-KiC,(2.10)TAτi-NτiT=0,(2.11)TAhi-NhiT=0,(2.12)TBi-Gi=0,(2.13)TRi-KiF=0,(2.14)E[FJ]=0,(2.15)THi-KiJ=0.For description brevity, (2.8) can be written as
(2.16)e˙(t)=Nμe(t)+Nμτe˙(t-τ)+Nμhe(t-h),
where Nμ=∑i=1Mμi(ξ(t))Ni, Nμh=∑i=1Mμi(ξ(t))Nhi, and Nμτ=∑i=1Mμi(ξ(t))Nτi.Remark 2.1.
It is noting that ifE is determined, we get T from (2.7) and deduce directly Gi, Nhi, and Nτi 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 e(t)=(I+EC)x(t)-z(t).
In the following, LMIs design conditions satisfyinge(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 (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 Q-j>0, j=1,…,4, S-1,U-l>0, l=1,…,3, and P-2,P-3,N-i,N-hi,N-di such that the conditions (2.9)–(2.15) and the following hold for all i∈IM:
(3.1)[Σ^11iΣ^12iΣ^13i-Q-3+Q-4TN-τi(*)Σ^22N-hi0N-τi(*)(*)Σ^3300(*)(*)(*)Σ^440(*)(*)(*)(*)-U-2]<0,
where Σ^11i=sym{N-i+Q-1+Q-3}+U-1+S-1, Σ^12i=-P-2T+N-iT, Σ^13i=N-hi-Q-1+Q-2T, 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 (2.8), we define a class of Lyapunov-Krasovskii functions as follows:
(3.2)V(t)=∑i=13‍Vi(t),
where
(3.3)V1(t)=eT(t)P1e(t),V2(t)=∫t-ht‍eT(θ)S1e(θ)dθ+∫-h0‍∫t+τt‍e˙T(θ)S2e˙(τ)dθdτ,V3(t)=∫t-τt‍eT(θ)U1e(θ)dθ+∫t-τt‍e˙T(s)U2e˙(s)ds+∫-τ0‍∫t+τt‍e˙T(θ)U3e˙(τ)dθdτ.
Time derivative of V1(t) along the system trajectory (2.16) becomes
(3.4)V˙1(t)=2eT(t)P1(Nμe(t)+Nμτe˙(t-τ)+Nμhe(t-h)),
Then by taking the time derivative of V2(t) and V3(t), one can read
(3.5)V˙2(t)=&eT(t)S1e(t)-eT(t-h)S1e(t-h)+he˙T(t)S1e˙(t)-∫t-ht‍e˙T(θ)S1e˙(θ)dθ,(3.6)V˙3(t)=eT(t)U1e(t)-eT(t-τ)U1e(t-τ)+e˙T(t)U2e˙(t)-e˙T(t-τ)U2e˙(t-τ)+τe˙T(t)U3e˙(t)-∫t-τt‍e˙T(τ)U3e˙(τ)dτ.
Moreover, from (2.16), the following equation holds for any matrices P2 and P3 with appropriate dimensions:
(3.7)2(eT(t)P2T+e˙T(t)P3T)×(-e˙(t)+Nμe(t)+Nμτe˙(t-τ)+Nμhe(t-h))=0
Furthermore, from the Leibniz-Newton formula, the following equations hold for any matrices Qj,j=1,…,4, with appropriate dimensions:
(3.8)2(eT(t)Q1+eT(t-h)Q2)(e(t)-e(t-h)-∫t-ht‍e˙(s)ds)=0,2(eT(t)Q3+eT(t-τ)Q4)(e(t)-e(t-τ)-∫t-τt‍e˙(s)ds)=0.
From the obtained derivative terms in (3.4)–(3.6) and adding the left-hand side of (3.7)–(24) into V˙(t), we obtain the following result for V˙(t):
(3.9)V˙(t)=χT(t)Πχ(t)-∫t-ht‍(χT(t)ν1+e˙(s)S1)S-1(χT(t)ν1+e˙(s)S1)Tds-∫t-τt‍(χT(t)ν2+e˙(s)U3)U-3(χT(t)ν2+e˙(s)U3)Tds,
or equivalently,
(3.10)V˙(t)=χT(t)Πχ(t),
with Π:=Ξ+hν1S-1ν1T+τν2U-3ν2T, S-1:=S1-1, U-3:=U3-1, χ(t)=col{e(t),e˙(t),e(t-h),e(t-τ),e˙(t-τ)}, ν1=col{Q1,0,Q2,0,0}, ν2=col{Q3,0,0,Q4,0}, and
(3.11)Ξ=[Σ11Σ12Σ13-Q3+Q4TΣ15(*)Σ22P3TNμh0P3TNμτ(*)(*)Σ3300(*)(*)(*)Σ440(*)(*)(*)(*)-U2],
where Σ11=sym{(P1+P2T)Nμ+Q1+Q3}+U1+S1, Σ12=-P2T+NμTP3, Σ13=(P1+P2T)Nμh-Q1+Q2T, Σ15=(P1+P2T)Nμτ, Σ22=sym{-P3}+U2+τU3+hS1, Σ33=-S1-sym{Q2}, and Σ44=-U1-sym{Q4}.
If a constant scalarκ>0 satisfies the following condition:
(3.12)Π-=Π+Π~<0,
where
(3.13)Π~:=[κI0⋯000⋯0⋮⋮⋱⋮00⋯0],
then
(3.14)V˙(t)≤χT(t)Πχ(t)=χT(t)(Π--Π~)χ(t)<-χT(t)Π~χ(t).
From the previous inequality, we can easily obtain, for all e^(t)≠0,
(3.15)V˙(t)<-κ∥e^(t)∥2<0.
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 Π-<0 yields (by Schur complements)
(3.16)∑i=1M‍μi(ξ(t))[Ξi+Π~hν1τν2(*)-hS10(*)(*)-τU3]<0,
with
(3.17)Ξi=[Σ11iΣ12iΣ13i-Q3+Q4TΣ15i(*)Σ22P3TNhi0P3TNτi(*)(*)Σ3300(*)(*)(*)Σ440(*)(*)(*)(*)-U2],
where Σ11i=sym{(P1+P2T)Ni+Q1+Q3}+U1+S1, Σ12i=-P2T+NiTP3, Σ13i=(P1+P2T)Nhi-Q1+Q2T, and Σ15i=(P1+P2T)Nτi. It is clear that Ξi<0 result in LMI stabilization conditions.
It can be easily seen that the matrices(P1+P2)T and P3 are nonsingular. Let ς:=diag{((P1+P2)T)-1,P3-1,((P1+P2)T)-1,((P1+P2)T)-1,P3-1}. By premultiplying ς, postmultiplying ςT to Ξi<0, and using the definitions N-i:=Ni(P1+P2)-1, N-hi:=Nhi(P1+P2)-1, N-τi:=NτiP3-1, Q-i:=(P1+P2T)-1Qi(P1+P2)-1, U-1:=(P1+P2T)-1U1(P1+P2)-1, U-2:=P3-1U1(P3T)-1, P-2:=P3-1P2(P1+P2)-1, and P-3:=P3-1, 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:
(3.18)[EK][CRCH[FJ]-F-J0]=[-R-H0],
with
(3.19)H=[H1H2⋯HM],K=[K1K2⋯KM],R=[R1R2⋯RM],F=[F0⋯00F⋯⋮⋮⋮⋱⋮0⋯⋯F]},J=[J0⋯00J⋯⋮⋮⋮⋱⋮0⋯⋯J]}Mtimes,
where F∈RM·p×M·qd, H∈Rn×M·qw, R∈Rn×M·qd, and K∈Rn×M·p. A necessary and sufficient condition for the existence of a solution [E,K] to (3.18) is [33]
(3.20)rank[CRCH[FJ]-F-J0-R-H0]=rank[CRCH[FJ]-F-J0].
Note that under this condition, a solution is obtained by
(3.21)[EK]=[-R-H0][CRCH[FJ]-F-J0]+-Z(I-[CRCH[FJ]-F-J0][CRCH[FJ]-F-J0]+),
where Z is an arbitrary matrix [33].Remark 3.3.
According to Theorem3.1, the observer parameters can be calculated in the following: (1)
calculateNτi=N-τiP3,(2)
computeT by solving (2.10) and calculate E from T=I+EC,(3)
computeNhi by solving (2.11),(4)
computeGi by solving (2.12),(5)
computeNi=N-iN-hi-1Nhi,(6)
computeKi by solving (2.9),(7)
computeLi from Li=Ki-NiE.Remark 3.4.
It is worth noting that the number of the variables to be determined in the LMI (3.1) is 3n2(M+2)+4n.Remark 3.5.
The reduced conservatism of Theorem3.1 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:
(4.1)A1=[-0.40.20.30.3-0.60.30.40.20.6],A2=[-0.450.3750.3750.15-0.4500.750.75-0.45],Ah1=[-0.010.030.020.01-0.00500.0050.07-0.03],Ah2=[-0.0050.010.040.01-0.0250.020.050.03-0.01],Aτ1=[0.0045-0.0037-0.0037-0.00150.00450-0.0075-0.00750.0045],Aτ2=[0.0170-0.0115-0.0135-0.00900.0210-0.0060-0.0230-0.0190-0.0030],B1=[1.0-0.5-0.5],B2=[-0.51.0-0.5],F=[11.2],R1=[0.4-0.40.4],R2=[0.40.4-0.8],C=[011101],J=0,H=0,τ=0.05,h=0.1,Aτ1=0.1I3,Aτ2=0.7I3.In the light of Theorem3.1 and Remark 3.2, we solved LMIs and obtained the following observer gains by using Matlab LMI Control Toolbox [34]
(4.2)E=[0.1945-0.1508-0.74740.6300-0.55240.4682],G1=[0.73630.56590.2927],G2=[-0.2518-0.0032-1.2461],N1=[-0.2699-0.21490.0543-0.2053-0.17780.0849-0.0775-0.0530-0.2330],N2=[-0.2734-0.1996-0.0930-0.4198-0.3263-0.14221.19090.92040.3656],L1=[0.15650.09730.2284-0.10710.61590.1294],L2=[0.32210.05570.03350.34800.8122-1.3116].
For simulation purpose, we simply choose w(t)=t/(1+t2) as the disturbance, u(t)=e-tsin(t) as the input signal, and μ1(ξ)=(1-sin(x2(t)))/2 and μ2(ξ)=(1+sin(x2(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.Figure 1
Time behaviour of states and corresponding error signals.


## 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]).



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