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

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 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 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}, 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+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_𝑖=1𝜇_𝑖(𝜉(𝑡))𝐴_𝑖𝑥(𝑡)+𝐴_𝜏𝑖𝑥̇(𝑡−𝜏)+𝐴_ℎ𝑖𝑥(𝑡−ℎ)+𝐵_𝑖𝑢(𝑡)+𝑅_𝑖𝑑(𝑡)+𝐻_𝑖𝑤(𝑡)𝑥(𝑡)(𝑡)[],𝑦(𝑡)(𝑡)(𝑡)(𝑡),=𝜙,𝑡∈−𝜅,0=𝐶𝑥+𝐹𝑑+𝐽𝑤 (2.1) with 𝜇_𝑖(𝜉(𝑡))≥0,^𝑀_𝑖=1𝜇_𝑖(𝜉(𝑡))=1, (2.2) 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^𝑞𝑑 is the unknown input, 𝑤(𝑡)∈𝐑^𝑞𝑤 is the external disturbance vector, and 𝑦∈𝐑^𝑝 is the measured output. 𝐴_𝑖∈𝐑^𝑛×𝑛, 𝐴_𝜏𝑖∈𝐑^𝑛×𝑛, 𝐵_𝑖∈𝐑^𝑛×𝑚, and 𝐶∈𝐑^𝑝×𝑛 define the 𝑖th local model. Matrices 𝑅_𝑖∈𝐑^{𝑛×𝑞𝑑} and 𝐹∈𝐑^{𝑝×𝑞𝑑} represent the influence matrices of the unknown inputs, and 𝐻_𝑖∈𝐑^{𝑛×𝑞𝑤} and 𝐽∈𝐑^{𝑝×𝑞𝑤} 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 (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: 𝑧̇(𝑡)=^𝑀_𝑖=1𝜇_𝑖(𝜉(𝑡))𝑁_𝑖𝑧(𝑡)+𝑁_𝜏𝑖𝑧̇(𝑡−𝜏)+𝑁_ℎ𝑖𝑧(𝑡−ℎ)+𝐺_𝑖𝑢(𝑡)+𝐿_𝑖𝑦(𝑡)𝑧(𝑡)(𝑡)[],𝑥(𝑡)(𝑡)(𝑡),=𝜑,𝑡∈−𝜅,0=𝑧−𝐸𝑦 (2.3) where the observer considers the same activation functions 𝜇_𝑖(⋅) as used for the T-S model (2.1). The variables 𝑁_𝑖∈𝐑^𝑛×𝑛, 𝑁_𝜏𝑖∈𝐑^𝑛×𝑛, 𝐺_𝑖∈𝐑^𝑛×𝑚, 𝐿_𝑖∈𝐑^𝑛×𝑝, and 𝐸∈𝐑^𝑛×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 𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡). (2.4)

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 𝑒(𝑡)=(𝐼+𝐸𝐶)𝑥(𝑡)−𝑧(𝑡)+𝐸𝐹𝑑(𝑡)+𝐸𝐽𝑤(𝑡). (2.5)

The dynamic of state estimation error is then given by 𝑒̇(𝑡)=^𝑀_𝑖=1𝜇_𝑖(𝜉)𝑁_𝑖𝑒(𝑡)+𝑇𝐴_𝑖−𝐾_𝑖𝐶−𝑁_𝑖𝑥(𝑡)+𝑇𝐴_𝜏𝑖𝑥̇(𝑡−𝜏)−𝑁_𝜏𝑖𝑧̇(𝑡−𝜏)+𝑇𝐴_ℎ𝑖𝑥(𝑡−ℎ)−𝑁_ℎ𝑖𝑧(𝑡−ℎ)+𝑇𝐵_𝑖−𝐺_𝑖𝑢(𝑡)+𝑇𝑅_𝑖−𝐾_𝑖𝐹𝑑(𝑡)+𝑇𝐻_𝑖−𝐾_𝑖𝐽𝑤(𝑡)+𝐸𝐹𝑑̇(𝑡)+𝐸𝐽𝑤̇(𝑡), (2.6) with 𝑇=𝐼+𝐸𝐶,𝐾_𝑖=𝑁_𝑖𝐸+𝐿_𝑖. (2.7)

Then, we get 𝑒̇(𝑡)=^𝑀_𝑖=1𝜇_𝑖(𝜉(𝑡))𝑁_𝑖𝑒(𝑡)+𝑁_𝜏𝑖𝑒̇(𝑡−𝜏)+𝑁_ℎ𝑖𝑒(𝑡−ℎ), (2.8) if the following constraints hold 𝑁_𝑖=𝑇𝐴_𝑖−𝐾_𝑖𝐶,𝑇𝐴_𝜏𝑖−𝑁_𝜏𝑖𝑇=0,𝑇𝐴_ℎ𝑖−𝑁_ℎ𝑖𝑇=0,𝑇𝐵_𝑖−𝐺_𝑖=0,𝑇𝑅_𝑖−𝐾_𝑖𝐸[]𝐹=0,𝐹𝐽=0,𝑇𝐻_𝑖−𝐾_𝑖𝐽=0. (2.9)(2.10)(2.11)(2.12)(2.13)(2.14)(2.15)

For description brevity, (2.8) can be written as 𝑒̇(𝑡)=𝑁_𝜇𝑒(𝑡)+𝑁_𝜇𝜏𝑒̇(𝑡−𝜏)+𝑁_𝜇ℎ𝑒(𝑡−ℎ), (2.16) where 𝑁_𝜇=∑^𝑀_𝑖=1𝜇_𝑖(𝜉(𝑡))𝑁_𝑖, 𝑁_𝜇ℎ=∑^𝑀_𝑖=1𝜇_𝑖(𝜉(𝑡))𝑁_ℎ𝑖, and 𝑁_𝜇𝜏=∑^𝑀_𝑖=1𝜇_𝑖(𝜉(𝑡))𝑁_𝜏𝑖.

Remark 2.1. It is noting that if 𝐸 is determined, we get 𝑇 from (2.7) and deduce directly 𝐺_𝑖, 𝑁_ℎ𝑖, and 𝑁_𝜏𝑖 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 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 (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, 𝑆₁,𝑈_𝑙>0, 𝑙=1,…,3, and 𝑃₂,𝑃₃,𝑁_𝑖,𝑁_ℎ𝑖,𝑁_𝑑𝑖 such that the conditions (2.9)–(2.15) and the following hold for all 𝑖∈𝐼_𝑀: Σ_11𝑖Σ_12𝑖Σ_13𝑖−𝑄₃+𝑄^𝑇₄𝑁_𝜏𝑖(∗)Σ₂₂𝑁_ℎ𝑖0𝑁_𝜏𝑖(∗)(∗)Σ₃₃(∗∗∗)Σ00)()(₄₄0(∗)(∗)(∗)(∗)−𝑈₂<0, (3.1) where Σ_11𝑖=sym{𝑁_𝑖+𝑄₁+𝑄₃}+𝑈₁+𝑆₁, Σ_12𝑖=−𝑃^𝑇₂+𝑁^𝑇_𝑖, Σ_13𝑖=𝑁_ℎ𝑖−𝑄₁+𝑄^𝑇₂, Σ₂₂=sym{−𝑃₃}+𝑈₂+𝑑𝑈₃+ℎ𝑆₁, Σ₃₃=−𝑆₁−sym{𝑄₂}, and Σ₄₄=−𝑈₁−sym{𝑄₄}.

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: 𝑉(𝑡)=³_𝑖=1𝑉_𝑖(𝑡), (3.2) where 𝑉₁(𝑡)=𝑒^𝑇(𝑡)𝑃₁𝑒(𝑡),𝑉₂(𝑡)=^𝑡_𝑡−ℎ𝑒^𝑇(𝜃)𝑆₁𝑒(𝜃)𝑑𝜃+⁰_−ℎ^𝑡_𝑡+𝜏𝑒^𝑇̇(𝜃)𝑆₂𝑒̇(𝜏)𝑉𝑑𝜃𝑑𝜏,₃(𝑡)=^𝑡_𝑡−𝜏𝑒^𝑇(𝜃)𝑈₁𝑒(𝜃)𝑑𝜃+^𝑡_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑈₂𝑒̇(𝑠)𝑑𝑠+⁰_−𝜏^𝑡_𝑡+𝜏𝑒^𝑇̇(𝜃)𝑈₃𝑒̇(𝜏)𝑑𝜃𝑑𝜏. (3.3) Time derivative of 𝑉₁(𝑡) along the system trajectory (2.16) becomes 𝑉̇₁(𝑡)=2𝑒^𝑇(𝑡)𝑃₁𝑁_𝜇𝑒(𝑡)+𝑁_𝜇𝜏𝑒̇(𝑡−𝜏)+𝑁_𝜇ℎ𝑒(𝑡−ℎ), (3.4) Then by taking the time derivative of 𝑉₂(𝑡) and 𝑉₃(𝑡), one can read 𝑉̇₂(𝑡)=&𝑒^𝑇(𝑡)𝑆₁𝑒(𝑡)−𝑒^𝑇(𝑡−ℎ)𝑆₁𝑒(𝑡−ℎ)+ℎ𝑒^𝑇̇(𝑡)𝑆₁𝑒̇(𝑡)−^𝑡_𝑡−ℎ𝑒^𝑇̇(𝜃)𝑆₁𝑒̇(𝜃)𝑉̇𝑑𝜃,₃(𝑡)=𝑒^𝑇(𝑡)𝑈₁𝑒(𝑡)−𝑒^𝑇(𝑡−𝜏)𝑈₁𝑒(𝑡−𝜏)+𝑒^𝑇̇(𝑡)𝑈₂𝑒̇(𝑡)−𝑒^𝑇̇(𝑡−𝜏)𝑈₂𝑒̇()𝑒̇𝑡−𝜏+𝜏^𝑇(𝑡)𝑈₃𝑒̇(𝑡)−^𝑡_𝑡−𝜏𝑒^𝑇̇(𝜏)𝑈₃𝑒̇(𝜏)𝑑𝜏. (3.5)(3.6) Moreover, from (2.16), the following equation holds for any matrices 𝑃₂ and 𝑃₃ with appropriate dimensions: 2𝑒^𝑇(𝑡)𝑃^𝑇₂+𝑒^𝑇̇(𝑡)𝑃^𝑇₃×−𝑒̇(𝑡)+𝑁_𝜇𝑒(𝑡)+𝑁_𝜇𝜏𝑒̇(𝑡−𝜏)+𝑁_𝜇ℎ𝑒(𝑡−ℎ)=0 (3.7) Furthermore, from the Leibniz-Newton formula, the following equations hold for any matrices 𝑄_𝑗,𝑗=1,…,4, with appropriate dimensions: 2𝑒^𝑇(𝑡)𝑄₁+𝑒^𝑇(𝑡−ℎ)𝑄₂𝑒(𝑡)−𝑒(𝑡−ℎ)−^𝑡_𝑡−ℎ𝑒̇(𝑠)2𝑒𝑑𝑠=0,_𝑇(𝑡)𝑄₃+𝑒^𝑇(𝑡−𝜏)𝑄₄𝑒(𝑡)−𝑒(𝑡−𝜏)−^𝑡_𝑡−𝜏𝑒̇(𝑠)𝑑𝑠=0. (3.8) From the obtained derivative terms in (3.4)–(3.6) and adding the left-hand side of (3.7)–(24) into 𝑉̇(𝑡), we obtain the following result for 𝑉̇(𝑡): 𝑉̇(𝑡)=𝜒^𝑇(𝑡)Π𝜒(𝑡)−^𝑡_𝑡−ℎ𝜒^𝑇(𝑡)𝜈₁+𝑒̇(𝑠)𝑆₁𝑆₁𝜒^𝑇(𝑡)𝜈₁+𝑒̇(𝑠)𝑆₁^𝑇−𝑑𝑠_{𝑡𝑡−𝜏}𝜒^𝑇(𝑡)𝜈₂+𝑒̇(𝑠)𝑈₃𝑈₃𝜒^𝑇(𝑡)𝜈₂+𝑒̇(𝑠)𝑈₃^𝑇𝑑𝑠, (3.9) or equivalently, 𝑉̇(𝑡)=𝜒^𝑇(𝑡)Π𝜒(𝑡), (3.10) with Π∶=Ξ+ℎ𝜈₁𝑆₁𝜈^𝑇₁+𝜏𝜈₂𝑈₃𝜈^𝑇₂, 𝑆₁∶=𝑆₁⁻¹, 𝑈₃∶=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 ΣΞ=¹¹Σ₁₂Σ₁₃−𝑄₃+𝑄^𝑇₄Σ₁₅(∗)Σ₂₂𝑃^𝑇₃𝑁_𝜇ℎ0𝑃^𝑇₃𝑁_𝜇𝜏(∗)(∗)Σ₃₃(∗∗∗)Σ00)()(₄₄0(∗)(∗)(∗)(∗)−𝑈₂, (3.11) where Σ₁₁=sym{(𝑃₁+𝑃^𝑇₂)𝑁_𝜇+𝑄₁+𝑄₃}+𝑈₁+𝑆₁, Σ₁₂=−𝑃^𝑇₂+𝑁^𝑇_𝜇𝑃₃, Σ₁₃=(𝑃₁+𝑃^𝑇₂)𝑁_𝜇ℎ−𝑄₁+𝑄^𝑇₂, Σ₁₅=(𝑃₁+𝑃^𝑇₂)𝑁_𝜇𝜏, Σ₂₂=sym{−𝑃₃}+𝑈₂+𝜏𝑈₃+ℎ𝑆₁, Σ₃₃=−𝑆₁−sym{𝑄₂}, and Σ₄₄=−𝑈₁−sym{𝑄₄}.
If a constant scalar 𝜅>0 satisfies the following condition: Π=Π+Π<0, (3.12) where Π∶=𝜅𝐼0⋯000⋯0⋮⋮⋱⋮00⋯0, (3.13) then 𝑉̇(𝑡)≤𝜒^𝑇(𝑡)Π𝜒(𝑡)=𝜒^𝑇(𝑡)Π−Π𝜒(𝑡)<−𝜒^𝑇(𝑡)Π𝜒(𝑡). (3.14) From the previous inequality, we can easily obtain, for all 𝑒(̂𝑡)≠0, 𝑉̇(𝑡)<−𝜅‖𝑒̂(𝑡)‖²<0. (3.15) 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) ^𝑀_𝑖=1𝜇_𝑖(𝜉(𝑡Ξ))^𝑖+Πℎ𝜈₁𝜏𝜈₂(∗)−ℎ𝑆₁0(∗)(∗)−𝜏𝑈₃<0, (3.16) with Ξ_𝑖=Σ_11𝑖Σ_12𝑖Σ_13𝑖−𝑄₃+𝑄^𝑇₄Σ_15𝑖(∗)Σ₂₂𝑃^𝑇₃𝑁_ℎ𝑖0𝑃^𝑇₃𝑁_𝜏𝑖(∗)(∗)Σ₃₃(∗∗∗)Σ00)()(₄₄0(∗)(∗)(∗)(∗)−𝑈₂, (3.17) where Σ_11𝑖=sym{(𝑃₁+𝑃^𝑇₂)𝑁_𝑖+𝑄₁+𝑄₃}+𝑈₁+𝑆₁, Σ_12𝑖=−𝑃^𝑇₂+𝑁^𝑇_𝑖𝑃₃, Σ_13𝑖=(𝑃₁+𝑃^𝑇₂)𝑁_ℎ𝑖−𝑄₁+𝑄^𝑇₂, and Σ_15𝑖=(𝑃₁+𝑃^𝑇₂)𝑁_𝜏𝑖. It is clear that Ξ_𝑖<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 𝜍^𝑇 to Ξ_𝑖<0, and using the definitions 𝑁_𝑖∶=𝑁_𝑖(𝑃₁+𝑃₂)⁻¹, 𝑁_ℎ𝑖∶=𝑁_ℎ𝑖(𝑃₁+𝑃₂)⁻¹, 𝑁_𝜏𝑖∶=𝑁_𝜏𝑖𝑃₃⁻¹, 𝑄_𝑖∶=(𝑃₁+𝑃^𝑇₂)⁻¹𝑄_𝑖(𝑃₁+𝑃₂)⁻¹, 𝑈₁∶=(𝑃₁+𝑃^𝑇₂)⁻¹𝑈₁(𝑃₁+𝑃₂)⁻¹, 𝑈₂∶=𝑃₃⁻¹𝑈₁(𝑃^𝑇₃)⁻¹, 𝑃₂∶=𝑃₃⁻¹𝑃₂(𝑃₁+𝑃₂)⁻¹, and 𝑃₃∶=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: []𝐸𝐊=𝐶𝐑𝐶𝐇𝐹𝐽−𝐅−𝐉0−𝐑−𝐇0, (3.18) with 𝐇=𝐻₁𝐻₂⋯𝐻_𝑀,𝐊=𝐾₁𝐾₂⋯𝐾_𝑀,𝐑=𝑅₁𝑅₂⋯𝑅_𝑀,𝐅=𝐹0⋯00𝐹⋯⋮⋮⋮⋱⋮0⋯⋯𝐹,𝐉=𝐽0⋯00𝐽⋯⋮⋮⋮⋱⋮0⋯⋯𝐽𝑀times, (3.19) where 𝐅∈𝐑^{𝑀⋅𝑝×𝑀⋅𝑞𝑑}, 𝐇∈𝐑^{𝑛×𝑀⋅𝑞𝑤}, 𝐑∈𝐑^{𝑛×𝑀⋅𝑞𝑑}, and 𝐊∈𝐑^{𝑛×𝑀⋅𝑝}. A necessary and sufficient condition for the existence of a solution [𝐸,𝐊] to (3.18) is [33] [][]rank𝐶𝐑𝐶𝐇𝐹𝐽−𝐅−𝐉0−𝐑−𝐇0=rank𝐶𝐑𝐶𝐇𝐹𝐽−𝐅−𝐉0. (3.20) Note that under this condition, a solution is obtained by =[]𝐸𝐊−𝐑−𝐇0𝐶𝐑𝐶𝐇𝐹𝐽−𝐅−𝐉0⁺[][]−𝑍𝐼−𝐶𝐑𝐶𝐇𝐹𝐽−𝐅−𝐉0𝐶𝐑𝐶𝐇𝐹𝐽−𝐅−𝐉0⁺, (3.21) where 𝑍 is an arbitrary matrix [33].

Remark 3.3. According to Theorem 3.1, the observer parameters can be calculated in the following: (1)calculate 𝑁_𝜏𝑖=𝑁_𝜏𝑖𝑃₃, (2)compute 𝑇 by solving (2.10) and calculate 𝐸 from 𝑇=𝐼+𝐸𝐶, (3)compute 𝑁_ℎ𝑖 by solving (2.11), (4)compute 𝐺_𝑖 by solving (2.12), (5)compute 𝑁_𝑖=𝑁_𝑖𝑁_ℎ⁻¹_𝑖𝑁_ℎ𝑖, (6)compute 𝐾_𝑖 by solving (2.9), (7)compute 𝐿_𝑖 from 𝐿_𝑖=𝐾_𝑖−𝑁_𝑖𝐸.

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 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: 𝐴₁=,𝐴−0.40.20.30.3−0.60.30.40.20.6₂=,𝐴−0.450.3750.3750.15−0.4500.750.75−0.45_ℎ1=,𝐴−0.010.030.020.01−0.00500.0050.07−0.03_ℎ2=,𝐴−0.0050.010.040.01−0.0250.020.050.03−0.01_𝜏1=,𝐴0.0045−0.0037−0.0037−0.00150.00450−0.0075−0.00750.0045_𝜏2=,𝐵0.0170−0.0115−0.0135−0.00900.0210−0.0060−0.0230−0.0190−0.0030₁=1.0−0.5−0.5,𝐵₂=1,𝑅−0.51.0−0.5,𝐹=1.2₁=0.4−0.40.4,𝑅₂=,0.40.4−0.8,𝐶=011101𝐽=0,𝐻=0,𝜏=0.05,ℎ=0.1,𝐴_𝜏1=0.1𝐼₃,𝐴_𝜏2=0.7𝐼₃. (4.1)

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] ,𝐺𝐸=0.1945−0.1508−0.74740.6300−0.55240.4682₁=0.73630.56590.2927,𝐺₂=,𝑁−0.2518−0.0032−1.2461₁=,𝑁−0.2699−0.21490.0543−0.2053−0.17780.0849−0.0775−0.0530−0.2330₂=,𝐿−0.2734−0.1996−0.0930−0.4198−0.3263−0.14221.19090.92040.3656₁=0.15650.09730.2284−0.10710.61590.1294,𝐿₂=.0.32210.05570.03350.34800.8122−1.3116 (4.2) 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]).

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.