Delay-Dependent Fuzzy Hyperbolic Model Based on Data-Driven Guaranteed Cost Control for a Class of Nonlinear Continuous-Time Systems with Uncertainties

Abstract

This paper develops the fuzzy hyperbolic model with time-varying delays guaranteed cost controller design via state-feedback for a class of nonlinear continuous-time systems with parameter uncertainties. A nonlinear quadratic cost function is developed as a performance measurement of the closed-loop fuzzy system based on fuzzy hyperbolic model with time-varying delays. Some sufficient conditions for the existence of such a fuzzy hyperbolic model based on data-driven guaranteed cost controller via state feedback are presented by a set of linear matrix inequalities (LMIs). A simulation example is provided to illustrate the effectiveness of the proposed approach.

1. Introduction

Since time delays are frequently encountered in various areas such as engineering systems, biology, and economics, and the existence of time delays is often the main cause of instability and poor performance of a control system, considerable attention has been paid to the problem of stability analysis and controller synthesis for time-delay systems [1, 2].

Recently, the problem of designing guaranteed cost controllers for uncertain time-delay systems has attracted a number of researchers’ attention [3–6]. Guaranteed cost control (GCC) for time-delay systems can also be categorized into delay-independent methods and delay-dependent methods. The recent research trend has been focus on delay-dependent methods. In [3], delay-dependent GCC was first proposed by utilizing model transformation. It was first illustrated that delay-dependent GCC can provide even less guaranteed cost than the delay-independent GCC methods. Reference [4] considered both state delays and input delays, and formulated the optimal guaranteed cost control problem which minimizes the upper bound of the closed-loop cost function. Reference [5] extend the delay-dependent method into the stabilization for time-delay T-S fuzzy systems, delay-dependent GCC problem for nonlinear systems with time-delays represented by the Takagi-Sugeno fuzzy mode was studied.

A novel continuous-time fuzzy model, called fuzzy hyperbolic model with time-varying delays (DFHM), has been proposed in [7]. Fuzzy Hyperbolic Model is essentially a data-driven model. The DFHM based on data-driven has its own distinguishing characteristics. Firstly, neither structure identification nor completeness design of premise variables space is required when the DFHM is used to approximate the nonlinear systems, therefore the computational effort of modeling the DFHM is lower than modeling the T-S models. Secondly, less computational effort is required when DFHM is used since only one LMI needs to be solved. Thirdly, the DFHM based on data-driven we designed is naturally fuzzy nonlinear saturated controller, which is suitable for applying to practical systems. Last but not least, DFHM is a new kind of fuzzy neural networks, whose nodes have clear physical meanings. Therefore, the advantages of the DFHM based on data-driven are more obvious.

In this paper, delay-dependent fuzzy guaranteed cost controller via state feedback design based on DFHM, called delay-dependent fuzzy hyperbolic model based on data-driven guaranteed cost controller (DD-DFHMGCC), is addressed. To the best of our knowledge, this is the first time to study the guaranteed cost controller problem of DFHM. By using the LMI technique, the DD-DFHMGCC design problem is converted into a feasible problem of LMI, which makes the prescribed attenuation level as small as possible, subject to some LMI constraints. A simulation example is finally presented to illustrate the effectiveness of the proposed design procedures.

2. System Description and Preliminaries

The DFHM modeling method for nonlinear systems was given in [7]. The following definition is addressed.

Definition 2.1. Given a plant with 𝑛 state variables 𝑥(𝑡)=[𝑥₁(𝑡),…,𝑥_𝑛(𝑡)]^𝑇, we call the fuzzy rule base a hyperbolic type fuzzy rule base (HFRB) if it satisfies the following conditions.(1) For each output variable 𝑥̇_𝑟(𝑡), 𝑟=1,2,…,𝑛, the 𝑘th fuzzy rule has the following form: 𝑅^𝑘. If 𝑥₁(𝑡) is 𝐴₀^1,𝑠 and 𝑥₁(𝑡−𝜏₁₁(𝑡)) is 𝐴₁^1,𝑠 and … 𝑥₁(𝑡−𝜏_1𝑑1(𝑡)) is 𝐴_𝑑^1,𝑠₁ … and 𝑥_𝑛(𝑡) is 𝐴₀^𝑛,𝑠 and 𝑥_𝑛(𝑡−𝜏_𝑛1(𝑡)) is 𝐴₁^𝑛,𝑠 and …𝑥_𝑛(𝑡−𝜏_𝑛𝑑𝑛(𝑡)) is 𝐴^𝑛,s_𝑑𝑛, then 𝑥̇_𝑟(𝑡)=𝑐^𝑟_𝐴01,𝑠+⋯+𝑐^𝑟_{𝐴𝑑11,𝑠}+𝑐^𝑟_𝐴02,𝑠+⋯+𝑐^𝑟_{𝐴𝑑22,𝑠}+⋯+𝑐^𝑟_𝐴0^𝑛,𝑠+⋯+𝑐^𝑟_𝐴𝑑𝑛^𝑛,𝑠+𝑢^𝑟,𝑠∈{+,−}, (2.1) where 𝐴_𝑖^𝑗,𝑠_𝑗 is the fuzzy set of 𝑥_𝑗(𝑡−𝜏_𝑗𝑖𝑗(𝑡)), which include 𝑃^𝑗_𝑖𝑗 (positive) and 𝑁^𝑗_𝑖𝑗 (negative), respectively, 𝑐^𝑟_{𝐴𝑖𝑗𝑗,𝑠} is a constant term, 𝑑_𝑗 represents the number of transmission delays associated with 𝑥_𝑗, 𝜏_𝑗𝑖𝑗(𝑡)>0 is the time-varying transmission delay with 𝜏_𝑗0(𝑡)=0, 𝑖_𝑗=0,1,…,𝑑_𝑗,  𝑗=1,2,…,𝑛.(2) The state variables in the If-part are optional, the same as the constant terms in the Then-part. That is the constant term 𝑐^𝑟_{𝐴𝑖𝑗𝑗,𝑠} in the Then-part is corresponding to 𝐴_𝑖^𝑗,𝑠_𝑗 in the If-part.(3) There are 2^{𝑛+∑𝑛𝑗=1𝑑𝑗} fuzzy rules in every 𝑥̇_𝑟(𝑡), that is, all the possible 𝑃^𝑗_𝑖𝑗 and 𝑁^𝑗_𝑖𝑗 combinations of input variables in the “If” part and all the linear combinations of constants in the “Then” part (not including 𝑢^𝑟). So there are total 𝑛2^{𝑛+∑𝑛𝑗=1𝑑𝑗} fuzzy rules in the rule base.(4) For every 𝑥̇_𝑟(𝑟=1,2,…,𝑛), we construct the same premise fuzzy subsets, but the conclusion parameters are different.

Lemma 2.2. Given 𝑛 HFRBs, if the membership functions of 𝑃^𝑗_𝑖𝑗 and 𝑁^𝑗_𝑖𝑗 are defined as 𝜇_𝑃^𝑗_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑗𝑖𝑗(𝑡)=𝑒^{−(1/2)(𝑥𝑗(𝑡−𝜏𝑗𝑗𝑖(𝑡))−𝑘𝑗𝑗𝑖)2},𝜇_𝑁^𝑗_𝑖𝑗𝑥_𝑗𝑡−𝜏_𝑗𝑖𝑗(𝑡)=𝑒^{−(1/2)(𝑥𝑗(𝑡−𝜏𝑗𝑗𝑖(𝑡))+𝑘𝑗𝑗𝑖)2}, (2.2) where 𝑖_𝑗=0,1,…,𝑑_𝑗, 𝑗=1,2,…,𝑛. 𝑘_𝑗𝑖𝑗>0 is a positive constant. Then the system can be derived as 𝑥̇(𝑡)=𝐴tanh(𝐾𝑥(𝑡))+^𝐽_𝑖=1𝐴_𝑖tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)+𝐼, (2.3) where 𝐼 is a constant vector, 𝐴 and 𝐴_𝑖 are constant matrix, (3) is called a fuzzy hyperbolic model with time-varying delays (DFHM).
From Definition 2.1, if we set 𝑐^𝑟_{𝐴𝑖𝑗𝑗,+} and 𝑐^𝑟_{𝐴𝑖𝑗𝑗,−} negative to each other, we can obtain a homogeneous DFHM: 𝑥̇(𝑡)=𝐴tanh(𝐾𝑥(𝑡))+^𝐽_𝑖=1𝐴_𝑖tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡). (2.4)
Since the difference between (2.3) and (2.4) is only the constant vector term in (2.3), there is essentially no difference between the control of (2.3) and (2.4). In this paper, we will design a fuzzy 𝐻∞ guaranteed cost controller based on DFHM described in (2.4).

3. Fuzzy Hyperbolic with Time-Varying Delays Guaranteed Cost Control Design via State-Feedback

The DFHM for the nonlinear time-delay systems with parameter uncertainty is proposed as the following form: 𝑥̇(𝑡)=(𝐴+Δ𝐴)tanh(𝐾𝑥(𝑡))+^𝐽_𝑖=1𝐴_𝑖+Δ𝐴_𝑖tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)+()𝑢(𝑡),𝑥(𝑡)(𝑡)[−𝐵+Δ𝐵=𝜙,∀𝑡∈𝜏,0], (3.1) where 𝑥(𝑡)∈𝑅^𝑛 and 𝑢(𝑡)∈𝑅^𝑚 denote the state vector and input vector, respectively; 𝐴∈𝑅^𝑛×𝑛, 𝐴_𝑖∈𝑅^𝑛×𝑛 and 𝐵∈𝑅^𝑛×𝑚, are known real constant matrices; 𝜏_𝑖(𝑡)=[𝜏_1𝑖(𝑡),𝜏_2𝑖(𝑡),…,𝜏_𝑛𝑖(𝑡)]^𝑇 is the bounded time-varying delay and is assumed to satisfy 0<𝜏_𝑗𝑖(𝑡)≤𝜏<∞ and 𝜏̇_𝑗𝑖(𝑡)≤ℎ, where 𝜏 and ℎ are known constant scalars, 𝑗=1,2,…,𝑛, 𝑖=1,2,…,𝐽. The initial condition 𝜙(𝑡) is given by initial vector function, which is continuous for −𝜏≤𝑡≤0; Δ𝐴(𝑡)∈𝑅^𝑛×𝑛, Δ𝐴_𝑖(𝑡)∈𝑅^𝑛×𝑛 and Δ𝐵(𝑡)∈𝑅^𝑛×𝑚 are time-varying parameter uncertainty matrices and satisfy the condition Δ𝐴(𝑡)Δ𝐴_𝑖(𝑡)Δ𝐵(𝑡)=𝑀𝐹(𝑡)𝑁𝑁_𝑖𝑁_𝑏, (3.2) where 𝑀,𝑁,𝑁_𝑖(𝑖=1,2,…,𝐽) and 𝑁_𝑏 are known real constant matrices of appropriate dimension, and 𝐹(𝑡) is an unknown matrix function satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼 (𝐼 is an identity matrix with appropriate dimension). Such parametric uncertainties are said to be admissible.

Definition 3.1. Consider system (3.1) with the following cost function: 𝐽=^∞₀tanh^𝑇(𝐾𝑥(𝑡))𝑆tanh(𝐾𝑥(𝑡))−𝑢^𝑇(𝑡)(𝑡𝑢(𝑡)((𝑡,𝑅𝑢)𝑑𝑡,=𝐺tanh𝐾𝑥)) (3.3)(3.4) where 𝑆 and 𝑅 are symmetric, positive-definite matrices; 𝐺 is the feedback gain. The controller is called fuzzy hyperbolic model with time-varying delays guaranteed cost controller (DFHMGCC) if there exist a fuzzy hyperbolic control 𝑢(𝑡) as in (3.4) and a scalar 𝐽₀ such that the closed-loop system is asymptotically stable and the closed-loop value of the cost function (3.3) satisfies 𝐽≤𝐽₀. 𝐽₀ is said to be a guaranteed cost and control law 𝑢(𝑡) is said to be a fuzzy hyperbolic with time-varying delays guaranteed cost control law for system (3.1).
With the control law (3.4) the overall closed-loop system can be written as: 𝑥̇(𝑡)=(𝐴+Δ𝐴+(𝐵+Δ𝐵)𝐺)tanh(𝐾𝑥(𝑡))+^𝐽_𝑖=1𝐴_𝑖+Δ𝐴_𝑖tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡),𝑥(𝑡)=𝜙(𝑡),∀𝑡∈[−𝜏,0]. (3.5)
For convenience, let Δ𝐴∶=Δ𝐴(𝑡),Δ𝐴_𝑖∶=Δ𝐴_𝑖(𝑡)(𝑖=1,…,𝐽),Δ𝐵∶=Δ𝐵(𝑡).

Lemma 3.2 (see [8]). Given appropriate dimension matrices 𝑀, 𝐸, and 𝐹 satisfying 𝐹^𝑇𝐹≤𝐼, for any real scalar 𝜀>0, the following result holds: 𝑀𝐹𝐸+𝐸^𝑇𝐹^𝑇𝑀^𝑇≤𝜀𝑀𝑀^𝑇+𝜀⁻¹𝐸^𝑇𝐸. (3.6)

Lemma 3.3 (see [9]). For any constant positive definite matrix 𝑊∈𝑅^𝑚×𝑚, a scalar 𝛽>0, a function 𝜂∶[0,𝛽]→𝑅⁺, and the vector function 𝜈∶[𝛽−𝜂(𝛽),𝛽]→𝑅^𝑚 such that the integrations in the following are well defined, then 𝜂𝛽^𝛽_{𝛽𝛽−𝜂}𝜈^𝑇(𝑠)𝑊𝜈(𝑠)𝑑𝑠≥^𝛽_{𝛽𝛽−𝜂}𝜈(𝑠)𝑑𝑠^𝑇𝑊^𝛽_{𝛽𝛽−𝜂}𝜈(𝑠)𝑑𝑠. (3.7)

Lemma 3.4 (see [10]). For any vector 𝜍∈𝑅^𝑛, 𝜍^𝑇=[𝜍₁,𝜍₂,…,𝜍_𝑛], and diagonal positive definite matrix 𝑋, the following result holds: taṅh^𝑇(𝜍)𝑋taṅh(𝜍)≤𝜍Ẋ𝜍.̇ (3.8)

Then, we have the following results.

Theorem 3.5. Given scalars 𝜏 and ℎ, for the nonlinear system (3.1) and associated cost function (3.3), if there exist positive scalars 𝜀₀₁,𝜀₀₂,𝜀₀₃,𝜀₀₄,𝜀_i(𝑖=1,2,…,𝐽), a positive definite diagonal matrix 𝑃=diag[𝑝₁,…,𝑝_𝑛], matrices 𝑄_𝜆>0(𝜆=1,2) and 𝐿 such that the matrix inequality Ξ₁₁0Ξ₁₃Ξ₁₄𝑃𝑁^𝑇+𝐿^𝑇𝑁^𝑇_𝑏0𝐿^𝑇𝑁^𝑇_𝑏𝑃𝑁^𝑇0∗Ξ₂₂Ξ₂₃000000∗∗Ξ₃₃Ξ₃₄0Ξ₃₆00Ξ₃₉∗∗∗Ξ₄₄00000∗∗∗∗−𝜀₀₄𝐼0000∗∗∗∗∗−𝜀₀₃𝐼000∗∗∗∗∗∗−𝜀₀₂𝐼00∗∗∗∗∗∗∗−𝜀₀₁𝐼0∗∗∗∗∗∗∗∗−Ξ₉₉<0 (3.9) holds, then, the control law, 𝑢(𝑡)=𝐺tanh(𝐾𝑥(𝑡)) is a fuzzy hyperbolic with time-varying delays guaranteed cost control law and 𝐽₀=2^𝑛_𝑖=1𝑝_𝑖⁻¹lncosh𝑘_𝑖0𝜙_𝑖(0)+^𝐽_𝑖=1⁰_{−𝜏𝑖(0)}tanh^𝑇𝐾_𝑖𝜙(𝑠)𝑃⁻¹𝑄₁𝑃tanh𝐾_𝑖𝜙(𝑠)+𝑑𝑠_{𝐽𝑖=10−𝜏0𝛽}taṅh^𝑇𝐾_𝑖𝑥(𝛼)𝑃𝑄₂𝑃taṅh𝐾_𝑖𝑥(𝛼)𝑑𝛼𝑑𝛽, (3.10) where Ξ₁₁=𝐾𝐴𝑃+𝑃𝐴^𝑇𝐾+𝐾𝐵𝐿+𝐿^𝑇𝐵^𝑇𝐾+𝜀₀₁𝐾𝑀𝑀^𝑇𝐾+𝜀₀₂𝐾𝑀𝑀^𝑇𝐾+𝑆+𝐿^𝑇𝑅𝐿+^𝐽_𝑖=1𝜀_𝑖𝐾𝑀𝑀^𝑇Ξ𝐾,₁₃=𝐾𝐴₁𝑃𝐾𝐴₂𝑃⋯𝐾𝐴_𝐽𝑃,Ξ₁₄=𝐾₁𝐴𝑃+𝐾₁𝐵𝐿^𝑇⋯𝐾_𝐽𝐴𝑃+𝐾_𝐽𝐵𝐿^𝑇,Ξ₂₂=diag𝑄₁−𝜏⁻¹𝑄₂𝑄₁−𝜏⁻¹𝑄₂⋯𝑄₁−𝜏⁻¹𝑄_2𝐽,Ξ₂₃=diag𝜏⁻¹𝑄₂𝜏⁻¹𝑄₂⋯𝜏⁻¹𝑄_2𝐽,Ξ₃₃=diag−(1−ℎ)𝑄₁−𝜏⁻¹𝑄₂⋯−(1−ℎ)𝑄₁−𝜏⁻¹𝑄_2𝐽,Ξ₃₄=diag𝑃𝐴^𝑇₁𝐾₁𝑃𝐴^𝑇₂𝐾₂⋯𝑃𝐴^𝑇_𝐽𝐾_𝐽,Ξ₃₆=diag𝑃𝑁^𝑇₁𝑃𝑁^𝑇₂⋯𝑃𝑁^𝑇_𝐽,Ξ₃₉=diag𝑃𝑁^𝑇₁𝑃𝑁^𝑇₂⋯𝑃𝑁^𝑇_𝐽,Ξ₄₄=diag−𝜏⁻¹𝑄₂+𝜀₀₃𝐾₁𝑀𝑀^𝑇𝐾₁⋯−𝜏⁻¹𝑄₂+𝜀₀₃𝐾_𝐽𝑀𝑀^𝑇𝐾_𝐽+𝜀₀₄𝑀^𝑇𝐾₁⋯𝑀^𝑇𝐾_𝐽^𝑇𝑀^𝑇𝐾₁⋯𝑀^𝑇𝐾_𝐽,Ξ₉₉=diag𝜀₁𝐼𝜀₂𝐼⋯𝜀_𝐽𝐼, (3.11) for any time-varying delay 𝜏_𝑖(𝑡)=[𝜏_1𝑖(𝑡),𝜏_2𝑖(𝑡),…,𝜏_𝑛𝑖(𝑡)]^𝑇 satisfying 0<𝜏_𝑖(𝑡)≤𝜏<∞ and 𝜏̇_𝑖(𝑡)≤ℎ, 𝑖=1,2,…,𝐽, and ∗ denotes the entries induced by symmetry.

Proof. A Lyapunov-Krasovskii functional candidate for the time-varying delay system (3.5) is chosen as follows: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (3.12) where 𝑉₁(𝑡)=2^𝑛_𝑖=1𝑝_𝑖lncosh𝑘_𝑖0𝑥_𝑖(𝑡),𝑉₂(𝑡)=^𝐽_𝑖=1^𝑡_{𝑡−𝜏𝑖(𝑡)}tanh^𝑇𝐾_𝑖𝑥(𝑠)𝑄₁tanh𝐾_𝑖𝑥(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝐽_𝑖=1⁰_−𝜏^𝑡_𝑡+𝛽taṅh^𝑇𝐾_𝑖𝑥(𝛼)𝑄₂taṅh𝐾_𝑖𝑥(𝛼)𝑑𝛼𝑑𝛽, (3.13) and scalars 𝑝_𝑖>0(𝑖=1,2,…,𝑛), matrices 𝑄_𝜆>0 (𝜆=1,2), 𝑃=diag[𝑝₁,𝑝₂,…,𝑝_𝑛] is diagonal positive definite matrix, and 𝑘_𝑖0 (𝑖=1,2,…,𝑛) are defined in (2.2).
Taking the derivative of 𝑉(𝑡) with respect to 𝑡 along the trajectory of (3.5) yields 𝑉̇(𝑡)_(3.5)=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡), (3.14) where 𝑉̇₁(𝑡)=2tanh^𝑇((𝑡𝑥̇(𝑡),𝑉̇𝐾𝑥))𝐾𝑃₂(𝑡)≤^𝐽_𝑖=1tanh^𝑇𝐾_𝑖𝑥(𝑡)𝑄₁tanh𝐾_𝑖𝑥(𝑡)−(1−ℎ)tanh^𝑇𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)𝑄₁tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡),𝑉̇₃(𝑡)=^𝐽_𝑖=1𝜏taṅh^𝑇𝐾_𝑖𝑥(𝑡)𝑄₂taṅh𝐾_𝑖𝑥(𝑡)−^𝑡_𝑡−𝜏taṅh^𝑇𝐾_𝑖𝑥(𝑠)𝑄₂taṅh𝐾_𝑖𝑥(𝑠)𝑑𝑠. (3.15)(3.16)(3.17)
According to 0<𝜏_𝑖(𝑡)≤𝜏<∞ and Lemma 3.3, we have −^𝑡_𝑡−𝜏taṅh^𝑇𝐾_𝑖𝑥(𝑠)𝑄₂taṅh𝐾_𝑖𝑥(𝑠)𝑑𝑠≤−𝜏⁻¹tanh𝐾_𝑖𝑥(𝑡)−tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)^𝑇𝑄₂tanh𝐾_𝑖𝑥(𝑡)−tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡). (3.18)
From (3.17), (3.18), and Lemma 3.4, we have 𝑉̇₃(𝑡)≤^𝐽_𝑖=1𝜏𝑥̇^𝑇(𝑡)𝐾_𝑖𝑄₂𝐾_𝑖𝑥̇(𝑡)−𝜏⁻¹tanh𝐾_𝑖𝑥(𝑡)−tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)^𝑇×𝑄₂tanh𝐾_𝑖𝑥(𝑡)−tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡). (3.19)
From Lemma 3.2 and (3.2), we have 2tanh^𝑇(𝐾𝑥(𝑡))𝐾𝑃(Δ𝐴+Δ𝐵𝐺)tanh(𝐾𝑥(𝑡))≤tanh^𝑇(𝐾𝑥(𝑡))𝜀₀₁𝐾𝑃𝑀𝑀^𝑇𝑃𝐾+𝜀⁻¹₀₁𝑁^𝑇𝑁tanh(𝐾𝑥(𝑡))+tanh^𝑇(𝐾𝑥(𝑡))𝜀₀₂𝐾𝑃𝑀𝑀^𝑇𝑃𝐾+𝜀⁻¹₀₂𝑁_𝑏𝐺^𝑇𝑁_𝑏𝐺tanh(𝐾𝑥(𝑡)),2tanh^𝑇(𝐾𝑥(𝑡))𝐾𝑃^𝐽_𝑖=1Δ𝐴_𝑖tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)≤^𝐽_𝑖=1𝜀_𝑖tanh^𝑇(𝐾𝑥(𝑡))𝐾𝑃𝑀𝑀^𝑇𝑃𝐾tanh(𝐾𝑥(𝑡))+𝜀_𝑖⁻¹tanh^𝑇𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)𝑁^𝑇_𝑖𝑁_𝑖tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡). (3.20)
So, we have 𝑉̇(𝑡)_(3.5)≤𝜁^𝑇(𝑡)𝜁(𝑡)−tanh^𝑇(𝐾𝑥(𝑡))𝑆tanh(𝐾𝑥(𝑡))−𝑢^𝑇(𝑡)𝑅𝑢(𝑡), (3.21) where 𝜁^𝑇(𝑡)=tanh^𝑇𝐾(𝑥(𝑡))𝜓^𝑇(𝑡)𝜂^𝑇(𝑡),𝜓^𝑇(𝑡)=tanh^𝑇𝐾₁𝑥(𝑡)tanh^𝑇𝐾₂𝑥(𝑡)⋯tanh^𝑇𝐾_𝐽𝑥(𝑡),𝜂^𝑇(𝑡)=tanh^𝑇𝐾₁𝑥𝑡−𝜏₁(𝑡)tanh^𝑇𝐾₂𝑥𝑡−𝜏₂(𝑡)⋯tanh^𝑇𝐾_𝐽𝑥𝑡−𝜏_𝐽(𝑡),∏=¹¹+𝜀⁻¹₀₁𝑁^𝑇𝑁+𝜀⁻¹₀₂𝑁_𝑏𝐺^𝑇𝑁_𝑏𝐺0∏₁₃∗∏₂₂∏₂₃∗∗∏₃₃+Ψ₃+𝜏Ψ^𝑇₁𝑄₂Ψ₁0𝜏Ψ^𝑇₁𝑄₂Ψ₂∗0𝜏Ψ^𝑇₂𝑄₂Ψ₂,∗∗0₁₁=𝐾𝑃(𝐴+𝐵𝐺)+(𝐴+𝐵𝐺)^𝑇𝑃𝐾+𝜀₀₁+𝜀₀₂𝐾𝑃𝑀𝑀^𝑇𝑃𝐾+^𝐽_𝑖=1𝜀_𝑖𝐾𝑃𝑀𝑀^𝑇𝑃𝐾+𝑆+𝐺^𝑇𝑅𝐺,₁₃=𝐾𝑃𝐴₁𝐾𝑃𝐴₂⋯𝐾𝑃𝐴_𝐽,₂₂=diag𝑄₁−𝜏⁻¹𝑄₂𝑄₁−𝜏⁻¹𝑄₂⋯𝑄₁−𝜏⁻¹𝑄_2𝐽,₂₃=diag𝜏⁻¹𝑄₂𝜏⁻¹𝑄₂⋯𝜏⁻¹𝑄_2𝐽,₃₃=diag−(1−ℎ)𝑄₁−𝜏⁻¹𝑄₂−(1−ℎ)𝑄₁−𝜏⁻¹𝑄₂⋯−(1−ℎ)𝑄₁−𝜏⁻¹𝑄_2𝐽,Ψ₁=𝐴+Δ𝐴+(𝐵+Δ𝐵𝐺)^𝑇𝐾₁⋯𝐴+Δ𝐴+(𝐵+Δ𝐵𝐺)^𝑇𝐾_𝐽,Ψ₂=𝐴₁+Δ𝐴₁^𝑇𝐾₁⋯𝐴_𝐽+Δ𝐴_𝐽^𝑇𝐾_𝐽,Ψ₃=diag𝜀₁⁻¹𝑁^𝑇₁𝑁₁𝜀₂⁻¹𝑁^𝑇₂𝑁₂⋯𝜀_𝐽⁻¹𝑁^𝑇_𝐽𝑁_𝐽. (3.22) Let 𝜃<0. Then 𝑉̇(𝑡)|_(3.5)≤−tanh^𝑇(𝐾𝑥(𝑡))𝑆tanh(𝐾𝑥(𝑡))−𝑢^𝑇(𝑡)𝑅𝑢(𝑡)≤−𝜆_𝑚(𝑆)‖tanh(𝐾𝑥(𝑡))‖²<0.
Thus, the closed-loop system is asymptotically stable. Furthermore, integrating 𝑉̇(𝑡)|_(3.5)≤ −tanh^𝑇(𝐾𝑥(𝑡)) 𝑆tanh(𝐾𝑥(𝑡))− 𝑢^𝑇(𝑡)𝑅𝑢(𝑡) from 0 to 𝑡_𝑓 yields ^𝑡𝑓₀tanh^𝑇(𝐾𝑥(𝑡))𝑆tanh(𝐾𝑥(𝑡))−𝑢^𝑇(𝑡)𝑅𝑢(𝑡)𝑑𝑡≤𝑉(0)−𝑉𝑡_𝑓. (3.23)
Since 𝑉(𝑡)≥0 and 𝑉̇(𝑡)<0, lim_{𝑡𝑓→∞}𝑉(𝑡_𝑓)=𝜇 which is a nonnegative constant. When 𝑡_𝑓→∞, (3.23) becomes ^𝑡𝑓₀tanh^𝑇(𝐾𝑥(𝑡))𝑆tanh(𝐾𝑥(𝑡))−𝑢^𝑇(𝑡)𝑅𝑢(𝑡)𝑑𝑡≤𝑉(0)=2^𝑛_𝑖=1𝑝_𝑖lncosh𝑘_𝑖0𝜙_𝑖(0)+^𝐽_𝑖=1⁰_−𝜏i(0)tanh^𝑇𝐾_𝑖𝜙(𝑠)𝑄₁tanh𝐾_𝑖𝜙(𝑠)+𝑑𝑠_{𝐽𝑖=10−𝜏0𝛽}taṅh^𝑇𝐾_𝑖𝑥(𝛼)𝑄₂taṅh𝐾_𝑖𝑥(𝛼)𝑑𝛼𝑑𝛽. (3.24) By Schur complement, ∏ can be rewritten as the following form: ∏₁₁+𝜀⁻¹₀₁𝑁^𝑇𝑁+𝜀⁻¹₀₂𝑁_𝑏𝐺^𝑇𝑁_𝑏𝐺0∏₁₃∏₁₄∗∏₂₂∏₂₃0∗∗∏₃₃+Ψ₃∏₃₄∗∗∗∏₄₄+000∏₁₄∗000∗∗0∏₃₄∗∗∗0<0, (3.25) where ₁₄=(𝐴+𝐵𝐺)^𝑇𝐾₁(𝐴+𝐵𝐺)^𝑇𝐾₂⋯(𝐴+𝐵𝐺)^𝑇𝐾_𝐽,₁₄=(Δ𝐴+Δ𝐵𝐺)^𝑇𝐾₁(Δ𝐴+Δ𝐵𝐺)^𝑇𝐾₂⋯(Δ𝐴+Δ𝐵𝐺)^𝑇𝐾_𝐽,₃₄=diag𝐴^𝑇₁𝐾₁𝐴^𝑇₂𝐾₂⋯𝐴^𝑇_𝐽𝐾_𝐽,₃₄=diagΔ𝐴^𝑇₁𝐾₁Δ𝐴^𝑇₂𝐾₂⋯Δ𝐴^𝑇_𝐽𝐾_𝐽,₄₄=diag−𝜏⁻¹𝑄₂−𝜏⁻¹𝑄₂⋯−𝜏⁻¹𝑄_2𝐽. (3.26)
Comparing inequality (3.25) with Lemma 3.2, we can obtain 000∏₁₄∗000∗∗0∏₃₄∗∗∗0≤𝜀₀₄000∏₁₄^𝑇000∏₁₄+𝜀⁻¹₀₄𝑁+𝑁_𝑏𝐺000^𝑇𝑁+𝑁_𝑏𝐺000+𝜀₀₃000∏₃₄^𝑇000∏₃₄+𝜀⁻¹₀₃00∏^′₃₄0^𝑇00∏^′₃₄0, (3.27) where ∏₁₄=𝑀^𝑇𝐾₁𝑀^𝑇𝐾₂⋯𝑀^𝑇𝐾_𝐽,∏₁₄=diag𝑀^𝑇𝐾₁𝑀^𝑇𝐾₂⋯𝑀^𝑇𝐾_𝐽,∏^′₃₄=diag𝑁₁𝑁₂⋯𝑁_𝐽. (3.28)
Thus, the necessary and sufficient condition for inequality (3.25) to hold is that there exists a positive constant 𝜀₀₃>0 and 𝜀₀₄>0 such that ∏₁₁+𝜀⁻¹₀₁𝑁^𝑇𝑁+𝜀⁻¹₀₂𝑁_𝑏𝐺^𝑇𝑁_𝑏𝐺0∏₁₃∏₁₄∗∏₂₂∏₂₃0∗∗∏₃₃+Ψ₃∏₃₄∗∗∗∏₄₄+𝜀⁻¹₀₄𝑁+𝑁_𝑏𝐺000^𝑇𝑁+𝑁_𝑏𝐺000+𝜀⁻¹₀₃00∏₄^′0^𝑇00∏₄^′0<0, (3.29) where ₄₄=₄₄+𝜀₀₄∏^𝑇₁₄∏₁₄+𝜀₀₃∏^𝑇₃₄∏₃₄. (3.30)
By Schur complement, (3.29) is equivalent to ∏₁₁0∏₁₃∏₁₄𝑁+𝑁_𝑏𝐺^𝑇0𝑁_𝑏𝐺^𝑇𝑁^𝑇0∗∏₂₂∏₂₃000000∗∗∏₃₃∏₃₄0∏₃₄^′𝑇00∏₃₄^′𝑇∗∗∗∏₄₄00000∗∗∗∗−𝜀₀₄𝐼0000∗∗∗∗∗−𝜀₀₃𝐼000∗∗∗∗∗∗−𝜀₀₂𝐼00∗∗∗∗∗∗∗−𝜀₀₁𝐼0∗∗∗∗∗∗∗∗−Θ<0, (3.31) where Θ=diag𝜀₁𝐼𝜀₂𝐼⋯𝜀_𝐽𝐼. (3.32) Pre- and post-multiplying diag[_𝑃⁻¹_𝑃⁻¹_⋯𝑃⁻¹_2𝐽+1,^𝐼⋯𝐼_3𝐽+3] to (3.31), and letting 𝑃=𝑃⁻¹, 𝑄_𝜆=𝑃⁻¹𝑄_𝜆𝑃⁻¹ (𝜆=1,2), 𝑆=𝑃⁻¹𝑆𝑃⁻¹ and 𝐿=𝐺𝑃⁻¹, (3.9) can be obtained by Schur complement. When LMI (3.9) is feasible, the guaranteed cost controller designed ensures the closed-loop system to be asymptotically stable and an upper bound of the closed-loop cost function given by 𝐽₀=2^𝑛_𝑖=1𝑝_𝑖lncosh𝑘_𝑖0𝜙_𝑖(0)+^𝐽_𝑖=1⁰_{−𝜏𝑖(0)}tanh^𝑇𝐾_𝑖𝜙(𝑠)𝑄₁tanh𝐾_𝑖𝜙(𝑠)+𝑑𝑠_{𝐽𝑖=10−𝜏0𝛽}taṅh^𝑇𝐾_𝑖𝑥(𝛼)𝑄₂taṅh𝐾_𝑖𝑥(𝛼)𝑑𝛼𝑑𝛽=2^𝑛_𝑖=1𝑝_𝑖⁻¹lncosh𝑘_𝑖0𝜙_𝑖(0)+^𝐽_𝑖=1⁰_{−𝜏𝑖(0)}tanh^𝑇𝐾_𝑖𝜙(𝑠)𝑃⁻¹𝑄₁𝑃⁻¹tanh𝐾_𝑖𝜙(𝑠)+𝑑𝑠_{𝐽𝑖=10−𝜏0𝛽}taṅh^𝑇𝐾_𝑖𝑥(𝛼)𝑃⁻¹𝑄₂𝑃⁻¹taṅh𝐾_𝑖𝑥(𝛼)𝑑𝛼𝑑𝛽. (3.33)
The proof is completed.

In fact, any feasible solution to (3.9) yields a suitable robust guaranteed cost controller. A better robust guaranteed cost control law minimizes the upper bound 𝐽₀. Then, we can obtain Theorem 3.6.

Theorem 3.6. Consider the nonlinear system (3.1) and its associated cost function (3.3). If the optimization problem min_{𝜀𝑖,𝑃,𝑄1,𝑄2,𝑉,𝐺}(𝑉)(1)(),(2)𝛿+𝛼⋅Tr𝑆𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜LMIS3.5−𝑉Φ^𝑇Ω^𝑇∗−𝑃⁻¹𝑄₁𝑃0∗∗−𝑃⁻¹𝑄₂𝑃𝜀̂<0ℎ𝑎𝑠𝑎𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛_𝑖,𝑃,𝑄₁,𝑄₂,𝑉,𝐺,𝑤ℎ𝑒𝑟𝑒ΦΦ^𝑇=⁰_−𝜏i(0)tanh𝐾_𝑖𝜙(𝑠)tanh^𝑇𝐾_𝑖𝜙(𝑠)𝑑𝑠,ΩΩ^𝑇=^𝐽_𝑖=1⁰_−𝜏⁰_𝛽taṅh^𝑇𝐾_𝑖𝑥(𝛼)taṅh𝐾_𝑖𝑥(𝛼)𝑑𝛼𝑑𝛽. (3.34)
Tr(⋅) denotes the trace of the matrix (⋅), 𝛿=2∑^𝑛_𝑖=1𝑝_𝑖⁻¹ln(cosh(𝑘_𝑖0𝜙_𝑖(0))), then, the corresponding guaranteed cost control law, 𝑢(𝑡)=𝐺tanh(𝐾𝑥(𝑡)) is an optimal guaranteed cost control. Under this control law the closed-loop cost function (3.2) is minimized.

Proof. By Theorem 3.6, the control law constructed in terms of any feasible solution of (3.9) is a guaranteed cost control law. According to Schur complement, the condition (2)) is equivalent to Φ^𝑇𝑃⁻¹𝑄₁𝑃Φ+Ω^𝑇𝑃⁻¹𝑄₂𝑃Ω<𝑉.
Since Tr(𝐴𝐵)=Tr(𝐵𝐴), we have ^𝐽_𝑖=1⁰_{−𝜏𝑖(0)}tanh^𝑇𝐾_𝑖𝜙(𝑠)𝑃⁻¹𝑄₁𝑃tanh𝐾_𝑖𝜙(𝑠)+𝑑𝑠_{𝐽𝑖=10−𝜏0𝛽}taṅh^𝑇𝐾_𝑖𝑥(𝛼)𝑃𝑄₂𝑃taṅh𝐾_𝑖𝑥(𝛼)=𝑑𝛼𝑑𝛽_{𝐽𝑖=10−𝜏i(0)}Trtanh^𝑇𝐾_𝑖𝜙(𝑠)𝑃⁻¹𝑄₁𝑃tanh𝐾_𝑖𝜙(𝑠)+𝑑𝑠_{𝐽𝑖=10−𝜏0𝛽}Trtaṅh^𝑇𝐾_𝑖𝑥(𝛼)𝑃𝑄₂𝑃taṅh𝐾_𝑖𝑥(𝛼)𝑑𝛼𝑑𝛽=TrΦΦ^𝑇𝑃⁻¹𝑄₁𝑃+TrΩΩ^𝑇𝑃⁻¹𝑄₂𝑃=TrΦ^𝑇𝑃⁻¹𝑄₁𝑃Φ+TrΩ^𝑇𝑃⁻¹𝑄₂(𝑉).𝑃Ω<Tr (3.35)
So it follows that 𝐽₀=2^𝑛_𝑖=1𝑝_𝑖⁻¹lncosh𝑘_𝑖0𝜙_𝑖(0)+^𝐽_𝑖=1⁰_−𝜏i(0)tanh^𝑇𝐾_𝑖𝜙(𝑠)𝑃⁻¹𝑄₁𝑃⁻¹tanh𝐾_𝑖𝜙(𝑠)+𝑑𝑠_{𝐽𝑖=10−𝜏0𝛽}taṅh^𝑇𝐾_𝑖𝑥(𝛼)𝑃⁻¹𝑄₂𝑃⁻¹taṅh𝐾_𝑖𝑥(𝛼)(𝑉).𝑑𝛼𝑑𝛽≤𝛿+Tr (3.36)
Therefore, the guaranteed cost controller subject to (3.34) is an optimal guaranteed cost control. Under this controller the closed-loop cost function (3.3) is minimized.
This completes the proof.

4. Simulation

In the following, we will give an example to demonstrate the effectiveness of the obtained results.

Example 4.1. We apply the above analysis technique to a continuous stirred tank reactor (CSTR) in which the first-order irreversible exothermic reaction 𝐴→𝐵 occurs [11, 12]. The material and energy balance equations are 𝑥̇₁(𝑡)=−1𝜆+Δ𝐴₀𝑥₁(𝑡)+1𝜆−1𝑥₁𝑡−𝜏₁₁(𝑡)+𝐷_𝛼1−𝑥₁(𝑡)𝑥exp²(𝑡)1+𝑥₂(𝑡)/𝛾₀,𝑥̇₂(𝑡)1=−𝜆+𝛽𝑥₂(𝑡)+1𝜆−1𝑥₂𝑡−𝜏₂₁(𝑡)+𝐻𝐷_𝛼1−𝑥₁(𝑡)𝑥exp²(𝑡)1+𝑥₂(𝑡)/𝛾₀+𝛽𝑢(𝑡), (4.1) where Δ𝐴₀ denotes the parameter uncertainty of original nonlinear systems. Constants 𝐻,𝛽,𝐷_𝛼, and 𝛾₀ are all positive. Here, the model parameters are given as 𝛾₀=20,𝐻=8,𝛽=0.3,𝐷_𝛼=0.072,𝜆=0.8. (4.2)

Suppose that we have the following hyperbolic type fuzzy rule bases: If 𝑥₁(𝑡) is 𝐴₀^1,+, 𝑥₁(𝑡−𝜏₁₁(𝑡)) is 𝐴₁^1,+, 𝑥₁(𝑡−𝜏₁₂(𝑡)) is 𝐴₂^1,+, 𝑥₂(𝑡) is 𝐴₀^2,+, then 𝑥̇₁=𝐶¹_𝑥1+𝐶_𝑑𝑥11+ 𝐶_𝑑𝑥12+𝐶¹_𝑥2; … If 𝑥₁(𝑡) is 𝐴₀^1,−, 𝑥₁(𝑡−𝜏₁₁(𝑡)) is 𝐴₁^1,−, 𝑥₁(𝑡−𝜏₁₂(𝑡)) is 𝐴₂^1,−,𝑥₂(𝑡) is 𝐴₀^2,−, then 𝑥̇₁=−𝐶¹_𝑥1−𝐶_𝑑𝑥11 −𝐶_𝑑𝑥12−𝐶¹_𝑥2; If 𝑥₁(𝑡) is 𝐴₀^1,+, 𝑥₂(𝑡) is 𝐴₀^2,+, 𝑥₂(𝑡−𝜏₂₁(𝑡)) is 𝐴₁^2,+, 𝑥₂(𝑡−𝜏₂₂(𝑡)) is 𝐴₂^2,+, then 𝑥̇₂=𝐶²_𝑥1+𝐶²_𝑥2+ 𝐶_𝑑𝑥21+𝐶_𝑑𝑥22; … If 𝑥₁(𝑡) is 𝐴₀^1,−, 𝑥₂(𝑡) is 𝐴₀^2,−, 𝑥₂(𝑡−𝜏₂₁(𝑡)) is 𝐴₁^2,−, 𝑥₂(𝑡−𝜏₂₂(𝑡)) is 𝐴₂^2,−, then 𝑥̇₂=−𝐶²_𝑥1−𝐶²_𝑥2− 𝐶_𝑑𝑥21−𝐶_𝑑𝑥22.

Here we choose membership functions of 𝑃^𝑗_𝑖𝑗 and 𝑁^𝑗_𝑖𝑗 (𝑖_𝑗=0,1,2,𝑗=1,2) as (2.2).

Then, we have the following model: 𝑥̇(𝑡)=𝐴tanh(𝐾𝑥(𝑡))+²_𝑖=1𝐴_𝑖tanh𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)+𝐵𝑢(𝑡), (4.3) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡)]^𝑇,  𝑥(𝑡−𝜏_𝑖(𝑡))=[𝑥₁(𝑡−𝜏_1𝑖(𝑡)), 𝑥₂(𝑡−𝜏_2𝑖(𝑡))]^𝑇, 𝐶𝐴=^1𝑥1𝐶¹_𝑥2𝐶²_𝑥1𝐶²_𝑥1,𝐴₁=𝐶_𝑑𝑥1100𝐶_𝑑𝑥21,𝐴₂=𝐶_𝑑𝑥1200𝐶_𝑑𝑥22,𝐾=diag𝑘₁₀𝑘₂₀,𝐾_𝑖=diag𝑘_1𝑖𝑘_2𝑖,𝐶=0𝐷_𝑥2,𝑖=1,2. (4.4)

We choose that 𝜏_𝑖1(𝑡)=0.8|1.1sin²(𝑡)−0.6|, 𝜏_𝑖2(𝑡)=0.5|1.1sin²(𝑡)−0.6|,𝑖=1,2, and the initial condition 𝜙(𝑡)=[1−1]^𝑇, for all 𝑡∈[−0.80]. The parameters of DFHM (4.3) can be obtained using neural network BP algorithm [13, 14] as follows: 𝐴=−3.61170.49225.08210.1119,𝐴₁=0.519300−2.3141,𝐴₂=,0.058900−0.1868𝐾=diag0.31650.2801,𝐾₁=diag0.62110.1207,𝐾₂=diag0.05620.0003. (4.5)

Other parameters are given as follows: 𝐵=0001. (4.6) Consider the modeling errors, we assume 𝑀=[0.20.2]^𝑇, 𝑁=[0.10.2], 𝑁₁=[0.20.1], 𝐹(𝑡)=sin𝑡. Solve the LMI problem in (3.34). We obtain 𝑢=((𝑡−0.0536−0.1082−6.0117−33.9723tanh𝐾𝑥)) (4.7) and corresponding 𝐽=80.9812. Figure 1 depicts the behavior of the closed-loop system based on the DFHM for the initial conditions 𝑥(0)=[1−1]^𝑇. Figure 2 shows that the control input 𝑢. Simulation result demonstrates the effectiveness of the fuzzy hyperbolic with time-varying delays guaranteed control approach.

Figure 1 

States response of the closed-loop system.

Figure 2 

The trajectories of control input.

5. Conclusion

In this paper, the delay-dependent fuzzy hyperbolic guaranteed cost control for nonlinear uncertain systems with time delay using DFHM has been considered. The design problem of DD-DFHMGCC is converted into linear matrix inequalities. The controller designed achieves closed-loop asymptotic stability and provides an upper bound on the closed-loop value of cost function. Simulation example is provided to illustrate the design procedure of the proposed method.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61104021), the National High Technology Research and Development Program of China (2012AA040104), and the Fundamental Research Funds for the Central Universities of China (N100304007).