State-Feedback 𝐻_∞ Control for LPV System Using T-S Fuzzy Linearization Approach

Abstract

This paper discusses the linear parameter varying (LPV) gain scheduling control problem based on the Takagi-Sugeno (T-S) fuzzy linearization approach. Firstly, the affine nonlinear parameter varying (ANPV) description of a class of nonlinear dynamic processes is defined; that is, at any scheduling parameter, the corresponding system is affine nonlinear as usual. For such a class of ANPV systems, a kind of developed T-S fuzzy modeling procedure is proposed to deal with the nonlinearity, instead of the traditional Jacobian linearization approach. More concretely, the evaluation system for the approximation ability of the novelly developed T-S fuzzy modeling procedure is established. Consequently, the LPV T-S fuzzy system is obtained which can approximate the ANPV system with required accuracy. Secondly, the notion of piecewise parameter-dependent Lyapunov function is introduced, and then the stabilization problem and the state-feedback 𝐻_∞ control problem of the LPV T-S fuzzy system are studied. The sufficient conditions are given in linear matrix inequalities (LMIs) form. Finally, a numerical example is provided to demonstrate the availability of the above approaches. The simulation results show the high approximation accuracy of the LPV T-S fuzzy system to the ANPV system and the effectiveness of the LPV T-S fuzzy gain scheduling control.

1. Introduction

It is well known that the gain scheduling control is an efficient solution for the control of nonlinear dynamic processes [1, 2]. In particular, due to the advantage to carry forward the stability and dynamic performance analysis, the LPV gain scheduling control has been popularly studied [3–12]. Currently, there are mainly two ways to realize the LPV gain scheduling control: the linear fractional transformation (LFT) gain scheduling technique based on a scaled version of the small gain theorem [5–8] and the quadratic gain scheduling technique based on Lyapunov theory [9–12]. However, both of them adopt the Jacobian linearization approach to deal with the nonlinearity around each steady operating point which means that the number of scheduling parameters is equal to the number of variables relevant to the nonlinearity. As a result, too many scheduling parameters are brought in and too much computation burden is caused for the control design; that is, while the gridding process of scheduling parameters and the parameterization of decision variables in LMIs are executed, the number of LMIs would have a rapid increase. Besides, due to the local linearization around each steady operating point, the control performance is restricted within the local region which would become a weakness when the controlled variables vary widely. Also, it is hard to guarantee the system stability and control performance during the scheduling process. Hence, there requires a further study on the appropriate linearization approach for the ANPV system which can both efficiently reduce the number of scheduling parameters and enlarge the linearization range to extend the effective region of control performance.

On the other hand, the T-S fuzzy model was firstly proposed by Takagi and Sugeno in 1985 [13] and developed by Sugeno and Kang [14], which was data based. To linearize the nonlinearity, the research on the model-based T-S fuzzy modeling procedure was proposed by Kawamoto et al. [15] via the nonlinear sector method which was an exact modeling procedure with constant consequent parts and nonlinear membership functions. Then, Kluska [16] exploited the local approximation method to establish T-S fuzzy model with homogeneous linear consequent parts. In order to evaluate the approximation ability, Abonyi and Babuska [17] discussed the ways to analyze the relation between the local and global approximation performances of the T-S fuzzy model. Subsequently, Teixeira and Zak [18] gave the T-S fuzzy modeling procedure with homogeneous linear consequent parts by solving a convex optimization problem. And Tanaka and Wang [19] also realized the similar results through uniform partition to the input space of premise variables and proved the universal approximation ability. However, as far as we know, the T-S fuzzy linearization to the ANPV system has been rarely studied in most of the literature.

If the T-S fuzzy linearization is applied to the ANPV system, the choosing of the scheduling parameters and the premise variables can be separated. The nonlinearity of scheduling parameters can remain and that of premise variables should be linearized within their varying regions. As a result, the number of scheduling parameters can be reduced. Meanwhile, because the T-S fuzzy linearization range can be extended to any optional one, the nonlinearity with widely varying state variables can be dealt with for the nonlinear dynamic processes. Due to the above points, it is valuable to study the T-S fuzzy modeling procedure for the ANPV system. Furthermore, in order to efficiently reduce the number of T-S fuzzy rules, an ununiform partition method is utilized, and the evaluation system is also established to adjust the approximation performance. Then, the ANPV system can be linearized on demand and the LPV T-S fuzzy system with required accuracy can be obtained. However, a challenging control problem for such a class of systems is correspondingly formulated. For the T-S fuzzy control, the piecewise Lyapunov functions have been brought in to improve the solvability compared with that based on the common Lyapunov function [20–22]. In addition, considering the approximation error to the ANPV system, the T-S fuzzy 𝐻_∞ control has been also studied [23–26]. Hence, the T-S fuzzy control needs to be generalized to the LPV system while improving the solvability by introducing the notion of piecewise Lyapunov function. In addition, for the sake of improving the control performance, the approximation error of LPV T-S fuzzy system to the ANPV system should be considered.

In the paper, the T-S fuzzy modeling procedure with homogeneous linear consequent parts is discussed for the ANPV system while an ununiform partition method is utilized to reduce the number of T-S fuzzy rules. To adjust the approximation accuracy, the evaluation system for the approximation performance of the LPV T-S fuzzy system is established. Then, in order to improve the solvability of the LPV T-S fuzzy gain scheduling control, the notion of piecewise parameter-dependent Lyapunov function is introduced and the LPV T-S fuzzy gain scheduling control design based on the piecewise parameter-dependent Lyapunov functions is studied. Meanwhile, while taking the approximation error to the ANPV system in consideration, the sufficient conditions of the stabilization problem and the state-feedback 𝐻_∞ control problem of the LPV T-S fuzzy system are given in LMIs form. More concretely, the main contributions of the paper are listed as follows.(i)The T-S fuzzy approximation with required accuracy aiming at the ANPV system is studied, and then the nonlinearity of the ANPV system can be dealt with and the LPV T-S fuzzy system with required accuracy can be obtained. (ii)The stabilization problem of the LPV T-S fuzzy system is studied by bringing in the piecewise parameter-dependent Lyapunov functions while the sufficient conditions are given in LMIs form. (iii)Considering the approximation error to the ANPV system, the state-feedback 𝐻_∞ control problem of the LPV T-S fuzzy system is studied based on the piecewise parameter-dependent Lyapunov functions. The sufficient conditions are given in both Riccati inequalities form and LMIs form.

The rest of the paper is organized as follows. In Section 2, the developed T-S fuzzy modeling procedure utilizing un-uniform partition method is proposed aiming at the ANPV system. And the evaluation system for the approximation performance of the LPV T-S fuzzy system with homogeneous consequent parts is established. Then, the LPV T-S fuzzy system with required accuracy can be obtained by applying the above ways to the ANPV system. In Section 3, the notion of piecewise parameter-dependent Lyapunov function is introduced and the stabilization problem of the LPV T-S fuzzy system is studied. In Section 4, the state-feedback 𝐻_∞ control problem of the LPV T-S fuzzy system is studied and the sufficient conditions are given in both Riccati inequalities form and LMIs form. In Section 5, a numerical example is provided to demonstrate the availability of the above approaches. Section 6 concludes the paper.

2. Establishment of LPV T-S Fuzzy System

In this section, the ANPV description of a class of nonlinear dynamic processes is defined aiming at which kind of novelly developed T-S fuzzy modeling procedure is proposed and the corresponding evaluation system for the approximation performance is established. Both can be combined to deal with the nonlinearity of the ANPV system to get the LPV T-S fuzzy system with required accuracy.

For a class of nonlinear dynamic processes, if we can achieve their nonlinear descriptions in mathematical models, the ANPV system can be obtained by choosing the proper variables as scheduling parameters. The general form of it can be defined as 𝑥̇𝑧=𝑓(𝑥,𝜃)+𝑑₀(𝑥,𝜃)𝜔+𝑔₀()𝑢𝜑()𝑥,𝜃𝑥,𝜃+𝑔₁()𝑢,𝑥,𝜃 (1) where 𝜃 is a vector of the scheduling parameters which may be system parameters, external inputs, or other parameters; 𝑥 is a column vector of the state variables with 𝑛 dimensions; and 𝑓(𝑥,𝜃),…,𝑔₁(𝑥,𝜃) are the nonlinear functions of 𝑥 and 𝜃. Then, referring to the way in [18], the T-S fuzzy modeling procedure aiming at (1) can be developed to linearize the nonlinearity of the ANPV system.

Considering (1), around the zero state, the Jacobian linearization approach can be easily executed to get homogeneous linear model in the local region. However, for the nonzero states, it would be unavailable. Assume the nonzero operating state 𝑥^𝑖₀ which can be steady or transient one and corresponds to the ith fuzzy rule, 𝑖=1,2,…,𝐿. It should be noted that 𝑥^𝑖₀ may include the part or all of the state variables and it is gotten from the partition to the input space of premise variables. Firstly, we establish the homogeneous linear model in the vicinity of 𝑥^𝑖₀. Because the local linear model should approximate the local dynamics in the vicinity of 𝑥^𝑖₀, we can get 𝑓(𝑥,𝜃)+𝑑₀(𝑥,𝜃)𝜔+𝑔₀()𝑢𝜑()𝑥,𝜃𝑥,𝜃+𝑔₁()𝑢≈𝐴𝑥,𝜃_𝑖(𝜃)𝐵^𝑖₁(𝜃)𝐵^𝑖₂(𝜃)𝐶^𝑖₁(𝜃)0𝐷^𝑖₁₂(𝜃)𝑥𝜔𝑢,𝑓𝑥^𝑖₀,𝜃+𝑑₀𝑥^𝑖₀,𝜃𝜔+𝑔₀𝑥^𝑖₀𝑢𝜑𝑥,𝜃_𝑖0,𝜃+𝑔₁𝑥^𝑖₀𝑢=𝐴,𝜃_𝑖(𝜃)𝐵^𝑖₁(𝜃)𝐵^𝑖₂(𝜃)𝐶^𝑖₁(𝜃)0𝐷^𝑖₁₂(𝜃)𝑥^𝑖₀𝜔𝑢, (2) where 𝜔 and 𝑢 are arbitrarily changed variables and 𝐴_𝑖(𝜃),…,𝐷^𝑖₁₂(𝜃) are parameter-dependent matrices.

Take the first equation 𝑥̇=𝑓(𝑥,𝜃)+𝑑₀(𝑥,𝜃)𝜔+𝑔₀(𝑥,𝜃)𝑢 for example. Due to that, 𝑓(𝑥,𝜃),𝑑₀(𝑥,𝜃), and 𝑔₀(𝑥,𝜃) are nonlinear functions of operating state 𝑥 at any 𝜃, and the matrices 𝐴_𝑖(𝜃), 𝐵^𝑖₁(𝜃), and 𝐵^𝑖₂(𝜃) only depend on 𝑥. Moreover, because 𝜔 and 𝑢 are arbitrarily changed, we can uniquely determine that 𝑑₀𝑥^𝑖₀,𝜃=𝐵^𝑖₁(𝜃),𝑔₀𝑥^𝑖₀,𝜃=𝐵^𝑖₂(𝜃),𝑓𝑥^𝑖₀,𝜃=𝐴_𝑖(𝜃)𝑥^𝑖₀,𝑓(𝑥,𝜃)≈𝐴_𝑖(𝜃)𝑥. (3)

Define 𝑎^𝑖_𝑙(𝜃)^𝑇, a row vector with 𝑛 dimensions, as the 𝑙th row of the matrix 𝐴_𝑖(𝜃). Then, condition (3) can be equivalently represented as 𝑓_𝑙(𝑥,𝜃)≈𝑎^𝑖_𝑙(𝜃)^𝑇𝑥,𝑙=1,2,…,𝑛, (4) where 𝑓_𝑙(𝑥,𝜃) is the 𝑙th row of 𝑓(𝑥,𝜃). Additionally, assume 𝑓_𝑙(0,𝜃)=0 and 𝑓_𝑙(𝑥,𝜃)∈𝐶¹, which means that 𝑓_𝑙 and 𝜕𝑓_𝑙/𝜕𝑥 are continuous functions on the compact set Θ⊂𝑅^𝑛. Meanwhile, 𝑓(𝑥^𝑖₀,𝜃)=𝐴_𝑖(𝜃)𝑥^𝑖₀ can be equivalently represented as 𝑓_𝑙𝑥^𝑖₀,𝜃=𝑎^𝑖_𝑙(𝜃)^𝑇𝑥^𝑖₀,𝑙=1,2,…,𝑛. (5)

Expanding 𝑓_𝑙(𝑥,𝜃) of (4) around the operating state 𝑥^𝑖₀ and neglecting second- and higher-order terms, we can get 𝑓_𝑙𝑥^𝑖₀,𝜃+∇^𝑇𝑓_𝑙𝑥^𝑖₀,𝜃𝑥−𝑥^𝑖₀≈𝑎^𝑖_𝑙(𝜃)^𝑇𝑥, (6) where ∇𝑓_𝑙(𝑥^𝑖₀,𝜃) is the gradient, a column vector, of 𝑓_𝑙(𝑥,𝜃) evaluated at 𝑥^𝑖₀. Substituting (5) into (6), we can get ∇^𝑇𝑓_𝑙𝑥^𝑖₀,𝜃𝑥−𝑥^𝑖₀≈𝑎^𝑖_𝑙(𝜃)^𝑇𝑥−𝑥^𝑖₀, (7) where 𝑥 is arbitrarily close to 𝑥^𝑖₀. From (7), it can be found that the coefficient vector 𝑎^𝑖_𝑙(𝜃) needs to be estimated. And in the vicinity of 𝑥^𝑖₀, an optimization problem can be constructed by defining an optimal index to evaluate the estimation. Here, define that the notation 𝐶^𝑚 represents the set of functions which is 𝑚-order continuous and differentiable on the domain of input variables.

Lemma 1. Consider the following constrained optimization problem: (𝑥)(𝑥)𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒𝑓𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜ℎ=0, (8) where 𝑓(𝑥) is 𝑓∶𝑅^𝑛→𝑅 and 𝑓∈𝐶¹; it is a convex function on the feasible set Ω={𝑥∈𝑅^𝑛∶ℎ(𝑥)=0}, where ℎ∶𝑅^𝑛→𝑅^𝑚 and ℎ∈𝐶¹. Assume that Ω is convex and there exist 𝑥^∗∈Ω and 𝜆∈𝑅^𝑚 such that ∇_Ω𝑓𝑥^∗+𝜆∇_Ωℎ𝑥^∗=0, (9) where 𝜆 is the Lagrange multiplier. Then, 𝑥^∗ is the optimal solution of 𝑓 over Ω and 𝜆 is the sufficient condition for the convex optimization problem [27]. Particularly, when ℎ(𝑥) is affine linear, Ω is convex.

Considering the estimation of 𝑎^𝑖_𝑙(𝜃) around 𝑥^𝑖₀≠0 in (7), the optimal index for 𝑓_𝑙(𝑥,𝜃) corresponding to the 𝑖th fuzzy rule can be defined as 𝐼^𝑖_𝑓𝑙𝑎^𝑖_𝑙(𝜃=1)2∇𝑓_𝑙𝑥^𝑖₀,𝜃−𝑎^𝑖_𝑙(𝜃)²₂. (10)

Then, the optimal problem can be constructed as minimize_{Ω𝑖𝑓𝑙}𝐼^𝑖_𝑓𝑙𝑎^𝑖_𝑙(𝜃)subjecttoℎ^𝑖_𝑓𝑙𝑎^𝑖_𝑙(𝜃)=0, (11) where (5) should be fulfilled as an equality constraint and ℎ^𝑖_𝑓𝑙(𝑎^𝑖_𝑙(𝜃))=𝑓_𝑙(𝑥^𝑖₀,𝜃)−𝑎^𝑖_𝑙(𝜃)^𝑇𝑥^𝑖₀.

Transform the objective function in (10) into the form 𝐼^𝑖_𝑓𝑙𝑎^𝑖_𝑙(𝜃=1)2∇𝑓_𝑙𝑥^𝑖₀,𝜃−𝑎^𝑖_𝑙(𝜃)²₂=12∇𝑓_𝑙𝑥^𝑖₀,𝜃−𝑎^𝑖_𝑙(𝜃)^𝑇∇𝑓_𝑙𝑥^𝑖₀,𝜃−𝑎^𝑖_𝑙(𝜃), (12) which is a quadratic function of the unknown coefficient vector 𝑎^𝑖_𝑙(𝜃) on 𝑅^𝑛. Obviously, it means that 𝐼^𝑖_𝑓𝑙(𝑎^𝑖_𝑙(𝜃)) is a convex function. Considering the equality constraint ℎ^𝑖_𝑓𝑙(𝑎^𝑖_𝑙(𝜃)), it is the linear function of 𝑎^𝑖_𝑙(𝜃), so the feasible set Ω^𝑖_𝑓𝑙={𝑎^𝑖_𝑙(𝜃)∈𝑅^𝑛∶ℎ(𝑎^𝑖_𝑙(𝜃))=0} is convex. Therefore, the optimal problem is a convex optimization problem and can be solved according to Lemma 1.

Computing the derivative of 𝐼^𝑖_𝑓𝑙(𝑎^𝑖_𝑙(𝜃)) and ℎ^𝑖_𝑓𝑙(𝑎^𝑖_𝑙(𝜃)) about 𝑎^𝑖_𝑙(𝜃), we can get ∇_Ω^𝑖_𝑓𝑙𝐼^𝑖_𝑓𝑙𝑎^𝑖_𝑙(𝜃)=𝑎^𝑖_𝑙(𝜃)−∇𝑓_𝑙𝑥^𝑖₀,∇,𝜃_{Ω𝑖𝑓𝑙}ℎ^𝑖_𝑓𝑙𝑎^𝑖_𝑙(𝜃)=𝑥^𝑖₀. (13) Then, substituting (13) into (9), we can get 𝑎^𝑖_𝑙(𝜃)−∇𝑓_𝑙𝑥^𝑖₀,𝜃+𝜆^𝑖_𝑓𝑙(𝜃)𝑥^𝑖₀=0. (14) Considering the equality constraint (5), the Lagrange multiplier 𝜆^𝑖_𝑓𝑙(𝜃) can be represented as 𝜆^𝑖_𝑓𝑙(𝜃)=∇^𝑇𝑓_𝑙𝑥^𝑖₀,𝜃𝑥^𝑖₀−𝑎^𝑖_𝑙(𝜃)^𝑇𝑥^𝑖₀𝑥^𝑖₀²₂=∇^𝑇𝑓_𝑙𝑥^𝑖₀,𝜃𝑥^𝑖₀−𝑓_𝑙𝑥^𝑖₀,𝜃𝑥^𝑖₀²₂. (15) As a result, we can get the column vector 𝑎^𝑖_𝑙(𝜃)=∇𝑓_𝑙𝑥^𝑖₀+𝑓,𝜃_𝑙𝑥^𝑖₀,𝜃−∇^𝑇𝑓_𝑙𝑥^𝑖₀,𝜃𝑥^𝑖₀𝑥^𝑖₀²₂𝑥^𝑖₀,𝑥^𝑖₀≠0. (16) Then, the ith T-S fuzzy rule corresponding to the operating state 𝑥^𝑖₀ can be chosen as Rule𝑖:IF𝑥₁is𝐴^𝑖₁𝑥₁,…,and𝑥_𝑛is𝐴^𝑖_𝑛𝑥_𝑛,THEN𝑓^𝑖_𝑙(𝑥,𝜃)=𝑎^𝑖_𝑙(𝜃)^𝑇𝑥, (17) where 𝑓^𝑖_𝑙(𝑥,𝜃) represents the 𝑖th local estimation of the 𝑙th row of 𝑓(𝑥,𝜃), 𝑎^𝑖_𝑙(𝜃) is the corresponding coefficient vector of the 𝑖th fuzzy rule, a column vector, and 𝐴^𝑖_𝑗(𝑥_𝑗) represents the fuzzy set and membership function of the 𝑗th premise variable, 𝑥_𝑗, in the 𝑖th fuzzy rule and 𝑗=1,2,…,𝑛.

The activated possibility of the 𝑖th fuzzy rule under a group of premise variables 𝑥 is computed by 𝜔_𝑖(𝑥)=^𝑛_𝑗=1𝐴^𝑖_𝑗𝑥_𝑗,𝑖=1,2,…,𝐿,𝑗=1,2,…,𝑛. (18)

The weight coefficients are computed and the center-of-gravity method is used for defuzzification: 𝛼_𝑖(𝑥)=𝜔_𝑖(𝑥)∑^𝐿_𝑖=1𝜔_𝑖(𝑥),𝑖=1,2,…,𝐿, (19) which fulfills that ^𝐿_𝑖=1𝛼_𝑖(𝑥)𝛼=1,_𝑖(𝑥)≥0. (20)

Finally, the overall T-S fuzzy model can be obtained as 𝑓_𝑙(𝑥,𝜃)=^𝐿_𝑖=1𝛼_𝑖(𝑥)𝑓^𝑖_𝑙(𝑥,𝜃)=^𝐿_𝑖=1𝛼_𝑖(𝑥)𝑎^𝑖_𝑙(𝜃)^𝑇𝑥. (21)

Usually, based on (21), the evaluation of the overall approximation performance about the nonlinear function 𝑓_𝑙(𝑥,𝜃) is defined as 𝑒_𝑓𝑙(𝑥,𝜃)₂=𝑓_𝑙(𝑥,𝜃)−𝑓_𝑙(𝑥,𝜃)₂. (22)

Subsequently, a kind of un-uniform partition method is utilized for the above T-S fuzzy modeling procedure which can efficiently reduce the number of T-S fuzzy rules. Meanwhile, define the subset Θ₀={𝑥∣|𝑥_𝑗|<𝜒₀} on Θ, where 𝜒₀ is a predefined positive scalar and 𝑎⁰_𝑙=𝜕𝑓_𝑙(𝑥,𝜃)/𝜕𝑥∣_𝑥=0 can be gotten at the zero state 𝑥=0. Considering the nonzero states, they can be represented as 𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀=(𝑥^𝑘1₀₁,𝑥^𝑘2₀₂,⋯,𝑥^𝑘𝑛_0𝑛)^𝑇, where 𝜒^𝑘𝑗_0𝑗=𝑥^𝑘𝑗+1_0𝑗−𝑥^𝑘𝑗_0𝑗 is a positive scalar and 𝑘_𝑗∈𝑁⁺ is the serial number of partition for 𝑥_𝑗. Define the subregion Θ_{𝑘1𝑘2⋯𝑘𝑛} on Θ (as shown in Figure 1): Θ_{𝑘1𝑘2⋯𝑘𝑛}=𝑥∣𝑥∈Θ,𝑥^𝑘𝑗_0𝑗≤𝑥_𝑗≤𝑥^𝑘𝑗+1_0𝑗,𝜒^𝑘𝑗_0𝑗=𝑥^𝑘𝑗+1_0𝑗−𝑥^𝑘𝑗_0𝑗. (23) Then, the coefficient vector in (16) can be represented as 𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)=∇𝑓_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀−𝑓,𝜃_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝜃−∇^𝑇𝑓_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝜃𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀²₂×𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀≠0. (24)

Figure 1 

Projection of Θ𝑘1𝑘2⋯𝑘𝑛 on 𝑥1𝑥2 plane.

Consequently, the T-S fuzzy rules can be chosen as follows: Rule0:IF𝑥₁isabout0,…,and𝑥_𝑛isabout0,THEN𝑓⁰_𝑙(𝑥,𝜃)=𝑎⁰_𝑙(𝜃)^𝑇𝑥,Rule𝑘₁𝑘₂⋯𝑘_𝑛:IF𝑥₁isabout𝑥^𝑘1₀₁,…,and𝑥_𝑛isabout𝑥^𝑘𝑛_0𝑛,THEN𝑓^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝑥,𝜃)=𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥. (25)

For Rule 0, the activated possibility 𝛼₀(𝑥) is 1 inside Θ₀ and 0 outside Θ₀. And the activated possibility of the Rule 𝑘₁𝑘₂⋯𝑘_𝑛 under a group of premise variables 𝑥 is computed by 𝜔_{𝑘1𝑘2⋯𝑘𝑛}(𝑥)=^𝑛_𝑗=1𝐴^{𝑘1𝑘2⋯𝑘𝑛}_𝑗𝑥_𝑗,𝛼_{𝑘1𝑘2⋯𝑘𝑛}=𝜔_{𝑘1𝑘2⋯𝑘𝑛}(𝑥)∑_{𝑘1𝑘2⋯𝑘𝑛}𝜔_{𝑘1𝑘2⋯𝑘𝑛}(𝑥), (26) where the membership function for (26) is given as 𝐴^{𝑘1𝑘2⋯𝑘𝑛}_𝑗𝑥_𝑗=𝑥1−^𝑗−𝑥^𝑘𝑗_0𝑗𝜒^𝑘𝑗_0𝑗,0≤𝑥_𝑗−𝑥^𝑘𝑗_0𝑗≤𝜒^𝑘𝑗_0𝑗,𝑥1+^𝑗−𝑥^𝑘𝑗_0𝑗𝜒^𝑘𝑗−1_0𝑗,0≤𝑥^𝑘𝑗_0𝑗−𝑥_𝑗≤𝜒^𝑘𝑗−1_0𝑗,0,elsewhere. (27) Then, 𝑓_𝑙(𝑥,𝜃) can be written as 𝑓_𝑙(𝑥,𝜃)=𝛼₀(𝑥)𝑎⁰_𝑙(𝜃)^𝑇𝑥+_{𝑘1𝑘2⋯𝑘𝑛}𝛼_{𝑘1𝑘2⋯𝑘𝑛}𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥. (28)

Here, the notion of the nonlinear measure along 𝑥_𝑗 in the partition region 𝜒^𝑘𝑗_0𝑗 can be defined as 𝑀^𝑘𝑗_0𝑗𝜒^𝑘𝑗_0𝑗,𝜃=∇𝑓_𝑙𝑥^{𝑘1⋯𝑘𝑗+1⋯𝑘𝑛}₀,𝜃−∇𝑓_𝑙𝑥^{𝑘1⋯𝑘𝑗⋯𝑘𝑛}₀,𝜃₂>0. (29)

Assuming the boundary of 𝑀^𝑘𝑗_0𝑗 is 𝜅_0𝑗, the value of 𝜅_0𝑗 can be set and the proper length of 𝜒^𝑘𝑗_0𝑗 can be chosen to guarantee 𝑀^𝑘𝑗_0𝑗≤𝜅_0𝑗. Thus, the length of the partition regions, 𝜒^𝑘𝑗_0𝑗 (𝑘_𝑗∈𝑁⁺), of 𝑥_𝑗 can be different with the same 𝜅_0𝑗.

So far, all of the methods show us how to establish the T-S fuzzy system. However, the approximation performance of the T-S fuzzy system is still unknown. Subsequently, the approximation performance of the T-S fuzzy system will be evaluated. Combining with the above T-S fuzzy modeling procedure, the T-S fuzzy system with required approximation accuracy can be obtained at last.

Then, using (24), the approximation performance of the T-S fuzzy system can be evaluated by 𝑒_𝑓𝑙(𝑥,𝜃)₂=𝑓_𝑙(𝑥,𝜃)−𝑓_𝑙(𝑥,𝜃)₂=𝑓_𝑙(𝑥,𝜃)−_{𝑘1𝑘2⋯𝑘𝑛}𝛼_{𝑘1𝑘2⋯𝑘𝑛}(𝑥)𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥₂=𝑓_𝑙(𝑥,𝜃)−_{𝑘1𝑘2⋯𝑘𝑛}𝛼_{𝑘1𝑘2⋯𝑘𝑛}(𝑥)𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀−_{𝑘1𝑘2⋯𝑘𝑛}𝛼_{𝑘1𝑘2⋯𝑘𝑛}(𝑥)𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀₂=𝑓_𝑙(𝑥,𝜃)−_{𝑘1𝑘2⋯𝑘𝑛}𝛼_{𝑘1𝑘2⋯𝑘𝑛}(𝑥)𝑓_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀−,𝜃_{𝑘1𝑘2⋯𝑘𝑛}𝛼_{𝑘1𝑘2⋯𝑘𝑛}𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀₂≤_{𝑘1𝑘2⋯𝑘𝑛}𝛼_{𝑘1𝑘2⋯𝑘𝑛}(𝑥)𝑓_𝑙(𝑥,𝜃)−𝑓_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝜃₂+_{𝑘1𝑘2⋯𝑘𝑛}𝛼_{𝑘1𝑘2⋯𝑘𝑛}(𝑥)𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀₂≤max_{𝑘1𝑘2⋯𝑘𝑛}𝑓_𝑙(𝑥,𝜃)−𝑓_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝜃₂+max_{𝑘1𝑘2⋯𝑘𝑛}𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀₂, (30) where 𝑥∈Θ_{𝑘1𝑘2⋯𝑘𝑛} and 𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)^𝑇𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀=∇^𝑇𝑓_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀×,𝜃𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀−𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀²₂𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀+𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀²₂𝑓_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝜃. (31) Because 𝑥∈Θ_{𝑘1𝑘2⋯𝑘𝑛}, the maximum distance between 𝑥 and any vertex point of Θ_{𝑘1𝑘2⋯𝑘𝑛} is less than max_{𝑘1𝑘2⋯𝑘𝑛}√𝑛𝜒^𝑘𝑗_0𝑗, which means ‖𝑥−𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀‖₂≤max_{𝑘1𝑘2⋯𝑘𝑛}√𝑛𝜒^𝑘𝑗_0𝑗; besides, in the boundary of 𝑀^𝑘𝑗_0𝑗, the max value of ‖𝑓_𝑙(𝑥)−𝑓_𝑙(𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀)‖₂ can be adjusted to fulfill the required accuracy.

Thus, the approximation error, 𝑒_𝑓𝑙(𝑥,𝜃), can be befittingly minimized by properly setting 𝜒^𝑘𝑗_0𝑗, which can be shown as 𝑒_𝑓𝑙(𝑥,𝜃)₂≤max_{𝑘1𝑘2⋯𝑘𝑛}𝑓_𝑙(𝑥,𝜃)−𝑓_𝑙𝑥^{𝑘1𝑘2⋯𝑘𝑛}₀,𝜃₂+max_{𝑘1𝑘2⋯𝑘𝑛}𝑎^{𝑘1𝑘2⋯𝑘𝑛}_𝑙(𝜃)₂√𝑛𝜒^𝑘𝑗_0𝑗. (32)

As a result, the evaluation system for the approximation performance of the LPV T-S fuzzy system is established. It provides a useful way to attain the required accuracy of the T-S fuzzy approximation. Then, the nonlinearity in (1) can be dealt with completely. Note that at each known operating state 𝑥^𝑖₀, one fuzzy rule is established, which approximates the local dynamics around 𝑥^𝑖₀. And the ith T-S fuzzy rule can be represented as Rule𝑖:IF𝑥₁is𝐴^𝑖₁𝑥₁,…,and𝑥_𝑛is𝐴^𝑖_𝑛𝑥_𝑛,THEN𝑥̇𝑧=𝐴_𝑖(𝜃)𝐵^𝑖₁(𝜃)𝐵^𝑖₂(𝜃)𝐶^𝑖₁(𝜃)0𝐷^𝑖₁₂(𝜃)𝑥𝜔𝑢, (33) where 𝑖=1,2,…,𝐿. More concretely, it is known that 𝑖=𝑘₁𝑘₂⋯𝑘_𝑛 and 𝐿=max{𝑘₁⋅𝑘₂⋯𝑘_𝑛}.

Finally, the overall LPV T-S fuzzy model can be represented as 𝑥̇𝑧=𝐴(𝛼,𝜃)𝐵₁(𝛼,𝜃)𝐵₂()𝐶𝛼,𝜃₁(𝛼,𝜃)0𝐷₁₂()𝑥𝜔𝑢=𝛼,𝜃_𝐿𝑖=1𝛼_𝑖(𝑥)𝐴_𝑖(𝜃)^𝐿_𝑖=1𝛼_𝑖(𝑥)𝐵^𝑖₁(𝜃)^𝐿_𝑖=1𝛼_𝑖(𝑥)𝐵^𝑖₂(𝜃)_𝐿𝑖=1𝛼_𝑖(𝑥)𝐶^𝑖₁(𝜃)0^𝐿_𝑖=1𝛼_𝑖(𝑥)𝐷^𝑖₁₂(𝜃)𝑥𝜔𝑢=^𝐿_𝑖=1𝛼_𝑖(𝑥)𝐴_𝑖(𝜃)𝐵^𝑖₁(𝜃)𝐵^𝑖₂(𝜃)𝐶^𝑖₁(𝜃)0𝐷^𝑖₁₂(𝜃)𝑥𝜔𝑢. (34)

Remark 2. It is noted that the T-S fuzzy modeling procedure utilizes an un-uniform partition method to reduce the number of T-S fuzzy rules while guaranteeing the approximation accuracy. And the evaluation system for the approximation performance of the LPV T-S fuzzy system can be used to balance the number of T-S fuzzy rules and the approximation performance of the LPV T-S fuzzy system.

3. Piecewise Parameter-Dependent Quadratic Stabilization of LPV T-S Fuzzy System

The LPV T-S fuzzy system (34) represents a class of complex continuous-time systems in a novel form which has both fuzzy inference and locally analytic linear models. In this section, the notion of piecewise parameter-dependent Lyapunov function is introduced for the stabilization problem of the LPV T-S fuzzy system. Firstly, the 𝑟th subspace in the state space can be defined as 𝑋_𝑟=𝑋_𝑟∪𝑋_𝑟, (35) where 𝑋_𝑟=𝑥∣𝛼_𝑖(𝑥)=𝛼_𝑘(𝑥),𝑋,𝑖,𝑘=1,2,…,𝐿,𝑖≠𝑘_𝑟=𝑥∣𝛼_𝑖(𝑥)>𝛼_𝑘(𝑥),𝑖,𝑘=1,2,…,𝐿,𝑖≠𝑘. (36)

Note that two subspaces are generated around the 𝑖th T-S fuzzy rule for the single state 𝑥_𝑗. The schematic about the state 𝑥_𝑗 can be shown in Figure 2. And then, the relation between the 𝑟th subspace and the 𝑖th T-S fuzzy rule is 𝑖=𝑘₁𝑘₂⋯𝑘_𝑛 and 𝑟=(2𝑘₁−2/2𝑘₁−1)(2𝑘₂−2/2𝑘₂−1)⋯(2𝑘_𝑛−2/2𝑘_𝑛−1) (𝑖=1,2,…,𝐿; word 𝑟=1,2,…,𝐿), where 𝐿=max{𝑘₁⋅𝑘₂⋯𝑘_𝑛} and 𝐿=max{2^𝑛⋅(𝑘₁−1)⋅(𝑘₂−1)⋯(𝑘_𝑛−1)}. Then, the overall model of the LPV T-S system in the 𝑟th subspace can be represented as 𝑥̇𝑧=𝐴_𝑟(𝛼,𝜃)𝐵^𝑟₁(𝛼,𝜃)𝐵^𝑟₂(𝛼,𝜃)𝐶^𝑟₁(𝛼,𝜃)0𝐷^𝑟₁₂()𝑥𝜔𝑢=𝐴𝛼,𝜃_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)𝐵^𝑖₁(𝜃)+𝐵^𝑟₁(𝛼,𝜃)𝐵^𝑖₂(𝜃)+𝐵^𝑟₂()𝐶𝛼,𝜃_𝑖1(𝜃)+𝐶^𝑟₁(𝛼,𝜃)0𝐷^𝑖₁₂(𝜃)+𝐷^𝑟₁₂()𝑥𝜔𝑢𝛼,𝜃 (37) for 𝑥∈𝑋_𝑟, where 𝐴_𝑟(𝛼,𝜃)=_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐴_𝑖𝑘(𝜃),𝐵^𝑟₁(𝛼,𝜃)=_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐵₁^𝑖𝑘(𝜃),𝐵^𝑟₂(𝛼,𝜃)=_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐵₂^𝑖𝑘(𝜃),𝐶^𝑟₁(𝛼,𝜃)=_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐶₁^𝑖𝑘(𝜃),𝐷^𝑟₁₂(𝛼,𝜃)=_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐷^𝑖𝑘₁₂(𝜃),𝐴_𝑖𝑘(𝜃)=𝐴_𝑘(𝜃)−𝐴_𝑖(𝜃),𝐵₁^𝑖𝑘=𝐵^𝑘₁(𝜃)−𝐵^𝑖₁(𝜃),𝐵₂^𝑖𝑘=𝐵^𝑘₂(𝜃)−𝐵^𝑖₂(𝜃),𝐶₁^𝑖𝑘=𝐶^𝑘₁(𝜃)−𝐶^𝑖₁(𝜃),𝐷^𝑖𝑘₁₂=𝐷^𝑘₁₂(𝜃)−𝐷^𝑖₁₂(𝜃),𝜙_𝑟=𝑘∣𝛼_𝑘(𝑥)≠0,𝛼_𝑖(𝑥)≥𝛼_𝑘(𝑥), (38) and 𝜙_𝑟 is a set including the indexes of the membership functions which are nonzero and less than 𝛼_𝑖(𝑥) around the 𝑖th rule in the 𝑟th subspace. For the overall model, 𝐴_𝑟(𝛼,𝜃), 𝐵^𝑟_∗(𝛼,𝜃), 𝐶^𝑟_∗(𝛼,𝜃), and 𝐷^𝑟_∗(𝛼,𝜃) represent the interpolation terms produced by interactions between the 𝑖th rule and the other rules in the 𝑟th subspace.

Figure 2 

Piecewise space partition of 𝑥𝑗.

In order to find the piecewise parameter-dependent Lyapunov function which is continuous across the 𝑟th subspace boundary at a fixed 𝜃, the following constant matrix 𝐹_𝑟 is established from the structure information of the 𝑟th subspace, which fulfills 𝐹_𝑟𝑥=𝐹_𝑝𝑥,𝑥∈𝑋_𝑟∩𝑋_𝑝,𝑟,𝑝=1,2,…,𝐿. (39)

Then, the piecewise parameter-dependent Lyapunov function candidates that are continuous across the 𝑟th subspace boundary can be parameterized as 𝑉(𝜃)=𝑥^𝑇𝑃_𝑟(𝜃)𝑥,𝑥∈𝑋_𝑟, (40) with 𝑃_𝑟(𝜃)=𝐹^𝑇_𝑟𝑇(𝜃)𝐹_𝑟, (41) where 𝑇(𝜃) is the symmetric matrix and characterizes 𝑃_𝑟(𝜃) with 𝐹_𝑟 together.

In order to carry forward the control design, the following upper bounds for the interpolation terms in (37) can be defined as 𝐴_𝑟(𝛼,𝜃)𝐴_𝑟(𝛼,𝜃)^𝑇≤𝐸_𝑟𝐴(𝛼,𝜃)𝐸^𝑇_𝑟𝐴(),𝐵𝛼,𝜃_𝑟1(𝛼,𝜃)𝐵^𝑟₁(𝛼,𝜃)^𝑇≤𝐸_𝑟𝐵1(𝛼,𝜃)𝐸^𝑇_𝑟𝐵1(),𝐵𝛼,𝜃_𝑟2(𝛼,𝜃)𝐵^𝑟₂(𝛼,𝜃)^𝑇≤𝐸_𝑟𝐵2(𝛼,𝜃)𝐸^𝑇_𝑟𝐵2(),𝐶𝛼,𝜃_𝑟1(𝛼,𝜃)𝐶^𝑟₁(𝛼,𝜃)^𝑇≤𝐸_𝑟𝐶1(𝛼,𝜃)𝐸^𝑇_𝑟𝐶1(),𝐷𝛼,𝜃_𝑟12(𝛼,𝜃)𝐷^𝑟₁₂(𝛼,𝜃)^𝑇≤𝐸_𝑟𝐷12(𝛼,𝜃)𝐸^𝑇_𝑟𝐷12(𝛼,𝜃). (42)

Since all the information of the interpolation terms in (42) is a priori knowledge, there are many ways to acquire these upper bounds. For example, one simple way is 𝐸_𝑟𝐴(𝛼,𝜃)𝐸^𝑇_𝑟𝐴()=𝛼,𝜃_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐴_𝑖𝑘(𝜃)_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐴_𝑖𝑘(𝜃)^𝑇, (43) where 𝑥 is the state such that 𝛼_𝑘(𝑥)=0.5.

Definition 3. On the compact set Ψ⊂𝑅^𝑠, one has finite nonnegative numbers {𝜐_𝑞}^𝑠_𝑞=1. Then the bounded variations set of scheduling parameters can be defined as Γ^𝜐_Ψ=𝜃∈𝐶¹𝑅,𝑅^𝑠∣𝜃∈Ψ,𝜃̇_𝑞2≤𝜐_𝑞,𝑞=1,…,𝑠, (44) where 𝐶¹ represents the class of functions which are piecewise continuous and one-order differential.

Definition 4. On the compact Ψ∈𝑅^𝑠, one has the LPV system 𝑥̇=𝐴(𝜃)𝑥, (45) where 𝜃∈Γ^𝜐_Ψ. If there exists a continuously differentiable symmetric function 𝑃(𝜃) such that 𝑃(𝜃)>0 and 𝐴^𝑇(𝜃)𝑃(𝜃)+𝑃(𝜃)𝐴(𝜃)+𝑃̇(𝜃)<0, (46) for all 𝜃∈Γ^𝜐_Ψ, then the continuous function 𝐴(𝜃) is parametrically dependent quadratically stable (or PDQ stable, for short).

Theorem 5. Consider the LPV T-S fuzzy system (34) with 𝜔=𝑢=0. If there exists a symmetric matrix 𝑇(𝜃) satisfying 𝑃_𝑟(𝜃)>0,𝑃_𝑟(𝜃)𝐴^𝑇_𝑖(𝜃)+𝐴_𝑖(𝜃)𝑃_𝑟(𝜃)−𝑃̇_𝑟(𝜃)𝐴+𝜀_𝑟𝑃𝑟(𝜃)𝐸_𝑟𝐴(𝛼,𝜃)𝐸^𝑇_𝑟𝐴()𝑃𝛼,𝜃^𝑟(𝜃)𝑃_𝑟(𝜃)𝐴−𝜀_𝑟𝑃𝑟(𝜃)𝐼<0 (47)(48) with 𝑃_𝑟(𝜃)=𝐹^𝑇_𝑟𝐹_𝑟⁻¹𝐹^𝑇_𝑟𝑇(𝜃)𝐹_𝑟𝐹^𝑇_𝑟𝐹_𝑟⁻¹ (49) for 𝑟=1,2,…,𝐿, then the fuzzy system is globally asymptotically stable.

Proof. Define the following Lyapunov function 𝑉(𝜃): 𝑉(𝜃)=𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝑥,𝑥∈𝑋_𝑟. (50) From (41), we can get 𝑃_𝑟⁻¹(𝜃)=𝐹^𝑇_𝑟𝑇⁻¹(𝜃)𝐹_𝑟. (51) From (50) and (51), there exist constants 𝜎>0 and 𝛽>0 such that 𝜎‖𝑥‖²₂≤𝑉(𝜃)≤𝛽‖𝑥‖²₂. (52) Thus, using the conditions (47) and (49), 𝑉(𝜃) is positive and continuous across the subspace boundary. If it can guarantee that the system (37) is asymptotically stable in each subspace, the global asymptotical stability for the system (34) can be attained. Next, we will demonstrate that 𝑉(𝜃) can guarantee the asymptotical stability in each subspace.
If there exist a constant 𝜀_𝑋𝑌>0 and matrices 𝑋 and 𝑌 with appropriate dimensions, the following matrix inequality can be gotten: 𝑋^𝑇𝑌+𝑌^𝑇1𝑋≤𝜀_𝑋𝑌𝑋^𝑇𝑋+𝜀𝑌^𝑇𝑌. (53)
From (53), define the parameter-dependent positive scalar 𝜀_𝑏𝑐(𝜃), where the subscript 𝑏𝑐 means the relation between 𝑏 and 𝑐 just like 𝜀_𝑋𝑌 means the relation between 𝑋 and 𝑌. In many cases, 𝜀_𝑎𝑏(𝜃) can be set as a constant value. Note that 𝑏 and 𝑐 are just used for example and do not have any special significance. However, subscripts of all 𝜀_∗(𝜃) are marked as follows to clearly show the meaning of 𝜀_∗(𝜃).
Then, from (53) we can get 𝑃_𝑟(𝜃)𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)^𝑇+𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)𝑃_𝑟(𝜃)−𝑃̇_𝑟(𝜃)=𝑃_𝑟(𝜃)𝐴^𝑇_𝑖(𝜃)+𝑃_𝑟(𝜃)𝐴^𝑇_𝑟(𝛼,𝜃)+𝐴_𝑖(𝜃)𝑃_𝑟(𝜃)+𝐴_𝑟(𝛼,𝜃)𝑃_𝑟(𝜃)−𝑃̇_𝑟(𝜃)≤𝑃_𝑟(𝜃)𝐴^𝑇_𝑖(𝜃)+𝐴_𝑖(𝜃)𝑃_𝑟(𝜃)+1𝜀𝐴_𝑟𝑃𝑟(𝜃)𝑃_𝑟(𝜃)𝑃_𝑟(𝜃)𝐴+𝜀_𝑟𝑃𝑟(𝜃)𝐴^𝑇_𝑟(𝛼,𝜃)𝐴_𝑟(𝛼,𝜃)−𝑃̇_𝑟(𝜃)≤𝑃_𝑟(𝜃)𝐴^𝑇_𝑖(𝜃)+𝐴_𝑖(𝜃)𝑃_𝑟(𝜃)+1𝜀𝐴_𝑟𝑃𝑟(𝜃)𝑃_𝑟(𝜃)𝑃_𝑟(𝜃)𝐴+𝜀_𝑟𝑃𝑟(𝜃)𝐸^𝑇_𝑟𝐴(𝛼,𝜃)𝐸_𝑟𝐴(𝛼,𝜃)−𝑃̇_𝑟(𝜃), (54) for the system (37). Besides, via the Schur complement lemma, it follows from (48) that 𝑃_𝑟(𝜃)𝐴^𝑇_𝑖(𝜃)+𝐴_𝑖(𝜃)𝑃_𝑟(𝜃)+1𝜀𝐴_𝑟𝑃𝑟(𝜃)𝑃_𝑟(𝜃)𝑃_𝑟(𝜃)𝐴+𝜀_𝑟𝑃𝑟(𝜃)𝐸_𝑟𝐴(𝛼,𝜃)𝐸^𝑇_𝑟𝐴(𝛼,𝜃)−𝑃̇_𝑟(𝜃)<0. (55) Then, from (54) and (55), we have 𝑃_𝑟(𝜃)𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)^𝑇+𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)𝑃_𝑟(𝜃)−𝑃̇_𝑟(𝜃)<0, (56) which suggests that 𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)^𝑇𝑃_𝑟⁻¹(𝜃)+𝑃_𝑟⁻¹(𝜃)𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)+𝑃̇_𝑟⁻¹(𝜃)<0, (57) where 𝑃̇_𝑟⁻¹(𝜃)=−𝐹^𝑇_𝑟𝑇⁻¹(𝜃)𝑇̇(𝜃)𝑇⁻¹(𝜃)𝐹_𝑟=−𝑃_𝑟⁻¹(𝜃)𝑃̇(𝜃)𝑃_𝑟⁻¹(𝜃).
Moreover, there exists a constant 𝛿>0 such that 𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)^𝑇𝑃_𝑟⁻¹(𝜃)+𝑃_𝑟⁻¹(𝜃)𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)+𝑃̇_𝑟⁻¹(𝜃)+𝛿𝐼<0. (58)
As a result, we have 𝑉̇(𝜃)=𝑥^𝑇𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)^𝑇𝑃_𝑟⁻¹(𝜃)+𝑃_𝑟⁻¹(𝜃)×𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)+𝑃̇_𝑟⁻¹(𝜃)𝑥≤𝑥^𝑇(−𝛿𝐼)𝑥=−𝛿‖𝑥‖²₂, (59) which completes the proof of this theorem.

Remark 6. It is noted that the sufficient conditions in LMIs form are easy to be solved. At a fixed 𝜃, conditions (47) and (49) guarantee that the Lyapunov function is positive and piecewise continuous across each subspace. And the Lyapunov functions solved from condition (48) guarantee that the system (37) is asymptotical in each subspace. All the conditions guarantee that the system (34) is globally asymptotically stable. Besides, in many cases, for the T-S fuzzy control based on the common Lyapunov function, it was hard to find the common Lyapunov function or such a Lyapunov function did not exist at all. Thus, the previous approach can improve the solvability of the LPV T-S fuzzy gain scheduling control.

4. State-Feedback 𝐻_∞ Control Design of LPV T-S Fuzzy System

In this section, the state-feedback 𝐻_∞ control of the LPV T-S fuzzy system is studied. Considering the approximation error, a controller synthesis approach with 𝐻_∞ performance is presented. In the rth subspace, the state feedback controller can be represented as 𝑢=𝐾_𝑟(𝜃)𝑥,𝑥∈𝑋_𝑟. (60)

Substituting (60) into (34), the closed-loop system can be obtained as 𝑥̇=𝐴_𝐶(𝛼,𝜃)𝑥+𝐵_𝐶(𝛼,𝜃)𝜔,𝑧=𝐶_𝐶(𝛼,𝜃)𝑥, (61) where 𝐴_𝐶(𝛼,𝜃)=𝐴(𝛼,𝜃)+𝐵₂(𝛼,𝜃)𝐾(𝜃),𝐵_𝐶(𝛼,𝜃)=𝐵₁(),𝐶𝛼,𝜃_𝐶(𝛼,𝜃)=𝐶₁(𝛼,𝜃)+𝐷₁₂(𝛼,𝜃)𝐾(𝜃). (62)

In the 𝑟th subspace, the system (61) can be represented as 𝑥̇=𝐴_𝐶𝑟(𝛼,𝜃)𝑥+𝐵_𝐶𝑟(𝛼,𝜃)𝜔,𝑧=𝐶_𝐶𝑟(𝛼,𝜃)𝑥,𝑥∈𝑋_𝑟, (63) where 𝐴_𝐶𝑟(𝛼,𝜃)=𝐴_𝑖(𝜃)+𝐴_𝑟()+𝐵𝛼,𝜃_𝑖2(𝜃)+𝐵^𝑟₂(𝛼,𝜃)𝐾_𝑟(𝜃),𝐵_𝐶𝑟(𝛼,𝜃)=𝐵^𝑖₁(𝜃)+𝐵^𝑟₁(),𝐶𝛼,𝜃_𝐶𝑟(𝛼,𝜃)=𝐶^𝑖₁(𝜃)+𝐶^𝑟₁()+𝐷𝛼,𝜃_𝑖12(𝜃)+𝐷^𝑟₁₂(𝛼,𝜃)𝐾_𝑟(𝜃). (64)

When substituting (37) into (1) and considering the approximating error, the closed-loop nonlinear system in the 𝑟th subspace can be represented as 𝑥̇𝑧=𝑓(𝑥,𝜃)+𝑑₀(𝑥,𝜃)𝜔+𝑔₀()𝑢𝜑()𝑥,𝜃𝑥,𝜃+𝑔₁()𝑢=𝑥,𝜃𝐴_𝑟(𝛼,𝜃)𝐵^𝑟₁(𝛼,𝜃)𝐵^𝑟₂(𝛼,𝜃)𝐶^𝑟₁(𝛼,𝜃)0𝐷^𝑟₁₂()𝑥𝜔𝑢+𝑓()−𝛼,𝜃𝑥,𝜃𝐴_𝑟(𝛼,𝜃)𝑥+𝑑₀(𝑥,𝜃)−𝐵^𝑟₁()𝜔+𝑔𝛼,𝜃₀(𝑥,𝜃)−𝐵^𝑟₂()𝑢𝜑()−𝛼,𝜃𝑥,𝜃𝐶^𝑟₁(𝛼,𝜃)𝑥+𝑔₁(𝑥,𝜃)−𝐷^𝑟₁₂()𝑢=𝛼,𝜃𝐴_𝑟(𝛼,𝜃)+𝐵^𝑟₂(𝛼,𝜃)𝐾_𝑟(𝜃)𝐵^𝑟₁(𝛼,𝜃)𝐶^𝑟₁(𝛼,𝜃)+𝐷^𝑟₁₂(𝛼,𝜃)𝐾_𝑟(𝜃)0𝑥𝜔+𝑓(𝑥,𝜃)−𝐴_𝑟(𝛼,𝜃)𝑥+𝑔₀(𝑥,𝜃)−𝐵^𝑟₂(𝛼,𝜃)𝐾_𝑟(𝜃)𝑥+𝑑₀(𝑥,𝜃)−𝐵^𝑟₁()𝜔𝜑()−𝛼,𝜃𝑥,𝜃𝐶^𝑟₁(𝛼,𝜃)𝑥+𝑔₁(𝑥,𝜃)−𝐷^𝑟₁₂(𝛼,𝜃)𝐾_𝑟(𝜃)𝑥=𝐴_𝐶𝑟(𝛼,𝜃)𝐵_𝐶𝑟()𝐶𝛼,𝜃_𝐶𝑟()0𝑥𝜔+𝛼,𝜃Δ𝑓_𝑟(𝜃)+Δ𝑔^𝑟₀(𝜃)+Δ𝑑^𝑟₀(𝜃)Δ𝜑_𝑟(𝜃)+Δ𝑔^𝑟₁(𝜃), (65) where Δ𝑓_𝑟(𝜃)=𝑓(𝑥,𝜃)−𝐴_𝑟(𝑥,𝜃)𝑥,Δ𝑔^𝑟₀(𝜃)=𝑔₀(𝑥,𝜃)−𝐵^𝑟₂(𝛼,𝜃)𝐾_𝑟(𝜃)𝑥,Δ𝑑^𝑟₀(𝜃)=𝑑₀(𝑥,𝜃)−𝐵^𝑟₁(𝛼,𝜃)𝜔,Δ𝜑_𝑟(𝜃)=𝜑(𝑥,𝜃)−𝐶^𝑟₁(𝛼,𝜃)𝑥,Δ𝑔^𝑟₁(𝜃)=𝑔₁(𝑥,𝜃)−𝐷^𝑟₁₂(𝛼,𝜃)𝐾_𝑟(𝜃)𝑥. (66)

According to (37) and (65), the approximation errors, Δ𝑓_𝑟(𝜃), Δ𝑔^𝑟₀(𝜃), Δ𝑑^𝑟₀(𝜃), Δ𝜑_𝑟(𝜃), and Δ𝑔^𝑟₁(𝜃), of the LPV T-S fuzzy system can be represented as follows: Δ𝑓_𝑟(𝜃)₂=𝑓(𝑥,𝜃)−𝐴_𝑟(𝛼,𝜃)𝑥₂=𝑓(𝑥,𝜃)−𝐴_𝑖(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐴_𝑖𝑘(𝜃)𝑥₂≤Δ𝐴_𝑖(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐴_𝑖𝑘(𝜃)𝑥₂=Δ𝐴_𝑟(𝜃)𝑥₂, (67) where Δ𝐴_𝑟(𝜃)=Δ𝐴_𝑖(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐴_𝑖𝑘(𝜃)=Δ𝐴_𝑖(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐴_𝑘(𝜃)−Δ𝐴_𝑖(𝜃)=𝛼_𝑖(𝑥)Δ𝐴_𝑖(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐴_𝑘(𝜃)=𝛼_𝑖(𝑥)𝛿_𝐴^𝑖_𝑟(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝛿_𝐴^𝑘_𝑟(𝜃)𝐴_𝑟(𝜃)=𝛿_𝐴𝑟(𝜃)𝐴_𝑟(𝜃),Δ𝑔^𝑟₀(𝜃)₂=𝑔₀(𝑥,𝜃)−𝐵^𝑟₂(𝛼,𝜃)𝐾_𝑟(𝜃)𝑥₂=𝑔₀(𝑥,𝜃)−𝐵^𝑖₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐵₂^𝑖𝑘(𝜃)₂×𝐾_𝑟(𝜃)𝑥₂≤Δ𝐵^𝑖₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐵₂^𝑖𝑘(𝜃)𝐾_𝑟(𝜃)𝑥₂=Δ𝐵^𝑟₂(𝜃)𝐾_𝑟(𝜃)𝑥₂, (68) where Δ𝐵^𝑟₂(𝜃)=Δ𝐵^𝑖₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐵₂^𝑖𝑘(𝜃)=Δ𝐵^𝑖₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐵^𝑘₂(𝜃)−Δ𝐵^𝑖₂(𝜃)=𝛼_𝑖(𝑥)Δ𝐵^𝑖₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐵^𝑘₂(𝜃)=𝛼_𝑖(𝑥)𝛿_𝐵2^𝑖,𝑟(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝛿_𝐵2^𝑘,𝑟(𝜃)𝐵^𝑟₂(𝜃)=𝛿_𝐵^𝑟₂𝐵^𝑟₂(𝜃),Δ𝑑^𝑟₀(𝜃)₂=𝑑₀(𝑥,𝜃)−𝐵^𝑟₁(𝛼,𝜃)𝜔₂=𝑑₀(𝑥,𝜃)−𝐵^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐵₁^𝑖𝑘(𝜃)𝜔₂≤Δ𝐵^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐵₁^𝑖𝑘(𝜃)𝜔₂=Δ𝐵^𝑟₁(𝜃)𝜔₂, (69) where Δ𝐵^𝑟₁(𝜃)=Δ𝐵^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝜃)Δ𝐵₁^𝑖𝑘(𝜃)=Δ𝐵^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐵^𝑘₁(𝜃)−Δ𝐵^𝑖₁(𝜃)=𝛼_𝑖(𝑥)Δ𝐵^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐵^𝑘₁(𝜃)=𝛼_𝑖(𝑥)𝛿_𝐵1^𝑖,𝑟(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝛿_𝐵1^𝑘,𝑟(𝜃)𝐵^𝑟₁(𝜃)=𝛿_𝐵^𝑟₁𝐵^𝑟₁(𝜃),Δ𝜑_𝑟(𝜃)₂=𝜑(𝑥,𝜃)−𝐶^𝑟₁(𝛼,𝜃)𝑥₂=𝜑(𝑥,𝜃)−𝐶^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐶₁^𝑖𝑘(𝜃)𝑥₂≤Δ𝐶^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐶₁^𝑖𝑘(𝜃)𝑥₂=Δ𝐶^𝑟₁(𝜃)𝑥₂, (70) where Δ𝐶^𝑟₁(𝜃)=Δ𝐶^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐶₁^𝑖𝑘(𝜃)=Δ𝐶^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐶^𝑘₁(𝜃)−Δ𝐶^𝑖₁(𝜃)=𝛼_𝑖(𝑥)Δ𝐶^𝑖₁(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐶^𝑘₁(𝜃)=𝛼_𝑖(𝑥)𝛿_𝐶1^𝑖,𝑟(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝛿_𝐶1^𝑘,𝑟(𝜃)×𝐶^𝑟₁(𝜃)=𝛿_𝐶^𝑟₁(𝜃)𝐶^𝑟₁(𝜃),Δ𝑔^𝑟₁(𝜃)₂=𝑔₁(𝑥,𝜃)−𝐷^𝑟₁₂(𝛼,𝜃)𝐾_𝑟(𝜃)𝑥₂=𝑔₁(𝑥,𝜃)−𝐷^𝑖₁₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝐷^𝑖𝑘₁₂(𝜃)×𝐾_𝑟(𝜃)𝑥₂≤Δ𝐷^𝑖₁₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐷^𝑖𝑘₁₂(𝜃)×𝐾_𝑟(𝜃)𝑥₂=Δ𝐷^𝑟₁₂(𝜃)𝐾_𝑟(𝜃)𝑥₂, (71) where Δ𝐷^𝑟₁₂(𝜃)=Δ𝐷^𝑖₁₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐷^𝑖𝑘₁₂(𝜃)=Δ𝐷^𝑖₁₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐷^𝑘₁₂(𝜃)−Δ𝐷^𝑖₁₂(𝜃)=𝛼_𝑖(𝑥)Δ𝐷^𝑖₁₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)Δ𝐷^𝑘₁₂(𝜃)=𝛼_𝑖(𝑥)𝛿_𝐷^𝑖,𝑟₁₂(𝜃)+_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)𝛿_𝐷^𝑘,𝑟₁₂(𝜃)𝐷^𝑟₁₂(𝜃)=𝛿_𝐷^𝑟₁₂(𝜃)𝐷^𝑟₁₂(𝜃). (72)

Note that not all of the nonlinearities can be given such bounds of approximation error in a linear form. However, for the nonlinearities satisfying the Lipschitz condition or even the one-order derivable condition, we can usually give the bounds of the approximation error in such a linear form. In addition, it should be noted that 𝛼_𝑖(𝑥)+∑_{𝑘∈𝜙𝑟}𝛼_𝑘(𝑥)=1 and ‖𝛿_𝐴^∗_𝑟(𝜃)‖₂≤1,…, ‖𝛿_𝐷^∗,𝑟₁₂(𝜃)‖₂≤1 for 𝑖=1,2,…,𝐿 and 𝑟=1,2,…,𝐿. Then, we can get Δ𝑓_𝑟(𝜃)₂Δ𝑑^𝑟₀(𝜃)₂Δ𝑔^𝑟₀(𝜃)₂Δ𝜑_𝑟(𝜃)₂0Δ𝑔^𝑟₁(𝜃)₂≤Δ𝐴_𝑟(𝜃)𝑥₂Δ𝐵^𝑟₁(𝜃)𝜔₂Δ𝐵^𝑟₂(𝜃)𝐾_𝑟(𝜃)𝑥₂Δ𝐶^𝑟₁(𝜃)𝑥₂0Δ𝐷^𝑟₁₂(𝜃)𝐾_𝑟(𝜃)𝑥₂, (73) for all the 𝑥, and describe the bounding matrices as Δ𝐴_𝑟(𝜃)Δ𝐵^𝑟₁(𝜃)Δ𝐵^𝑟₂(𝜃)Δ𝐶^𝑟₁(𝜃)0Δ𝐷^𝑟₁₂(𝜃)=𝛿_𝐴𝑟(𝜃)𝐴_𝑟(𝜃)𝛿_𝐵^𝑟₁(𝜃)𝐵^𝑟₁(𝜃)𝛿_𝐵^𝑟₂(𝜃)𝐵^𝑟₂(𝜃)𝛿_𝐶^𝑟₁(𝜃)𝐶^𝑟₁(𝜃)0𝛿_𝐷^𝑟₁₂(𝜃)𝐷^𝑟₁₂(𝜃), (74) where ‖𝛿_𝐴𝑟(𝜃)‖₂≤1,… and ‖𝛿_𝐷^𝑟₁₂(𝜃)‖₂≤1 for 𝑖=1,2,…,𝐿 and 𝑟=1,2,…,𝐿.

Theorem 7. For the closed-loop system (61), while considering the approximation error and a positive scalar 𝛾>0, if there exist symmetric matrixes 𝑇(𝜃) and 𝑃_𝑟(𝜃)=(𝐹^𝑇_𝑟𝐹_𝑟)⁻¹𝐹^𝑇_𝑟𝑇(𝜃)𝐹_𝑟(𝐹^𝑇_𝑟𝐹_𝑟)⁻¹ satisfying 𝑃_𝑟(𝜃)>0,𝑃_𝑟(𝜃)𝐴^𝑇_𝐶𝑟(𝛼,𝜃)+𝐴_𝐶𝑟(𝛼,𝜃)𝑃_𝑟(𝜃)−𝑃̇_𝑟(𝜃)+1𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)𝑃_𝑟(𝜃)𝐴^𝑇_𝑟(𝜃)𝐴_𝑟(𝜃)𝑃_𝑟(𝜃)+1𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)𝑃_𝑟(𝜃)𝐵^𝑟₂(𝜃)𝐾_𝑟(𝜃)^𝑇𝐵^𝑟₂(𝜃)𝐾_𝑟(𝜃)𝑃_𝑟(𝜃)+𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)+𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)+1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)×𝑃_𝑟(𝜃)𝐶^𝑇_𝐶𝑟(𝛼,𝜃)𝐶_𝐶𝑟(𝛼,𝜃)𝑃_𝑟(𝜃)+11+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+1𝜀_{Δ𝜑𝑟Δ𝑔}^𝑟₁(𝜃)×𝑃_𝑟(𝜃)𝐶^𝑟₁(𝛼,𝜃)^𝑇𝐶^𝑟₁(𝛼,𝜃)𝑃_𝑟(𝜃)+11+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)+1𝜀_{Δ𝜑𝑟Δ𝑔}^𝑟₁(𝜃)𝑃_𝑟(𝜃)×𝐷^𝑟₁₂(𝛼,𝜃)^𝑇𝐷^𝑟₁₂(𝛼,𝜃)𝑃_𝑟(𝜃)+𝛾²1𝐼−𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)^𝑇𝐵^𝑟₁(𝜃)⁻¹×𝐵_𝐶𝑟(𝛼,𝜃)𝐵^𝑇_𝐶𝑟(𝛼,𝜃)<0,𝛾²1𝐼−𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)^𝑇𝐵^𝑟₁(𝜃)>0, (75)(76)(77) the closed-loop system (61) while considering the approximation error has the 𝐻_∞ performance with disturbance attenuation 𝛾. And it is globally asymptotically stable when 𝜔=0.

Proof. From (63), the sufficient condition for (46) can be gotten: 𝐴^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)+𝑃_𝑟⁻¹(𝜃)𝐴_𝐶𝑟(𝛼,𝜃)+𝑃̇_𝑟⁻¹<0. (78) According to Theorem 5, if the condition (78) is fulfilled, the system (63) is asymptotically stable. For the system (65) with the approximation error, we can get 𝐴^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)+𝑃_𝑟⁻¹(𝜃)𝐴_𝐶𝑟()+𝑃̇𝛼,𝜃_𝑟−1(𝜃)+1𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)𝐴^𝑇_𝑟(𝜃)𝐴_𝑟(𝜃)+1𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₂(𝜃)𝐾_𝑟(𝜃)^𝑇𝐵^𝑟₂(𝜃)𝐾_𝑟(𝜃)+𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)+𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝑃_𝑟⁻¹(𝜃)𝑃_𝑟⁻¹(𝜃)+1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)𝐶^𝑇_𝐶𝑟(𝛼,𝜃)𝐶_𝐶𝑟()+1𝛼,𝜃1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+1𝜀_{Δ𝜑𝑟Δ𝑔}^𝑟₁(𝜃)𝐶^𝑟₁(𝛼,𝜃)^𝑇𝐶^𝑟₁()+1𝛼,𝜃1+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)+1𝜀_{Δ𝜑𝑟Δ𝑔}^𝑟₁(𝜃)𝐷^𝑟₁₂(𝛼,𝜃)^𝑇𝐷^𝑟₁₂()+𝛾𝛼,𝜃₂1𝐼−𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)^𝑇𝐵^𝑟₁(𝜃)⁻¹𝑃_𝑟⁻¹(𝜃)𝐵_𝐶𝑟(𝛼,𝜃)×𝐵^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)<0 (79) from (76), which suggests that the condition (78) is fulfilled. Here, the condition (77) needs to be fulfilled. Then, the system (65) is asymptotically stable. Next, we will show the disturbance attenuation performance.
Considering the Lyapunov function 𝑉(𝜃)=𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝑥 which is piecewise continuous across the rth subspace boundary, we differentiate 𝑉(𝜃) and then integrate it from zero to infinity in the rth subspace. Then, we can get 𝑉̇^∞₀(𝜃)=𝑑𝑡_∞0𝑥̇^𝑇𝑃_𝑟⁻¹(𝜃)𝑥+𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝑥̇+𝑥^𝑇𝑃̇_𝑟⁻¹(𝜃)𝑥=𝑑𝑡_∞0𝑥^𝑇𝐴^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)+𝑃_𝑟⁻¹(𝜃)𝐴_𝐶𝑟()+𝑃̇𝛼,𝜃_𝑟−1(𝜃)𝑥+Δ𝑓^𝑇_𝑟(𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+Δ𝑔^𝑟₀(𝜃)^𝑇𝑃_𝑟⁻¹(𝜃)𝑥+Δ𝑑^𝑟₀(𝜃)^𝑇𝑃_𝑟⁻¹(𝜃)𝑥+𝑥^𝑇𝑃_𝑟⁻¹(𝜃)Δ𝑓_𝑟(𝜃)+𝑥^𝑇𝑃_𝑟⁻¹(𝜃)Δ𝑔^𝑟₀(𝜃)+𝑥^𝑇𝑃_𝑟⁻¹(𝜃)Δ𝑑^𝑟₀(𝜃)+𝜔^𝑇𝐵^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝐵_𝐶𝑟()𝜔≤𝛼,𝜃𝑑𝑡_∞0𝑥^𝑇𝐴^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)+𝑃_𝑟⁻¹(𝜃)𝐴_𝐶𝑟()+𝑃̇𝛼,𝜃_𝑟−1(𝜃)𝑥+1𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)Δ𝑓^𝑇_𝑟(𝜃)Δ𝑓_𝑟(𝜃)+𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+1𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)Δ𝑔^𝑟₀(𝜃)^𝑇Δ𝑔^𝑟₀(𝜃)+𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+1𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)Δ𝑑^𝑟₀(𝜃)^𝑇Δ𝑑^𝑟₀(𝜃)+𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+𝜔^𝑇𝐵^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝐵_𝐶𝑟(𝛼,𝜃)𝜔𝑑𝑡≤^∞₀𝑥^𝑇𝐴^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)+𝑃_𝑟⁻¹(𝜃)𝐴_𝐶𝑟()+𝑃̇𝛼,𝜃_𝑟−1(𝜃)𝑥+1𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)𝐴_𝑟(𝜃)𝑥^𝑇𝐴_𝑟(𝜃)1𝑥+𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)×𝐵^𝑟₂(𝜃)𝐾_𝑟(𝜃)𝑥^𝑇𝐵^𝑟₂(𝜃)𝐾_𝑟(𝜃)𝑥+1𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)𝜔^𝑇𝐵^𝑟₁(𝜃)𝜔+𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)+𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+𝜔^𝑇𝐵^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝐵_𝐶𝑟()𝜔≤𝛼,𝜃𝑑𝑡_∞0−𝑥^𝑇1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)×𝐶^𝑇_𝐶𝑟(𝛼,𝜃)𝐶_𝐶𝑟()+1𝛼,𝜃1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+1𝜀_{Δ𝜑𝑟Δ𝑔}^𝑟₁(𝜃)𝐶^𝑟₁(𝛼,𝜃)^𝑇𝐶^𝑟₁()+1𝛼,𝜃1+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)+1𝜀_{Δ𝜑𝑟Δ𝑔}^𝑟₁(𝜃)𝐷^𝑟₁₂(𝛼,𝜃)^𝑇𝐷^𝑟₁₂()+𝛾𝛼,𝜃₂1𝐼−𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)^𝑇𝐵^𝑟₁(𝜃)⁻¹𝑃_𝑟⁻¹(𝜃)×𝐵_𝐶𝑟(𝛼,𝜃)𝐵^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+1𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)𝜔^𝑇𝐵^𝑟₁(𝜃)𝜔+𝜔^𝑇𝐵^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)𝑥+𝑥^𝑇𝑃_𝑟⁻¹(𝜃)𝐵_𝐶𝑟(𝛼,𝜃)𝜔+𝛾²𝜔^𝑇𝜔−𝛾²𝜔^𝑇𝜔≤𝑑𝑡_∞0−𝑧^𝑇𝑧+𝛾²𝜔^𝑇𝜔−𝛾²1𝐼−𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)^𝑇𝐵^𝑟₁(𝜃)^1/2𝜔−𝛾²1𝐼−𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)^𝑇𝐵^𝑟₁(𝜃)^−1/2×𝐵^𝑇_𝐶𝑟(𝛼,𝜃)𝑃_𝑟⁻¹(𝜃)𝑥²₂≤𝑑𝑡_∞0−𝑧^𝑇𝑧+𝛾²𝜔^𝑇𝜔𝑑𝑡. (80) That is 𝑉(𝑥(∞),𝜃)−𝑉(𝑥(0),𝜃)≤^∞₀−𝑧^𝑇𝑧+𝛾²𝜔^𝑇𝜔𝑑𝑡, (81) which suggests that, with 𝑥(0)=0, ‖𝑧‖₂≤𝛾‖𝜔‖₂. (82)
Overall, combining with the condition (75) and 𝑃_𝑟(𝜃)=(𝐹^𝑇_𝑟𝐹_𝑟)⁻¹𝐹^𝑇_𝑟𝑇(𝜃)𝐹_𝑟(𝐹^𝑇_𝑟𝐹_𝑟)⁻¹, the system (61), while considering the approximation error, has the 𝐻_∞ performance with disturbance attenuation 𝛾 and it is globally asymptotically stable when 𝜔=0. Then, the proof of this theorem is completed.

Remark 8. Considering the approximation error of the LPV T-S fuzzy system and bringing in the piecewise parameter-dependent Lyapunov functions to the LPV T-S fuzzy gain scheduling control, the state-feedback 𝐻_∞ control design is discussed here. And the sufficient conditions are given in Riccati inequalities form which can guarantee the 𝐻_∞ performance boundary 𝛾 of the closed-loop system.
As follows, based on Theorem 7, the equivalent conditions can be obtained in LMIs form which are easier to be solved.

Theorem 9. For the closed-loop system (61), while considering the approximation error and a positive scalar 𝛾>0, if there exist a symmetric matrix 𝑇(𝜃), a set of matrices 𝑄_𝑟(𝜃)=𝐾_𝑟(𝜃)𝑃_𝑟(𝜃) and 𝑃_𝑟(𝜃)=(𝐹^𝑇_𝑟𝐹_𝑟)⁻¹𝐹^𝑇_𝑟𝑇(𝜃)𝐹_𝑟(𝐹^𝑇_𝑟𝐹_𝑟)⁻¹ satisfying 𝑃_𝑟(𝜃)𝑊>0,_𝑟𝑃_𝑟(𝜃)𝑄^𝑇_𝑟(𝜃)𝑃_𝑟(𝜃)−𝑀⁻¹_𝑃𝑟0𝑄_𝑟(𝜃)0−𝑀⁻¹_𝑄𝑟<0, (83) where 𝑊_𝑟=𝑃_𝑟(𝜃)𝐴^𝑇_𝑖(𝜃)+𝐴_𝑖(𝜃)𝑃_𝑟(𝜃)−𝑃̇_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐵^𝑖₂(𝜃)^𝑇+𝐵^𝑖₂(𝜃)𝑄_𝑟(𝜃)𝐴+𝜀_𝑟𝑃𝑟(𝜃)𝐸_𝑟𝐴(𝛼,𝜃)𝐸^𝑇_𝑟𝐴()𝐵𝛼,𝜃+𝜀^𝑟_2𝑃𝑟(𝜃)𝐸_𝑟𝐵2(𝛼,𝜃)𝐸^𝑇_𝑟𝐵2()+𝛾𝛼,𝜃₂1𝐼−𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)^𝑇𝐵^𝑟₁(𝜃)⁻¹×11+𝜀_𝐵𝑖1𝐵^𝑟₁(𝜃)𝐵^𝑖₁(𝜃)𝐵^𝑖₁(𝜃)^𝑇+𝛾²1𝐼−𝜀_Δ𝑑^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₁(𝜃)^𝑇𝐵^𝑟₁(𝜃)⁻¹×1+𝜀_𝐵𝑖1𝐵^𝑟₁(𝜃)𝐸_𝑟𝐵1(𝛼,𝜃)𝐸_𝑟𝐵1(𝛼,𝜃)^𝑇,𝑀_𝑃𝑟=1𝜀𝐴_𝑟𝑃𝑟(𝜃)+1𝜀_{Δ𝑓𝑟𝑃𝑟}(𝜃)𝐴^𝑇_𝑟(𝜃)𝐴_𝑟(𝜃)+11+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+1𝜀_{Δ𝜑𝑟Δ𝑔}^𝑟₁(𝜃)×𝐶^𝑟₁(𝛼,𝜃)^𝑇𝐶^𝑟₁()+1𝛼,𝜃1+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)+1𝜀_{Δ𝜑𝑟Δ𝑔}^𝑟₁(𝜃)×𝐷^𝑟₁₂(𝛼,𝜃)^𝑇𝐷^𝑟₁₂()+𝛼,𝜃1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)×11+𝜀_𝐶𝑖1𝐶^𝑟₁(𝜃)+1𝜀_𝐶^𝑖_1𝐷^𝑖₁₂(𝜃)+1𝜀_𝐶𝑖1𝐷^𝑟₁₂(𝜃)×𝐶^𝑖₁(𝜃)^𝑇𝐶^𝑖₁(𝜃)+1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃×)1+𝜀_𝐶𝑖1𝐶^𝑟₁(𝜃)+𝜀_𝐷𝑖12𝐶^𝑟₁(𝜃)+1𝜀𝐶^𝑟₁𝐷^𝑟₁₂(𝜃)𝐸^𝑇_𝑟𝐶1×𝐸^𝑇_𝑟𝐶1(𝛼,𝜃)𝐸_𝑟𝐶1(),𝑀𝛼,𝜃_𝑄𝑟=1𝜀𝐵^𝑟₂^𝑃_𝑟(𝜃)+1𝜀_Δ𝑔^𝑟_0𝑃𝑟(𝜃)𝐵^𝑟₂(𝜃)^𝑇𝐵^𝑟₂(𝜃)+1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)×1+𝜀_𝐶^𝑖_1𝐷^𝑖₁₂(𝜃)+1𝜀_𝐷𝑖12𝐶^𝑟₁(𝜃)+1𝜀_𝐷𝑖12𝐷^𝑟₁₂(𝜃)×𝐷^𝑖₁₂(𝜃)^𝑇𝐷^𝑖₁₂(𝜃)+1+𝜀_{Δ𝜑𝑟𝐶𝐶𝑟}(𝜃)+𝜀_Δ𝑔^𝑟_{1𝐶𝐶𝑟}(𝜃)×1+𝜀_𝐶𝑖1𝐷^𝑟₁₂(𝜃)𝐶+𝜀^𝑟₁𝐷^𝑟₁₂(𝜃)+𝜀_𝐷𝑖12𝐷^𝑟₁₂(𝜃)×𝐸^𝑇_𝑟𝐷12(𝛼,𝜃)𝐸_𝑟𝐷12(𝛼,𝜃), (84) and condition (77), the closed-loop system (61), while considering the approximation error, has the 𝐻_∞ performance with disturbance attenuation 𝛾 and it is globally asymptotically stable when 𝜔=0. Moreover, the controller gain in the rth subspace is given by 𝐾_𝑟(𝜃)=𝑄_𝑟(𝜃)𝑃_𝑟⁻¹(𝜃). (85)

Proof. According to condition (76) of Theorem 7, we know that the closed-loop system (61), while considering the approximation error, has the 𝐻_∞ performance boundary 𝛾.
Then, substituting (37) into (76), we can get 𝑃_𝑟(𝜃)𝐴^𝑇_𝐶𝑟(𝛼,𝜃)+𝐴_𝐶𝑟(𝛼,𝜃)𝑃_𝑟(𝜃)=𝑃_𝑟(𝜃)𝐴_𝑖(𝜃)+𝐴_𝑟()+𝐵𝛼,𝜃_𝑖2(𝜃)+𝐵^𝑟₂(𝛼,𝜃)𝐾_𝑟(𝜃)^𝑇+𝐴_𝑖(𝜃)+𝐴_𝑟(𝛼,𝜃)+𝐵^𝑖₂(𝜃)+𝐵^𝑟₂(𝛼,𝜃)×𝐾_𝑟(𝜃)𝑃_𝑟(𝜃)=𝑃_𝑟(𝜃)𝐴^𝑇_𝑖(𝜃)+𝐴_𝑖(𝜃)𝑃_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐵^𝑖₂(𝜃)^𝑇+𝐵^𝑖₂(𝜃)𝑄_𝑟(𝜃)+𝑃_𝑟(𝜃)𝐴^𝑇_𝑟()+𝐴𝛼,𝜃_𝑟(𝛼,𝜃)𝑃_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐵^𝑟₂(𝛼,𝜃)^𝑇+𝐵^𝑟₂(𝛼,𝜃)𝑄_𝑟(𝜃)≤𝑃_𝑟(𝜃)𝐴^𝑇_𝑖(𝜃)+𝐴_𝑖(𝜃)𝑃_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐵^𝑖₂(𝜃)^𝑇+𝐵^𝑖₂(𝜃)𝑄_𝑟(𝜃)𝐴+𝜀_𝑟𝑃𝑟(𝜃)𝐸_𝑟𝐴(𝛼,𝜃)𝐸^𝑇_𝑟𝐴()𝐵𝛼,𝜃+𝜀^𝑟_2𝑃𝑟(𝜃)𝐸_𝑟𝐵2(𝛼,𝜃)𝐸^𝑇_𝑟𝐵2()+1𝛼,𝜃𝜀𝐴_𝑟𝑃𝑟(𝜃)𝑃_𝑟(𝜃)𝑃_𝑟(𝜃)+1𝜀𝐵^𝑟_2𝑃𝑟(𝜃)𝑄^𝑇_𝑟(𝜃)𝑄_𝑟(𝜃). (86) Besides, we also have 𝑃_𝑟(𝜃)𝐶^𝑇_𝐶𝑖(𝛼,𝜃)𝐶_𝐶𝑖(𝛼,𝜃)𝑃_𝑟(𝜃)=𝑃_𝑟(𝜃)𝐶^𝑖₁(𝜃)^𝑇𝐶^𝑖₁(𝜃)+𝐶^𝑖₁(𝜃)^𝑇𝐶^𝑟₁()+𝐶𝛼,𝜃_𝑟1(𝛼,𝜃)^𝑇𝐶^𝑖₁(𝜃)+𝐶^𝑟₁(𝛼,𝜃)^𝑇𝐶^𝑟₁(𝛼,𝜃)𝑃_𝑟(𝜃)+𝑃_𝑟(𝜃)𝐶^𝑖₁(𝜃)^𝑇𝐷^𝑖₁₂(𝜃)𝑄_𝑟(𝜃)+𝑃_𝑟(𝜃)𝐶^𝑖₁(𝜃)^𝑇𝐷^𝑟₁₂(𝛼,𝜃)𝑄_𝑟(𝜃)+𝑃_𝑟(𝜃)𝐶^𝑟₁(𝛼,𝜃)^𝑇𝐷^𝑖₁₂(𝜃)𝑄_𝑟(𝜃)+𝑃_𝑟(𝜃)𝐶^𝑟₁(𝛼,𝜃)^𝑇𝐷^𝑟₁₂(𝛼,𝜃)𝑄_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐷^𝑖₁₂(𝜃)^𝑇𝐶^𝑖₁(𝜃)𝑃_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐷^𝑖₁₂(𝜃)^𝑇𝐶^𝑟₁(𝛼,𝜃)𝑃_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐷^𝑟₁₂(𝛼,𝜃)^𝑇𝐶^𝑖₁(𝜃)𝑃_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐷^𝑟₁₂(𝛼,𝜃)^𝑇𝐶^𝑟₁(𝛼,𝜃)𝑃_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)𝐷^𝑖₁₂(𝜃)^𝑇𝐷^𝑖₁₂(𝜃)+𝐷^𝑖₁₂(𝜃)^𝑇𝐷^𝑟₁₂()+𝐷𝛼,𝜃_𝑟12(𝛼,𝜃)^𝑇𝐷^𝑖₁₂(𝜃)+𝐷^𝑟₁₂(𝛼,𝜃)^𝑇𝐷^𝑟₁₂(𝛼,𝜃)𝑄_𝑟(𝜃)≤𝑃_𝑟(𝜃)11+𝜀_𝐶𝑖1𝐶^𝑟₁(𝜃)+1𝜀_𝐶^𝑖_1𝐷^𝑖₁₂(𝜃)+1𝜀_𝐶𝑖1𝐷^𝑟₁₂(𝜃)𝐶^𝑖₁(𝜃)^𝑇𝐶^𝑖₁(𝜃)+1+𝜀_𝐶𝑖1𝐶^𝑟₁(𝜃)+𝜀_𝐷𝑖12𝐶^𝑟₁(𝜃)+1𝜀𝐶^𝑟₁𝐷^𝑟₁₂(𝜃)𝐸^𝑇_𝑟𝐶1×(𝛼,𝜃)𝐸_𝑟𝐶1(𝛼,𝜃)𝑃_𝑟(𝜃)+𝑄^𝑇_𝑟(𝜃)1+𝜀_𝐶^𝑖_1𝐷^𝑖₁₂(𝜃)+1𝜀_𝐷𝑖12𝐶^𝑟₁(𝜃)+1𝜀_𝐷𝑖12𝐷^𝑟₁₂(𝜃)𝐷^𝑖₁₂(𝜃)^𝑇𝐷^𝑖₁₂(𝜃)+1+𝜀_𝐶𝑖1𝐷^𝑟₁₂(𝜃)𝐶+𝜀^𝑟₁𝐷^𝑟₁₂(𝜃)+𝜀_𝐷𝑖12𝐷^𝑟₁₂(𝜃)×𝐸^𝑇_𝑟𝐷12(𝛼,𝜃)𝐸_𝑟𝐷12(𝛼,𝜃)𝑄_𝑟(𝜃),𝐵_𝐶𝑟(𝛼,𝜃)𝐵^𝑇_𝐶𝑟(𝛼,𝜃)=𝐵^𝑖₁(𝜃)𝐵^𝑖₁(𝜃)^𝑇+𝐵^𝑖₁(𝜃)𝐵^𝑟₁(𝛼,𝜃)^𝑇+𝐵^𝑟₁(𝛼,𝜃)𝐵^𝑖₁(𝜃)^𝑇+𝐵^𝑟₁(𝛼,𝜃)𝐵^𝑟₁(𝛼,𝜃)^𝑇≤11+𝜀_𝐵𝑖1𝐵^𝑟₁(𝜃)𝐵^𝑖₁(𝜃)𝐵^𝑖₁(𝜃)^𝑇+1+𝜀_𝐵𝑖1𝐵^𝑟₁(𝜃)𝐸_𝑟𝐵1(𝛼,𝜃)𝐸^𝑇_𝑟𝐵1(𝛼,𝜃). (87)(88)
Substituting (86), (87), and (88) into (76), the complete representation of (76) can be gotten. Then, by defining 𝑊_𝑟, 𝑀_𝑃𝑟, and 𝑀_𝑄𝑟 and using the Schur complement lemma, we can get the sufficient conditions in LMIs form as (83) and (80). The proof of this theorem is completed.

Remark 10. Based on Theorem 7, the conveniently solvable conditions are obtained in LMIs form in Theorem 9. For some parameter-dependent variables, such as 𝜀_∗(𝜃), they can be set as constant values in many cases by which the computation can be further simplified. However, subscripts of all 𝜀_∗(𝜃) are marked clearly in Theorem 5, Theorem 7, and Theorem 9 in order to show the meaning of them.

Remark 11. From Theorem 9, the sufficient conditions in LMIs form have been given. Though the solving of the LPV T-S fuzzy gain scheduling control design is different from the solving of the existing LPV gain scheduling control design, most of the procedures are similar. According to [9, 28, 29], the solution of the existing LPV gain scheduling control design has been given. Then, the main procedures of the LPV gain scheduling control design can be shown as follows. Firstly, according to the property of 𝜃, the gridding process may be executed for the nonlinear form of 𝜃 or not. In particular, for the affine linear form of 𝜃, the multiconvexity property can be used to guarantee the continuous properties of continuously scheduled 𝜃 instead of the gridding process and the number of the LMIs can be reduced which can simplify the computation. Secondly, the parametrization of decision variables in LMIs needs to be executed. Through the two steps, infinite LMIs with infinite decision variables can be gotten. By solving the LMIs, the piecewise parameter-dependent functions can be obtained and then the controller can be determined.

5. Numerical Example

In order to show the availability of the approaches, a numerical example is provided in this subsection. Consider the ANPV system as 𝑥̇=𝑓(𝑥,𝜃)+𝐵₁𝜔+𝐵₂(),𝑢,𝑧=𝜑𝑥,𝜃 (89) where 𝑓(𝑥,𝜃)=^{0.0005𝜃𝑥21+0.0008𝑥2}_{𝑥1−0.5𝑥2}, 𝜑(𝑥,𝜃)=0.5𝜃−𝑥₁, 𝐵₁=0.00010^𝑇, 𝐵₂=01^𝑇, and 𝜔=sin(𝑡) is the sinusoidal interference signal.

Here, define that 𝜃=𝜐𝑡+𝜏 is a linear function about time 𝑡 and set the variation rate ‖𝜐‖₂≤1. Assume that 𝜃 changes in the range [−2,2] and the variation range of 𝑥₁ is [−1,1]. From (89), it can be found that the T-S fuzzy linearization is only used to deal with the nonlinearity about 𝑥₁. Using the T-S fuzzy modeling procedure given before, the input space of 𝑥₁ can be partitioned into [−1,0] and [0,1]. Then, three T-S fuzzy rules can be obtained as follows: Rule1:If𝑥₁isabout−1,theṅ001𝑥=𝑢−0.0005𝜃0.00081−0.5𝑥+0.0001𝜔+𝑧=−0.5𝜃−10𝑥,Rule2:If𝑥₁isabout0,theṅ001𝑥=𝑢00.00081−0.5𝑥+0.0001𝜔+𝑧=−10𝑥,Rule3:If𝑥₁isabout1,theṅ001𝑥=𝑢0.0005𝜃0.00081−0.5𝑥+0.0001𝜔+𝑧=0.5𝜃−10𝑥. (90)(91)(92)

In most cases, the approximation of the LPV T-S fuzzy system to the ANPV system usually has high accuracy, so the approximation error can often be ignored. Here, three T-S fuzzy rules are used at −1, 0, and 1 of 𝑥₁ to approximate the nonlinear term 𝑥²₁; the maximum absolute error is 1.1102𝑒−16. Thus, the approximation error can be ignored.

Meanwhile, using the definition in (35)-(36) and the triangular membership functions, four piecewise subspaces are obtained which are named 𝑋₁,𝑋₂,𝑋₃, and 𝑋₄ (as shown in Figure 3). Then, the T-S fuzzy model in each subspace can be represented as 𝑋₁∶𝑥̇=−0.0005𝜃0.00081−0.5+𝛼₂𝑥₁𝑥+001𝑢0.0005𝜃0000.0001𝜔+𝑧=−0.5𝜃−10+𝛼₂𝑥₁0.5𝜃0𝑥,𝑥₁∈[−1,−0.5],𝑋₂∶𝑥̇=00.00081−0.5+𝛼₁𝑥₁𝑥+001𝑢−0.0005𝜃0000.0001𝜔+𝑧=−10+𝛼₁𝑥₁−0.5𝜃0𝑥,𝑥₁∈[−0.5,0],𝑋₃∶𝑥̇=00.00081−0.5+𝛼₃𝑥₁𝑥+001𝑢0.0005𝜃0000.0001𝜔+𝑧=−10+𝛼₃𝑥₁0.5𝜃0𝑥,𝑥₁∈[0,0.5],𝑋₄∶𝑥̇=0.0005𝜃0.00081−0.5+𝛼₂𝑥₁𝑥+001𝑢−0.0005𝜃0000.0001𝜔+𝑧=0.5𝜃−10+𝛼₂𝑥₁−0.5𝜃0𝑥,𝑥₁∈[0.5,1]. (93)(94)(95)(96)

Figure 3 

Piecewise space partition of 𝑥1.

According to the method in [22], it can be set that 𝛼_𝑘(𝑥,𝜃)=0.5 and the matrices of 𝐸_𝑟𝐴(𝜃) and 𝐸_𝑟𝐶1(𝜃) for the 𝑖th (𝑖=1,2,3) fuzzy rule in the 𝑟th (𝑟=1,2,3,4,𝑘∈𝜙_𝑟) subspace can be obtained. Using the detailed method in [20], the constant matrices 𝐹_𝑟 can be given.

When the LPV T-S fuzzy system is linear with the scheduling parameter 𝜃, we can parameterize the system matrices 𝐴_𝑟(𝜃) and 𝐶^𝑟₁(𝜃) and decision variables 𝑃_𝑟(𝜃) and 𝑄_𝑟(𝜃) in the rth subspace as 𝐴_𝑟(𝜃)=𝐴⁰_𝑟+𝜃𝐴¹_𝑟,𝐶^𝑟₁(𝜃)=𝐶^𝑟_1,0+𝜃𝐶^𝑟_1,1,𝑃_𝑟(𝜃)=𝑃⁰_𝑟+𝜃𝑃¹_𝑟,𝑄_𝑟(𝜃)=𝑄⁰_𝑟+𝜃𝑄¹_𝑟, (97) where 𝐴⁰_𝑟, 𝐴¹_𝑟, 𝐶^𝑟_1,0, 𝐶^𝑟_1,1, 𝑃⁰_𝑟, 𝑃¹_𝑟, 𝑄⁰_𝑟, and 𝑄¹_𝑟 are constant values and 𝑟=1,2,3,4.

Besides, the gridding process is avoided when solving the parameter-dependent Lyapunov functions 𝑃_𝑟(𝜃) in 𝑟th subspace 𝑋_𝑟. Utilizing the multiconvexity property in [30], it just needs to validate the LMIs at the vertexes of 𝜃. Here, the two vertexes at 𝜃=−2 and 𝜃=2 are tested, respectively.

Following the design algorithm with 𝛾=0.9, the solution to the LMIs when 𝜀_∗(𝜃)=10 can be obtained: 𝑃₁(𝜃)=,𝑄0.0265+0.0777𝜃−0.2+0.996𝜃−0.2+0.996𝜃4.475+1.5562𝜃₁(𝜃)=,𝑃0.1476+0.0840𝜃10.9778+0.0191𝜃₂(𝜃)=,𝑄−0.0126+0.3006𝜃−0.6849+0.0309𝜃−0.6849+0.0309𝜃3.6378+0.8595𝜃₂(𝜃)=,𝑃0.0997+0.0373𝜃7.2065−0.1058𝜃₃(𝜃)=,𝑄−0.0126+0.3006𝜃−0.6849+0.0309𝜃−0.6849+0.0309𝜃3.6378+0.8595𝜃₃(𝜃)=,𝑃0.0997+0.0373𝜃7.2065−0.1058𝜃₄(𝜃)=,𝑄−0.0176+0.0651𝜃−0.1457−1.0275𝜃−0.1457−1.0275𝜃6.5439+1.5280𝜃₄(𝜃)=0.0423−0.0114𝜃10.8734+0.0179𝜃. (98)

Then, the state-feedback controller can be determined as 𝑢=𝐾_𝑟(𝜃)𝑥, where 𝐾_𝑟(𝜃)=𝑄_𝑟(𝜃)𝑃_𝑟⁻¹(𝜃) and 𝑟=1,2,3,4.

Because the effective performance region of the controller about 𝑥₁ is set as [−1,1], the scheduling values of 𝜃 are chosen to linearly vary from 0.3 to 1.3 with the interval of 0.1 to validate the effectiveness of the controller. The state responses of 𝑥₁ and 𝑥₂ and the evaluation response of 𝑧 are shown in Figures 4, 5, and 6, respectively.

Figure 4 

State response of 𝑥1.

Figure 5 

State response of 𝑥2.

Figure 6 

Evaluation response of 𝑧.

Meanwhile, in order to validate the effectiveness of the controller during the scheduling process of 𝜃, 𝜃 is scheduled from 0.5 to 1 at 0.5 seconds, which guarantees that the variation rate of 𝜃 is 1. The responses of 𝑥₁, 𝑥₂, and 𝑧 are shown in Figures 7, 8, and 9, respectively.

Figure 7 

State response of 𝑥1 in scheduling.

Figure 8 

State response of 𝑥2 in scheduling.

Figure 9 

Evaluation response of 𝑧 in scheduling.

From Figures 4, 5, and 6, they suggest the effectiveness of the parameter-dependent controller. At different scheduling points of 𝜃, the system can be stabilized by the state-feedback controller with 𝐻_∞ performance boundary 𝛾=0.9. From Figures 7, 8, and 9, the performance when switching at different 𝜃 points is validated. The time-domain response curves illustrate the stability and control performance of the system when continuously scheduling. Obviously, all the above results suggest the availability of sufficient conditions in Theorems 5, 7, and 9 for the control design of the LPV T-S fuzzy system.

6. Conclusion

In order to improve the existing LPV gain scheduling control, the T-S fuzzy modeling procedure was adopted to deal with the nonlinearity and relevant control design was studied in the paper. For the ANPV description of the nonlinear dynamic processes, a kind of developed T-S fuzzy modeling procedure with homogeneous linear consequent parts was proposed to carry forward the linearization. As a result, greater flexibility was obtained by which the number of scheduling parameters could be reduced and the approximation accuracy could be raised in any optional region. Moreover, the un-uniform partition method was utilized and the evaluation system for the approximation ability of the novelly developed T-S fuzzy modeling procedure was established. Through this way, the number of fuzzy rules could be decreased while guaranteeing the approximation performance, and the LPV T-S fuzzy system was obtained with required accuracy. For the LPV T-S fuzzy gain scheduling control, the notion of piecewise parameter-dependent Lyapunov function was introduced to improve the solvability. Subsequently, the stabilization problem and the state-feedback 𝐻_∞ control problem of the LPV T-S fuzzy system were discussed. The sufficient conditions were given in LMIs form and the approximation error was considered for the 𝐻_∞ control. Note that the novel LPV T-S fuzzy system can represent a great class of nonlinear dynamic processes. Based on the LPV T-S fuzzy system framework, more problems could be further studied following such a line, such as the output feedback control design and the corresponding applications.

Acknowledgments

This work is partially supported by the National Basic Research Program of China (973 Program) (Grant no. 2012CB215203), the National Natural Science Foundation of China (no. 51036002 and no. 61203043), and the Fundamental Research Funds for the Central Universities (no. 13XS14).