# State-FeedbackH∞ Control for LPV System Using T-S Fuzzy Linearization Approach

Authors: Jizhen Liu; Yang Hu; Zhongwei Lin
Journal: Mathematical Problems in Engineering
 (2013)

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## 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-feedbackH∞ 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.


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## Body

## 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-feedbackH∞ 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-feedbackH∞ 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 Section2, 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(1)[x˙z]=[f(x,θ)+d0(x,θ)ω+g0(x,θ)uφ(x,θ)+g1(x,θ)u],
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 x0i which can be steady or transient one and corresponds to the ith fuzzy rule, i=1,2,…,L. It should be noted that x0i 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 x0i. Because the local linear model should approximate the local dynamics in the vicinity of x0i, we can get
(2)[f(x,θ)+d0(x,θ)ω+g0(x,θ)uφ(x,θ)+g1(x,θ)u]≈[Ai(θ)B1i(θ)B2i(θ)C1i(θ)0D12i(θ)][xωu],[f(x0i,θ)+d0(x0i,θ)ω+g0(x0i,θ)uφ(x0i,θ)+g1(x0i,θ)u]=[Ai(θ)B1i(θ)B2i(θ)C1i(θ)0D12i(θ)][x0iωu],
where ω and u are arbitrarily changed variables and Ai(θ),…,D12i(θ) are parameter-dependent matrices.Take the first equationx˙=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 Ai(θ), B1i(θ), and B2i(θ) only depend on x. Moreover, because ω and u are arbitrarily changed, we can uniquely determine that
(3)d0(x0i,θ)=B1i(θ),g0(x0i,θ)=B2i(θ),f(x0i,θ)=Ai(θ)x0i,f(x,θ)≈Ai(θ)x.Defineali(θ)T, a row vector with n dimensions, as the lth row of the matrix Ai(θ). Then, condition (3) can be equivalently represented as
(4)fl(x,θ)≈ali(θ)Tx,l=1,2,…,n,
where fl(x,θ) is the lth row of f(x,θ). Additionally, assume fl(0,θ)=0 and fl(x,θ)∈C1, which means that fl and ∂fl/∂x are continuous functions on the compact set Θ⊂Rn. Meanwhile, f(x0i,θ)=Ai(θ)x0i can be equivalently represented as
(5)fl(x0i,θ)=ali(θ)Tx0i,l=1,2,…,n.Expandingfl(x,θ) of (4) around the operating state x0i and neglecting second- and higher-order terms, we can get
(6)fl(x0i,θ)+∇Tfl(x0i,θ)(x-x0i)≈ali(θ)Tx,
where ∇fl(x0i,θ) is the gradient, a column vector, of fl(x,θ) evaluated at x0i. Substituting (5) into (6), we can get
(7)∇Tfl(x0i,θ)(x-x0i)≈ali(θ)T(x-x0i),
where x is arbitrarily close to x0i. From (7), it can be found that the coefficient vector ali(θ) needs to be estimated. And in the vicinity of x0i, an optimization problem can be constructed by defining an optimal index to evaluate the estimation. Here, define that the notation Cm 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:(8)minimizef(x)subjecttoh(x)=0,
where f(x) is f:Rn→R and f∈C1; it is a convex function on the feasible set Ω={x∈Rn:h(x)=0}, where h:Rn→Rm and h∈C1. Assume that Ω is convex and there exist x*∈Ω and λ∈Rm such that
(9)∇Ωf(x*)+λ∇Ωh(x*)=0,
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 ofali(θ) around x0i≠0 in (7), the optimal index for fl(x,θ) corresponding to the ith fuzzy rule can be defined as
(10)Ifli(ali(θ))=12∥∇fl(x0i,θ)-ali(θ)∥22.Then, the optimal problem can be constructed as(11)minimizeΩfliIfli(ali(θ))subjecttohfli(ali(θ))=0,
where (5) should be fulfilled as an equality constraint and hfli(ali(θ))=fl(x0i,θ)-ali(θ)Tx0i.Transform the objective function in (10) into the form
(12)Ifli(ali(θ))=12∥∇fl(x0i,θ)-ali(θ)∥22=12(∇fl(x0i,θ)-ali(θ))T(∇fl(x0i,θ)-ali(θ)),
which is a quadratic function of the unknown coefficient vector ali(θ) on Rn. Obviously, it means that Ifli(ali(θ)) is a convex function. Considering the equality constraint hfli(ali(θ)), it is the linear function of ali(θ), so the feasible set Ωfli={ali(θ)∈Rn:h(ali(θ))=0} is convex. Therefore, the optimal problem is a convex optimization problem and can be solved according to Lemma 1.Computing the derivative ofIfli(ali(θ)) and hfli(ali(θ)) about ali(θ), we can get
(13)∇ΩfliIfli(ali(θ))=ali(θ)-∇fl(x0i,θ),∇Ωflihfli(ali(θ))=x0i.
Then, substituting (13) into (9), we can get
(14)ali(θ)-∇fl(x0i,θ)+λfli(θ)x0i=0.
Considering the equality constraint (5), the Lagrange multiplier λfli(θ) can be represented as
(15)λfli(θ)=∇Tfl(x0i,θ)x0i-ali(θ)Tx0i∥x0i∥22λfli(θ)=∇Tfl(x0i,θ)x0i-fl(x0i,θ)∥x0i∥22.
As a result, we can get the column vector
(16)ali(θ)=∇fl(x0i,θ)ali(θ)=+fl(x0i,θ)-∇Tfl(x0i,θ)x0i∥x0i∥22x0i,x0i≠0.
Then, the ith T-S fuzzy rule corresponding to the operating state x0i can be chosen as
(17)Rulei:IFx1isA1i(x1),…,andxnisAni(xn),THENf^li(x,θ)=ali(θ)Tx,
where f^li(x,θ) represents the ith local estimation of the lth row of f(x,θ), ali(θ) is the corresponding coefficient vector of the ith fuzzy rule, a column vector, and Aji(xj) represents the fuzzy set and membership function of the jth premise variable, xj, in the ith fuzzy rule and j=1,2,…,n.The activated possibility of theith fuzzy rule under a group of premise variables x is computed by
(18)ωi(x)=∏j=1nAji(xj),i=1,2,…,L,j=1,2,…,n.The weight coefficients are computed and the center-of-gravity method is used for defuzzification:(19)αi(x)=ωi(x)∑i=1Lωi(x),i=1,2,…,L,
which fulfills that
(20)∑i=1L‍αi(x)=1,αi(x)≥0.Finally, the overall T-S fuzzy model can be obtained as(21)f^l(x,θ)=∑i=1L‍αi(x)f^li(x,θ)=∑i=1L‍αi(x)ali(θ)Tx.Usually, based on (21), the evaluation of the overall approximation performance about the nonlinear function fl(x,θ) is defined as
(22)∥efl(x,θ)∥2=∥fl(x,θ)-f^l(x,θ)∥2.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 subsetΘ0={x∣|xj|<χ0} on Θ, where χ0 is a predefined positive scalar and al0=∂fl(x,θ)/∂x∣x=0 can be gotten at the zero state x=0.  Considering  the nonzero states, they can be represented as x0k1k2⋯kn=(x01k1,x02k2,⋯,x0nkn)T, where χ0jkj=x0jkj+1-x0jkj is a positive scalar and kj∈N+ is the serial number of partition for xj. Define the subregion Θk1k2⋯kn on Θ (as shown in Figure 1):
(23)Θk1k2⋯kn={x∣x∈Θ,x0jkj≤xj≤x0jkj+1,χ0jkj=x0jkj+1-x0jkj}.
Then, the coefficient vector in (16) can be represented as
(24)alk1k2⋯kn(θ)=∇fl(x0k1k2⋯kn,θ)-fl(x0k1k2⋯kn,θ)-∇Tfl(x0k1k2⋯kn,θ)x0k1k2⋯kn∥x0k1k2⋯kn∥22×x0k1k2⋯kn,x0k1k2⋯kn≠0.Figure 1
Projection ofΘk1k2⋯kn on x1x2 plane.Consequently, the T-S fuzzy rules can be chosen as follows:(25)Rule0:IFx1isabout0,…,andxnisabout0,THENf^l0(x,θ)=al0(θ)Tx,Rulek1k2⋯kn:IFx1isaboutx01k1,…,andxnisaboutx0nkn,THENf^lk1k2⋯kn(x,θ)=alk1k2⋯kn(θ)Tx.For Rule 0, the activated possibilityα0(x) is 1 inside Θ0 and 0 outside Θ0. And the activated possibility of the Rule  k1k2⋯kn under a group of premise variables x is computed by
(26)ωk1k2⋯kn(x)=∏j=1nAjk1k2⋯kn(xj),αk1k2⋯kn=ωk1k2⋯kn(x)∑k1k2⋯knωk1k2⋯kn(x),
where the membership function for (26) is given as
(27)Ajk1k2⋯kn(xj)={1-(xj-x0jkj)χ0jkj,0≤xj-x0jkj≤χ0jkj,1+(xj-x0jkj)χ0jkj-1,0≤x0jkj-xj≤χ0jkj-1,0,+(xj-x0jkj)χ0jkj-1,elsewhere.
Then, f^l(x,θ) can be written as
(28)f^l(x,θ)=α0(x)al0(θ)Tx+∑k1k2⋯kn‍αk1k2⋯knalk1k2⋯kn(θ)Tx.Here, the notion of the nonlinear measure alongxj in the partition region χ0jkj can be defined as
(29)M0jkj(χ0jkj,θ)=∥∇fl(x0k1⋯kj+1⋯kn,θ)M0jkj(χ0jkj,θ)=l-∇fl(x0k1⋯kj⋯kn,θ)∥2>0.Assuming the boundary ofM0jkj is κ0j, the value of κ0j can be set and the proper length of χ0jkj can be chosen to guarantee M0jkj≤κ0j. Thus, the length of the partition regions, χ0jkj (kj∈N+), of xj can be different with the same κ0j.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
(30)∥efl(x,θ)∥2=∥fl(x,θ)-f^l(x,θ)∥2=∥fl(x,θ)-∑k1k2⋯kn‍αk1k2⋯kn(x)alk1k2⋯kn(θ)Tx∥2=∥fl(x,θ)-∑k1k2⋯kn‍αk1k2⋯kn(x)alk1k2⋯kn(θ)Tx0k1k2⋯kn111111-∑k1k2⋯kn‍αk1k2⋯kn(x)alk1k2⋯kn(θ)T(x-x0k1k2⋯kn)∥2=∥fl(x,θ)-∑k1k2⋯kn‍αk1k2⋯kn(x)f^l(x0k1k2⋯kn,θ)111111-∑k1k2⋯kn‍αk1k2⋯knalk1k2⋯kn(θ)T(x-x0k1k2⋯kn)∥2≤∑k1k2⋯kn‍αk1k2⋯kn(x)∥fl(x,θ)-f^l(x0k1k2⋯kn,θ)∥2+∑k1k2⋯kn‍αk1k2⋯kn(x)∥alk1k2⋯kn(θ)T(x-x0k1k2⋯kn)∥2≤maxk1k2⋯kn∥fl(x,θ)-f^l(x0k1k2⋯kn,θ)∥2+maxk1k2⋯kn∥alk1k2⋯kn(θ)T(x-x0k1k2⋯kn)∥2,
where x∈Θk1k2⋯kn and
(31)alk1k2⋯kn(θ)T(x-x0k1k2⋯kn)=∇Tfl(x0k1k2⋯kn,θ)×[-〈(x-x0k1k2⋯kn),x0k1k2⋯kn〉∥x0k1k2⋯kn∥22x0k1k2⋯kn](x-x0k1k2⋯kn)11111-〈(x-x0k1k2⋯kn),x0k1k2⋯kn〉∥x0k1k2⋯kn∥22x0k1k2⋯kn]+〈(x-x0k1k2⋯kn),x0k1k2⋯kn〉∥x0k1k2⋯kn∥22fl(x0k1k2⋯kn,θ).
Because x∈Θk1k2⋯kn, the maximum distance between x and any vertex point of Θk1k2⋯kn is less than maxk1k2⋯knnχ0jkj, which means ∥x-x0k1k2⋯kn∥2≤maxk1k2⋯knnχ0jkj; besides, in the boundary of M0jkj, the max value of ∥fl(x)-f^l(x0k1k2⋯kn)∥2 can be adjusted to fulfill the required accuracy.Thus, the approximation error,efl(x,θ), can be befittingly minimized by properly setting χ0jkj, which can be shown as
(32)∥efl(x,θ)∥2≤maxk1k2⋯kn∥fl(x,θ)-f^l(x0k1k2⋯kn,θ)∥2∥efl(x,θ)∥2≤+maxk1k2⋯kn∥alk1k2⋯kn(θ)∥2nχ0jkj.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 x0i, one fuzzy rule is established, which approximates the local dynamics around x0i. And the ith T-S fuzzy rule can be represented as
(33)Rulei:IFx1isA1i(x1),…,andxnisAni(xn),THEN[x˙z]=[Ai(θ)B1i(θ)B2i(θ)C1i(θ)0D12i(θ)][xωu],
where i=1,2,…,L. More concretely, it is known that i=k1k2⋯kn and L=max{k1·k2⋯kn}.Finally, the overall LPV T-S fuzzy model can be represented as(34)[x˙z]=[A(α,θ)B1(α,θ)B2(α,θ)C1(α,θ)0D12(α,θ)][xωu][x˙z]=[∑i=1Lαi(x)Ai(θ)∑i=1Lαi(x)B1i(θ)∑i=1Lαi(x)B2i(θ)∑i=1Lαi(x)C1i(θ)0∑i=1Lαi(x)D12i(θ)][xωu][x˙z]=∑i=1L‍αi(x)[Ai(θ)B1i(θ)B2i(θ)C1i(θ)0D12i(θ)][xωu].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
(35)X¯r=Xr∪X~r,
where
(36)Xr={x∣αi(x)=αk(x),i,k=1,2,…,L,i≠k},X~r={x∣αi(x)>αk(x),i,k=1,2,…,L,i≠k}.Note that two subspaces are generated around theith T-S fuzzy rule for the single state xj. The schematic about the state xj can be shown in Figure 2. And then, the relation between the rth subspace and the ith T-S fuzzy rule is i=k1k2⋯kn and r=(2k1-2/2k1-1)(2k2-2/2k2-1)⋯(2kn-2/2kn-1)  (i=1,2,…,L; word r=1,2,…,L¯), where L=max{k1·k2⋯kn} and L¯=max{2n·(k1-1)·(k2-1)⋯(kn-1)}. Then, the overall model of the LPV T-S system in the rth subspace can be represented as
(37)[x˙z]=[A¯r(α,θ)B¯1r(α,θ)B¯2r(α,θ)C¯1r(α,θ)0D¯12r(α,θ)][xωu]=[Ai(θ)+A~r(α,θ)B1i(θ)+B~1r(α,θ)B2i(θ)+B~2r(α,θ)C1i(θ)+C~1r(α,θ)0D12i(θ)+D~12r(α,θ)][xωu]
for x∈X¯r, where
(38)A~r(α,θ)=∑k∈ϕr‍αk(x)A~ik(θ),B~1r(α,θ)=∑k∈ϕr‍αk(x)B~1ik(θ),B~2r(α,θ)=∑k∈ϕr‍αk(x)B~2ik(θ),C~1r(α,θ)=∑k∈ϕr‍αk(x)C~1ik(θ),D~12r(α,θ)=∑k∈ϕr‍αk(x)D~12ik(θ),A~ik(θ)=Ak(θ)-Ai(θ),B~1ik=B1k(θ)-B1i(θ),B~2ik=B2k(θ)-B2i(θ),C~1ik=C1k(θ)-C1i(θ),D~12ik=D12k(θ)-D12i(θ),ϕr={k∣αk(x)≠0,αi(x)≥αk(x)},
and ϕr is a set including the indexes of the membership functions which are nonzero and less than αi(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 ofxj.In order to find the piecewise parameter-dependent Lyapunov function which is continuous across therth subspace boundary at a fixed θ, the following constant matrix Fr is established from the structure information of the rth subspace, which fulfills
(39)Frx=Fpx,x∈X¯r∩X¯p,r,p=1,2,…,L¯.Then, the piecewise parameter-dependent Lyapunov function candidates that are continuous across therth subspace boundary can be parameterized as
(40)V(θ)=xTPr(θ)x,x∈X¯r,
with
(41)Pr(θ)=FrTT(θ)Fr,
where T(θ) is the symmetric matrix and characterizes Pr(θ) with Fr together.In order to carry forward the control design, the following upper bounds for the interpolation terms in (37) can be defined as
(42)[A~r(α,θ)][A~r(α,θ)]T≤ErA(α,θ)ErAT(α,θ),[B~1r(α,θ)][B~1r(α,θ)]T≤ErB1(α,θ)ErB1T(α,θ),[B~2r(α,θ)][B~2r(α,θ)]T≤ErB2(α,θ)ErB2T(α,θ),[C~1r(α,θ)][C~1r(α,θ)]T≤ErC1(α,θ)ErC1T(α,θ),[D~12r(α,θ)][D~12r(α,θ)]T≤ErD12(α,θ)ErD12T(α,θ).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
(43)ErA(α,θ)ErAT(α,θ)=(∑k∈ϕr‍αk(x¯)A~ik(θ))(∑k∈ϕr‍αk(x¯)A~ik(θ))T,
where x¯ is the state such that αk(x¯)=0.5.Definition 3.
On the compact setΨ⊂Rs, one has finite nonnegative numbers {υq}q=1s. Then the bounded variations set of scheduling parameters can be defined as
(44)ΓΨυ={θ∈C1(R,Rs)∣θ∈Ψ,∥θ˙q∥2≤υq,q=1,…,s},
where C1 represents the class of functions which are piecewise continuous and one-order differential.Definition 4.
On the compactΨ∈Rs, one has the LPV system
(45)x˙=A(θ)x,
where θ∈ΓΨυ. If there exists a continuously differentiable symmetric function P(θ) such that P(θ)>0 and
(46)AT(θ)P(θ)+P(θ)A(θ)+P˙(θ)<0,
for all θ∈ΓΨυ, 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
(47)Pr(θ)>0,(48)[Pr(θ)AiT(θ)+Ai(θ)Pr(θ)-P˙r(θ)+εA~rPr(θ)ErA(α,θ)ErAT(α,θ)Pr(θ)Pr(θ)-εA~rPr(θ)I]<0
with
(49)Pr(θ)=(FrTFr)-1FrTT(θ)Fr(FrTFr)-1
for r=1,2,…,L¯, then the fuzzy system is globally asymptotically stable.Proof.
Define the following Lyapunov functionV(θ):
(50)V(θ)=xTPr-1(θ)x,x∈X¯r.
From (41), we can get
(51)Pr-1(θ)=FrTT-1(θ)Fr.
From (50) and (51), there exist constants σ>0 and β>0 such that
(52)σ∥x∥22≤V(θ)≤β∥x∥22.
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 constantεXY>0 and matrices X and Y with appropriate dimensions, the following matrix inequality can be gotten:
(53)XTY+YTX≤1εXYXTX+εYTY.
From (53), define the parameter-dependent positive scalar εbc(θ), where the subscript bc means the relation between b and c just like εXY means the relation between X and Y. In many cases, εab(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
(54)Pr(θ)(Ai(θ)+A~r(α,θ))T+(Ai(θ)+A~r(α,θ))Pr(θ)-P˙r(θ)=Pr(θ)AiT(θ)+Pr(θ)A~rT(α,θ)+Ai(θ)Pr(θ)+A~r(α,θ)Pr(θ)-P˙r(θ)≤Pr(θ)AiT(θ)+Ai(θ)Pr(θ)+1εA~rPr(θ)Pr(θ)Pr(θ)+εA~rPr(θ)A~rT(α,θ)A~r(α,θ)-P˙r(θ)≤Pr(θ)AiT(θ)+Ai(θ)Pr(θ)+1εA~rPr(θ)Pr(θ)Pr(θ)+εA~rPr(θ)ErAT(α,θ)ErA(α,θ)-P˙r(θ),
for the system (37). Besides, via the Schur complement lemma, it follows from (48) that
(55)Pr(θ)AiT(θ)+Ai(θ)Pr(θ)+1εA~rPr(θ)Pr(θ)Pr(θ)+εA~rPr(θ)ErA(α,θ)ErAT(α,θ)-P˙r(θ)<0.
Then, from (54) and (55), we have
(56)Pr(θ)(Ai(θ)+A~r(α,θ))T+(Ai(θ)+A~r(α,θ))Pr(θ)-P˙r(θ)<0,
which suggests that
(57)(Ai(θ)+A~r(α,θ))TPr-1(θ)+Pr-1(θ)(Ai(θ)+A~r(α,θ))+P˙r-1(θ)<0,
where P˙r-1(θ)=-FrTT-1(θ)T˙(θ)T-1(θ)Fr=-Pr-1(θ)P˙(θ)Pr-1(θ).
Moreover, there exists a constantδ>0 such that
(58)(Ai(θ)+A~r(α,θ))TPr-1(θ)+Pr-1(θ)(Ai(θ)+A~r(α,θ))+P˙r-1(θ)+δI<0.
As a result, we have(59)V˙(θ)=xT[(Ai(θ)+A~r(α,θ))TPr-1(θ)+Pr-1(θ)V˙(θ)=xT1×(Ai(θ)+A~r(α,θ))+P˙r-1(θ)[(Ai(θ)+A~r(α,θ))TPr-1(θ)]xV˙(θ)≤xT(-δI)x=-δ∥x∥22,
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 fixedθ, 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-FeedbackH∞ Control Design of LPV T-S Fuzzy System

In this section, the state-feedbackH∞ 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
(60)u=Kr(θ)x,x∈X¯r.Substituting (60) into (34), the closed-loop system can be obtained as
(61)x˙=AC(α,θ)x+BC(α,θ)ω,z=CC(α,θ)x,
where
(62)AC(α,θ)=A(α,θ)+B2(α,θ)K(θ),BC(α,θ)=B1(α,θ),CC(α,θ)=C1(α,θ)+D12(α,θ)K(θ).In therth subspace, the system (61) can be represented as
(63)x˙=ACr(α,θ)x+BCr(α,θ)ω,z=CCr(α,θ)x,x∈X¯r,
where
(64)ACr(α,θ)=Ai(θ)+A~r(α,θ)ACr(α,θ)=+(B2i(θ)+B~2r(α,θ))Kr(θ),BCr(α,θ)=B1i(θ)+B~1r(α,θ),CCr(α,θ)=C1i(θ)+C~1r(α,θ)CCr(α,θ)=+(D12i(θ)+D~12r(α,θ))Kr(θ).When substituting (37) into (1) and considering the approximating error, the closed-loop nonlinear system in the rth  subspace can be represented as
(65)[x˙z]=[f(x,θ)+d0(x,θ)ω+g0(x,θ)uφ(x,θ)+g1(x,θ)u][x˙z]=[A¯r(α,θ)B¯1r(α,θ)B¯2r(α,θ)C¯1r(α,θ)0D¯12r(α,θ)][xωu][x˙z]=+[(f(x,θ)-A¯r(α,θ)x)+(d0(x,θ)-B¯1r(α,θ))ω+(g0(x,θ)-B¯2r(α,θ))u(φ(x,θ)-C¯1r(α,θ)x)+(g1(x,θ)-D¯12r(α,θ))u][x˙z]=[A¯r(α,θ)+B¯2r(α,θ)Kr(θ)B¯1r(α,θ)C¯1r(α,θ)+D¯12r(α,θ)Kr(θ)0][xω][x˙z]=+[(f(x,θ)-A¯r(α,θ)x)+(g0(x,θ)-B¯2r(α,θ))Kr(θ)x+(d0(x,θ)-B¯1r(α,θ))ω(φ(x,θ)-C¯1r(α,θ)x)+(g1(x,θ)-D¯12r(α,θ))Kr(θ)x][x˙z]=[ACr(α,θ)BCr(α,θ)CCr(α,θ)0][xω]+[Δfr(θ)+Δg0r(θ)+Δd0r(θ)Δφr(θ)+Δg1r(θ)],
where
(66)Δfr(θ)=f(x,θ)-A¯r(x,θ)x,Δg0r(θ)=(g0(x,θ)-B¯2r(α,θ))Kr(θ)x,Δd0r(θ)=(d0(x,θ)-B¯1r(α,θ))ω,Δφr(θ)=φ(x,θ)-C¯1r(α,θ)x,Δg1r(θ)=(g1(x,θ)-D¯12r(α,θ))Kr(θ)x.According to (37) and (65), the approximation errors, Δfr(θ), Δg0r(θ), Δd0r(θ), Δφr(θ), and Δg1r(θ), of the LPV T-S fuzzy system can be represented as follows:
(67)∥Δfr(θ)∥2=∥f(x,θ)-A¯r(α,θ)x∥2=∥f(x,θ)-(Ai(θ)+∑k∈ϕr‍αk(x)A~ik(θ))x∥2≤∥(ΔAi(θ)+∑k∈ϕr‍αk(x)ΔA~ik(θ))x∥2=∥ΔA¯r(θ)x∥2,
where
(68)ΔA¯r(θ)=ΔAi(θ)+∑k∈ϕr‍αk(x)A~ik(θ)=ΔAi(θ)+∑k∈ϕr‍αk(x)(ΔAk(θ)-ΔAi(θ))=αi(x)ΔAi(θ)+∑k∈ϕr‍αk(x)ΔAk(θ)=(αi(x)δAri(θ)+∑k∈ϕr‍αk(x)δArk(θ))A^r(θ)=δA¯r(θ)A^r(θ),∥Δg0r(θ)∥2=∥(g0(x,θ)-B¯2r(α,θ))Kr(θ)x∥2=∥(g0(x,θ)-(B2i(θ)+∑k∈ϕr‍αk(x)B~2ik(θ)))2×Kr(θ)x(g0(x,θ)-(B2i(θ)+∑k∈ϕr‍αk(x)B~2ik(θ)))∥2≤∥(ΔB2i(θ)+∑k∈ϕr‍αk(x)ΔB~2ik(θ))Kr(θ)x∥2=∥ΔB¯2r(θ)Kr(θ)x∥2,
where
(69)ΔB¯2r(θ)=ΔB2i(θ)+∑k∈ϕr‍αk(x)ΔB~2ik(θ)=ΔB2i(θ)+∑k∈ϕr‍αk(x)(ΔB2k(θ)-ΔB2i(θ))=αi(x)ΔB2i(θ)+∑k∈ϕr‍αk(x)ΔB2k(θ)=(αi(x)δB2i,r(θ)+∑k∈ϕr‍αk(x)δB2k,r(θ))B^2r(θ)=δB¯2rB^2r(θ),∥Δd0r(θ)∥2=∥(d0(x,θ)-B¯1r(α,θ))ω∥2=∥(d0(x,θ)-(B1i(θ)+∑k∈ϕr‍αk(x)B~1ik(θ)))ω∥2≤∥(ΔB1i(θ)+∑k∈ϕr‍αk(x)ΔB~1ik(θ))ω∥2=∥ΔB¯1r(θ)ω∥2,
where
(70)ΔB¯1r(θ)=ΔB1i(θ)+∑k∈ϕr‍αk(θ)ΔB~1ik(θ)=ΔB1i(θ)+∑k∈ϕr‍αk(x)(ΔB1k(θ)-ΔB1i(θ))=αi(x)ΔB1i(θ)+∑k∈ϕr‍αk(x)ΔB1k(θ)=(αi(x)δB1i,r(θ)+∑k∈ϕr‍αk(x)δB1k,r(θ))B^1r(θ)=δB¯1rB^1r(θ),Δ∥φr(θ)∥2=∥φ(x,θ)-C¯1r(α,θ)x∥2=∥φ(x,θ)-(C1i(θ)+∑k∈ϕr‍αk(x)C~1ik(θ))x∥2≤∥(ΔC1i(θ)+∑k∈ϕr‍αk(x)ΔC~1ik(θ))x∥2=∥ΔC¯1r(θ)x∥2,
where
(71)ΔC¯1r(θ)=ΔC1i(θ)+∑k∈ϕr‍αk(x)ΔC~1ik(θ)ΔC¯1r(θ)=ΔC1i(θ)+∑k∈ϕr‍αk(x)(ΔC1k(θ)-ΔC1i(θ))ΔC¯1r(θ)=αi(x)ΔC1i(θ)+∑k∈ϕr‍αk(x)ΔC1k(θ)ΔC¯1r(θ)=(αi(x)δC1i,r(θ)+∑k∈ϕr‍αk(x)δC1k,r(θ))ΔC¯1r(θ)(θ)×C^1r(θ)ΔC¯1r(θ)=δC¯1r(θ)C^1r(θ),∥Δg1r(θ)∥2=∥(g1(x,θ)-D¯12r(α,θ))Kr(θ)x∥2=∥(g1(x,θ)-(+∑k∈ϕr‍αk(x)D~12ik(θ)D12i(θ)11111111111111+∑k∈ϕr‍αk(x)D~12ik(θ)))×Kr(θ)x(g1(x,θ)-(+∑k∈ϕr‍αk(x)D~12ik(θ)D12i(θ)∥2≤∥(ΔD12i(θ)+∑k∈ϕr‍αk(x)ΔD~12ik(θ))×Kr(θ)x(ΔD12i(θ)+∑k∈ϕr‍αk(x)ΔD~12ik(θ))∥2=∥ΔD¯12r(θ)Kr(θ)x∥2,
where
(72)ΔD¯12r(θ)=ΔD12i(θ)+∑k∈ϕr‍αk(x)ΔD~12ik(θ)=ΔD12i(θ)+∑k∈ϕr‍αk(x)(ΔD12k(θ)-ΔD12i(θ))=αi(x)ΔD12i(θ)+∑k∈ϕr‍αk(x)ΔD12k(θ)=(αi(x)δD12i,r(θ)+∑k∈ϕr‍αk(x)δD12k,r(θ))D^12r(θ)=δD¯12r(θ)D^12r(θ).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 thatαi(x)+∑k∈ϕrαk(x)=1 and ∥δAr*(θ)∥2≤1,…, ∥δD¯12*,r(θ)∥2≤1 for i=1,2,…,L and r=1,2,…,L¯. Then, we can get
(73)[∥Δfr(θ)∥2∥Δd0r(θ)∥2∥Δg0r(θ)∥2∥Δφr(θ)∥20∥Δg1r(θ)∥2]≤[∥ΔA¯r(θ)x∥2∥ΔB¯1r(θ)ω∥2∥ΔB¯2r(θ)Kr(θ)x∥2∥ΔC¯1r(θ)x∥20∥ΔD¯12r(θ)Kr(θ)x∥2],
for all the x, and describe the bounding matrices as
(74)[ΔA¯r(θ)ΔB¯1r(θ)ΔB¯2r(θ)ΔC¯1r(θ)0ΔD¯12r(θ)]=[δA¯r(θ)A^r(θ)δB¯1r(θ)B^1r(θ)δB¯2r(θ)B^2r(θ)δC¯1r(θ)C^1r(θ)0δD¯12r(θ)D^12r(θ)],
where ∥δA¯r(θ)∥2≤1,… and ∥δD¯12r(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 Pr(θ)=(FrTFr)-1FrTT(θ)Fr(FrTFr)-1 satisfying
(75)Pr(θ)>0,(76)Pr(θ)ACrT(α,θ)+ACr(α,θ)Pr(θ)-P˙r(θ)+1εΔfrPr(θ)Pr(θ)A^rT(θ)A^r(θ)Pr(θ)+1εΔg0rPr(θ)Pr(θ)(B^2r(θ)Kr(θ))T(B^2r(θ)Kr(θ))Pr(θ)+(εΔfrPr(θ)+εΔg0rPr(θ)+εΔd0rPr(θ))+(1+εΔφrCCr(θ)+εΔg1rCCr(θ))×Pr(θ)CCrT(α,θ)CCr(α,θ)Pr(θ)+(1+1εΔφrCCr(θ)+1εΔφrΔg1r(θ))×Pr(θ)C^1r(α,θ)TC^1r(α,θ)Pr(θ)+(1+1εΔg1rCCr(θ)+1εΔφrΔg1r(θ))Pr(θ)×D^12r(α,θ)TD^12r(α,θ)Pr(θ)+(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))-1×BCr(α,θ)BCrT(α,θ)<0,(77)γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ)>0,
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:
(78)ACrT(α,θ)Pr-1(θ)+Pr-1(θ)ACr(α,θ)+P˙r-1<0.
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
(79)ACrT(α,θ)Pr-1(θ)+Pr-1(θ)ACr(α,θ)+P˙r-1(θ)+1εΔfrPr(θ)A^rT(θ)A^r(θ)+1εΔg0rPr(θ)(B^2r(θ)Kr(θ))TB^2r(θ)Kr(θ)+(εΔfrPr(θ)+εΔg0rPr(θ)+εΔd0rPr(θ))Pr-1(θ)Pr-1(θ)+(1+εΔφrCCr(θ)+εΔg1rCCr(θ))CCrT(α,θ)CCr(α,θ)+(1+1εΔφrCCr(θ)+1εΔφrΔg1r(θ))C^1r(α,θ)TC^1r(α,θ)+(1+1εΔg1rCCr(θ)+1εΔφrΔg1r(θ))D^12r(α,θ)TD^12r(α,θ)+(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))-1Pr-1(θ)BCr(α,θ)×BCrT(α,θ)Pr-1(θ)<0
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 functionV(θ)=xTPr-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
(80)∫0∞‍V˙(θ)dt=∫0∞‍(x˙TPr-1(θ)x+xTPr-1(θ)x˙+xTP˙r-1(θ)x)dt=∫0∞‍[xT(+P˙r-1(θ))ACrT(α,θ)Pr-1(θ)+Pr-1(θ)ACr(α,θ)1111111111+P˙r-1(θ))x+ΔfrT(θ)Pr-1(θ)x1111111+Δg0r(θ)TPr-1(θ)x+Δd0r(θ)TPr-1(θ)x1111111+xTPr-1(θ)Δfr(θ)+xTPr-1(θ)Δg0r(θ)1111111+xTPr-1(θ)Δd0r(θ)+ωTBCrT(α,θ)Pr-1(θ)x1111111+xTPr-1(θ)BCr(α,θ)ωxT(+P˙r-1(θ))ACrT(α,θ)Pr-1(θ)+Pr-1(θ)ACr(α,θ)]dt≤∫0∞‍[xT(+P˙r-1(θ))ACrT(α,θ)Pr-1(θ)+Pr-1(θ)ACr(α,θ)1111111111+P˙r-1(θ))x1111111+1εΔfrPr(θ)ΔfrT(θ)Δfr(θ)1111111+εΔfrPr(θ)xTPr-1(θ)Pr-1(θ)x1111111+1εΔg0rPr(θ)Δg0r(θ)TΔg0r(θ)1111111+εΔg0rPr(θ)xTPr-1(θ)Pr-1(θ)x1111111+1εΔd0rPr(θ)Δd0r(θ)TΔd0r(θ)1111111+εΔd0rPr(θ)xTPr-1(θ)Pr-1(θ)x1111111+ωTBCrT(α,θ)Pr-1(θ)x+xTPr-1(θ)BCr(α,θ)ωxT(ACrT(α,θ)Pr-1(θ)+Pr-1(θ)ACr(α,θ)+P˙r-1(θ))x]dt≤∫0∞‍[xT(+P˙r-1(θ))ACrT(α,θ)Pr-1(θ)+Pr-1(θ)ACr(α,θ)1111111111+P˙r-1(θ))x1111111+1εΔfrPr(θ)(A^r(θ)x)TA^r(θ)x+1εΔg0rPr(θ)1111111×(B^2r(θ)Kr(θ)x)TB^2r(θ)Kr(θ)x1111111+1εΔd0rPr(θ)(B^1r(θ)ω)TB^1r(θ)ω1111111+(εΔfrPr(θ)+εΔg0rPr(θ)1111111111+εΔd0rPr(θ))xTPr-1(θ)Pr-1(θ)x1111111+ωTBCrT(α,θ)Pr-1(θ)x+xTPr-1(θ)BCr(α,θ)ωxT(ACrT(α,θ)Pr-1(θ)+Pr-1(θ)ACr(α,θ)+P˙r-1(θ))x]dt≤∫0∞‍[-xT((1+εΔφrCCr(θ)+εΔg1rCCr(θ))111111111111×CCrT(α,θ)CCr(α,θ)111111111111+(1+1εΔφrCCr(θ)1111111111111111+1εΔφrΔg1r(θ))C^1r(α,θ)TC^1r(α,θ)111111111111+(1+1εΔg1rCCr(θ)1111111111111111+1εΔφrΔg1r(θ))D^12r(α,θ)TD^12r(α,θ)111111111111+(γ2I-1εΔd0rPr(θ)(B^1r(θ))TB^1r(θ))-1Pr-1(θ)111111111111×BCr(α,θ)BCrT(α,θ)Pr-1(θ))x1111111+1εΔd0rPr(θ)(B^1r(θ)ω)TB^1r(θ)ω1111111+ωTBCrT(α,θ)Pr-1(θ)x1111111+xTPr-1(θ)BCr(α,θ)ω+γ2ωTω-γ2ωTω[-xT((1+εΔφrCCr(θ)+εΔg1rCCr(θ))CCrT(α,θ)CCr(α,θ)]dt≤∫0∞‍[×∥(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))12ωBCrT(α,θ)Pr-1(θ)x)∥22]-zTz+γ2ωTω111111111-∥×∥(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))12ωBCrT(α,θ)Pr-1(θ)x)∥22(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))1/2ω1111111111111-(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))-1/21111111111111×∥(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))12ωBCrT(α,θ)Pr-1(θ)x∥22]dt≤∫0∞‍[-zTz+γ2ωTω]dt.
That is
(81)V(x(∞),θ)-V(x(0),θ)≤∫0∞‍(-zTz+γ2ωTω)dt,
which suggests that, with x(0)=0,
(82)∥z∥2≤γ∥ω∥2.
Overall, combining with the condition (75) and Pr(θ)=(FrTFr)-1FrTT(θ)Fr(FrTFr)-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-feedbackH∞ 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 Theorem7, 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 Qr(θ)=Kr(θ)Pr(θ) and Pr(θ)=(FrTFr)-1FrTT(θ)Fr(FrTFr)-1 satisfying
(83)Pr(θ)>0,[WrPr(θ)QrT(θ)Pr(θ)-MPr-10Qr(θ)0-MQr-1]<0,
where
(84)Wr=Pr(θ)AiT(θ)+Ai(θ)Pr(θ)-P˙r(θ)Wr=+QrT(θ)B2i(θ)T+B2i(θ)Qr(θ)Wr=+εA~rPr(θ)ErA(α,θ)ErAT(α,θ)Wr=+εB~2rPr(θ)ErB2(α,θ)ErB2T(α,θ)Wr=+(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))-1Wr=×(1+1εB1iB~1r(θ))B1i(θ)B1i(θ)TWr=+(γ2I-1εΔd0rPr(θ)B^1r(θ)TB^1r(θ))-1Wr=×(1+εB1iB~1r(θ))ErB1(α,θ)ErB1(α,θ)T,MPr=1εA~rPr(θ)+1εΔfrPr(θ)A^rT(θ)A^r(θ)MPr=+(1+1εΔφrCCr(θ)+1εΔφrΔg1r(θ))MPr=×C^1r(α,θ)TC^1r(α,θ)MPr=+(1+1εΔg1rCCr(θ)+1εΔφrΔg1r(θ))MPr=×D^12r(α,θ)TD^12r(α,θ)MPr=+(1+εΔφrCCr(θ)+εΔg1rCCr(θ))MPr=×(1+1εC1iC~1r(θ)+1εC1iD12i(θ)+1εC1iD~12r(θ))MPr=×C1i(θ)TC1i(θ)MPr=+(1+εΔφrCCr(θ)+εΔg1rCCr(θ))MPr=×(1+εC1iC~1r(θ)+εD12iC~1r(θ)+1εC~1rD~12r(θ))ErC1TMPr=×ErC1T(α,θ)ErC1(α,θ),MQr=1εB~2rPr(θ)+1εΔg0rPr(θ)B^2r(θ)TB^2r(θ)MQr=+(1+εΔφrCCr(θ)+εΔg1rCCr(θ))MQr=×(1+εC1iD12i(θ)+1εD12iC~1r(θ)+1εD12iD~12r(θ))MQr=×D12i(θ)TD12i(θ)MQr=+(1+εΔφrCCr(θ)+εΔg1rCCr(θ))MQr=×(1+εC1iD~12r(θ)+εC~1rD~12r(θ)+εD12iD~12r(θ))MQr=×ErD12T(α,θ)ErD12(α,θ),
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
(85)Kr(θ)=Qr(θ)Pr-1(θ).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
(86)Pr(θ)ACrT(α,θ)+ACr(α,θ)Pr(θ)=Pr(θ)(Ai(θ)+A~r(α,θ)1111111111+(B2i(θ)+B~2r(α,θ))Kr(θ))T1111+(Ai(θ)+A~r(α,θ)+(B2i(θ)+B~2r(α,θ))1111111×Kr(θ)(Ai(θ)+A~r(α,θ)+(B2i(θ)+B~2r(α,θ)))Pr(θ)=Pr(θ)AiT(θ)+Ai(θ)Pr(θ)+QrT(θ)B2i(θ)T1111+B2i(θ)Qr(θ)+Pr(θ)A~rT(α,θ)1111+A~r(α,θ)Pr(θ)+QrT(θ)B~2r(α,θ)T1111+B~2r(α,θ)Qr(θ)≤Pr(θ)AiT(θ)+Ai(θ)Pr(θ)1111+QrT(θ)B2i(θ)T+B2i(θ)Qr(θ)1111+εA~rPr(θ)ErA(α,θ)ErAT(α,θ)1111+εB~2rPr(θ)ErB2(α,θ)ErB2T(α,θ)1111+1εA~rPr(θ)Pr(θ)Pr(θ)+1εB~2rPr(θ)QrT(θ)Qr(θ).
Besides, we also have
(87)Pr(θ)CCiT(α,θ)CCi(α,θ)Pr(θ)=Pr(θ)[C1i(θ)TC1i(θ)+C1i(θ)TC~1r(α,θ)111111111+C~1r(α,θ)TC1i(θ)111111111+C~1r(α,θ)TC~1r(α,θ)]Pr(θ)1111+Pr(θ)C1i(θ)TD12i(θ)Qr(θ)1111+Pr(θ)C1i(θ)TD~12r(α,θ)Qr(θ)1111+Pr(θ)C~1r(α,θ)TD12i(θ)Qr(θ)1111+Pr(θ)C~1r(α,θ)TD~12r(α,θ)Qr(θ)1111+QrT(θ)D12i(θ)TC1i(θ)Pr(θ)1111+QrT(θ)D12i(θ)TC~1r(α,θ)Pr(θ)1111+QrT(θ)D~12r(α,θ)TC1i(θ)Pr(θ)1111+QrT(θ)D~12r(α,θ)TC~1r(α,θ)Pr(θ)1111+QrT(θ)[D12i(θ)TD12i(θ)111111111111+D12i(θ)TD~12r(α,θ)111111111111+D~12r(α,θ)TD12i(θ)111111111111+D~12r(α,θ)TD~12r(α,θ)]Qr(θ)≤Pr(θ)[(1+1εC1iC~1r(θ)+1εC1iD12i(θ)111111111111+1εC1iD~12r(θ))C1i(θ)TC1i(θ)1111111111+(+1εC~1rD~12r(θ))1+εC1iC~1r(θ)+εD12iC~1r(θ)11111111111111+1εC~1rD~12r(θ))ErC1T1111111111×(α,θ)ErC1(α,θ)+1εC~1rD~12r(θ))]Pr(θ)1111+QrT(θ)[(1+εC1iD12i(θ)+1εD12iC~1r(θ)11111111111111+1εD12iD~12r(θ))D12i(θ)TD12i(θ)111111111111+(+εD12iD~12r(θ))1+εC1iD~12r(θ)+εC~1rD~12r(θ)111111111111111+εD12iD~12r(θ))111111111111×ErD12T(α,θ)ErD12(α,θ)[(1+εC1iD12i(θ)+1εD12iC~1r(θ)]Qr(θ),(88)BCr(α,θ)BCrT(α,θ)=B1i(θ)B1i(θ)T+B1i(θ)B~1r(α,θ)T+B~1r(α,θ)B1i(θ)T+B~1r(α,θ)B~1r(α,θ)T≤(1+1εB1iB~1r(θ))B1i(θ)B1i(θ)T+(1+εB1iB~1r(θ))ErB1(α,θ)ErB1T(α,θ).
Substituting (86), (87), and (88) into (76), the complete representation of (76) can be gotten. Then, by defining Wr, MPr, and MQr 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 Theorem7, 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 Theorem9, 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(89)x˙=f(x,θ)+B1ω+B2u,z=φ(x,θ),
where f(x,θ)=[0.0005θx12+0.0008x2x1-0.5x2], φ(x,θ)=0.5θ-x1, B1=[0.00010]T, B2=[01]T, and ω=sin(t) is the sinusoidal interference signal.Here, define thatθ=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:
(90)Rule1:Ifx1isabout-1,then{x˙=[-0.0005θ0.00081-0.5]x+[0.00010]ω+[01]uz=[-0.5θ-10]x,(91)Rule2:Ifx1isabout0,then{x˙=[00.00081-0.5]x+[0.00010]ω+[01]uz=[-10]x,(92)Rule3:Ifx1isabout1,then{x˙=[0.0005θ0.00081-0.5]x+[0.00010]ω+[01]uz=[0.5θ-10]x.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 x12; 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 X¯1,X¯2,X¯3, and X¯4 (as shown in Figure 3). Then, the T-S fuzzy model in each subspace can be represented as
(93)X¯1:{x˙=([-0.0005θ0.00081-0.5]+α2(x1)[0.0005θ000])x+[0.00010]ω+[01]uz=([-0.5θ-10]+α2(x1)[0.5θ0])x,x1∈[-1,-0.5],(94)X¯2:{x˙=([00.00081-0.5]+α1(x1)[-0.0005θ000])x+[0.00010]ω+[01]uz=([-10]+α1(x1)[-0.5θ0])x,x1∈[-0.5,0],(95)X¯3:{x˙=([00.00081-0.5]+α3(x1)[0.0005θ000])x+[0.00010]ω+[01]uz=([-10]+α3(x1)[0.5θ0])x,x1∈[0,0.5],(96)X¯4:{x˙=([0.0005θ0.00081-0.5]+α2(x1)[-0.0005θ000])x+[0.00010]ω+[01]uz=([0.5θ-10]+α2(x1)[-0.5θ0])x,x1∈[0.5,1].Figure 3
Piecewise space partition ofx1.According to the method in [22], it can be set that αk(x¯,θ)=0.5 and the matrices of ErA(θ) and ErC1(θ) for the ith (i=1,2,3) fuzzy rule in the rth (r=1,2,3,4,k∈ϕr) subspace can be obtained. Using the detailed method in [20], the constant matrices Fr can be given.When the LPV T-S fuzzy system is linear with the scheduling parameterθ, we can parameterize the system matrices A¯r(θ) and C¯1r(θ) and decision variables Pr(θ) and Qr(θ) in the rth subspace as
(97)A¯r(θ)=A¯r0+θA¯r1,C¯1r(θ)=C¯1,0r+θC¯1,1r,P¯r(θ)=P¯r0+θP¯r1,Q¯r(θ)=Q¯r0+θQ¯r1,
where A¯r0, A¯r1, C¯1,0r, C¯1,1r, P¯r0, P¯r1, Q¯r0, and Q¯r1 are constant values and r=1,2,3,4.Besides, the gridding process is avoided when solving the parameter-dependent Lyapunov functionsPr(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 withγ=0.9, the solution to the LMIs when ε*(θ)=10 can be obtained:
(98)P1(θ)=[0.0265+0.0777θ-0.2+0.996θ-0.2+0.996θ4.475+1.5562θ],Q1(θ)=[0.1476+0.0840θ10.9778+0.0191θ],P2(θ)=[-0.0126+0.3006θ-0.6849+0.0309θ-0.6849+0.0309θ3.6378+0.8595θ],Q2(θ)=[0.0997+0.0373θ7.2065-0.1058θ],P3(θ)=[-0.0126+0.3006θ-0.6849+0.0309θ-0.6849+0.0309θ3.6378+0.8595θ],Q3(θ)=[0.0997+0.0373θ7.2065-0.1058θ],P4(θ)=[-0.0176+0.0651θ-0.1457-1.0275θ-0.1457-1.0275θ6.5439+1.5280θ],Q4(θ)=[0.0423-0.0114θ10.8734+0.0179θ].Then, the state-feedback controller can be determined asu=Kr(θ)x, where Kr(θ)=Qr(θ)Pr-1(θ) and r=1,2,3,4.Because the effective performance region of the controller aboutx1 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 ofx1.Figure 5
State response ofx2.Figure 6
Evaluation response ofz.Meanwhile, in order to validate the effectiveness of the controller during the scheduling process ofθ, 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 ofx1 in scheduling.Figure 8
State response ofx2 in scheduling.Figure 9
Evaluation response ofz in scheduling.From Figures4, 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-feedbackH∞ 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.



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*Source: 169454-2013-09-30.xml*