State-Feedback H ∞ Control for LPV System Using TS Fuzzy Linearization Approach

This paper discusses the linear parameter varying (LPV) gain scheduling control problem based on the Takagi-Sugeno (T-S) fuzzy linearization approach. Firstly, the affinenonlinear 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.


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][4][5][6][7][8][9][10][11][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][6][7][8] and the quadratic gain scheduling technique based on Lyapunov theory [9][10][11][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][21][22].In addition, considering the approximation error to the ANPV system, the T-S fuzzy  ∞ control has been also studied [23][24][25][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  ∞ 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 parameterdependent Lyapunov functions while the sufficient conditions are given in LMIs form.
(iii) Considering the approximation error to the ANPV system, the state-feedback  ∞ 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 ununiform 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  ∞ 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.

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 where  is a vector of the scheduling parameters which may be system parameters, external inputs, or other parameters;  is a column vector of the state variables with  dimensions; and (, ), . . .,  1 (, ) are the nonlinear functions of  and .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   0 which can be steady or transient one and corresponds to the ith fuzzy rule,  = 1, 2, . . ., .It should be noted that   0 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 where  and  are arbitrarily changed variables and   (), . . .,   12 () are parameter-dependent matrices.Take the first equation ẋ = (, ) +  0 (, ) +  0 (, ) for example.Due to that, (, ),  0 (, ), and  0 (, ) are nonlinear functions of operating state  at any , and the matrices   (),   1 (), and   2 () only depend on .Moreover, because  and  are arbitrarily changed, we can uniquely determine that Define    ()  , a row vector with  dimensions, as the th row of the matrix   ().Then, condition (3) can be equivalently represented as where   (, ) is the th row of (, ).Additionally, assume   (0, ) = 0 and   (, ) ∈ Expanding   (, ) of (4) around the operating state   0 and neglecting second-and higher-order terms, we can get where ∇  (  0 , ) is the gradient, a column vector, of   (, ) evaluated at   0 .Substituting (5) into (6), we can get where  is arbitrarily close to   0 .From (7), it can be found that the coefficient vector    () needs to be estimated.And in the vicinity of   0 , an optimization problem can be constructed by defining an optimal index to evaluate the estimation.Here, define that the notation   represents the set of functions which is -order continuous and differentiable on the domain of input variables.

Lemma 1. Consider the following constrained optimization problem:
minimize  () where () is  :   →  and  ∈  1 ; it is a convex function on the feasible set Ω = { ∈   : ℎ() = 0}, where ℎ :   →   and ℎ ∈  1 .Assume that Ω is convex and there exist  * ∈ Ω and  ∈   such that where  is the Lagrange multiplier.Then,  * is the optimal solution of  over Ω and  is the sufficient condition for the convex optimization problem [27].Particularly, when ℎ() is affine linear, Ω is convex.
Considering the estimation of    () around   0 ̸ = 0 in (7), the optimal index for   (, ) corresponding to the th fuzzy rule can be defined as Then, the optimal problem can be constructed as minimize where ( 5) should be fulfilled as an equality constraint and Transform the objective function in (10) into the form which is a quadratic function of the unknown coefficient vector    () on   .Obviously, it means that     (   ()) is a convex function.Considering the equality constraint ℎ    (   ()), it is the linear function of    (), so the feasible set Ω    = {   () ∈   : ℎ(   ()) = 0} is convex.Therefore, the optimal problem is a convex optimization problem and can be solved according to Lemma 1.
Computing the derivative of     (   ()) and ℎ    (   ()) about    (), we can get Then, substituting ( 13) into ( 9), we can get Considering the equality constraint ( 5), the Lagrange multiplier     () can be represented as As a result, we can get the column vector Then, the ith T-S fuzzy rule corresponding to the operating state   0 can be chosen as Rule : where f  (, ) represents the th local estimation of the th row of (, ),    () is the corresponding coefficient vector of the th fuzzy rule, a column vector, and    (  ) represents the fuzzy set and membership function of the th premise variable,   , in the th fuzzy rule and  = 1, 2, . . ., .
The activated possibility of the th fuzzy rule under a group of premise variables  is computed by x 2 x The weight coefficients are computed and the center-ofgravity method is used for defuzzification: which fulfills that Finally, the overall T-S fuzzy model can be obtained as Usually, based on (21), the evaluation of the overall approximation performance about the nonlinear function   (, ) is defined as 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 Θ 0 = { | |  | <  0 } on Θ, where  0 is a predefined positive scalar and  0  =   (, )/| =0 can be gotten at the zero state  = 0. Considering the nonzero states, they can be represented as is a positive scalar and   ∈  + is the serial number of partition for   .Define the subregion Θ  1  2 ⋅⋅⋅  on Θ (as shown in Figure 1): Then, the coefficient vector in ( 16) can be represented as Consequently, the T-S fuzzy rules can be chosen as follows: Rule 0: For Rule 0, the activated possibility  0 () is 1 inside Θ 0 and 0 outside Θ 0 .And the activated possibility of the Rule  1  2 ⋅ ⋅ ⋅   under a group of premise variables  is computed by where the membership function for ( 26) is given as Then, f (, ) can be written as Here, the notion of the nonlinear measure along   in the partition region    0 can be defined as Assuming the boundary of 0 is  0 , the value of  0 can be set and the proper length of    0 can be chosen to guarantee    0 ≤  0 .Thus, the length of the partition regions, 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 where  ∈ Θ  1  2 ⋅⋅⋅  and Because  ∈ Θ  1  2 ⋅⋅⋅  , the maximum distance between  and any vertex point of ( 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   0 , one fuzzy rule is established, which approximates the local dynamics around   0 .And the ith T-S fuzzy rule can be represented as Rule : (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.

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 where Note that two subspaces are generated around the th T-S fuzzy rule for the single state   .The schematic about the state   can be shown in Figure 2.And then, the relation between the th subspace and the th T-S fuzzy rule is Then, the overall model of the LPV T-S system in the th subspace can be represented as (37) for  ∈   , where and   is a set including the indexes of the membership functions which are nonzero and less than   () around the th rule in the th subspace.For the overall model, Ã (, ), B * (, ), C * (, ), and D * (, ) represent the interpolation terms produced by interactions between the ith rule and the other rules in the th subspace.
In order to find the piecewise parameter-dependent Lyapunov function which is continuous across the th subspace boundary at a fixed , the following constant matrix   is established from the structure information of the th subspace, which fulfills    =   ,  ∈   ∩   , ,  = 1, 2, . . ., . (39) Then, the piecewise parameter-dependent Lyapunov function candidates that are continuous across the th subspace boundary can be parameterized as with where () is the symmetric matrix and characterizes   () with   together.
In order to carry forward the control design, the following upper bounds for the interpolation terms in (37) can be defined as ( 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 where  is the state such that   () = 0.5.
Definition 3. On the compact set Ψ ⊂   , one has finite nonnegative numbers {  }  =1 .Then the bounded variations set of scheduling parameters can be defined as where  1 represents the class of functions which are piecewise continuous and one-order differential.
Proof.Define the following Lyapunov function (): From (41), we can get From ( 50) and ( 51), there exist constants  > 0 and  > 0 such that Thus, using the conditions ( 47) and (49), () 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 () can guarantee the asymptotical stability in each subspace.
If there exist a constant   > 0 and matrices  and  with appropriate dimensions, the following matrix inequality can be gotten: From (53), define the parameter-dependent positive scalar   (), where the subscript  means the relation between  and  just like   means the relation between  and .In many cases,   () can be set as a constant value.Note that  and  are just used for example and do not have any special significance.However, subscripts of all  * () are marked as follows to clearly show the meaning of  * ().
Then, from (53) we can get for the system (37).Besides, via the Schur complement lemma, it follows from (48) that Then, from (54) and (55), we have which suggests that where Moreover, there exists a constant  > 0 such that As a result, we have 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 , 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.

State-Feedback 𝐻 ∞ Control Design of LPV T-S Fuzzy System
In this section, the state-feedback  ∞ control of the LPV T-S fuzzy system is studied.Considering the approximation error, a controller synthesis approach with  ∞ performance is presented.In the rth subspace, the state feedback controller can be represented as Substituting (60) into (34), the closed-loop system can be obtained as where In the th subspace, the system (61) can be represented as where When substituting (37) into (1) and considering the approximating error, the closed-loop nonlinear system in the th subspace can be represented as (66) According to (37) and ( 65 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  * (), they can be set as constant values in many cases by which the computation can be further simplified.However, subscripts of all  * () 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 , the gridding process may be executed for the nonlinear form of  or not.In particular, for the affine linear form of , the multiconvexity property can be used to guarantee the continuous properties of continuously scheduled  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 parameterdependent functions can be obtained and then the controller can be determined.
Mathematical Problems in Engineering matrices   () and   1 () and decision variables   () and   () in the rth subspace as where  0  ,  Besides, the gridding process is avoided when solving the parameter-dependent Lyapunov functions   () in th subspace   .Utilizing the multiconvexity property in [30], it just needs to validate the LMIs at the vertexes of .Here, the two vertexes at  = −2 and  = 2 are tested, respectively.
Following the design algorithm with  = 0.9, the solution to the LMIs when  * () = 10 can be obtained: Then, the state-feedback controller can be determined as  =   (), where   () =   () −1  () and  = 1, 2, 3, 4. Because the effective performance region of the controller about  1 is set as [−1, 1], the scheduling values of  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  1 and  2 and the evaluation response of  are shown in Figures 4, 5, and 6, respectively.
Meanwhile, in order to validate the effectiveness of the controller during the scheduling process of ,  is scheduled from 0.5 to 1 at 0.5 seconds, which guarantees that the variation rate of  is 1.The responses of  1 ,  2 , and  are shown in Figures 7, 8, and 9, respectively.
From Figures 4, 5, and 6, they suggest the effectiveness of the parameter-dependent controller.At different scheduling points of , the system can be stabilized by the state-feedback controller with  ∞ performance boundary  = 0.9.From Figures 7, 8, and 9, the performance when switching at different  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.

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 parameterdependent Lyapunov function was introduced to improve the solvability.Subsequently, the stabilization problem and the = { |   () =   () , ,  = 1, 2, . . ., ,  ̸ = } , X = { |   () >   () , ,  = 1, 2, . . ., ,  ̸ = } .