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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 169454, 18 pages
State-Feedback Control for LPV System Using T-S Fuzzy Linearization Approach
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China
Received 2 April 2013; Revised 1 July 2013; Accepted 24 July 2013
Academic Editor: Xiao He
Copyright © 2013 Jizhen Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 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.
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  and developed by Sugeno and Kang , 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.  via the nonlinear sector method which was an exact modeling procedure with constant consequent parts and nonlinear membership functions. Then, Kluska  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  discussed the ways to analyze the relation between the local and global approximation performances of the T-S fuzzy model. Subsequently, Teixeira and Zak  gave the T-S fuzzy modeling procedure with homogeneous linear consequent parts by solving a convex optimization problem. And Tanaka and Wang  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 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 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 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 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 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 are the nonlinear functions of and . Then, referring to the way in , 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 which can be steady or transient one and corresponds to the ith fuzzy rule, . It should be noted that 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 . Because the local linear model should approximate the local dynamics in the vicinity of , we can get where and are arbitrarily changed variables and are parameter-dependent matrices.
Take the first equation for example. Due to that, , and are nonlinear functions of operating state at any , and the matrices , , and 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 and , which means that and are continuous functions on the compact set . Meanwhile, can be equivalently represented as
Expanding of (4) around the operating state and neglecting second- and higher-order terms, we can get where is the gradient, a column vector, of evaluated at . Substituting (5) into (6), we can get where is arbitrarily close to . From (7), it can be found that the coefficient vector needs to be estimated. And in the vicinity of , 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: where is and ; it is a convex function on the feasible set , where and . Assume that is convex and there exist and such that where is the Lagrange multiplier. Then, is the optimal solution of over and is the sufficient condition for the convex optimization problem . Particularly, when is affine linear, is convex.
Considering the estimation of around in (7), the optimal index for corresponding to the th fuzzy rule can be defined as
Then, the optimal problem can be constructed as 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 , it is the linear function of , so the feasible set is convex. Therefore, the optimal problem is a convex optimization problem and can be solved according to Lemma 1.
Computing the derivative of and 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 can be chosen as where 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 .
The activated possibility of the th fuzzy rule under a group of premise variables is computed by
The weight coefficients are computed and the center-of-gravity 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 on , where is a predefined positive scalar and can be gotten at the zero state . Considering the nonzero states, they can be represented as , where is a positive scalar and is the serial number of partition for . Define the subregion on (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:
For Rule 0, the activated possibility is 1 inside and 0 outside . And the activated possibility of the Rule under a group of premise variables is computed by where the membership function for (26) is given as Then, can be written as
Here, the notion of the nonlinear measure along in the partition region can be defined as
Assuming the boundary of is , the value of can be set and the proper length of can be chosen to guarantee . Thus, the length of the partition regions, (), of can be different with the same .
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 and Because , the maximum distance between and any vertex point of is less than , which means ; besides, in the boundary of , the max value of can be adjusted to fulfill the required accuracy.
Thus, the approximation error, , can be befittingly minimized by properly setting , which can be shown as
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 , one fuzzy rule is established, which approximates the local dynamics around . And the ith T-S fuzzy rule can be represented as where . More concretely, it is known that and .
Finally, the overall LPV T-S fuzzy model can be represented as
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 th 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 and (; word ), where and . Then, the overall model of the LPV T-S system in the th subspace can be represented as 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, , , , and represent the interpolation terms produced by interactions between the th 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
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 .
Definition 3. On the compact set , one has finite nonnegative numbers . Then the bounded variations set of scheduling parameters can be defined as where represents the class of functions which are piecewise continuous and one-order differential.
Definition 4. On the compact , one has the LPV system where . If there exists a continuously differentiable symmetric function such that and for all , then the continuous function is parametrically dependent quadratically stable (or PDQ stable, for short).
Theorem 5. Consider the LPV T-S fuzzy system (34) with . If there exists a symmetric matrix satisfying with for , then the fuzzy system is globally asymptotically stable.
Proof. Define the following Lyapunov function :
From (41), we can get
From (50) and (51), there exist constants and 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 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 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.
4. State-Feedback 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
In the th subspace, the system (61) can be represented as where
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 and , for and . Then, we can get for all the , and describe the bounding matrices as where and for and .
Theorem 7. For the closed-loop system (61), while considering the approximation error and a positive scalar , if there exist symmetric matrixes and satisfying the closed-loop system (61) while considering the approximation error has the performance with disturbance attenuation . And it is globally asymptotically stable when .
Proof. From (63), the sufficient condition for (46) can be gotten:
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
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 which is piecewise continuous across the rth subspace boundary, we differentiate and then integrate it from zero to infinity in the rth subspace. Then, we can get