#### Abstract

This paper is devoted to solve the regulation problem on the basis of local regulators, which are combined using “new” membership functions. As a result, the exact tracking of references is achieved. The design of linear local regulators is suggested in this paper, but now adequate membership functions are computed in order to ensure the proper combination of the local regulators in the interpolation regions. These membership functions, which are given as mathematical expressions, solve the fuzzy regulation problem in a relative simple way. The form of the new membership functions is systematically derived for a class of Takagi-Sugeno (T-S) fuzzy systems. Some numerical examples are used to illustrate the viability of the proposed approach.

#### 1. Introduction

One of the most important problems in control theory consists of finding a controller capable of taking the outputs of a plant towards the reference signals generated by an external system, named exosystem.

This problem has been studied by several authors due to its wide applicability in mechanical systems, aeronautics, and telematics, just to name a few.

The works of Francis [1] and Francis and Wonham [2] have shown that the solvability of a multivariable linear regulator problem corresponds to the solvability of a system of two linear matrix equations, called Francis Equations. Also they have shown that, in the case of error feedback, the regulator which solves the problem includes the exosystem. This property is commonly know as Internal Model Principle.

Later, Isidori, and Byrnes [3] showed that the result established by Francis could be extended to the nonlinear field as a general case and that the equations of Francis represent a particular case of a set of nonlinear equations. They showed that the solvability for the nonlinear case depends on the solution of a set of nonlinear partial differential equations, called Francis-Isidori-Byrnes (FIB) equations. Unfortunately, such equations are too difficult to solve in a practical manner, in general.

On the other hand, some techniques have been proven to be an alternative to solve this problem by combining the theory of output regulation and the Takagi-Sugeno fuzzy modeling [4–9].

In [10], the authors propose an approach based on the weighted summation of local linear regulators in order to synchronize chaotic systems by means of regulation theory. However, to ensure the exact output regulation, two conditions need to be fulfilled: (1) the same input matrix for all subsystems, that is, , where is the number of rules in the fuzzy model, and (2) same zero error manifold for every local subsystem.

Later in [11], the exact output regulator is directly designed for the overall T-S fuzzy model. Although such a controller achieves the exact output regulation where the weighted summation of linear local regulators fails, its expression may be very large.

For that reason, in the present paper the simplicity of the fuzzy regulator obtained from the weighted summation of linear local regulators is exploited; the effectiveness of the controller given in this approach guarantees at least for a class of T-S fuzzy models the exact output regulation. To this end, new membership functions will be systematically computed in order to adequately combine the linear local regulators, guaranteeing in this way the proper performance of the fuzzy regulator in the interpolation regions. A preliminary result has been given in [6], where the new membership functions are approximated by soft computing techniques.

The main idea comes from the fact that each local controller is valid, at least, for its corresponding local system, while the fuzzy interpolation regions require more attention at the moment of evaluating the performance of the overall fuzzy controller. For that reason, the proposed approach consists of finding new membership functions capable of adequately combine adjacent local controllers in order to achieve the control goal.

So, the main contribution of the present work is to find a control law for a class of Takagi-Sugeno fuzzy models, in order to achieve exact output regulation on the basis of local regulators and computing of new membership functions, even if different input matrices appear in the linear local subsystems. Consequently, one of the restrictions given in [10] is avoided, and there is no need of verifying the existence of the fuzzy regulator for all [11]. Besides, the new membership functions, allowing the proper combination the local regulators, are given as a mathematical expressions.

The rest of the paper is organized as follows. In Section 2 the nonlinear regulation problem formulation is given with a brief review of the Takagi-Sugeno models and the fuzzy regulation problem. The main result is developed in Section 3. In Section 4 some examples are presented and finally, in Section 5, some conclusions are drawn.

#### 2. The Output Regulation Problem

Consider a nonlinear system defined bywhere is the state vector of the plant, is the state vector of the exosystem, which generates the reference and/or the perturbation signals, and is the input signal. Equation (5) refers to difference between output system of the plant () and the reference signal (), that is, , taking into account that . Besides, it is assumed that , and are functions (for some large ) of their arguments and also that , , and [12].

Clearly, by linearizing (1)–(5) around one getsThus, the Nonlinear Regulator Problem [3, 13] consists of finding a controller , such that the closed-loop system, , has an asymptotically stable equilibrium point, and the solution of the system (6) satisfies .

So, by defining as the steady-state zero error manifold and as the steady-state input, the following theorem gives the conditions for the solution of nonlinear regulation problem.

Theorem 1. *Suppose that is Poisson stable and a gain exists such that the matrix is stable and there exist mappings and with and satisfying**Then the signal control for the nonlinear regulation is given by*

Nonlinear partial differential equations (7) are known as Francis-Isidori-Byrnes (FIB) equations and their linear counterparts are obtained when the mappings and become into and , respectively. Thus, the linear problem is reduced to solve a set of linear matrix equations (Francis equations) [1]:

##### 2.1. The Exact Output Fuzzy Regulation Problem

Takagi and Sugeno proposed a fuzzy model composed by a set of linear subsystem with IF-THEN rules capable of relating physical knowledge, linguistic characteristics, and properties of the system. Such a model successfully represents a nonlinear system at least in a predefined region of phase space [14]. It is important to remark that in this work the exosystem is purely “linear,” because the computation of new membership functions for the general case is still an open problem.

The T-S model for system (1)–(5) is given by [15].

*Model*. Rule : IF is and and is , THENwhere is the number of rules in the model and is the fuzzy sets defined on the basis of the knowledge of the system.

Then, the regulation problem defined by (1)–(5) can be represented through the T-S fuzzy model; that is, [10]where is the state vector of the plant, is the state vector of the exosystem, is the input signal, and and is the normalized weight of each rule, which depends on the membership function for the premise variable in , wherewith as a function of , , and .

The Exact Fuzzy Regulator Problem consists of finding a controller , such that the closed-loop system, has an asymptotically stable equilibrium point, and the solution of system (11) satisfies .

From [10, 12, 13] the desired overall fuzzy controller can be represented asConsidering that approximations for mappings and can be obtained by then, the solution of the fuzzy regulation problem requires to obtain and from the linear local regulators problems included in (11) and defined by for .

Thus, the following controller is obtained:However, according to [10], the exact fuzzy regulation is achieved with this controller only when , and (1)input matrix is equal for all subsystems; that is, , or(2)the steady-state zero error manifold is equal for all subsystems; that is, .

In the present work, restrictions 1 and 2 are avoided by computing new membership functions for a class of T-S fuzzy systems. On the other hand, in [11] the Exact Output Fuzzy Regulation Problem (EOFRP) was solved by finding the steady-state zero error manifold and the steady-state input for the overall T-S fuzzy problem through (steady-state error equation), resulting in and , where and are continuous time-variant matrices and is a function [12].

Within the next section, the exact output regulation, for a class of T-S fuzzy systems, is obtained by proposing a solution for on the basis of different membership functions in the regulator.

#### 3. The Exact Output Fuzzy Regulation by Local Regulators

In this section, a particular class of T-S fuzzy models is considered to solve the exact output regulation on the basis of linear local controllers. So, the main goal is to find an overall regulator based on the fuzzy summation of local regulators considering adequate membership functions. Clearly, such membership functions are not necessarily the same included in the fuzzy plant. Thus, the steady-state input can be defined aswhere are new membership functions, such that the fuzzy output regulator obtained from local regulators coincides with the controller given in [11], at least for the class of fuzzy models considered. This novel approach only requires the computation of the linear local controllers and the computation of the new membership functions which will be presented below. The regulator obtained in this way is valid for all , while in [11] this condition needs to be verified.

Consider the following matrices:where represents the elements of the last row of and represents the element of the last row of with and . It is worth mentioning that and are matrices of the same dimension, as shown in (19). Notice that matrices in the form of and can be used to generate a great variety of signals, ensuring, in that way, the applicability of the approach in a great number of cases. Therefore, from (14) the control input can be defined bybecause, as mentioned before, is a function of and in steady-state .

On the other hand, from , and from (6), it can be concluded that in steady-state, while from (19) the following exosystem can be easily derived:Besides, each subsystem can be rewritten as follows:

By equating the time derivative of the fuzzy output with the time derivative of the reference signal, it results in . Furthermore, from (21) and (22), it is possible to conclude thatwhere . From this analysis, two results can be obtained. Firstand due to the shape of (19) is constant and common for all subsystems. Thus, . Second, the local steady-state input is defined byThe previous expression can be rewritten in a compact form as follows:and from (18) the fuzzy steady-state input is

From the previous analysis, each can be computed fromNow, it is possible to rewrite the fuzzy model as [11]where

For the sake of simplicity, in the following analysis it is considered that , , that is, . However, taking similar steps as those used below, the result can be easily extended to the case involving .

Thus, from (19) and (29) it can be observed that the last equation in (29) is given by where compact form isconsidering that in steady-state , the total steady-state input can be obtained from (23), resulting in Consequently, the steady-state input for the total fuzzy system can be constructed by

Now, a new set of fuzzy membership functions , satisfying with , , will be computed in order to obtain a similar steady-state input to (36) but formed from the fuzzy summation of local linear regulators. At this point, the local regulators and total regulator are defined as follows: respectively. Thus, the local fuzzy regulators will be multiplied by the new membership functions and the result will be equated to the global fuzzy regulator (36) as follows: So, the adequate membership functions arewith . It is important to remark that the new membership functions are given in terms of for simplicity. However, can be removed from the new membership functions (39), when is replaced by its corresponding expression. Besides, is directly obtained from by considering that in steady-state .

Notice that (39) fulfills conditions for all .

*Remark 2. *It can be observed that (39) is always valid when the entries of input matrix have the same sign; if this condition is not fulfilled the denominators of the new fuzzy membership functions will present singularities. Nevertheless, these singularities may appear outside the operation region, allowing more flexibility, but such cases require a particular study of the system, which is beyond the scope of this work.

The following theorem provides the conditions for the existence of the exact output fuzzy regulator for a class of T-S fuzzy models.

Theorem 3. *The exact fuzzy output regulation with full information for T-S fuzzy systems in the form of (11), defined by matrices (19), is solvable if (a) the sign of entries for corresponding position, inside input matrices for , is the same, (b) there exists a fuzzy stabilizer for the T-S fuzzy system [16], and (c) the exosystem is Poisson stable. Moreover, the Exact Output Fuzzy Regulation Problem is solvable by the controller:**where can be readily obtained from (39).*

*Proof. *From the previous analysis, the existence of mappings and is guaranteed when the local subsystems are defined by (19); that is, the form of corresponds to a diagonal matrix and will be the same for all subsystems, while can be directly substitute.

On the other hand, condition (a) avoids singularities in the new membership functions , while the inclusion of condition (b) has been thoroughly discussed in [12, 13, 17–19], and it implies that the steady-state manifold can be made asymptotically stable by the action of the fuzzy stabilizer.

Finally, condition (c) is introduced to avoid that the reference signal converges to zero, because if the reference tends to zero, then the regulation problem turns into a stabilization one, which can be solved by means of the fuzzy stabilizer. In addition, as mentioned before, this kind of matrix can be used to generate a great number of reference signals. The rest of the proof follows directly from the previous analysis.

*Remark 4. *Although, the PDC approach [14] can be used to compute the fuzzy stabilizer, more relaxed approaches can be considered to verify condition (b). For instance [16, 20–25].

#### 4. Exact Output Fuzzy Regulation Problem for a Class of T-S Fuzzy Systems with Multiple Inputs and Multiple Outputs (MIMO Case)

Now the problem will be extended to T-S fuzzy MIMO models defined by matrices:for , withwhere , , , , , , , , and . As before, this proposal is given because a great number of mechanical systems can be represented in this form.

Notice that this analysis cannot be omitted because the computation of the new membership functions is not the same as that developed in the previous section. From (42) the following exosystem can be easily derived:Moreover, each subsystem can be described byAs before, using expressions (44) and (45), considering , , and performing successive substitutions of with , the following results are obtained:where , and ; also , , and so forth. Notice that the values of repeat themselves every entry up to . Additionally, is constant and common for all subsystems while the local steady-state inputs are defined byfor . In compact form, (47) turns intoand from (27) the local matrices have the form

Now, again from the fuzzy plant given in (42) and the global fuzzy model (29), the expressions for areSo, by applying successive substitutions as before with (44), (45) and knowing that in steady-state , the following expressions are obtained: Therefore, the steady-state input for the overall fuzzy system is

But, the desired form of the steady-state is

Consequently, the new membership functions, , can be obtained after equating (52) and (53), resulting inwherefor all ; . It can be readily observed that the sum of the corresponding elements is equal to one and .

These new membership functions are organized in a matrix form; for that reason (where is the identity matrix), the arrangement of (54) results as a consequence of ensuring (20) and again this representation is valid when the values of the input matrices have the same sign at corresponding positions. Clearly, this condition avoids singularities in (55). As before, is directly obtained from by considering that in steady-state . At this point, the following theorem can be defined.

Theorem 5. *The exact fuzzy output regulation with full information for T-S fuzzy MIMO systems in the form of (11), defined by matrices (42), is solvable if (a) the sign of entries for corresponding position, inside input matrices for , is the same, (b) there exists a fuzzy stabilizer for the fuzzy system [16], and (c) the exosystem is Poisson stable.**Moreover, the Exact Output Fuzzy Regulation Problem for MIMO systems is solvable by the controller: **where can be readily obtained from (54) and (55).*

*Proof. *It follows directly from Theorem 3 and previous analysis.

*Remark 6. *Notice that is defined as a matrix and this is not a typical membership function; however, contains a set of membership functions related to each entry of . This form allows us to design controller (20).

#### 5. Numerical Examples

##### 5.1. Simple Input Case

Consider the problem of balancing and swing of an inverted pendulum on a cart. The motion equations for this system are [14]where denotes the angle (in radians) of the pendulum from the vertical and is the angular velocity; m/s^{2} is the gravity constant, is the mass of the pendulum, is the mass of the cart, is the length of the pendulum, and is the force applied to the cart (in newtons); , with kg, kg, and m in the simulations. The goal is to track the reference defined by the exosystem within the range . To this end, a fuzzy model representing the nonlinear dynamics is as follows.

*Rule 1. *
IF is about 0, THEN .

*Rule 2. *
IF is about , THEN .

Membership functions for Rules 1 and 2 are shown in Figure 1. In this example, the T-S fuzzy model of the plant is defined by matrices: while the exosystem can be constructed from the following matrices:which allows us to generate a great kind of sinusoidal references. Figure 2 shows the simulation results after applying the controller:defined in [10]. As expected, controller (61) is unable to achieve exact regulation, although tracking error remains bounded.

**(a)**

**(b)**

Now, using the method derived in this work, the local matrices can be readily obtained from (28), resulting in Besides, knowing that , matrix is obtained from (24): while the new membership functions computed from (39) are

On the other hand, the fuzzy stabilizer for this system is constructed by means of the PDC approach developed in [14], which is formed by

Figure 3 shows simulation with the controller defined in (20). Observe how the new membership functions are capable of combining the local regulators in the proper way to accomplish asymptotic tracking.

**(a)**

**(b)**

To verify the effectiveness of the latter regulator, the fuzzy controller is applied on the original system (57). The simulation results appear in Figure 4. Notice that the tracking error depicted in Figure 4(b) is due to the approximation provided by the T-S fuzzy model. Such an error can be reduced by considering a better approximation or an exact representation of the nonlinear system, as the one given by the nonlinear sector approach [14].

**(a)**

**(b)**

##### 5.2. Multiple Inputs and Multiple Outputs (MIMO) Case

Now, consider a T-S fuzzy system of two rules, defined by the following matrices:and an exosystem defined byMembership functions for Rules 1 and 2 are shown in Figure 5. Matrices defining the local regulators can be easily obtained by using (49). For this example they are

On the other hand, , , , , and by (46), and matrix is Figures 6 and 7 show the results obtained using linear regulators defined in (61).

**(a)**

**(b)**

**(a)**

**(b)**

Now, by computing the new membership functions for the fuzzy regulator through (54), one gets

As previously, the fuzzy stabilizer for this example is also constructed by means of the PDC approach developed in [14], resulting in

Thus, Figures 8 and 9 show the simulation results when the controller defined by (20) is applied.

**(a)**

**(b)**

**(a)**

**(b)**

Besides, Figure 10 shows the shape of the new membership functions, which are clearly different from the original ones given in Figure 5.

**(a)**

**(b)**

##### 5.3. More General Problems

Consider the T-S fuzzy MIMO system with multiple inputs defined by the following matrices:

The reference signals will be generated by an exosystem defined by

Membership functions for Rules 1 and 2 are and . Notice that for this case, and . This implies that and must be obtained by solving Francis Equations for each subsystem. However, due to form of the fuzzy plant it is possible to verify that .

So, by applying (9) for each local subsystem with , it results in

Again, the fuzzy stabilizer for this system is constructed by means of the PDC approach developed in [14], yielding

The behavior obtained by applying controller (61) is depicted in Figures 11 and 12. Notice the exact output regulation is not achieved.

**(a)**

**(b)**

**(a)**

**(b)**

Now, the adequate membership functions are computed from (54), resulting inWhen is replaced by its corresponding expressions, one gets

The results of applying control law (20) on the fuzzy plant are given in Figures 13 and 14.

**(a)**

**(b)**

**(a)**

**(b)**