Tuning of a TS Fuzzy Output Regulator Using the Steepest Descent Approach and ANFIS

Abstract

The exact output regulation problem for Takagi-Sugeno (TS) fuzzy models, designed from linear local subsystems, may have a solution if input matrices are the same for every local linear subsystem. Unfortunately, such a condition is difficult to accomplish in general. Therefore, in this work, an adaptive network-based fuzzy inference system (ANFIS) is integrated into the fuzzy controller in order to obtain the optimal fuzzy membership functions yielding adequate combination of the local regulators such that the output regulation error in steady-state is reduced, avoiding in this way the aforementioned condition. In comparison with the steepest descent method employed for tuning fuzzy controllers, ANFIS approximates the mappings between local regulators with membership functions which are not necessary known functions as Gaussian bell (gbell), sigmoidal, and triangular membership functions. Due to the structure of the fuzzy controller, Levenberg-Marquardt method is employed during the training of ANFIS.

1. Introduction

A fundamental problem in dynamic systems is to control a plant in order to have its outputs tracking some reference signals produced by an exosystem (external generator), while the stability property of the closed-loop system is guaranteed. In this regard, the output regulation problem [1] consists of finding the control law capable of(a)stabilizing the closed-loop system when the plant is not influenced by the exosystem,(b)taking the tracking error asymptotically to zero when the plant is under the action of the exosystem.

One of the first works on output regulation theory presented in the literature was developed by Francis and Wonham in 1975 [2]. This paper obtains structural criteria necessary to synthesize multivariable linear regulators with structural stability to small perturbations. The main contribution of this work is that structural stability requires feedback from the regulate variable, and these structural features are called “internal model principle.” Later in 1977 [3], the linear regulation theory for tracking a reference is proposed by Francis, by finding that the solution for the output regulation of an autonomous linear system subject to perturbations and reference signals can be obtained from a system of linear matrix equations.

Isidori and Byrnes [1] extend the results obtained by Francis to the general case in which the plant and/or the exosystem are nonlinear. They also show that Francis equations are a particular case of nonlinear output regulation equations. In addition, they also show, as in linear case, that if the plant is exponentially stabilizable via output feedback, the solution for the nonlinear case requires an error feedback regulator. Unfortunately, the so-called Francis-Isidori-Byrnes (FIB) equations used to solve the nonlinear output regulation problem involve a set of nonlinear partial differential equations which in many cases are too difficult, or even impossible to solve in a practical way (see Figure 1).

(a)

(b)

Figure 1 

Output regulation scheme for (a) linear systems and (b) nonlinear systems [4].

On the other hand, in the real world, there exists a wide range of problems that involve the analysis of uncertain and inaccurate information, and that is the main reason for the “soft computing.” The term soft computing, according to Zadeh, is “A collection of methodologies that aim to exploit the tolerance for imprecision and uncertainty to achieve tractability, robustness, and low solution cost.” Its principal constituents are fuzzy logic, neurocomputing, and probabilistic reasoning [5]. Each of these approaches has features that make them suitable to solve different problems. Fuzzy systems are characterized by their ability to handle and use inaccurate approximate reasoning, while neural networks have the property of learning. Many of these tools have been used with efficiency to solve complex real world problems as diagnosis, estimation, control, and autonomy of systems.

In that sense, some techniques have been developed to characterize nonlinear systems by means of linear local subsystems [5]. One of these approaches is the well-known Takagi Sugeno (TS) fuzzy modeling. This technique allows describing the nonlinear dynamics by means of a suitable “blending” of linear subsystems, each of them corresponding to different operation points. Basically, the “combination” is performed by a weighted summation of linear local subsystems. Thus, local controllers can be designed for each subsystem, obtaining the aggregate controller by using the same membership functions of the TS fuzzy plant.

As mentioned previously, the set of differential equations derived by Isidori and Byrnes in nonlinear regulation is difficult to solve, representing an interesting opportunity to apply soft computing: in particular neural networks and fuzzy logic.

For instance, in [6], an iterative algorithm with the structure of a recurrent neural network is presented. The network is introduced with the purpose of dealing with model errors caused by the estimation of parameters of an off-line training and the possible dynamic changes; an online learning approach providing the necessary parameters for the network is also used. In this sense, the network can capture the uncertainties of the system, and the regulator adjusts the control action in order to guarantee the regulation condition. That is an attempt to combine the identification capability of neural networks and the stability properties of the control theory. As a result, the control structure learns the system dynamics and ensures the asymptotic convergence of the error. Subsequently, this method was applied at the control scheme of a solar plant [7]. Zhang and Wang [8] propose an approximation in power series form, wherein a neural network carries out the synthesis and self-tuning of controller in real time. This method ensures regulation properties, namely, stability and asymptotic tracking.

Later in [9], the fuzzy output regulation is formulated, in which a nonlinear system described by a Takagi-Sugeno fuzzy model follows a reference signal generated by a fuzzy exosystem. It was also shown that the fuzzy regulation problem can be solved by means of local linear regulator only if the following conditions are satisfied.(i)The zero error manifold 𝜋(𝑤) is equal for every local linear subsystem (see Figure 1).(ii)Input matrices 𝐵_𝑖 are equal for every local linear subsystem, that is, (𝐵₁=𝐵₂=⋯=𝐵_𝑖).

Unfortunately, the aforementioned conditions are not satisfied in a great number of systems. A proposal to overcome this problem is presented in [5, 10], wherein the fuzzy regulator is designed on the overall fuzzy model instead of local linear subsystems. The advantage of this approach is the global convergence of the tracking error in the interest region, in contrast to local properties of classical method of nonlinear control. Moreover, the solution of the fuzzy regulation problem is given in terms of the “dynamical Francis equations,” and a very practical way to solve such a dynamical equation is also given. However, a disadvantage of this approach is that exact fuzzy controller is far more complex than the regulator designed from the fuzzy summation of local linear regulators, in general. For that reason, in the present work, an approach based on the local design of linear regulators is proposed, but considering that the nonlinearity of the controller may be different from the nonlinearity of the plant. In other words, it will be assumed that the membership functions of the fuzzy plant and the membership functions of fuzzy regulator are different. Therefore, the appropriate membership functions for the fuzzy regulators will be obtained by means of an adaptive network-based fuzzy inference system (ANFIS) and the steepest descent approach.

A hybrid intelligent system combines two or more soft computing techniques [10]. Such combination allows taking advantage of each tool, improving the behavior of the overall system. According to [11, 12], the following combinations can be found in the literature:(i)fuzzy controllers tuned by neural networks,(ii)fuzzy controller tuned by evolutionary computation,(iii)neural networks synthesized by evolutionary computation, (iv)neural networks controlled by fuzzy logic,(v)evolutionary computation controlled by fuzzy logic.

The Artificial Neural Networks (ANN) have shown great efficiency in different fields as identification, classification, or recognition patterns. On the other hand, a system based on fuzzy logic has the capability of working with uncertain information and taking decisions based on rules previously established by an expert system. A neurofuzzy system takes the inference under cognitive uncertainty from fuzzy logic and from ANNs their learning capability, adaption, parallelism, and generalization [13, 14].

The performance of a fuzzy controller is extensively linked to the shape of membership functions that represent linguistic variables [15], so as the binary operators [16]. These parameters conform the structure of fuzzy controller and generally are defined by an expert to trial and error [17]. ANN brings the possibility of avoiding the trial and error procedure [18–20]. Therefore, ANN gives to fuzzy controller the capability of learning, which means that fuzzy system can be tuned by itself [21]. Besides, the neural approach can not only be used to adjust the parameters of the fuzzy controller but also an ANN can be used as a fuzzy controller, in the sense that the fuzzy controller will have the structure of a neural network [22]. For instance, Moreno-Velo develops Xfuzzy which is a program where several proposed approaches are applied to learning process on neurofuzzy systems. Xfuzzy integrates a wide variety of supervised learning algorithms [23].

In [24, 25], the authors present methodologies that reduce the time of training for neurofuzzy systems with the aim of using them on real-time applications.

(Adaptive Neural-Fuzzy Network Inference System) (ANFIS) proposed by Jang [26] builds a mapping between input-output pairs, using IF-THEN rules. ANFIS employs hybrid training, combining backpropagation and the least-squares estimator, and it has been used for modeling nonlinear functions, identification of nonlinear components in control systems, and for prediction of chaotic series.

So, the main contribution of the present work is to develop an approach based on ANFIS that is capable of approximating the appropriate membership functions, such that tracking error for the overall fuzzy system is reduced when the fuzzy controller is obtained from the fuzzy summation of local linear regulators, while avoiding the restriction of proposing a priori form of the membership functions, which is the main disadvantage of the steepest descent approach.

The rest of the paper is organized as follows. In Section 2, a review of basic concepts of fuzzy output regulation is presented. In Section 3, an overview of ANFIS and an introductory example using the steepest descent in fuzzy regulation are given. In Section 4, some cases of study are solved and their respective simulation results are depicted. Finally, in Section 5, some conclusions are drawn.

2. The Output Regulation Problem

From a given linear system 𝑥̇=𝐴𝑥+𝐵𝑢,𝑦=𝐶𝑥 (1) and an exosystem ̇𝑦𝜔=𝑆𝜔,_ref=𝑄𝜔, (2) it is desired to design a controller that stabilizes asymptotically the system (1) when the exosystem (2) is disconnected, that is, (𝜔=0), and achieving 𝑒(𝑡)→0 as 𝑡→∞ when (1) is under the influence of (2). According to [26], the controller is 𝑢=𝐾(𝑥−Π𝜔)+Γ𝜔 and the tracking error is 𝑒_ss=𝑥−𝑥_ss, where 𝑥_ss=Π𝜔𝑒_ss=𝑥−Π𝜔∴𝑒̇_ss=𝑥̇−Π𝜔̇⟶𝑥̇=𝑒̇_ss+Π𝜔̇. (3) By substituting the previous equations in (1), one gets 𝑒̇_ss=𝐴𝑒_ss+𝐴Π𝜔+𝐵𝐾𝑒_sṡ+𝐵𝐾Π𝜔−𝐵𝐾Π𝜔+𝐵Γ𝜔−Π𝜔. (4) But 𝐾 has to be designed such that (𝐴+𝐵𝐾) is asymptotically stable. Therefore, 𝑒(𝑡)→0 as 𝑡→∞ if and only if Π𝑆=𝐴Π+𝐵Γ,0=𝐶Π−𝑄. (5) The set of (5) is called Francis equations. Dimensions of each matrix are 𝐴∈ℝ^𝑛×𝑛,𝐵∈ℝ^𝑛×𝑚,𝐶∈ℝ^𝑚×𝑛,Π∈ℝ^𝑛×𝑟,𝑆∈ℝ^𝑟×𝑟,Γ∈ℝ^𝑚×𝑟,𝑃∈ℝ^𝑛×𝑟,𝑄∈ℝ^𝑚×𝑟. (6) According to Isidori and Byrnes, the extension of the regulation problem to the nonlinear case is ̇(),(𝑥),̇(𝜔),𝑦𝑥=𝑓𝑥,𝜔,𝑢𝑦=ℎ𝜔=𝑠_ref=𝑞(𝜔). (7) Thus, the control signal becomes 𝑢=𝑘[𝑥−𝜋(𝜔)]+𝛾(𝜔). Then, similar to the linear case, the error in steady-state is 𝑒̇_ss=𝑥̇−𝑥̇_ss with 𝑥_ss=Π𝜔: 𝑒_ss()−𝑑=𝑓𝑥,𝜔,𝑢[𝜋(𝜔,𝑒𝑑𝑡)]_ss=𝑓(𝑥,𝜔,𝑢)−𝜕𝜋̇𝑒𝜕𝜔𝜔,_ss=𝑓(𝑥,𝜔,𝑢)−𝜕𝜋𝑠(𝜔),𝜕𝜔𝜕𝜋𝑠(𝜔)𝜋(𝜔)(𝜔),(𝜋(𝜔(𝜔).𝜕𝜔=𝑓,𝜔,𝛾0=ℎ))−𝑞 (8)(9)(10)(11) The set (11) is called Francis-Isidori-Byrnes (FIB) equations [1], subjected to 𝑓()ℎ(0)𝜋(0)𝛾(0)𝑆(0)𝑞(0)0,0,0=0,=0,=0,=0,=0,=0. (12)

2.1. Exact Fuzzy Output Fuzzy Regulation

The fuzzy model proposed by Takagi and Sugeno is a fuzzy system described by the rules of the type IF-THEN, where the consequent parts include the local dynamics represented by linear subsystems [27], that is.

Local linear subsystems: 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡),𝑦(𝑡)=𝐶_𝑖𝑥(𝑡)𝑖=1,2,…,𝑟. (13) Therefore, the nonlinear dynamics can be approximated by the following TS model: 𝑥̇=^𝑟1_𝑖=1ℎ¹_𝑖𝐴_𝑖𝑥+𝐵_𝑖𝑢+^𝑟1_𝑖=1ℎ¹_𝑖𝑃_𝑖,𝑦=^𝑟1_𝑖=1ℎ¹_𝑖𝐶_𝑖𝑥. (14) In a similar way, the exosystem can be approximated by a TS fuzzy system of 𝑟₂ rules as follows: 𝜔̇=^𝑟2_𝑖=1ℎ²_𝑖𝑆_𝑖𝑦𝜔,_ref=^𝑟2_𝑖=1ℎ²_𝑖𝑄_𝑖𝜔. (15) Note that the membership functions of the plant are not necessarily the same as the exosystem functions. Fuzzy plant (14) and fuzzy exosystem (15) can be rewritten as ̇𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑢(𝑡),𝐶(𝑡)𝑥(𝑡),̇𝑆(𝑡)𝜔(𝑡),𝑦𝑥=𝑦=𝜔=_ref=𝑄(𝑡)𝜔(𝑡). (16) And the desired fuzzy controller is 𝑢(𝑡)=^𝑟1_𝑖=1ℎ¹_𝑖𝑧₁(𝑡)𝐾_𝑖[𝑥(𝑡)−𝜋(𝜔(𝑡))]+𝛾(𝜔(𝑡)). (17) With this representation, the steady-state manifold is 𝜋(𝜔(𝑡))=Π(𝑡)𝜔(𝑡), while the steady-state input is 𝛾(𝜔(𝑡))=Γ(𝑡)𝜔(𝑡) [9, 28]. Π(𝑡) and Γ(𝑡) can be obtained by differentiating the steady-state error: 𝑒̇_ss=𝑥̇(𝑡)−̇Π(𝑡)𝜔(𝑡)−Π(𝑡)𝜔̇(𝑡). (18) In steady-state, one gets ̇Π(𝑡)+Π(𝑡)𝑆(𝑡)=𝐴(𝑡)Π(𝑡)+𝐵(𝑡)Γ(𝑡)+𝑃(𝑡). (19) Solving (19) in an analytical way may be very difficult. But its numerical solution is given in [4] through a very practical way. However, the resulting controller is very complex, in general.

3. Overview of Adaptive-Network-Bases Fuzzy Inference System (ANFIS)

In Figure 2, an Adaptive-Network-Based Fuzzy Inference System network is shown. ANFIS is a system that employs the structure of fuzzy Takagi-Sugeno-Kang systems (TSK) and learning algorithms of neural networks with the purpose of adjusting the consequent and antecedent parameters of the fuzzy system.

Figure 2 

ANFIS architecture for 2 input variables.

Basically, ANFIS architecture is formed by five layers, each of them is described in the following.

Layer??1. Fuzzification process, variables 𝑥 and 𝑦 are the inputs to membership functions 𝑂_1,𝑖. Generally, gbell functions are used in this stage, for example.

Rule 1. If 𝑥 is 𝑂_1,1 and 𝑦 is 𝑂_1,3, then 𝑓₁=𝑝₁𝑥+𝑞₁𝑦+𝑟₁.

Rule 2. If 𝑥 is 𝑂_1,2 and 𝑦 is 𝑂_1,4, then 𝑓₂=𝑝₂𝑥+𝑞₂𝑦+𝑟₂.
Consider𝑚_𝐴,𝑗=𝑂_1,𝑗(𝑥)=11+𝑥−𝑐_𝑖/𝑎_𝑖²𝑏_𝑗𝑗=1,2 (20) or 𝑚_𝐵,𝑗=𝑂_1,𝑗𝑦=11+𝑦−𝑐_𝑖/𝑎_𝑖²𝑏_𝑗𝑗=3,4. (21) Parameters 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖 are called premise parameters, where these premise parameters can be modified (Figure 3).

(a) Modifying parameter “a” 

(b) Modifying parameter “b”

(c) Modifying parameter “c”

(d) Modifying parameter “a” and “b”

Figure 3 

Shapes of gbell function with modified parameters.

Layer??2. Nodes denoted with Π. In these nodes, the inferences between fuzzy sets are performed. The 𝑇-norm is used for firing strengths: 𝑂_2,𝑗=𝑤_𝑗=𝑚_𝐴𝑗(𝑥)⋅𝑚_𝐵𝑗(𝑥),𝑗=1,2. (22)

Layer??3. Normalization of firing strengths 𝑂_3,𝑗=𝑤_𝑗=𝑤_𝑗𝑤₁+𝑤₂. (23)

Layer??4. The consequent part of fuzzy system is in nodes 𝑂⁴_𝑖, and it is given by a set of functions 𝑓_𝑖=𝑝_𝑖𝑥+𝑞_𝑖𝑦+𝑟_𝑖, and they are activated according to the normalize firing strengths: 𝑂⁴_𝑗=𝑤_𝑗𝑓_𝑗=𝑤_𝑗𝑝_𝑗𝑥+𝑞_𝑗𝑦+𝑟_𝑗. (24)

Layer??5. Summation of input signals is obtained as 𝑂⁵₁=_𝑗𝑂⁴_𝑗=_𝑗𝑤_𝑗𝑓_𝑗. (25)

3.1. Case of Study

Consider the following TS fuzzy system: 𝑥̇(𝑡)=²_𝑖=1ℎ_𝑖𝑥₁(𝑡)𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡,𝜔̇(𝑡)(𝑡),𝑒(𝑡)=)=𝑆𝜔_2𝑖=1ℎ_𝑖𝑥₁(𝑡)𝐶_𝑖𝑥(𝑡)−𝑄𝜔(𝑡), (26) where 𝐴₁=0120,𝐴₂=,𝐵0130₁=02,𝐵₂=01,𝐶₁=𝐶₂=𝑄=10,𝑆=,01−10 (27) with the control 𝑢(𝑡)=²_𝑖=1ℎ_𝑖𝑥_𝑖(𝑡)𝐾_𝑖𝑥(𝑡)−Π_𝑖𝜔(𝑡)_{fuzzystabilizer}+²_𝑖=1ℎ_𝑖𝑥_𝑖(𝑡)Γ_𝑖𝜔(𝑡)_{fuzzyregulator}. (28) It is important to remark that the fuzzy regulator included in (28) is obtained by using the same membership functions of the plant.

Observe that in this example, 𝑥₁ should track 𝜔₁. This can be easily deduced from matrices 𝐶₁, 𝐶₂, and 𝑄. Π_𝑖 and Γ_𝑖 are obtained from the solution of (5) for each of the two rules. Thus, a fuzzy regulator is designed on the basis of two local linear systems. The membership functions ℎ_𝑖 are proposed as triangular ones satisfying ℎ₁+ℎ₂=1.

The fuzzy steady-state manifold is 𝜋(𝜔(𝑡))=Π₁𝜔(𝑡)=Π₂𝜔(𝑡)=10𝜔01¹(𝑡)𝜔₂(𝑡)=Π𝜔(𝑡), (29) while the steady-state input is 𝛾(𝜔(𝑡))=²_𝑖=1ℎ_𝑖𝜔₁(𝑡)Γ_𝑖𝜔(𝑡)=−32ℎ₁𝜔₁(𝑡)−4ℎ₂𝜔₁(𝑡𝜔)¹(𝑡)𝜔₂(𝑡). (30) And the 𝐾_𝑖 gains are obtained from LMI approach [29] in order to ensure global stability: 𝐾₁=,𝐾−1.61−0.35₂=−4.22−0.71. (31) From the matrices considered in the system (26), it is clear that 𝐵₁ and 𝐵₂ are different from each other. Therefore, an asymptotical error cannot be expected when the fuzzy regulator is constructed by the fuzzy summation of the local linear regulators using the same membership functions of the plant [28]. So, Figure 4 shows the behavior of state variable 𝑥₁ of fuzzy plant and state 𝜔₁ of exosystem, while the tracking error is given in Figure 5.

Figure 4 

𝑥1 versus 𝜔1 when membership functions of the plant are used to fuzzily combine the local regulators.

Figure 5 

Tracking error when the membership functions of the plant are used to fuzzily combine the local regulators.

In order to overcome the problem of controller (28), a decoupling between the membership functions of fuzzy stabilizer and fuzzy regulator is proposed, that is, 𝑢(𝑡)=²_𝑖=1ℎ_𝑖𝑥_𝑖(𝑡)𝐾_𝑖[𝑥(𝑡)−Π𝜔(𝑡)]_{FuzzyStabilizer}+²_𝑖=1𝜇_𝑖𝑥_𝑖(𝑡)Γ_𝑖𝜔(𝑡)_{FuzzyRegulator}. (32) Observe that in this new controller, the membership functions of the regulation part are not the same as that considered in the fuzzy plant. In the following, these membership functions 𝜇_𝑖 will be estimated by means of the Steepest Descent approach and ANFIS.

According to [5], a fuzzy system with membership functions of type gbell is a universal approximator and consequently gbell functions are used to approximate fuzzy mapping in local regulators, subjected to condition 𝜇₂=1−𝜇₁, where 𝜇₁()=1=𝑓𝑥,𝑎,𝑏,𝑐1+|(𝑥−𝑐)/𝑎|^2𝑏. (33) The parameters 𝑎, 𝑏, and 𝑐 determine the shape of gbell function. As a consequence, the main goal is to estimate 𝑎, 𝑏, and 𝑐 such that the resulting membership functions 𝜇₁ and 𝜇₂ improve the performance of the fuzzy controller (32). One of the most useful methods for tuning task found in the literature is the “steepest descent” method: 𝑎_𝑖(𝑘+1)=𝑎_𝑖(𝑘)−𝜂_𝑎𝑖𝜕𝐸(𝑎,𝑏,𝑐)𝑏𝜕𝑎𝑖=1,2,_𝑖(𝑘+1)=𝑏_𝑖(𝑘)−𝜂_𝑏𝑖𝜕𝐸𝑎_𝑖,𝑏_𝑖,𝑐_𝑖𝜕𝑏_𝑖𝑐𝑖=1,2,_𝑖(𝑘+1)=𝑐_𝑖(𝑘)−𝜂_𝑐𝑖𝜕𝐸𝑎_𝑖,𝑏_𝑖,𝑐_𝑖𝜕𝑐_𝑖𝑖=1,2. (34) It can be readily observed that expressions (34) depend on error 𝑒_ee(𝑘)=𝑥(𝑘)−Π𝜔(𝑘), where 𝑥(𝑘) and 𝜔(𝑘) represent the 𝑘th training input-output pair for the fuzzy plant and the exosystem, respectively.

The “steepest descent” approach must reduce the error on the direction of the parameters of the controller. Therefore, (34) need to be adapted such that they consider the evolution of the states of the plant and exosystem (see Figure 6).

Figure 6 

Scheme of the iterative regulation.

In order to obtain the training pairs, the plant is discretized using the approximation 𝑥̇≈(𝑥_𝑘−𝑥_𝑘−1)/𝑇. Therefore, the following discrete plant can be readily obtained: 𝑥_𝑘=ℎ₁𝐴₁+ℎ₂𝐴₂𝑥_𝑘−1+ℎ₁𝐵₁+ℎ₂𝐵₂×ℎ₁𝐾₁𝑥_𝑘−1+ℎ₂𝐾₂𝑥_𝑘−1−ℎ₁𝐾₁Π𝜔_𝑘−ℎ₂𝐾₂Π𝜔_𝑘+𝜇₁Γ₁𝜔_𝑘+𝜇₂Γ₂𝜔_𝑘𝑇+𝑥_𝑘−1. (35) And the mean squared error is 𝑒_ss=𝑥−Π𝜔⟶𝑒_ss(𝑘)=𝑥_𝑘−Π𝜔_𝑘⟹𝐸_ss=12𝑥_𝑘−Π𝜔_𝑘². (36) The current value of state 𝑥_𝑘 is an explicit function of 𝜇₁ and 𝜇₂, and this implies that 𝑥_𝑘 depends on parameters 𝑎, 𝑏, and 𝑐. Therefore, the steepest descent method should minimize the error function with respect to independent variables [30–33] (see Figure 7), that is, 𝜕𝐸𝜕𝑎₁=𝑥_𝑘−Π𝜔_𝑘^𝑇ℎ₁𝐵₁+ℎ₂𝐵₂Γ₁−Γ₂𝜔_𝑘𝑇𝜕𝜇₁𝜕𝑎₁,𝜕𝐸𝜕𝑏₁=𝑥_𝑘−Π𝜔_𝑘^𝑇ℎ₁𝐵₁+ℎ₂𝐵₂Γ₁−Γ₂𝜔_𝑘𝑇𝜕𝜇₁𝜕𝑏₁,𝜕𝐸𝜕𝑐₁=𝑥_𝑘−Π𝜔_𝑘^𝑇ℎ₁𝐵₁+ℎ₂𝐵₂Γ₁−Γ₂𝜔_𝑘𝑇𝜕𝜇₁𝜕𝑐₁. (37) Recalling that partial derivatives for a generalized bell (gbell) membership function are [34] 𝜕𝜇₁𝜕𝑎₁=2𝑏𝑎𝜇₁1−𝜇₁,𝜕𝜇₁𝜕𝑏₁=𝜔−ln^𝑘−1−𝑐𝑎𝜇₁1−𝜇₁,if𝜔_𝑘−1≠𝑐,0,if𝜔_𝑘−1=𝑐,𝜕𝜇₁𝜕𝑐₁=2𝑏𝜔_𝑘−1𝜇−𝑐¹1−𝜇₁,if𝜔_𝑘−1≠𝑐,0,if𝜔_𝑘−1=𝑐. (38) The parameters that adjust the membership functions 𝜇_𝑖 are updated by the steepest descent method according to the following training law in the antecedent part: 𝑎₁(𝑘+1)=𝑎₁(𝑘)−𝜂_𝑎𝑥_𝑘−Π𝜔_𝑘^𝑇×ℎ₁𝐵₁+ℎ₂𝐵₂Γ₁−Γ₂𝜔_𝑘𝑇𝜕𝜇₁𝜕𝑎₁,𝑏₁(𝑘+1)=𝑏₁(𝑘)−𝜂_𝑏𝑥_𝑘−Π𝜔_𝑘^𝑇×ℎ₁𝐵₁+ℎ₂𝐵₂Γ₁−Γ₂𝜔_𝑘𝑇𝜕𝜇₁𝜕𝑏₁,𝑐₁(𝑘+1)=𝑐₁(𝑘)−𝜂_𝑐𝑥_𝑘−Π𝜔_𝑘^𝑇×ℎ₁𝐵₁+ℎ₂𝐵₂Γ₁−Γ₂𝜔_𝑘𝑇𝜕𝜇₁𝜕𝑐₁. (39)

Figure 7 

Training of the fuzzy controller.

The simulation results are shown in Figure 8, where state 𝑥₁ tracks to 𝜔₁. From Figure 9, it is observed that the error is minimized considerably in comparison with the linear controller (28) (Figure 5). When the regulator is fully trained, the reference tracking is achieved in a very acceptable way, which is depicted in Figures 10 and 11.

Figure 8 

𝑥1 versus 𝜔1 during the learning scheme.

Figure 9 

Tracking error during the learning scheme.

Figure 10 

𝑥1 versus 𝜔1 when the controller has been fully trained.

Figure 11 

Tracking error when the controller has been fully trained.

But according to [1], if the regulation problem has a solution, then it is unique. In fuzzy regulation, this means that the membership functions of the fuzzy regulator cannot have arbitrary shapes, and in many cases, their shape is unknown. Obviously, the steepest descent approach is not applicable if the shape of the membership functions is not given a priori. In the following, this disadvantage will be overcome by considering ANFIS.

3.2. ANFIS in Output Fuzzy Regulation

Considering (26), when ℎ₁ is equal to 1, the dynamics of the overall fuzzy system are described by the first local linear subsystem and the exact output regulator is obtained from Γ₁. In the same way, if ℎ₂ is equal to 1, the exact output regulator is obtained directly from Γ₂. With this in mind, the next fuzzy implications are obtained:?IF ℎ₁ is 1, THEN 𝜇₁=1→𝑢(𝑡)=ℎ₁𝐾₁[𝑥(𝑡)−Π𝜔(𝑡)]+𝜇₁Γ₁𝜔(𝑡),?IF ℎ₂ is 1, THEN 𝜇₂=1→𝑢(𝑡)=ℎ₂𝐾₂[𝑥(𝑡)−Π𝜔(𝑡)]+𝜇₂Γ₂𝜔(𝑡).

A set of fuzzy rules can be established, taking into account that the regulation problem is exactly solved by local linear regulator, that is,?If ℎ₁ is 1 then 𝜇₁=1⇒𝑢_reg(𝑡)=Γ₁,?If ℎ₂ is 1 then 𝜇₂=1⇒𝑢_reg(𝑡)=Γ₂.

In the fuzzy rules given previously, it is supposed that 𝜇_𝑖 is a function of 𝜔₁, while 𝜇₂ can be obtained from 𝜇₂=1−𝜇₁. Then, 𝜔₁ is chosen as the input variable in ANFIS. Figure 12 shows the ANFIS architecture with one input.

Figure 12 

ANFIS architecture for one input variable.

Layer??1. For this example, gbell membership functions are chosen, and 8 nodes are distributed uniformly (see Figure 13) according to [35] 𝑎₁=4,𝑏₁=2,𝑐₁𝑎=−30,₂=4,𝑏₂=2,𝑐₂𝑎=−21,₃=4,𝑏₃=2,𝑐₃𝑎=−12,₄=4,𝑏₄=2,𝑐₄𝑎=−3,₅=4,𝑏₅=2,𝑐₅𝑎=3,₆=4,𝑏₆=2,𝑐₆𝑎=12,₇=4,𝑏₇=2,𝑐₇𝑎=21,₈=4,𝑏₈=2,𝑐₈𝜇=30,_𝐴𝑗ℎ₁=11+ℎ₁−𝑐_𝑖/𝑎_𝑖²𝑏_𝑖𝑗=1,2,…,8;0≥ℎ₁≤1. (40)

Figure 13 

Distribution of the membership functions for input layer.

Layer??2. The weight of each function corresponds to its own membership function because only one input variable is considered: 𝑤₁=𝜇_𝐴1,𝑤₂=𝜇_𝐴2,𝑤₃=𝜇_𝐴3,𝑤₄=𝜇_𝐴4,𝑤₅=𝜇_𝐴5,𝑤₆=𝜇_𝐴6,𝑤₇=𝜇_𝐴7,𝑤₈=𝜇_𝐴8. (41)

Layer??3. Weight normalizing: 𝑤_𝑖=𝑤_𝑖𝑤₁+𝑤₂+𝑤₃+𝑤₄+𝑤₅+𝑤₆+𝑤₇+𝑤₈,𝑖=1,2,…,8. (42)

Layer??4. Takagi-Sugeno coefficients: 𝑓_𝑖=𝑤_𝑖𝑝_𝑖ℎ₁+𝑟_𝑖𝑖=1,2,…,8. (43)

Layer??5.𝑓=𝑓₁+𝑓₂+𝑓₃+𝑓₄+𝑓₅+𝑓₆+𝑓₇+𝑓₈. (44)

3.3. Adjustment of Input Layer Parameters

Thus, the goal is to generate the adequate membership functions for the fuzzy regulator by means of ANFIS regulators, but these values depend on the reference signal. The proposed scheme is depicted in Figure 14.

Figure 14 

Rule tuning including ANFIS.

As discussed in Section 3.1, the steepest descent method adjusts parameters 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖. Then, the chain rule used for the error function 𝑒_ee=𝑥₁−𝜔₁ depends on parameters 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖, that is,𝜕𝐸𝜕𝑎_𝑖=𝜕𝐸𝜕𝑥₁𝜕𝑥₁𝜕𝜇₁𝜕𝜇₁𝜕𝑓_𝑖𝜕𝑓_𝑖𝜕𝑤_𝑖𝜕𝑤_𝑖𝜕𝜇_𝐴𝑖𝜕𝜇_𝐴𝑖𝜕𝑎_𝑖,𝜕𝐸𝜕𝑏_𝑖=𝜕𝐸𝜕𝑥₁𝜕𝑥₁𝜕𝜇₁𝜕𝜇₁𝜕𝑓_𝑖𝜕𝑓_𝑖𝜕𝑤_𝑖𝜕𝑤_𝑖𝜕𝜇_𝐴𝑖𝜕𝜇_𝐴𝑖𝜕𝑏_𝑖,𝜕𝐸𝜕𝑐_𝑖=𝜕𝐸𝜕𝑥₁𝜕𝑥₁𝜕𝜇₁𝜕𝜇₁𝜕𝑓_𝑖𝜕𝑓_𝑖𝜕𝑤_𝑖𝜕𝑤_𝑖𝜕𝜇_𝐴𝑖𝜕𝜇_𝐴𝑖𝜕𝑐_𝑖,𝜕𝐸𝜕𝑥₁=𝑥₁−𝜔₁,𝜕𝑥₁𝜕𝜇₁=Γ₁−Γ₂ℎ₁𝑇,𝜕𝜇₁𝜕𝐹_𝑖=1,𝜕𝐹_𝑖𝜕𝑤_𝑖=𝑤_𝑖∑𝑤_𝑖𝑝_𝑖𝑥+𝑟_𝑖=𝑝_𝑖𝑥+𝑟_𝑖−𝑓∑𝑤_𝑖,𝜕𝑤_𝑖𝜕𝜇_𝐴1=1. (45) The terms 𝜕𝜇_𝐴𝑖/𝜕𝑎₁, 𝜕𝜇_𝐴𝑖/𝜕𝑎₁, and 𝜕𝜇_𝐴𝑖/𝜕𝑎₁ are calculated from (38), while 𝜕𝐸𝜕𝑎_𝑖=𝑥₁−𝜔₁Γ₁−Γ₂ℎ₁𝑇𝑝_𝑖ℎ₁+𝑟_𝑖−𝑓∑𝑤_𝑖𝜕𝜇_𝐴𝑖𝜕𝑎_𝑖,𝜕𝐸𝜕𝑏_𝑖=𝑥₁−𝜔₁Γ₁−Γ₂ℎ₁𝑇𝑝_𝑖ℎ₁+𝑟_𝑖−𝑓∑𝑤_𝑖𝜕𝜇_𝐴𝑖𝜕𝑏_𝑖,𝜕𝐸𝜕𝑐_𝑖=𝑥₁−𝜔₁Γ₁−Γ₂ℎ₁𝑇𝑝_𝑖ℎ₁+𝑟_𝑖−𝑓∑𝑤_𝑖𝜕𝜇_𝐴𝑖𝜕𝑐_𝑖. (46) Each one of the 𝑛 functions that are proposed in the input layer of ANFIS is fitted by using the previous expressions.

3.4. ANFIS Learning Using Levenberg-Marquardt Approach

The Levenberg-Marquardt (L-M) method is an effective nonlinear least-squares approach to nonlinear regression problems [36]: Θ_𝑘+1=Θ_𝑘−𝐽^𝑇𝐽+𝜆𝐼⁻¹𝑔_ℎ, (47) where 𝑔_ℎ≡(1/2)𝑔; 𝜆 is a nonnegative number.

It has been proved that the L-M method reduces the root mean squared error further than hybrid learning method. In this work, the mean square error is used, and the gradient takes into account 𝑇-𝑆 parameters of the fourth layer. Therefore, the following training rules can be obtained: 𝜕𝐸𝜕𝑝_𝑖=𝜕𝐸𝜕𝑥₁𝜕𝑥₁𝜕𝜇₁𝜕𝜇₁𝜕𝑓_𝑖𝜕𝑓_𝑖𝜕𝑝_𝑖𝑖=1,…,𝑛,𝜕𝐸𝜕𝑟_𝑖=𝜕𝐸𝜕𝑥₁𝜕𝑥₁𝜕𝜇₁𝜕𝜇₁𝜕𝑓_𝑖𝜕𝑓_𝑖𝜕𝑟_𝑖𝑖=1,…,𝑛. (48)

4. Numerical Results after Applying ANFIS

From Figures 15 and 16, it can be observed that the fuzzy tracking is improved and that the tracking error is smaller than the one obtained through the steepest descent approach.

Figure 15 

𝑥1 versus 𝜔1 when ANFIS is included in the fuzzy controller.

Figure 16 

Tracking error when ANFIS is included in the fuzzy controller.

Figure 17 shows membership function 𝜇₁ obtained from different methods: (1) analytically computed, (2) membership functions for fuzzy regulator are the same than ones of fuzzy plant (𝜇₁=ℎ₁), and (3) ANFIS estimation. Note that the membership function generated by ANFIS tends to the real one (analytically computed). It is important to recall that the 𝜇₁+𝜇₂=1, and therefore the missing membership function, 𝜇₂, can be easily obtained.

Figure 17 

Membership function 𝜇1 obtained from different methods.

The control signal appears in Figure 18.

Figure 18 

Control signal when ANFIS is included in the fuzzy controller.

Case 1. In some applications, it is desired that external reference signal has variable frequency, and thus a new exosystem is proposed: 𝑆=03−30. (49) The local steady-state manifolds can be obtained from Π₁=Π₂=,Γ1003₁=−1120,Γ₂=−130. (50) Figure 19 shows the system behavior during the training stage. As expected, the fuzzy plant tracks asymptotically the exosystem.

Figure 19 

𝑥1 versus 𝜔1 for Case 1.

Case 2. A practical example is the inverted pendulum (Figure 20), whose nonlinear model is given by [29] 𝑥̇₁(𝑡)=𝑥₂(𝑡)𝑥̇₂(𝑡)=𝑔sin𝑥₁(𝑡)−𝑎𝑚𝑙𝑥²₂(𝑡)sin2𝑥₁(𝑡)/2−𝑎cos𝑥₁(𝑡)𝑢(𝑡)4𝑙/3−𝑎𝑚𝑙cos²𝑥₁(𝑡), (51) where the mass of the cart is 𝑀=8?kg, the mass of the pendulum is 𝑚=2?kg, the length of the pendulum is 2𝑙=1?m, the gravity is 𝑔=9.81?m/s², and 𝑎=1/(𝑚+𝑀).

Figure 20 

Inverted pendulum on a cart.

This nonlinear model can be approximated by a TS fuzzy model of two rules with 𝐴₁=0117.311760,𝐴₂=,𝐵019.369570₁=0−0.17647,𝐵₂=0,−0.00523𝑆=01−10,𝐾₁=−,𝐾120.6667−22.6667₂=−2551.6−764, (52) and membership functions as those depicted in Figure 21.

Figure 21 

Membership functions for the two-rule model [29].

The regulation condition is not satisfied when the membership functions of the plant are used to combine the local regulators because matrices 𝐵₁ and 𝐵₂ are different (Figure 22).

Figure 22 

𝑥1 versus 𝜔1 when the regulator uses the same membership functions of the plant.

On the other hand, from Figures 23 and 24, it can be observed that the tracking is improved, that is, the tracking error is considerably reduced when ANFIS is used to obtain the adequate membership.

Figure 23 

𝑥1 versus 𝜔1 when ANFIS is included in the fuzzy controller.

Figure 24 

Tracking error when ANFIS is included in the fuzzy controller.

5. Conclusions

In this paper, it has been shown that the nonlinearity of the plant is not necessarily the same nonlinearity needed by the regulator to ensure reference tracking. In other words, the membership functions for fuzzy regulator and for the fuzzy plant are not the same, in general. In that sense, the steepest descent and ANFIS approaches have been used to estimate the adequate membership functions of fuzzy regulator in order to reduce the tracking error when the overall fuzzy controller is obtained from the fuzzy summation of local controllers.

It is also important to mention that although the Steepest Descent can be used to approximate the fuzzy membership functions, the approach can be trapped in a local minimum or local maximum. On the other hand, ANFIS due the hybrid training avoids this problem.

On the other hand, the use of ANFIS avoids the restriction of proposing a priori form of the membership functions, which is another disadvantage of the Steepest Descent approach.

Finally, it can be concluded that a suitable approximation of the nonlinearity of the fuzzy regulator is obtained when ANFIS is used as a tuning tool during the designing of fuzzy regulators.

Acknowledgments

This work was supported by CONACYT through scholarship SNI, by IPN through scholarships PIFI, COFAA, and EDI, and research Project 20130760.