Accepted Manuscript

Improved adaptive control for wing rock via fuzzy neural networkwith randomly assigned fuzzy membership function parameters

Hai-Jun Rong, Sai Han, Jian-Ming Bai, Yong-Qi Liang

PII: S1270-9638(14)00128-XDOI: 10.1016/j.ast.2014.06.009Reference: AESCTE 3083

To appear in: Aerospace Science and Technology

Received date: 18 March 2014Revised date: 23 June 2014Accepted date: 25 June 2014

Please cite this article in press as: H.-J. Rong et al., Improved adaptive control for wing rock via fuzzyneural network with randomly assigned fuzzy membership function parameters, Aerosp. Sci. Technol.(2014), http://dx.doi.org/10.1016/j.ast.2014.06.009

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Improved Adaptive Control for Wing Rock via Fuzzy Neural

Network with Randomly Assigned Fuzzy Membership Function

Parameters

Hai-Jun Ronga,1, Sai Hanb,2, Jian-Ming Baic,3, Yong-Qi Liang a,∗,4

a State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong

University, Xi’an, ShaanXi, 710049, China

b Institute of Avionics and Flight Control, AVIC Xi’an Aircraft Branch Company, Yanliang, ShaanXi, 710089,

China

c Optical Direction and Pointing Technique Research Department, Xi’an Institute of Optics and Precision

Mechanics of CAS, Xi’an, ShaanXi, 710119, China

Abstract: Two stable adaptive fuzzy-neural control schemes within the indirect and direct frameworks are

proposed to suppress the wing rock occurring at high angles of attack. In the two control strategies, a fuzzy neural

network (FNN) with any bounded nonconstant piecewise continuous membership function is used to approximate the

system nonlinear dynamics and external disturbances. Different from the existing techniques, the parameters of the

fuzzy membership functions are determined based on the recently developed fuzzy-neural algorithm named online

sequential fuzzy extreme learning machine (OS-Fuzzy-ELM) where the fuzzy membership function parameters

need not be adjusted and could randomly be generated according to any given continuous probability distribution

without any prior knowledge. This simplifies the design of the controllers. Furthermore to ensure stable control

performance, the tuning laws of the consequent parameters are derived using the projection algorithm and Lyapunov

stability theorem. The merits of the proposed control schemes lie in the simplicity, robustness and stability, which

manifests they can be applied for online learning and real-time control. In order to evaluate the performance of the

proposed two control schemes, a comparison between a neural control, a fuzzy control and a fuzzy-neural control

is carried out on various initial conditions. Results indicate the performance of the proposed controllers is superior

using the randomly assigned fuzzy membership function parameters.

Keywords: Wing Rock, Fuzzy Neural Network, Online Sequential Fuzzy Extreme Learning Machine, Adaptive

Control

1. Introduction

1

Modern high-performance fighters are operating at high angles of attack to capture the maximum maneuverability

and controllability. However, serious lateral directional stability problems have been encountered at high angles of

attack. One frequently encountered instability is the limit-cycle roll oscillation called wing rock that is a highly

nonlinear aerodynamic phenomenon. Some researchers have performed several theoretical studies to understand

the dynamics of wing rock and to predict the amplitude and frequency of oscillation of the limit cycle [1–4].

The onset of wing rock may cause a loss of stability in the lateral/directional mode due to the large amplitudes

and high frequencies of the rolling oscillations. This severely constrains the manoeuvring envelop of the high-

performance fighters. Therefore, suppression of wing-rock motion is a very important task. To achieve the goal,

some nonlinear control strategies including nonlinear optimal feedback control [5], the adaptive control based on the

feedback linearization [6] and the robust H∞ control [7] have been used to suppress the wing rock. An essential

characteristic of these controllers is their model dependence, i.e., the need for full knowledge of the nonlinear

dynamics a priori. It has been shown in [8, 9] that an aircraft’s wing rock is a complex, uncertain, and time-

varying nonlinear system, and therefore its precise analytical modelling is unavailable. This will bring significant

difficulties to the design of these controllers.

Recently an Extended State Observer (ESO) based robust control design [10] is proposed for suppressing the

wing rock motion by employing the ESO to estimate the uncertainty. An advantage of the mehtod is that it neither

requires accurate plant model nor any information about the uncertainty. Besides, some researchers have turned

to intelligent control as a means of explicitly accounting for nonlinear control problems [11–20]. Fuzzy logic

system is one of the popular intelligent control methods that can provide knowledge-based heuristic controllers

for ill-defined and complex systems without requiring a complete analytical model of dynamic systems. In the

design of fuzzy logic controllers, the selection of appropriate parameters of fuzzy membership functions existing

in the premise part of fuzzy rules is an important issue as the change of fuzzy membership functions may alter

the performance of the fuzzy controller significantly [9]. Many researchers [21, 22] have constructed model-free

fuzzy logic based controllers to suppress the oscillation of wing rock. In these methods the parameters of fuzzy

membership functions are determined by uniformly partitioning the fixed universe of the input variables, which

lacks the online learning ability to deal with the uncertainties and time-variations under the large angle of attack.

By applying a variable universe technique to modify the premise parameters of the fuzzy system, a reinforcement

adaptive fuzzy controller using a fuzzy logic system is proposed for suppressing or tracking wing rock phenomena

[9]. In the variable universe, the universe of the fuzzy sets can change along with changing of the input variable

using a contraction-expansion factor. This method can improve the interpolation precision of fuzzy systems, but

the improvement of the performance largely relies on the contraction-expansion factor chosen by trial-and-error.

2

By incorporating the fuzzy logic into a neural network, the fuzzy neural network (FNN) possesses an inherent

structure suitable for online learning and reconstructing complex nonlinear mappings, which provides an effective

mechanism for the control of wing rock. The authors [23] have addressed the wing rock problem using the FNN

and a sequential growing and pruning learning algorithm for FNN, called as Extended Sequential Adaptive Fuzzy

Inference System (ESAFIS). Although ESAFIS can determine the proper number of fuzzy membership functions

during learning, fuzzy membership function parameters are determined according to the novelty of incoming data

which requires many control parameters to be tuned and in the case of non-optimal values, the control performance

may be low. Another deficiency of ESAFIS is the utilization of one specific type of fuzzy membership function

(Gaussian) and not any type. In addition, there is no strict theory to guarantee the universal approximation for

ESAFIS. This can not provide justification in applying EASFIS to almost any nonlinear system control problems.

Lastly, in the control approaches based on ESAFIS, the knowledge of bound on uncertainties is necessary for

successful design of the controller.

The objective of the present work is to develop the simple and improved adaptive fuzzy-neural control strategies

within the indirect and direct frameworks for the wing rock problems by circumventing the deficiencies suffered from

the ESAFIS. As a first step in achieving the objective, a FNN with any bounded nonconstant piecewise continuous

membership function in a unified framework is used to approximate the system nonlinearities and uncertainties.

Then the parameters of any fuzzy membership functions are determined by the OS-Fuzzy-ELM algorithm [24]

developed based on the ELM [25, 26]. In OS-Fuzzy-ELM, the parameters of the fuzzy membership functions need

not be adjusted during learning and could be randomly generated according to any given continuous probability

distribution without any prior knowledge about the target function. This simplifies the controller design process

and also avoids tuning many control parameters in ESAFIS. Besides, it has been shown in our recent work [27]

that with the fuzzy membership function parameters chosen randomly, the FNN with any bounded nonconstant

piecewise continuous membership function can work as a universal approximator. To remove the requirement of a

priori knowledge about the bound on uncertainties, the estimated value which is tuned based on the adaptive laws

derived from the Lyapunov stability theorem and the projection algorithm is used for the controller design.

The performance of the proposed fuzzy-neural controllers are evaluated by comparing with a neural controller

using RBF network, a fuzzy controller based on TS fuzzy system and a fuzzy neural controller using ESAFIS at

various initial conditions. Simulation results demonstrate that the proposed control schemes achieve superior control

performance for suppressing the wing rock with the randomly assigned fuzzy membership function parameters.

The rest of the paper is organized as follows. Section 1 introduces the dynamic model of the wing rock under

consideration and the control objective. Section 2 describes the theoretical foundation of designing the proposed

3

controllers, that is a brief review about the OS-Fuzzy-ELM algorithm together with the structure of a five-layer FNN.

The proposed indirect and direct adaptive fuzzy-neural control schemes are introduced in Section 3. Convergence

and stability of the proposed controllers are proven using the Lyapunov theory and Barbalat’s lemma. Section 4

presents the simulation results and analysis in controlling the wing-rock system between various control methods.

Section 5 shows the conclusions from this study.

2. Wing Rock Dynamics and Control Objective

In this study, a nonlinear mathematical model for wing rock of 800 slender delta wings [2] is considered. The

dynamics of the wing rock system is a complex nonlinear function related with the roll angle φ, roll rate φ and

the angle of attack α. By choosing the state vector x = [x1, x2] = [φ, φ], the wing rock dynamics can be written

in the following state variable form:

x1 = x2

x2 = f(x) + u+ d(1)

where d is the disturbance, u is the control input of the system. f(x) is assumed to be bounded real continuous

nonlinear function and given as,

f(x) = −ω2φ+ μ1φ+ b1φ3 + μ2φ

2φ+ b2φφ2 (2)

The coefficients in the above equation are given by the following relations:

ω2 = −Ca1, μ1 = Ca2 −D, μ2 = Ca4,

b1 = Ca3, b2 = Ca5

(3)

These various coefficients are dependent on the variables C,D and the dimensionless aerodynamic parameters a1

to a5. C is a fixed constant and equal to 0.354. D is the damping coefficient and fixed as 0.0001. Variables a1 to

a5 are nonlinear functions of the angle of attack α and presented in Table 1.

TABLE 1: Coefficients of Rolling Moment with Angle of Attack α

α a1 a2 a3 a4 a5

15.0 -0.01026 -0.02117 -0.14181 0.99735 -0.83478

21.5 -0.04207 -0.01456 0.04714 -0.18583 0.24234

22.5 -0.04681 0.01966 0.05671 -0.22691 0.59065

25.0 -0.05686 0.03254 0.07334 -0.35970 1.46810

The values of the coefficients in equation (2) could be obtained for any angle of attack according to any

interpolation method. According to [2], the observed onset angle where μ1 is zero corresponding to the onset

4

of wing rock is “19-20 deg”. The aim of the present study is to suppress the wing rock, i.e., maintain the 800 delta

wing model at zero roll angle (φ) and zero roll rate (φ) condition.

Defining x = [x1, x2], equation (1) can be further written in the following state-space form,

x = Ax+ bf(x) + bu+ bd (4)

where

A =

⎡⎢⎣ 0 1

0 0

⎤⎥⎦ , b =

⎡⎢⎣ 0

1

⎤⎥⎦ (5)

By denoting the reference output vector φd = [φd, φd]T , the output tracking error e = φd−φ and the tracking error

vector e = [e, e]T = [e1, e2]T , the control objective is to design the control law u such that the state φ = [φ, φ]T

can track the desired command φd = [φd, φd]T as closely as possible.

Based on the feedback linearization method and system dynamics in equation (1), the desired control input u∗

can be obtained as follows:

u∗ = φd − f(x)− d+ k1e+ k2e (6)

where k1 and k2 are the real number. By substituting equation (6) into equation (1), the following equation is

obtained

e+ k1e+ k2e = 0 (7)

By properly choosing k1 and k2, all roots of the polynomial s2 + k1s + k2 = 0 are in the open left half plane,

which implies that the tracking error will converge to zero. Equation (7) also shows that the tracking of reference

command is asymptotically achieved from any initial conditions, i.e., limt→∞|e| = 0.

However, if the nonlinear function f(x) and external disturbances d are unknown, the ideal control law (6)

is difficult to implement in practical applications. Here, two adaptive fuzzy-neural control strategies within the

indirect and direct frameworks based on the FNN are designed. Also the antecedent parameters of the FNN are

determined using the OS-Fuzzy-ELM algorithm which is the theoretical foundation of designing the fuzzy-neural

control schemes. A brief description of the OS-Fuzzy-ELM algorithm is given in the following section.

3. Theoretical Foundation

OS-Fuzzy-ELM can handle the most commonly used Mamdani type of fuzzy models where the antecedent (if)

part and the consequent (then) part are both described by the fuzzy sets, and Takagi-Sugeno-Kang (TSK) type of

fuzzy models where only the antecedent part is described by fuzzy sets whereas the consequent part is described

by linear functions about input variables. TSK fuzzy model can achieve better performance with fewer rules and

5

has been widely applied into the control of the nonlinear systems [28, 29]. Thus in the study a FNN with TSK

fuzzy model is used to construct the control law.

The TSK fuzzy model is given by the following rules:

Rule i : if (x1 is A1i) AND (x2 is A2i) AND · · · AND (xn is Ani), then y is βi

where x(x = [x1, · · · , xn]T ) and y are the input and output of the system, respectively, Aji(j = 1, 2, · · · , n; i =1, 2, · · · , N) is the fuzzy set of the jth input variable xj in rule i, n is the dimension of the input vector, and N is

the number of fuzzy rules. βi(i = 1, 2, · · · , N) is the crisp value and a linear combination of input variables, that

is βi = qi0 + qi1x1 + · · · + qinxn. The fuzzy model can be realized by a five-layer fuzzy neural network whose

structure is illustrated in figure 1.

Fig. 1: Structure of Fuzzy Neural Network

Layer 1: In Layer 1, each node represents an input variable and directly transmits the input signal to Layer 2.

Layer 2: In this layer each node represents the membership value of each input variable. The degree to which the

given jth input variable xj satisfies the quantifier Aji in rule i is specified by its membership function μAji(xj).

In this paper, a general bounded nonconstant piecewise continuous membership function g(c, a) with parameters

(c, a) ∈ Rn−1×R is considered, which includes eight commonly used membership functions [30] and any Pseudo

Trapezoid-Shaped function [31]. The membership value μAji(xj) of the jth input variable xj in the ith rule can

be achieved by any one of bounded nonconstant piecewise continuous membership functions:

μAji(xj ; cji, ai) = g(xj ; cji, ai) (8)

6

where cji and ai are the parameters existing in the membership function g corresponding to the jth input variable

xj and the ith rule.

Layer 3: Each node in this layer represents the if part of if-then rules obtained by fuzzy logic AND operation,

which can be any type of T-norm such as the product composition. The firing strength (if part) of the ith rule is

given by

Ri(x; ci, ai) =μA1i(x1; c1i, ai)⊗ μA2i

(x2; c2i, ai)⊗ · · ·

⊗ μAni(xn; cni, ai)

(9)

where symbol ⊗ represents any type of T-norm operation. If the Gaussian membership function with the algebraic

product operation is employed it will be simply as

Ri(x; ci, ai) =

n∏j=1

μAji(xj ; cji, ai) =

n∏j=1

exp

((xj − cji)

2

ai

)(10)

Layer 4: The nodes in this layer are named as normalized nodes whose number is equal to the number of the

nodes in the third layer. The ith normalized node is equal to the following equation:

G(x; ci, ai) =Ri(x; ci, ai)

N∑i=1

Ri(x; ci, ai)

(11)

Layer 5: The node in this layer corresponds to the output variable. The system output is achieved by the weighted

sum of the output of each normalized rule. As such the system output y for given input x is calculated by

y =

N∑i=1

βiRi(x; ci, ai)

N∑i=1

Ri(x; ci, ai)

=

N∑i=1

βiG(x; ci, ai) = βTG(x; c,a) (12)

where c = [c1, · · · , cN ] and a = [a1, · · · , aN ]. For TSK fuzzy model, the consequent parameters are the linear

equation of the input variables and given by

βi = xTe qi (13)

where xe is the extended input vector by appending the input vector x with 1, that is [1,xT ]T ; qi is the parameter

vector existing in the TSK model for the ith fuzzy rule and given by

qi = [qi0, qi1, · · · , qin]T (14)

Thus the output equation (12) becomes as:

y =

N∑i=1

xTe qiG(x; ci, ai) (15)

7

The equation further is written in the following compact form,

y = QTH (16)

where H is the normalized firing strength vector weighted by the extended input and Q is the parameter vector,

respectively:

H(c1, · · · , cN , a1, · · · , aN ;x)

=[xTe G(x; c1, a1), · · · ,xT

e G(x; cN , aN )]T (17)

and

Q =

⎡⎢⎢⎢⎢⎣

q1

...

qN

⎤⎥⎥⎥⎥⎦ (18)

Lemma 1 [27]: Given any bounded nonconstant piecewise continuous membership function g : R �−→ R, for any

continuous target function yL and any randomly generated function sequence {Gi}, limi→∞ ‖y − yL‖ = 0 holds

with probability one.

The function sequence {Gi} is randomly generated when the corresponding membership function parameters

(ci, ai) are randomly generated based on a continuous sampling distribution probability. Thus in OS-Fuzzy-ELM

all the parameters of the fuzzy membership functions (c1, · · · , cN, a1, · · · , aN ) can be randomly generated unlike

the conventional implementation of the fuzzy neural networks where they need to be tuned. This will simplify

the controller design process. When the antecedent parameters (c and a) are randomly generated and based on

this training a FNN is simply equivalent to finding a least-square solution of the consequent parameters Q in

equation (16). In earlier OS-Fuzzy-ELM, the consequent parameters (Q) are modified based on a RLSE method.

However, when this method is applied for controller design, problems such as the stability of the overall scheme and

convergence of the approximation error arise. Thus in the study the tuning laws for the consequent parameters Q

are derived in the sense of projection algorithm and Lyapunov stability theorem to ensure the network convergence

as well as stable control performance. This will be introduced detailedly below.

The OS-Fuzzy-ELM algorithm offers a fast online fuzzy-neural learning algorithm, which can assign the random

values to the parameters of the fuzzy membership functions. Its outstanding computational efficiency about learning

speed, adaptability and generalization has been verified in the latest work [24, 27]. In essence, the learning capability

of OS-Fuzzy-ELM can be used as the basis for learning in the fuzzy-neural adaptive control. Generally fuzzy-neural

adaptive control approaches can be classified into two categories, namely, indirect and direct adaptive control

schemes. In the indirect control, the FNNs are used to identify the nonlinear dynamics and then the fuzzy-neural

8

control law is implemented using the identified models. In the direct adaptive control, a FNN is directly used to

approximate the control law without explicitly attempting to determine the model of the dynamics. To fully depict

the potential of OS-Fuzzy-ELM in the design of fuzzy-neural adaptive controller, the following section focuses on

showing how to achieve stable indirect/direct adaptive control for the wing-rock problem when OS-Fuzzy-ELM is

used as an online learning algorithm.

4. Adaptive Fuzzy-Neural Control Schemes

Both indirect and direct adaptive fuzzy-neural control schemes equipped with a robustifying control term to

eliminate the approximation error of the fuzzy neural network are developed sequentially in this section.

4.1. Indirect Adaptive Fuzzy-Neural Control Scheme

The proposed indirect adaptive fuzzy-neural control scheme is illustrated in figure 2. This controller structure

consists of two components. The first component uf represents the adaptive fuzzy-neural control law and provides

asymptotic convergence of the tracking error. The fuzzy-neural control term uf is constructed using the five-

layer FNN with any bounded nonconstant piecewise continuous membership function to learn the unknown system

dynamics and external disturbances. Different from the existing methods, the parameters (c,a) of fuzzy membership

functions in FNN are assigned randomly based on the OS-Fuzzy-ELM algorithm, which simplifies the controller

design procedure. Also the consequent parameters QI are updated based on the stable adaptive laws, which

guarantees the asymptotical stability of the system. The second component ur represents the robustifying control

law, which is used to compensate for approximation error between the nonlinear system dynamics and the FNN

with optimal parameters. Besides, to avert the priori knowledge about the approximation error bound in calculating

ur, it is estimated online using the estimation law. The working of this indirect control scheme is described in

detail below.

In the proposed indirect adaptive fuzzy-neural control scheme, the FNN is used as an estimator for learning the

system unknown dynamics and disturbances f(x) + d. Then the estimative value is used to indirectly develop the

fuzzy-neural control law that is given as,

uf = φd − fd(x) + k1e+ k2e (19)

where fd(x) is the estimated value of f(x) + d from FNN and represented as

fd(x) = QTI H (20)

9

Fig. 2: The proposed indirect adaptive fuzzy-neural control scheme

where H is the abbreviation of H(x; c,a). During the modelling process of the FNN estimator, the determination

of the antecedent parameters (c,a) in the fuzzy membership functions is based on the OS-Fuzzy-ELM algorithm

described above. Thus the design problem of the FNN estimator now becomes the problem of adapting the

consequent parameters (QI ) in order to make the approximation error as small as possible.

According to the universal approximation property of OS-Fuzzy-ELM described above, there exists an optimal

FNN estimator to approximate the nonlinear dynamic function f(x) + d in equation (6) such that

fd(x) = Q∗TI H+ εI (21)

where εI is the approximation error, Q∗I are the optimal parameters and defined as follows:

Q∗I

Δ= argmin

[supx∈Mx

‖fd(x)− fd(x)‖]

(22)

where Mx is the predefined compact set of input vector x.

It is worthy noting that the zero approximation error in OS-Fuzzy-ELM is hard to be obtained since infinite fuzzy

rules are required for the network. Instead in real applications, only limited rules exist, which results in the inherent

approximation error εI . However, the approximation error εI can be reduced arbitrarily when the number of fuzzy

rules increases. Thus it can be assumed that εI is bounded by the constant εI , i.e., |εI | ≤ εI . To compensate for

the approximation error, a robustifying control term ur is used to augment the fuzzy-neural controller above and

thus the overall control law is considered as,

u = uf + ur (23)

10

where

ur = εIsgn(eTPb) (24)

with

sgn(eTPb) =

⎧⎪⎨⎪⎩

1 eTPb > 0

−1 eTPb < 0(25)

In the robustifying control term, εI is the estimated value of the approximation error bound εI , which is to relax

the requirement of the error bound.

On the basis of equations (19)-(21) and (23), the tracking error of equation (4) becomes as,

e = Λe+ b[QTI H− εI − ur] (26)

where QTI (= QT

I −Q∗TI ) is the parameter error with dimension 3N × 1 and H is the normalized firing strength

with dimension 1× 3N . Λ represents the 2× 2 controllable canonical form matrix equaling to

Λ =

⎡⎢⎣ 0 1

0 0

⎤⎥⎦+

⎡⎢⎣ 0 0

−k2 −k1

⎤⎥⎦

=

⎡⎢⎣ 0 1

−k2 −k1

⎤⎥⎦

(27)

.

To realize the asymptotic tracking, the two parameters (QI , εI ) are tuned based on the stable adaptive laws

introduced from the stability analysis of the closed-loop system by using Lyapunov function method. To guarantee

the bound of the control signals (uf , ur), this requires that the parameters QI and εI must be bounded. In order

to avoid the problem that the parameters QI and εI may drift to infinity over time, the projection algorithm [32]

is applied by limiting their upper bound. The constraint set for QI and εI are defined respectively as follows:

Ω1 � {‖QI‖ ≤ QI0}

Ω2 � {εI ≤ εI0}(28)

for some QI0 > 0, εI0 > 0 and εI0 ≤ εI .

The adaption laws for the parameters QI and εI are defined as,

QI =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−η1HeTPb

if ‖QI‖ < QI0 or

if ‖QI‖ = QI0

and QTI HeTPb ≥ 0

−η1Ψ1

if ‖QI‖ = QI0

and QTI HeTPb < 0

(29)

11

˙εI =

⎧⎪⎨⎪⎩

η2|eTPb| if εI < εI0

0 if εI ≥ εI0

(30)

where Ψ1 =

(I − QIQ

TI

‖QI‖2)HeTPb. η1 and η2 are positive constants that appear in the adaptation laws and

referred to as the learning rate.

Concerning the stability of the closed-loop system, the following theorem is achieved.

Theorem 1: Consider the wing-rock dynamics represented by equation (1). The indirect fuzzy-neural control

law is designed as equation (19) with the adaptive law of FNN given in (29) and the robustifying control term is

designed as (24) with the bound estimation algorithm presented in (30). Then the convergence of tracking error

and the stability of the proposed indirect fuzzy-neural control system can be guaranteed.

Proof: See the Appendix.

The indirect adaptive control design requires one to carry the system dynamic approximation and then a controller

is developed based on the current best guess at the system dynamics. A direct adaptive control scheme without

requiring the system dynamic approximation, on the other hand, is also proposed by directly adjusting the parameters

of a controller to meet some performance specifications.

4.2. Direct Adaptive Fuzzy-Neural Control Scheme

The proposed direct adaptive fuzzy-neural control scheme is illustrated in figure 3 and is composed of two

controllers, viz, the fuzzy-neural controller uf and the robustifying controller ur. Different from the indirect control

strategy where the system dynamics is identified first and then the fuzzy-neural controller is developed based on

the identified system dynamics, the direct adaptive fuzzy-neural controller uf is built using the five-layer FNN with

any bounded nonconstant piecewise membership function by directly adjusting the parameters of the controller to

ensure asymptotic convergence of the tracking error. The robustifying control term ur is also required due to the

modeling error between the ideal feedback controller and the fuzzy-neural controller with optimal parameters. As

with the indirect adaptive scheme, the parameters (c,a) of the fuzzy-neural controller are assigned randomly based

on the OS-Fuzzy-ELM algorithm, which simplifies the controller design procedure. The consequent parameters QD

are updated based on the stable adaptive laws and the approximation error bound in calculating ur is estimated

online using the estimation law. The design details of this direct control scheme is described below.

In the direct control strategy, a fuzzy-neural controller is directly designed to imitate the perfect control law u∗

represented in equation (6), which is given as

uf = QTDH (31)

12

Fig. 3: The proposed direct adaptive fuzzy-neural control scheme

where H is the abbreviation of H(z; c,a). It is noted that the inputs z of the fuzzy-neural controller in the direct

control scheme are the desired command φd, the state vector x, and the error vector e. During modelling the

control input signal, the OS-Fuzzy-ELM algorithm is used to determine the parameters of the fuzzy membership

functions (c,a). Thus the design problem of the fuzzy-neural controller now becomes the problem of adapting the

consequent parameters (QD) in order to make the approximation error as small as possible.

According to the universal approximation property of OS-Fuzzy-ELM, there exists the optimal parameters Q∗D

to approximate the perfect control law, which is defined as

Q∗D

Δ= argmin

[supz∈Mz

‖u∗f − uf‖]

(32)

such that

u∗ = Q∗TD H+ εD (33)

where Mz is the predefined compact set of input vector z. εD is the approximation error due to the limited fuzzy

rules and assumed to be bounded by the constant εD with the increase of the fuzzy rules, i.e., |εD| ≤ εD.

A robustifying control term ur is required to compensate for the approximation error in modeling fuzzy-neural

controller uf with the FNN. Thus the overall control input equals as,

u = uf + ur (34)

where

ur = εDsgn(eTPb) (35)

13

with

sgn(eTPb) =

⎧⎪⎨⎪⎩

1 eTPb > 0

−1 eTPb < 0(36)

εD is the estimated value of the approximation error bound εD.

Adding and substructing bu∗ to equation (4) and applying equation (34), the system tracking error can be shown

to be

e = Λe+ b [u∗ − uf − ur] (37)

According to equations (31) and (33), the system error of equation (37) can further be expressed as

e = Λe+ b[QT

DH+ εD − ur

](38)

where QTD(= Q∗T

D −QTD) is the parameter error.

The parameters QD and εD are tuned based on the stable laws derived from the Lyapunov theorem to ensure the

stability of the control system. Also the projection algorithm is applied to guarantee the bound of the parameters

so that the control signals remain bounded. Define the constraint set for the two parameters as follows,

Ω3 � {‖QD‖ ≤ QD0}

Ω4 � {εD ≤ εD0}(39)

for some QD0 > 0, εD0 > 0 and εD0 ≤ εD.

The tuning laws are designed as,

QD =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

η3HeTPb

if ‖QD‖ < QD0 or

if ‖QD‖ = QD0

and QTDHeTPb ≤ 0

η3Ψ2

if ‖QD‖ = QD0

and QTDHeTPb > 0

(40)

˙εD =

⎧⎪⎨⎪⎩

η4|eTPb| if εD < εD0

0 if εD ≥ εD0

(41)

where Ψ2 = (I − QDQTD

‖QD‖2 )HeTPb. η3 and η4 are positive constants which represent the learning rate.

Concerning the stability of the closed-loop system, the following theorem is obtained.

Theorem 2: Consider the wing-rock dynamics represented by equation (1). If the direct fuzzy-neural control

law is designed as equation (31) with the parameter update law of the FNN given in (40) and the robustifying

14

control law are designed as equation (35) with the bound estimation algorithm presented in (41), the convergence

of tracking error and the stability of the proposed direct fuzzy-neural control system can be guaranteed.

The proof of Theorem 2 follows the same path as the one of Theorem 1 by setting a different Lyapunov function

V =1

2eTPe+

1

2η3QT

DQD +1

2η4ε2D with εD = εD − εD. For brevity, it is omitted here and one can refer to the

proof of Theorem 1.

Remark 1: The indirect and direct adaptive fuzzy-neural controllers both allow for the FNN with any bounded

nonconstant piecewise membership function, require the OS-Fuzzy-ELM algorithm, and provide the same stability

results for the wing rock system. However there are some implementation differences. The indirect and direct

adaptive controllers require a FNN for the wing rock system and there may be resulting simplified implementation.

But the inputs (φd,x, e) in the direct control are more than those (x) of the indirect control. This will cause more

consequent parameters to be adjusted, thereby maybe resulting in computational burden in the implementation. But

when the required rules as illustrated by the simulations below are quite small, the difference about the number of

the parameters is not evident and thus the resulting computational difference is not significant. Besides, the indirect

controller is comprised of a PD-like controller to generate a compensated control signal as given in equation (19),

which can enhance the tracking performance. Actually the direct adaptive technique can allows for the inclusion

of a conventional controller so that it may be used to enhance the performance of the fuzzy-neural controller.

Remark 2: Though some concepts within this paper are similar to those used within [23], there are many significant

differences. i) A general bounded nonconstant piecewise continuous membership function can be applied in the

proposed adaptive fuzzy-neural control schemes. The specific fuzzy membership function type named Gaussian

membership function used in [23] is only a special case of the general bounded nonconstant piecewise continuous

membership functions. ii) In [23], The parameters of fuzzy membership functions are determined based on the

novelty of incoming data, which is dependent on the approximated target function and requires many control

parameters to be tuned. This makes the controller design process become complex. However, in our current study,

these parameters are determined using the OS-Fuzzy-ELM algorithm where they need not be adjusted during

learning and could randomly be generated according to any given continuous probability distribution without any

prior knowledge about the target function. This makes the implementation simpler. iii) Unlike [23], the universal

approximation theorem for the OS-Fuzzy-ELM learning algorithm [27] provides us with justification for applying

it to almost any nonlinear system modeling problems.

The proposed indirect and direct controller are effective ways to solve the wing rock problem and both achieve

better tracking performance than the existing control technologies. The simulation results given below verify its

effectiveness whether in designing the indirect or direct adaptive controller.

15

5. Performance Evaluation

In the section, the proposed indirect and direct adaptive fuzzy-neural control schemes are evaluated by suppressing

the wing rock under α = 25o where limit-cycle roll oscillations occur as shown in figures 4(a) and 4(b). The

amplitude and frequency of oscillation of the limit cycle are consistent with those given in [2]. In the simulations,

the indirect and direct controller design parameters are chosen as follows: J = diag(50, 50), η1 = η3 = 50,

η2 = η4 = 0.01, QI0 = QD0 = 80, εI0 = εD0 = 0.6. To stabilize the system, the coefficients in equation (7) are

chosen as k1 = 2 and k2 = 1. Two initial conditions, viz, small initial condition (φ(0) = 11.46 degree, φ(0) = 2.865

degree/s) and large initial condition (φ(0) = 103.14 degree, φ(0) = 57.3 degree/s) are used to investigate the

effectiveness of the proposed control schemes. The reference trajectory vector is chosen as (φd, φd)T = (0, 0)T . In

order to evaluate the robustness of the proposed control schemes, the disturbance d = 0.2sin(4πt) is added to the

system.

Moreover OS-Fuzzy-ELM can be applied into any bounded nonconstant piecewise continuous membership

function and thus another two commonly used membership functions, viz., triangular and cauchy fuzzy membership

functions are studied in the problem to verify the effectiveness of the proposed controllers. When the triangular

membership function is utilized, the firing strength of equation (9) is given as,

Ri(x; ci, ai) =

n∏j=1

(1− |xj − cji|

ai

)(42)

For the cauchy membership function, the firing strength of equation (9) equals,

Ri(x; ci, ai) =

n∏j=1

1

1 +

(xj − cji

ai

) (43)

The parameters of the fuzzy membership functions (c,a) are chosen randomly in the intervals [−1, 1] and [0, 1]

respectively. The experimental results reported here are based on the average of 20 independent trials. All the

simulations have been conducted in MATLAB 2009a environment running in an ordinary PC with 2.4 GHZ CPU.

Also the control schemes based on the RBF network [8], ESAFIS [23] and TS fuzzy system [33] are applied here

to control the wing rock system for comparison.

5.1. Effect of Number of Fuzzy Rules and Different Fuzzy Membership Functions on Tracking

Here, the effect of number of fuzzy rules and different membership functions is investigated on the tracking

performance achieved by the proposed indirect and direct control schemes. Tables 2 and 3 separately give the

tracking performance of the proposed adaptive indirect and direct fuzzy-neural control methods under the small

and large initial states by changing the number of fuzzy rules from 2 to 10. The Eφ and Eφ shown in these tables

16

(a) Roll Angle (b) Derivative of Roll Angle

Fig. 4: Time process of uncontrolled roll angle and its derivative for α = 25 deg.

represent the RMS errors of roll angle and roll rate while Eave is the average of both. One can observe from these

tables that the tracking RMS errors are relatively stable regardless of the number of fuzzy rules used. This further

means that the designed indirect and direct fuzzy-neural controllers have good robustness to the number of fuzzy

rules. Thus in the following study, the number of fuzzy rules is set to 2. Besides, it can be found from the two

tables that the indirect control can achieve slightly smaller tracking errors than the direct control. The reason is

that a PD-like controller is included in the indirect controller to generate a compensated control signal, thereby

enhancing the tracking performance.

The evaluation of the tracking performance using different fuzzy membership functions for the proposed indirect

and direct fuzzy-neural control schemes under the small and large initial states are shown in tables 4 and 5,

respectively. Besides the Gaussian fuzzy membership function used above, another two commonly used fuzzy

membership functions, namely Triangular and Cauchy functions are studied. It can be seen from the two tables

that the simulation results achieved by the three membership functions are similar, and moreover, the proposed

fuzzy-neural control schemes using OS-Fuzzy-ELM produce smaller standard deviations (SD) of tracking errors,

indicating that random selection of antecedent parameters of membership functions does not affect the tracking

performance significantly. Also slightly better tracking performance is obtained by the indirect control than that of

the direct control.

5.2. Performance Comparison between Different Algorithms

17

TABLE 2: Effect of number of fuzzy rules on the tracking performance under the small initial condition

No. of Indirect Control Scheme Direct Control Scheme

Rules Eφ Eφ Eave Eφ Eφ Eave

2 2.0711 2.0763 2.0737 2.3060 2.3096 2.3078

4 2.5198 2.5358 2.5278 2.4965 2.5059 2.5012

6 2.4966 2.4996 2.4981 2.5534 2.5583 2.5559

8 2.5474 2.5653 2.5564 2.7575 2.7611 2.7593

10 2.5193 2.5437 2.5315 2.7556 2.7683 2.7620

TABLE 3: Effect of number of fuzzy rules on the tracking performance under the large initial condition

No. of Indirect Control Scheme Direct Control Scheme

Rules Eφ Eφ Eave Eφ Eφ Eave

2 18.9587 18.9565 18.9576 19.2384 19.2415 19.2400

4 18.9613 18.9838 18.9726 19.3495 19.3770 19.3633

6 18.9981 19.0078 19.0030 19.3179 19.3233 19.3206

8 19.0411 19.0926 19.0669 19.4490 19.4638 19.4564

10 19.0476 19.0641 19.0559 19.6046 19.5914 19.5980

In this section, the proposed fuzzy-neural control schemes using OS-Fuzzy-ELM are further evaluated by

comparing with other control technologies using RBF network, ESAFIS and TS fuzzy system. The simulation

results of the roll angle φ and the roll rate φ from the indirect and direct control schemes under the small initial

states are illustrated in figures 5 and 6 where a clear illustration for the initial transient portion between 0 and 5

second is given. Figures 7 and 8 depict the simulation results from the indirect and direct control schemes in the

case of the large initial states together with the clear illustration of the initial transience between 0 and 5 second.

For comparison purposes, OS-Fuzzy-ELM applies the Gaussian membership function used by other algorithms.

TABLE 4: Effect of different membership functions on the tracking performance under the small initial condition

Control Membership Eφ Eφ Eave

Schemes Function RMSE SD RMSE SD RMSE SD

Gaussian 2.0711 0.0013 2.0763 0.0020 2.0737 0.0017

Indirect Triangular 2.2678 0.0040 2.2386 0.0082 2.2532 0.0061

Cauchy 2.1567 0.0036 2.1628 0.0043 2.1598 0.0040

Gaussian 2.3060 0.0079 2.3096 0.0083 2.3078 0.0081

Direct Triangular 2.4404 0.0025 2.4536 0.0025 2.4470 0.0025

Cauchy 2.3614 0.0073 2.3643 0.0065 2.3629 0.0069

18

Also the simulation results from the RBF network, ESAFIS, TS fuzzy system and the reference zero signals are

also presented in these figures. These results show that in the proposed adaptive indirect and direct fuzzy-neural

control schemes the difference between the system output and the reference output is large at the beginning, but

the system still keeps stable and the roll oscillations are quickly suppressed after 5 sec. This largely decreases the

maneuverability burden caused by the undesired rolling oscillation. Compared with the control strategies based on

the RBF network and ESAFIS, the proposed fuzzy-neural control schemes achieve faster error convergence speed

and better transient results under the small and large initial conditions. From these figures, one can see that the TS

fuzzy system obtains a faster converge speed than the OS-Fuzzy-ELM. But the OS-Fuzzy-ELM has smaller roll

rate errors during the initial learning process than the TS fuzzy system.

TABLE 5: Effect of different membership functions on the tracking performance under the large initial condition

Control Membership Eφ Eφ Eave

Schemes Function RMSE SD RMSE SD RMSE SD

Gaussian 18.9587 0.0031 18.9565 0.0030 18.9576 0.0031

Indirect Triangular 19.9314 0.0068 19.9318 0.0068 19.9316 0.0068

Cauchy 18.9871 0.0017 18.9939 0.0016 18.9905 0.0017

Gaussian 19.2384 0.0034 19.2415 0.0054 19.2400 0.0044

Direct Triangular 19.9363 0.0096 19.9367 0.0087 19.9365 0.0092

Cauchy 19.7268 0.0068 19.7272 0.0049 19.7270 0.0059

Table 6 gives an overview of the comparison results from the four methods based on the RMS errors between

the reference and the actual values. It is clear that OS-Fuzzy-ELM overall achieves better tracking performance

compared with the others even with lesser fuzzy rules whether in the indirect or direct control. Also the indirect

control achieves smaller tracking errors than the direct control with the same network size. The simulation results

about the computation cost between different control methods are shown in Table 7. From the table, it can be

observed that the OS-Fuzzy-ELM has less computation time than the ESAFIS with the same number of fuzzy rules

in the indirect control scheme. The reason is that the antecedent parameters in the ESAFIS are determined according

to the adding and pruning of rules, which increases the computation complexity and causes more computation time.

Besides, the lesser the number of fuzzy rules is, the lesser consequent parameters the fuzzy system tends to have.

This results in smaller computation burden. Considering these, the OS-Fuzzy-ELM has less computation cost than

the ESAFIS and TS fuzzy system regardless of the indirect and direct control schemes. Compared to the RBF, the

computation time of the OS-Fuzzy-ELM is more but the tracking accuracy is much better. Also one can note from

the table that the indirect control scheme requires less computation time than the direct one. The main reason is

19

(a) φ(0)=11.46 degree (b) φ(0)=2.865 degree/s

Fig. 5: Roll angle and roll rate under the small initial states in the indirect control scheme.

TABLE 6: Comparison results between different control methods

Control Algorithms Small initial condition Large initial condition

Schemes Eφ Eφ Eave No. Rules Eφ Eφ Eave No. Rules

/Neurons /Neurons

OS-Fuzzy-ELM 2.07 2.08 2.08 2 18.96 18.96 18.96 2

Indirect ESAFIS 2.85 1.90 2.38 2 21.27 20.48 20.88 2

TS 1.27 5.33 3.30 9 10.57 42.61 26.59 9

OS-Fuzzy-ELM 2.31 2.31 2.31 2 19.24 19.24 19.24 2

Direct ESAFIS 2.33 3.22 2.78 3 20.96 19.49 20.22 3

TS 1.16 5.13 3.15 9 10.87 42.51 26.69 9

RBF 15.34 34.58 24.96 5 52.06 80.48 66.27 5

20

TABLE 7: Computation cost comparison between different control methods

Control Schemes Algorithms Small initial condition Large initial condition

(seconds) (seconds)

OS-Fuzzy-ELM 0.9636 1.0482

Indirect ESAFIS 3.7048 3.9550

TS 25.855 28.876

OS-Fuzzy-ELM 1.1283 1.2582

Direct ESAFIS 4.2897 4.4157

TS 144.69 256.83

RBF 0.2595 0.2786

that the inputs (φd,x, e) in the direct control are more than those (x) of the indirect control. This will cause more

consequent parameters to be adjusted, thereby resulting in larger computational burden.

Figures 9(a) and 9(b) illustrate that the consequent parameters (QI , QD) for the indirect and direct control schemes

in the small and large initial conditions, respectively. It can be seen that they are bounded throughout the control

process. This guarantees the bound of the indirect and direct control signals in the two initial conditions, which

are depicted in figures 10(a) and 10(b), respectively. Figure 11 depict the estimation process of the approximation

error bounds (εI , εD). One can observe from the figure the approximation error bounds gradually become stable

according to the control process.

6. Conclusions

An improved fuzzy-neural adaptive control scheme with the indirect and direct frameworks are developed for

suppressing the wing rock where the dynamic characteristics are nonlinear and uncertain. In the developed control

scheme, the FNN with any bounded nonconstant piecewise continuous membership function is used to approximate

the nonlinearities and uncertainties without requiring the knowledge of the mathematical model of the wing rock

system. The fuzzy membership function parameters are determined based on the idea of the OS-Fuzzy-ELM

algorithm by randomly assigning the values to them. Simulation results via various existing control methods are

provided in this paper for the purpose of comparison and also display the superior performance of the proposed

control schemes with the randomly assigned fuzzy membership function parameters. In addition, the indirect and

direct adaptive fuzzy-neural controllers both provide the same stability results for suppressing the wing rock and

have simplified implementation. However, the indirect control can achieve slightly smaller tracking errors than the

direct control with equal network size. Our adaptive schemes are only for one-degree-of-freedom wing rock system.

21

(a) φ(0)=11.46 degree (b) φ(0)=2.865 degree/s

Fig. 6: Roll angle and roll rate under the small initial states in the direct control scheme.

Extension to multiple-degree-of-freedom case is an important direction of the future work.

Appendix: Proof of Theorem 1

Lyapunov’s theory is used to prove the result. Consider the following Lyapunov function candidate,

V =1

2eTPe+

1

2η1QT

I QI +1

2η2ε2I (A-1)

where P is the symmetric and positive definite matrix, εI = εI − εI .

22

(a) φ(0)=103.14 degree (b) φ(0)=57.3 degree/s

Fig. 7: Roll angle and roll rate under the large initial states in the indirect control scheme.

Using equation (26), the derivative of the Lyapunov function is given as,

V =1

2[eTPe+ eTPe] +

1

η1QT

I˙QI − 1

η2(εI − εI) ˙εI

=1

2eT (ΛTP+PΛ)e+ QT

I (HeTPb+1

η1

˙QI)

− eTPbεI − eTPbur − 1

η2(εI − εI) ˙εI

= −1

2eTJe+ QT

I (HeTPb+1

η1QI)− eTPbεI

− eTPbur − 1

η2(εI − εI) ˙εI

(A-2)

where J = −(ΛTP+PΛ). J is a symmetric definite matrix and selected by the user.

When the condition 1 of equation (29) is satisfied, this term QTI (HeTPb +

1

η1QI) equals to zero. For the

23

(a) φ(0)=103.14 degree (b) φ(0)=57.3 degree/s

Fig. 8: Roll angle and roll rate under the large initial states in the direct control scheme.

condition 2 of equation (29), the term equals as

QTI (HeTPb+

1

η1QI) = (QI −Q∗

I)T (

QIQTI

‖QI‖2HeTPb)

= (1− Q∗TI QI

‖QI‖2 )QTI HeTPb

(A-3)

Since Q∗I ∈ Ω1 and 1− Q∗T

I QI

‖QI‖2 ≥ 0, equation (A-3) is always negative.

Thus equation (A-2) becomes,

V ≤ −1

2eTJe− eTPbεI − eTPbεIsgn(e

TPb)− 1

η2(εI − εI) ˙εI (A-4)

where equation (24) is utilized. Equation (A-4) further satisfies the following condition:

V ≤ −1

2eTJe+ |eTPb||εI | − eTPbεIsgn(e

TPb)− 1

η2(εI − εI) ˙εI (A-5)

24

(a) (φ, φ)T = (11.46, 2.865)T (b) (φ, φ)T = (103.14, 57.3)T

Fig. 9: Norm of consequent parameters under different initial states.

(a) (φ, φ)T = (11.46, 2.865)T (b) (φ, φ)T = (103.14, 57.3)T

Fig. 10: Control input u under different initial states.

Since |eTPb| = (eTPb)sgn(eTPb) and |εI | ≤ εI , equation (A-5) becomes

V ≤ −1

2eTJe+ |eTPb|(εI − εI)− 1

η2(εI − εI) ˙εI (A-6)

Under the condition 1 of equation (30), equation (A-6) equals

V ≤ −1

2eTJe ≤ 0 (A-7)

Using the condition 2 of equation (30) yields

V ≤ −1

2eTJe+ |eTPb|(εI − εI) (A-8)

25

(a) (φ, φ)T = (11.46, 2.865)T (b) (φ, φ)T = (103.14, 57.3)T

Fig. 11: Estimated value of the approximation error bound under different initial states.

Noting that εI − εI > 0, equation (A-8) is further expressed as,

V ≤ −1

2eTJe ≤ 0 (A-9)

Equations (A-1) and (A-9) show that V ≥ 0 and V ≤ 0. Thus

V (e(t), QI(t), εI(t)) ≤ V (e(0), QI(0), εI(0)) (A-10)

which implies that e, QI , εI are bounded. Denoting function

Vc(t) ≡ 1

2eTJe = −V (e(t), QI(t), εI(t)) (A-11)

Integrating (A-11) with respect to time can be shown that,∫ t

0Vc(τ)dτ ≤ V (e(0), QI(0), εI(0))− V (e(t), QI(t), εI(t)) (A-12)

Since V (e(0), QI(0), εI(0)) is bounded and also V (e(t), QI(t), εI(t)) is nonincreasing and bounded, the following

result can be obtained,

limt→∞

∫ t

0Vc(τ)dτ <∞ (A-13)

Because Vc(t) is bounded, it can be concluded that limt→∞ Vc(t) = 0 according to Barbalat’s lemma [32, 34]. This

implies that e→ 0 as t→∞. Therefore the indirect fuzzy-neural control system is asymptotically stable and the

tracking error of the system will converge to zero.

Acknowledgment

26

This work is funded in part by National Natural Science Foundation of China (Grant No. 61004055, 61105028),

National Science Council of ShaanXi Province (Grant No. 2014JM8337) and the Fundamental Research Funds for

the Central Universities.

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