Composite adaptive fuzzy H∞ tracking control of uncertain nonlinear systems

Neurocomputing 99 (2013) 15–24

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Composite adaptive fuzzy HN tracking control of uncertain nonlinear systems Yongping Pan a,n, Yu Zhou b, Tairen Sun c, Meng Joo Er a a

School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore School of Electric Power, North China Institute of Water Conservancy and Hydroelectric Power, Zhengzhou 450011, China c School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China b

a r t i c l e i n f o

abstract

Article history: Received 20 June 2011 Received in revised form 10 November 2011 Accepted 14 May 2012 Available online 30 May 2012 Communicated by W. Yu

In the HN tracking-based adaptive fuzzy controllers (HAFCs) of perturbed uncertain nonlinear systems, additional HN control terms would greatly degrade fuzzy approximation abilities, which violates the original intention of using fuzzy logic systems. To solve this problem, a composite HAFC (CHAFC), which combines the HAFC with composite adaptation technique, is proposed in this paper. Outside of the approximation region, a robust stabilization controller is developed to achieve semi-global stability of the closed-loop system. Within the approximation region, a series–parallel identification model is introduced into an indirect HAFC to construct a CHAFC that can simultaneously achieve fuzzy identification and HN tracking control. It is proved that the closed-loop system obtains HN tracking performance in the sense that both tracking and modeling errors converge to small neighborhoods of zero. Simulated applications of aircraft wing rock suppression and inverted pendulum tracking demonstrate that the proposed approach not only effectively solves the aforementioned approximation problem, but also obviously outperforms previous approaches. & 2012 Elsevier B.V. All rights reserved.

Keywords: Adaptive fuzzy control HN tracking control Composite adaptation Fuzzy identification

1. Introduction For a class of nonlinear systems with unknown dynamics, adaptive fuzzy controller (AFC) [1] design has aroused widespread concern over the past two decades. Some recent related works can be found in [2–5] and the references therein. Based on feedback linearization technique, the affine nonlinear system can be transformed into a controllable canonical form. Then, two categories of the AFC, namely an indirect AFC based on fuzzy modeling rules and a direct AFC based on fuzzy control rules [3], can be applied to control the system. Usually, control systems would be affected by uncertainties such as unmodeled dynamics and external disturbances, which may degrade tracking performance or even destroy stability of the closed-loop systems [6]. The HN tracking-based AFC (HAFC) [7], which contains a basic AFC and an additional HN control term, is an effective tool to reject those uncertainties. Generally, the direct HAFC [5–12] requires the control gain function to be known a priori. Yet, its indirect counterpart does not have this requirement and has been well-developed in recent years [7,8,13–18]. The indirect HAFC was firstly introduced for state feedback affine nonlinear systems in [7]. Next, it was extended to output feedback systems in [13–15]. Afterward, sliding-mode

n

Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Pan).

0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.05.011

control technique was combined to improve the robustness of the indirect HAFC in [8,16,17]. Note that certain plant bound functions are needed to be known a priori in [8]. Recently, the indirect HAFC was applied to the continuous stirred tank reactor (CSTR) to achieve HN temperature tracking control in [18]. Those HAFCs cannot only stabilize the closed-loop systems, but also guarantee L2-gains from lumped uncertainties to tracking errors are equal to or less than prescribed attenuation levels. However, fuzzy approximation abilities in those control structures would be greatly degraded while the HN control terms are added to the basic AFCs, which results in the sharp increase of modeling errors. Hence, it violates the original intention of using fuzzy logic systems (FLSs) since the HN control terms become major controllers and play more important roles than the basic AFCs. One possible solution for solving the aforementioned problem is to make the modeling errors as additional feedback information in the HAFCs. The modeling error feedback was firstly introduced into the indirect AFC to construct a composite AFC for a class of affine nonlinear systems in [19], where faster state tracking and better parameter convergence were obtained due to the quicker and smoother parameter adaptation. However, the nth derivative of the plant output is required to be known in [19], which is quite impractical in real-would applications, where n is the order of the plant. To relax this limitation, a novel composite AFC that contains a series–parallel identification model with a low-pass filter [20] was developed in [21] and implemented to real-world

16

Y. Pan et al. / Neurocomputing 99 (2013) 15–24

motor control problems in [22,23]. Comparative experiments showed that the controller in [20] outperforms the classical linear adaptive controller and the controller in [19]. Most recently, a continuous composite AFC guaranteed asymptotic tracking performance was presented in [4] to synchronize two uncertain generalized Lorenz systems. Nonetheless, to be the best of our knowledge, the modeling error feedback has not been introduced into the HAFC design yet. In this paper, to solve the fuzzy approximation problem caused by the HN control terms, a composite HAFC (CHAFC), which combines the HAFC with composite adaptation technique, is presented for a class of perturbed uncertain single-input single-output (SISO) nonlinear systems. The overall control scheme is comprised of an identification model, a stabilization controller and an indirect HAFC. The design procedure of the proposed approach is as follows: firstly, outside of the approximation region, a robust stabilization controller based on adaptive bounding technique is developed to achieve semi-global stability of the closed-loop system and boundness of an optimal fuzzy the approximation error (FAE); secondly, within the approximation region, the series–parallel identification model is introduced into the indirect HAFC to construct a CHAFC; finally, the composite adaptive laws, which utilize both tracking and modeling errors, are derived by the Lyapunov synthesis. Note that the proposed approach can be easily extended to the general multiinput multi-output (MIMO) non-affine system by the combination of the approaches in [16] and [24]. The rest of this paper is organized as follows. The preliminaries and problem under consideration are formulated in Section 2. The controller design procedure is proposed in Section 3. In Section 4, two illustrative examples are given. Finally, concluding remarks are shown in Section 5. Throughout this paper, R, R þ and Rn denote the real number, positive real number and real n-vector, respectively; subscripts min and max denote the minimum and maximum of corresponding variables, respectively; supðUÞ, lmin ðUÞ and lmax ðUÞ represent the functions of supremum, minimal eigenvalue and maximal eigenvalue, respectively. Moreover, 9 U9 and :U: represent the absolute value and standard Euclidean norm, respectively.

i

with

Pm

l1 ¼ 1 . . .

Pm

ln ¼ 1

Qn

i¼1

mAli ðxi Þ ¼ 1, where l1    ln ¼ j and i

j ¼ 1,    ,M. Lemma 1. [1]: The FLS in (2) is a universal function approximator, i.e., for any given real continuous function f(x) on D and an arbitrary small constant mf 40, there exists a FLS in the form of (2) such that supx A D 9f ðxÞf^ ðx9hÞ9 o mf . 2.2. System transformation Consider the following SISO affine nonlinear system [7]: (

z_ ¼ fðzÞ þ gðzÞu þdðtÞ y ¼ hðzÞ

,

ð3Þ

where z A Rp is the state vector, u A R and yA R are the input and output variables, respectively, fðzÞ A Rp and gðzÞ A Rp are unknown but bounded smooth vector fields, hðzÞ A R is a smooth function and dðtÞ ¼ ½d1 ðtÞ,    ,dn ðtÞT A Rp denotes a unknown but bounded external disturbance vector. Let Lfh(z) and Lgh(z) denote the Lie derivatives of h(z) with respect to f and g, respectively. Assumption 1. [25]. The system in (3) satisfies: (1) its strong relative degree is equal to n with n rp; (2) the one-dimensional distribution D ¼span{g(z)} is involutive. According to Assumption 1, there exist p  1 functions Z1(z),

Z2 ðzÞ,    , Zp1 ðzÞ that satisfies

(

rank½dZ1 ðzÞ, dZ2 ðzÞ,    ,dZp1 ðzÞ ¼ n1 dZi ðzÞgðzÞ ¼ 0 ði ¼ 1,2,    ,p1Þ 8zA Rp

:

hT and g ¼ ½Z1 ,    , Zpn T . Accordingly, there Let x ¼ ½h,Lf h,    ,Ln1 f exists a diffeomorphism: TðzÞ ¼ ½x; g that transforms (3) to the controllable canonical form [13]: 8 ðnÞ > < x ¼ f ðx, gÞ þ gðx, gÞu þdðtÞ g_ ¼ qðx, gÞ , > :y¼x

2. Preliminaries and problem formulation 2.1. Description of FLSs The applied FLS performs a mapping from an input vector x A D  Rn to a scale output yf  OY  R, where D ¼ OX1  OX2      OXn is a fuzzy approximation region, OXi  R, and i ¼ 1,    ,n. Let Ai ¼ fAlii g ðli ¼ 1,. . .,mÞ and B ¼ fBj g ðj ¼ 1,. . .,MÞ denote fuzzy partitions on OXi and OY, respectively, where Alii and Bj are linguistic variables. Thus, a fuzzy rule base can be constructed as follows: Rj : If x1 is Al11 and    and xn is Alnn , then yf is Bl1 ln

ð1Þ n

where j ¼ 1,    ,M, li ¼ 1,    ,p, i ¼ 1,    ,n, M¼m , and l1    ln l1 ln

denotes the permutation of elements l1 ,    ,ln . Let mAli and y i

denote the membership function of Alii and the peak point of Bl1 ln , respectively. By the use of the singleton fuzzifier, product inference engine and center-average defuzzifier, the applied FLS can be expressed as follows [1]: yf ¼ f^ ðx9hÞ ¼ hT nðxÞ

the elements in x(x) are given by Yn xl1 ln ðxÞ ¼ m l ðx Þ, i ¼ 1 Ai i

ð4Þ

ð5Þ

ðn1Þ T _ where x ¼ ½x1 ,x2 ,:::,xn T ¼ ½x, x,:::,x  is a new state vector, n f ðx, gÞ ¼ Lf hðzÞ is the nonlinear driving function, gðx, gÞ ¼ Lg Ln1 f hðzÞ is the control gain function, g_ ¼ qð0, gÞ is the zero dynamics Pn1 k1 nk and d(t)¼Ln1 hðzÞ is an external f þ gu þ d Ld hðzÞ þ k ¼ 1 Lf þ gu þ d Ld Lf disturbance. Clearly, the vector d in (3) is transformed into the scalar d in (5). If the zero dynamics in (5) is asymptotically stable, then during controller design, one can only consider the following system: ( xðnÞ ¼ f ðxÞ þ gðxÞu þ dðtÞ : ð6Þ y¼x

Assumption 2. [26]. There exist unknown bound functions f ðxÞ and gðxÞ, and constants g and d such that 9f ðxÞ9 r f ðxÞ, 0 og r gðxÞ r gðxÞ and 9dðtÞ9 rd hold, 8x A Rn . Remark 1. If the considered system is a general MIMO non-affine nonlinear system: ( z_ ¼ fðz, uÞ þ dðtÞ , ð7Þ y ¼ hðzÞ

ð2Þ

where h ¼ ½y1 ,    ,yM T is a vector of adjusting parameters, nðxÞ ¼ ½x1 ðxÞ,    , nM ðxÞT is a vector of fuzzy basic functions, and

where u A Rm and y A Rm are the input and output vectors, respectively, fðz, uÞ A Rp and h(z) A Rm are smooth vector fields, and z and d have the same definitions as in (3). Then from the

Y. Pan et al. / Neurocomputing 99 (2013) 15–24

approach of [27], (7) can be transformed into ( z_ ¼ fðz, 0Þ þ gðz, u0 Þu þ dðtÞ , y ¼ hðzÞ

ð8Þ

where gðz, u0 Þ ¼ ½g1 ðz, u0 Þ,    , gmðz, u0 Þ, gi ðz, u0 Þ ¼ ½g 1i ðz, u0 Þ,    , g mi ðz, u0 Þ, g ji ðz, u0 Þ ¼ @f j ðz, uÞ=@ui u ¼ u0 , u0 ¼ ½u01 ,    , u0m , u0i A ð0, ui Þ and i, j ¼ 1,    ,m. Let (8) be the MIMO form of (3), i.e., ( z_ ¼ fðzÞ þgðzÞu þdðtÞ : ð9Þ y ¼ hðzÞ From the approach in Appendix D of [16], (9) can be transformed into m SISO affine nonlinear systems in the form of (6). Thus, the following approach based on the system in (3) can be easily extended to the system in (7).

17

  Define the compact set D :¼ x : :x:r M x , where M x A R þ is a finite constant. Actually, the dynamic of d in (6) can also be captured by the FLS [27]. Thus, one can let Df T :¼ Df þ d denote a total nonlinear uncertainty. Under the indirect scheme, the dynamics Df T and Dg can be approximated by two FLSs in the form of (2): T f^ ðx9h^ f Þ ¼ h^ f nðxÞ,

ð15Þ

T ^ h^ g Þ ¼ h^ g nðxÞ, gðx9

ð16Þ

respectively, where h^ f ¼ ½y^ f 1 ,    , y^ f M T and h^ g ¼ ½y^ g1 ,    , y^ gM T are vectors of adjusting parameters. From (13), (15) and (16), one can choose the following certain control law: f 0 ðxÞf^ ðx9h^ f Þvf þ yðnÞ þk e d þuh , ^ ^ g ðxÞ þ gðx9hg Þ þ vg T

u¼

ð17Þ

0

2.3. Control objective _    ,yðn1Þ T and yd ¼ ½yd , y_ d ,    ,ydðn1Þ T , where yd is a Let y ¼½y, y, bounded desired input single which has the nth-order derivative. Define the output tracking error e:¼yd  y, the error vector _ . .,eðn1Þ T , and the generalized e :¼ yd y ¼ ½e1 ,e2 ,. . .,en T ¼ ½e, e,. error vector e~ :¼ ½e; eF , where eF is a filtered modeling error. As mentioned before, eF will be applied as additional feedback information to suppress itself into a small neighborhood of zero. Thus, the control objective of this study is to design a CHAFC based on the FLS in (2) such that the closed-loop system achieves: (1) semi-global stability in the sense that all involving singles are uniformly ultimately bounded (UUB); (2) the following HN tracking performance [7]: Z T Z T T ~ r 2V L ð0Þ þ r2 e~ Q E edt w2 dt, T A ½0,1Þ ð10Þ 0

0

where rAR þ is the prescribed attenuation level, VL is a Lyapunov function, wAL2[0,T] denotes the optimal FAE, and QE is a matrix satisfying Q E ¼ Q TE 4 0. Note that the terms eF, VL, w and QE will be defined in the following sections.

where vf and vg are robust stabilization terms applied when x A D, uh is the HN control term applied when x A D, and D :¼ Rn D. Form [7], uh can be design as follows: ^ h^ g ÞÞ, uh ¼ eT Pb=rðg 0 ðxÞ þ gðx9

ð18Þ T

þ

where r A R , and matrix P satisfying P¼P 40 is the solution of the following Riccati-like equation:   2 1 T  2 b P þ Q ¼ 0: PA þ AT PPb ð19Þ r r 3.2. Robust semi-global stabilization Since the fuzzy approximation property shown in Lemma 1 is only valid over D, one should first consider the stabilization problem of the closed-loop system within D. Let x(x)¼0 and uh ¼0, 8x A D. Then, one makes the following assumption. Assumption 3. [26]. The system uncertainties DfT and Dg satisfy U

U

U f ðhU f 9xÞ r Df T ðxÞ rf ðhf 9xÞ,

ð20Þ

U U g 0 ðxÞ og U ðhU g 9xÞ r DgðxÞ r g ðhg 9xÞ, n

hUf

T

M

¼ ½mg ,    ,mg  A RM .

8x A R with

3.1. Indirect adaptive control scheme

To avoid large control effects, the adaptive bounding technique in [29] instead of the bounds in (20) and (21) is applied to stabilize the system in (6). Generally, from (17)–(21), one has the following robust stabilization controller:

f ðxÞ ¼ f 0 ðxÞ þ Df ðxÞ,

ð11Þ

gðxÞ ¼ g 0 ðxÞ þ DgðxÞ,

ð12Þ

respectively, where f0 and g0 are known nominal estimations and Df and Dg are system uncertainties. Ideally, if Df and Dg are known and d¼ 0, a certain feedback controller can be determined to stabilize the system in (6). Generally, choose a vector k ¼ ½kn ,::,k1 T A Rn such that h(s)¼sn þ k1 sn1 þ    þ kn is a Hurwitz polynomial, where s is a complex variable. Then, one has the following ideal control law: un ¼

T f 0 ðxÞDf ðxÞ þyðnÞ þk e d

g 0 ðxÞ þ DgðxÞ

:

ð13Þ

Substituting (13) into (6) leads to ðnÞ

ðn1Þ

e þ k1 e þ    þkn e ¼ 0: ð14Þ By virtue of the selection of k, one has limt-1 :eðtÞ: ¼ 0. However, u* in (13) is unrealizable since Df and Dg are unknown and d a0 in this study.

and

T

3. Composite adaptive control design

In practice, the rough estimations of the nonlinearities in (6) can usually be obtained. By the incorporation of such a priori knowledge, f and g in (6) can be expressed into

¼ ½mf ,    ,mf  A R

ð21Þ

hUg

T

u ¼ uS ¼

f 0 ðxÞvf þ yðnÞ þk e d , g 0 ðxÞ þvg

ð22Þ

^ f , vg ¼ sgnðeT Pbua Þm ^ g , ua ¼ f 0 ðxÞvf where vf ¼ sgnðeT PbÞm T ðnÞ ^ ^ g are the estimations of mf and mg, þ yd þ k e, and mf and m respectively. From (22), one has T

ydðnÞ ¼ k e þ ðf 0 ðxÞ þ vf Þ þ ðg 0 ðxÞ þvg ÞuS :

ð23Þ

From (6), (11) and (12), one obtains yðnÞ ¼ ðf 0 ðxÞ þ Df T ðxÞÞþ ðg 0 ðxÞ þ DgðxÞÞuS :

ð24Þ

Subtracting (24) from (23) and making some transformations, one obtains the tracking error dynamics: e_ ¼ Ae þ bððvf Df T ðxÞÞ þðvg DgðxÞÞuS Þ, in which 2 0 6 6 ^ A¼6 6 0 4

kn

1



^

&

0



kn1



0

3

7 ^ 7 7, 1 7 5 k1

ð25Þ

2 3 0 6^7 6 7 b ¼ 6 7: 405 1

18

Y. Pan et al. / Neurocomputing 99 (2013) 15–24

From the selection of k, one knows that A is a stable matrix. Thus, there exists a unique positive definite symmetric matrix P for any given positive definite symmetric matrix Q such that T

A P þPA ¼ Q :

ð26Þ

^ f mf and m ~ g ¼m ^ g mg , and choose the Lyapunov ~ f ¼m Let m function candidate as follows: ~ 2f þ m ~ 2g Þ=2gm , V S ¼ eT Pe=2 þ ðm

ð27Þ

eF :¼ x^ ðn1Þ xðn1Þ ,

ð37Þ

and introduce the following series–parallel identification model with a low-pass filter [20]: 8 < x_^ i1 ¼ x^ i ði ¼ 2,    ,nÞ , ð38Þ : x^_ n ¼ aF eF þ ðf ðxÞ þ f^ ðx9h^ ÞÞ þ ðg ðxÞ þ gðx9 ^ h^ g ÞÞuT þ v f

0

where gm A R þ is a learning rate. Then, we start the first result of this study. Theorem 1. For the nonlinear system in (3) within D satisfying Assumptions 1–3, select (22) as the robust stabilization controller and design the boundary estimation laws as follows: _^ ¼ g ð9eT Pb9s m ð28Þ m m ^ f Þ, f m

0

where x^ i are the estimations of xi, aF A R þ is a user-defined filter parameter, v is a modeling compensation term, and i ¼ 1,    ,n. Subtracting (35) from the second line of (38) and noting e_ F ¼ x_^ n xðnÞ , one obtains the modeling error dynamics: T

T

e_ F ¼ aF eF þ h~ f nðxÞ þ h~ g nðxÞuT wþ v:

ð39Þ

Then, one has the following lemma.

_^ ¼ g ð9eT Pbu 9s m m g m ^ g Þ, S m

ð29Þ

where sm A R þ is a user-defined small constant. Then the closedloop system achieves semi-globally stability in the sense that all involving singles are UUB and e converges to a compact set: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio n ð30Þ Oes ¼ e : :e: r sm ðm2f þ m2g Þ=2lmin ðQ Þ : Proof. See in Appendix A.

As shown in Fig. 1, to achieve the composite adaptation, one defines the filtered modeling error [22]:

Lemma 2. The optimal FAE wALN, 8x A D , i.e., there exists a finite positive constant w rwsup such that w ¼ sup8x A D 9w9 holds. Proof. See in Appendix B.

&

Accordingly, v can be given by the following form [31]: v ¼ bsgnðeF Þ,

ð40Þ

where b Z w is a user-defined finite constant. Choose the Lyapunov function candidate as follows:

&

3.3. Regional modeling and tracking

VL ¼

Now, consider the regional modeling and tracking problems within D. Let vf ¼vg ¼0, 8x A D. From Assumption 3, one further makes the following assumption. Assumption 4. [30]. The parameter vectors h^ f and h^ g in (15) and (16) belong to compact sets Of and Og, respectively, which are defined as Of :¼ fh^ f : :h^ f : rM f g and Og :¼ fh^ g : kh^ g krMg}, where Mf, M g A R þ are user-defined finite constants.

1 T g 1 ~T ~ 1 ~T ~ e Peþ e e2f þ h h þ h hg , 2 2gf f f 2gg g 2

ð41Þ

where ge , gf , gg A R þ are learning rates. Now, we start the second result of this study. Theorem 2. For the nonlinear system in (3) within D satisfying Assumptions 1–4, choose (38) with (40) as the identification model, select (33) with (18) as the tracking controller, and design the parameter adaptive laws as follows: _ h^ f ¼ gf ðeT Pbþ ge eF ÞnðxÞ, ð42Þ

Define the FAE (i.e., the modeling error) wa as follows: ^ h^ g ÞÞu, wa :¼ ðDf T ðxÞf^ ðx9h^ f ÞÞ þ ðDgðxÞgðx9

_

ð31Þ

and the optimal FAE w as follows: ^ hg ÞÞu, w :¼ ðDf T ðxÞf^ ðx9hf ÞÞ þ ðDgðxÞgðx9 n

n

ð32Þ

where hnf and hng are HN optimal parameter vectors given by

hnf ¼ argminh^ f A Of ðsupx A D 9f ðxÞf^ ðx9h^ f Þ9Þ, ^ h^ g Þ9Þ, hng ¼ argminh^ g A Og ðsupx A D 9gðxÞgðx9

ð43Þ

The overall control scheme is shown in Fig. 2. Then the closed-loop system achieves: (1) stability in the sense that all involving singles are UUB; (2) the HN tracking performance of (10) in the sense that e and ef converge to small neighborhoods of zero: n o Oet ¼ e : :e:r rwðlmin ðQ ÞÞ1=2 , ð44Þ n

respectively. According to (17), one obtains the following indirect HAFC: T f 0 ðxÞf^ ðx9h^ f Þ þ ydðnÞ þk e þ uh : u ¼ uT ¼ ^ ^ hg Þ g ðxÞ þ gðx9

h^ g ¼ gg ðeT Pb þ ge eF ÞnðxÞuT :

o

Oe ¼ eF : 9eF 9 r rwðge aF Þ1=2 , respectively, where Q E ¼ diagðQ , ge aF =2Þ.

ð33Þ

0

From above expression, one obtains T ^ h^ g ÞÞuT uh : yðnÞ ¼ k eþ ðf 0 ðxÞ þ f^ ðx9h^ f ÞÞ þ ðg 0 ðxÞ þ gðx9 d

ð34Þ

From (6), (11), (12) and (32), one has ^ hng ÞÞuT þ w: yðnÞ ¼ ðf 0 ðxÞ þ f^ ðx9hnf ÞÞ þ ðg 0 ðxÞ þ gðx9

ð35Þ

Let h~ f ¼ h^ f hf and h~ g ¼ h^ g hg . Subtracting (35) from (34) and making some transformations, one obtains the tracking error dynamics as follows:  T T e_ ¼ Aeþ b h~ f nðxÞ þ h~ g nðxÞuT wuh : ð36Þ n

n

Fig. 1. Block diagram of identification model.

ð45Þ

Y. Pan et al. / Neurocomputing 99 (2013) 15–24

19

observer-based approaches; (2) the proposed approach utilizes both the tracking and the modeling errors to update the adaptive laws, whereas the observer-based approaches only utilize the observation error to update the adaptive laws. Note that if the state variables of the system in (6) are not completely measurable, the observers in [13–15] can be combined into the proposed approach to construct an output feedback CHAFC.

4. Illustrative examples 4.1. Example 1: aircraft wing rock suppression Fig. 2. Block diagram of CHAFC.

Consider a wing rock model in the form of (3) [33–35] with Proof. See in Appendix C.

&

Remark 2. From Theorems 1 and 2, one knows that two types of controllers are applied to the system in (3). To obtain smooth control input, we introduce the following overall controller: u ¼ aw uS þ ð1aw ÞuT ,

ð46Þ

where aw is a weight parameter defined as follows: 8 if x A D > <1 aw ¼ ðMx :x:Þ=ðMx M0 Þ if x A DD0 , ð47Þ > : 0 if x A D0   in which D0 :¼ x : :x:r M 0  D is a compact set, and M 0 A R þ is a user-defined finite constant. Remark 3. Theorem 2 shows that the boundness of the adjusting parameters h^ and h^ g can be guaranteed by the proposed approach. f

However, to ensure h^ f and h^ g to be retained inside the compact sets Of and Og, the smooth projection algorithm in [8] still need to be applied. Note that the selections of Mf and Mg depend on the rough estimations of plant functions f and g, respectively. Remark 4. The parameter selections of the control laws in Theorems 1 and 2 are summarized as follows: (1) for the selection of sm in (28) and (29), one obtains that decreasing sm improves the tracking error convergence, but sm that is too small may make ^ f and m ^ g too large; (2) for the selection of gm in (28) and (29), as m well as gf, gg and ge in (42) and (43), one obtains that increasing gm, gf and/or gg improves the tracking error convergence, increasing ge improves the modeling error convergence, but gm, gf, gg and/or ge that are too large may make the adjusting parameters oscillation [22] or even destroy the system stability [32]. Remark 5. From (31), (37) and (38), one obtains eF ¼ ðvwa Þ=ðs þ aF Þ,

ð48Þ

which implies that the filtered modeling error eF is a filtered output of a modified modeling error term (v wa). For the previous HAFC approaches of [5–18], the fuzzy approximation abilities are greatly degraded while the HN control terms in the form of (18) are added to the basic AFCs, which violates the original intention of using FLSs. In our approach, eF in (48) is introduced as additional feedback information to construct the CHAFC that can simultaneously achieve fuzzy identification and HN tracking control. Thus, it can not only solve the aforementioned problem, but also has the potential of improving control performance. Remark 6. In the state observer-based AFCs [13–15], the structure of adaptive fuzzy observers are somewhat similar with the identification model in (38). Yet, there exist two major differences between those observer-based approaches and the proposed approach, which are shown as follows: (1) the identification model in (38) does not contain the observation error feedback term which appears in all

8 2 3 z2 > > > > 6 o2 z þ m z þb z3 þ m z2 z þb z z2 þ b z 7 > > 1 1 2 2 1 2 0 35 1 2 2 1 2 > > fðzÞ ¼ 4 > > z3 =t < 2 3 2 3 , 0 d1 ðtÞ > > > > 6 7 6 7 > 0 5,dðtÞ ¼ 4 d2 ðtÞ 5,hðzÞ ¼ z1 , > > > gðzÞ ¼ 4 > > : 1=t d3 ðtÞ

ð49Þ

_ , d T , f is the aircraft roll angle (rad), where z ¼ ½z1 , z2 , z3 T ¼ ½f, f A f_ is the roll rate (rad/s), dA is the actuator output, b0 is the actuator gain, t is the aileron time constant, o2 ¼  c1a1, m1 ¼c1a2  c2, b1 ¼c1a3, m2 ¼c1a4, b2 ¼c1a5, c1 and c2 are certain constants, and ai(i ¼1,...,5) are time-varying parameters to be relative to free-to-roll experiment conditions. The wing rock phenomenon is a limit cycle roll oscillation which is experienced by the aircraft with slender delta wings and pointed forebodies at high angle of attack aa. Such phenomenon may lead to serious danger due to the potential of aircraft instability. The model in (49) satisfies Assumption 1 with n ¼3. Thus, using the diffeomorphism: 2 6 x ¼ TðzÞ ¼ 4

3

z1 2

o z1 þ m

z2 3 2 2 z þ b z þ 2 1 1 2 z1 z2 þ b2 z1 z2 þb0 z3 2

m

7 5,

ð50Þ

(49) can be transformed into the form of (6), where 8 f ðxÞ ¼ x2 ðo2 þ 2m2 x1 x2 þ b2 x22 Þ þ x3 ðm1 þ3b1 x22 þ m2 x21 þ 2b2 x1 x2 Þ > > < þ ðo2 x1 þ m1 x2 þ b1 x32 þ m2 x21 x2 þb2 x1 x22 x3 Þ=t, > P > 3k : z1 : gðxÞ ¼ b0 =t, dðtÞ ¼ L2f þ gu þ d Ld z1 þ 2k ¼ 1 Lfk1 þ gu þ d Ld Lf ð51Þ For simulation studies, let b0 ¼1.5, t ¼1/15, c1 ¼0.354, c2 ¼ 0.001, x(0) ¼[p/2,0,0]T, and dðtÞ ¼ 2sinð2tÞ þ sinð5tÞ after t ¼10 s. The parameters ai for i ¼1,...,5 varying with aa at 801 swept back wing are given in Table 1. The time-varying wing rock model is constructed by applying a cubic interpolation function to the date in Table 1. Note that during simulation, ai with i¼1,...,5 are assumed to be unknown and aa is set to be 20þ5 sin(0.1pt). The design procedure of controller parameters is as follows: (1) for constructing the FLS f^ in (15) (here only one needs to construct f^ since Dg ¼ 0 in (51)), select h^ f ¼ ½0,    ,0T and design the membership functions of xi as follows: 



mAli ðxi Þ ¼ exp ðxi 0:5pðli 2ÞÞ2 = 2ð0:65Þ2

,

i

where li, i¼1, 2, 3; (2) for designing the stabilization controller in (22) with (28) and (29), let k¼[3, 3, 1]T, gm ¼100 and sm ¼ 0.1; (3) for designing the tracking controller in (33) with (18), (42), and (43), let Q¼diag(10,10,10), r ¼0.1, r ¼2r2, gf ¼100, ge ¼30,

20

Y. Pan et al. / Neurocomputing 99 (2013) 15–24

Mf ¼50, and obtain 2

3

23:125

19:375

5:000

6 P ¼ 4 19:375 5:000

32:500

7 8:125 5

8:125

4:375

from (19); (4) for designing the identification model in (38) with (40), select aF ¼10 and b ¼1; (5) for designing the overall controller in (46) with (47), choose Mx ¼1.2 and M0 ¼0.8. Simulated experiment design is as follows: (1) the classic indirect AFC in [27] and the indirect HAFC in [7] are selected to be compared controllers, and same values are applied to all identical user-defined parameters of all controllers for making fair comparison; (2) the robust stabilization controller uS in (22) is applied to all controllers since the initial state vector xð0Þ A D; (3) the sample time and the running time are set to be 10 ms and 30 s, respectively; (4) the Rt Rt tracking indexes Je(ITAE) ¼ to t:ejðtÞ:dt and Je(IAE) ¼ to t:ejðtÞ:dt, Rt the identification index Je(IAE) ¼ to t9eðtÞ9dt, and the control energy 90 70

aa(1)

a1

a2

a3

a4

a5

15.0 21.5 22.5 25.0

 0.01026  0.04207  0.04681  0.05686

 0.02117  0.01456 0.01966 0.03254

 0.14181 0.04714 0.05671 0.07334

0.99735  0.18583  0.22691  0.35970

 0.83478 0.24234 0.59065 1.46810

5 0

60 50 40 30 20

-10 -15 -20 -25

10

-30

0

-35

-10

0

5

10

15 time(s)

20

25

-40

30

10

40

0

35

-10

30

-20

25

-40

0.1

-50

0

-60

-0.1

e2( /s)

-30

-80 0

5

10

15 time(s)

0

5

10

15 time(s)

25

30

15

-5

30

30

0.5 0 -0.5 -1 29

0 29.5

25

1

20

5

20

20

by indirect AFC in [27] by indirect HAFC in [7] by proposed CHAFC

10

-0.2 by indirect AFC in [27] by indirect HAFC in [7] -0.3 by proposed CHAFC -0.4 29

-70 -90

y'd x2

-5 Roll rate( /s)

Roll angle( )

Table 1 Variation of wing rock parameters with angle of attack.

yd x1

80

e1 ( )

Rt Ec ¼ 0 u2 ðtÞdt are chosen to evaluate control performance, where t0 denotes the time when x converges to D. Simulation trajectories of wing rock suppression are shown in Fig. 3. One observes that the proposed approach achieves favorable regulating performance (see Fig. 3(a) and (b)) with smooth control input (see Fig. 3(e)) and the best tracking accuracy (see Fig. 3(c) and (d)); the HN control term uh of the proposed approach is much smaller than that of the indirect HAFC in [7] (see Fig. 3(f)). Performance comparisons of aircraft wing rock suppression with

0

5

10

15 time(s)

29.5

20

30

25

30

0.2

0.15

0

0.1

-0.2

0.05 0

-0.4

-0.05

-0.6 -0.8

-0.1

-1

-0.15

-1.2

by indirect HAFC in [7] by proposed CHAFC

uh

u

0.4

0

5

10

15 time(s)

20

25

30

-0.2 0

5

10

15

20

25

30

time(s)

_ , (c) comparison of tracking error e , (d) comparison of Fig. 3. Suppression trajectories of aircraft wing rock. (a) Trajectories of roll angle j, (b) trajectories of roll rate f 1 tracking error e2, (e) overall control input u, and (f) comparison of HN control term uh.

Y. Pan et al. / Neurocomputing 99 (2013) 15–24

various controllers are shown in Example 1 of Table 2. One observes that the classic indirect AFC in [27] cannot obtain satisfied tracking performance; with the aid of the HN control term uh, the indirect Table 2 Performance comparisons of various controllers. Indexes

Example 1

Example 2

21

HAFC in [7] obviously improves tracking performance at the cost of greatly losing identification accuracy; the proposed approach obtains the best tracking and identification performance without additional control efforts. 4.2. Example 2: inverted pendulum tracking

The indirect AFC in [27]

The indirect HAFC in [7]

The proposed CHAFC

Je(ITAE) Je(IAE) Je(IAE) Ec

13.63 2.179 0.2552 0.4625

7.563 1.824 3.057 0.4268

4.135 1714 0.08876 0.4298

Je(ITAE) Je(IAE) Je(IAE) Ec

11.06 1.065 0.4804 2286

1.993 0.3912 2.515 2274

1.079 0.3279 0.1590 2186

Consider a pendulum model in the form of (6) [7] with 8   m lp x2 cosx1 sinx1 4l l m cos2 x > = 3p  pmcl þ ml 1 , < f ðxÞ ¼ g v sinx1  l m2 c þ ml  4l l m cos2 x > 1 : gðxÞ ¼ mcosx = 3p  pmcl þ ml 1 , dðtÞ ¼ 5sinðtÞ, c þ ml

ð52Þ

where x1 is the angular position of the pendulum, x2 is the corresponding angular velocity, gv ¼ 9.8 m/s2 is the gravitational acceleration, mc is the mass of the cart, ml is the mass of the pendulum, lp is the half-length of the pendulum, and d(t) represents the

80 yd

20 0

10 0 -10

-30 0

5

10

15 time(s)

20

25

-40 0

30

5

10

15 time(s)

50

10 0

by indirect AFC in [27] by indirect HAFC in [7] by proposed CHAFC

40

-10 -20

e2( /s)

e1( )

20

-20

-20 -40

x2

30 Velocity( /s)

Angle( )

40

y’d

40

x1

60

0.3

-30

0.2

-40 by indirect AFC in [27] by indirect HAFC in [7] by proposed CHAFC

-60 0

5

10

0

30

4 2 0

30

-2

20

-4 29

29.5

30

0

-0.1 29

15 time(s)

25

10

0.1

-50

20

29.5

20

25

30

-10

30

0

5

10

15 time(s)

20

25

30

20

15

by indirect HAFC in [7] by proposed CHAFC

10 15

5 0

10 uh

-5 u

-10

5

-15 0

-20 -25

-5

-30 -35

-10 0

5

10

15 time(s)

20

25

30

0

5

10

15 time(s)

20

25

30

Fig. 4. Tracking trajectories of inverted pendulum. (a) Trajectories of angle tracking, (b) Trajectories of velocity tracking, (c) Comparison of tracking error e1, (d) Comparison of tracking error e2, (e) Overall control input u, (f) Comparison of HNcontrol term uh.

22

Y. Pan et al. / Neurocomputing 99 (2013) 15–24

external disturbance. For simulation, select mc ¼1 kg, ml ¼0.1 kg, lp ¼0.5 m and x(0) ¼[p/3,0]T. The control objective is to ensure that the output y tracks yd ¼ ðp=6ÞsinðtÞ. The design procedure of controller parameters is as follows: (1) for constructing the FLSs f^ and g^ in (15) and (16), respectively, select h^ f ¼ ½0,    ,0T and h^ g ¼ ½0,    ,0T , and design the membership functions of xi as follows:



mAli ðxi Þ ¼ expð xi ðp=6Þðli 3Þ 2 =ð2ð0:12Þ2 ÞÞ, i

where li ¼ 1,    ,5 and i¼1, 2; (2) for designing the stabilization controller in (22) with (28) and (29), let k¼[2, 1]T, gm ¼100 and sm ¼0.1; (3) for designing the tracking controller in (33) with (18), (42), and (43), let Q¼diag(10,10), r ¼0.1, r ¼2r2, gf ¼100, ge ¼30, Mf ¼10, Mg ¼1, and obtain

15 5 P¼ 5 5

Appendix A. Proof of Theorem 1 Differentiating (27) along (25) and using (26), one obtains _^ þ m ~ fm ~ gm ^_ g Þ=gm V_ S ¼ eT Q e=2 þ ðm f 

 

T þe Pb vf Df T ðxÞ þ vg DgðxÞ uS :

ðA:1Þ

Substituting (28) and (29) into (A.1) leads to



 ^ f þm ^g ~ f 9eT Pb9sm m ~ g 9eT PbuS 9sm m V_ S ¼ eT Q e=2 þ m 

 

^ f þ Df T ðxÞ þ sgnðeT Pbua Þm ^ g þ DgðxÞ uS eT Pb sgnðeT PbÞm U

^ f ðm ^ f mf Þ ¼ eT Q e=2eT PbðsgnðeT PbÞf Df T ðxÞÞsm m

 ^ g ðm ^ g mg Þ: eT PbuS sgnðeT PbuS Þg U DgðxÞ sm m From (A.1) and Assumption 3, one obtains ^ 2f m ^ f mf Þsm ðm ^ 2g m ^ g mg Þ: V_ S reT Q e=2sm ðm

from (19); (4) for designing the identification model in (38) with (40), select aF ¼10 and b ¼1; (5) for designing the overall control input in (46) with (47), choose Mx ¼0.8 and M0 ¼ 0.7. Simulated experiment design here is the same as that in Example 1 except that the sample time is set to be 1 ms. Simulation trajectories of inverted pendulum tracking are shown in Fig. 4. One observes that the proposed CHAFC also achieves favorable tracking performance (see Fig. 4(a) and (b)) with smooth control input (see Fig. 4(e)) and the best tracking accuracy (see Fig. 4(c) and (d)); the HN control term uh of the proposed CHAFC is much smaller than that of the indirect HAFC in [7] (see Fig. 4(f)). Performance comparisons of inverted pendulum tracking with various controllers are shown in Example 2 of Table 2. The qualitative analysis of these results is very similar with that in Example 1. The major difference is that the superiority of the proposed approach in Example 2 is more obvious than that in Example 1.

5. Conclusion This paper has successfully developed the CHAFC for a class of perturbed uncertain nonlinear systems. The overall control scheme is comprised of a series–parallel identification model, a robust stabilization controller and an indirect HAFC. Compared with the previous HAFCs, the novelties of this controller are as follows: (1) a robust stabilization controller is developed to obtain semi-global stability of the closed-loop system and boundness of the optimal FAE; (2) the composite adaptation is combined into HAFC design to solve the fuzzy approximation problem caused by the additional HN control term; (3) the advantages of the proposed approach are achieved at a negligible increase of computational complexity. Simulation studies of aircraft wing rock suppression and inverted pendulum tracking have demonstrated that the proposed approach not only effectively solves the aforementioned approximation problem, but also outperforms the classic indirect AFC and the indirect HAFC under external disturbances and parameter variations. Further study would focus on state observer-based output feedback design of the proposed controller.

ðA:2Þ

Since Q is a positive definite symmetric matrix, it can be decomposed as Q¼UTLU where U is a unitary matrix satisfying UTU ¼I, and L is a diagonal matrix containing the eigenvalues of Q [28]. Thus, one obtains 2

2

lmin ðQ Þ:e: reT Q er lmin ðQ Þ:e: =2:

ðA:3Þ

From (A.2) and (A.3), one obtains 2 V_ S rlmin ðQ Þ:e: =2 þ sm m2f =4 þ sm m2g =4, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which is negative for :e: 4 sm ðm2f þm2g Þ=2lmin ðQ Þ. Thus one has eALN and e converges to the compact set Oes defined in (30). According to the definition of e, one obtains xALN. Form the ^ f, m ^ g ALN. From s-modifications in (28) and (29), one obtains m Assumption 2, (11) and (12), one obtains f0, g0ALN. Combining with ydðnÞ A L1 , one knows that all terms in the right side of (22) are bounded. Thus, one obtains uSALN. Consequently, the closed-loop system is semi-globally stable [28] in the sense that all involving ^ f, m ^ g and uS are UUB. singles x, m

B. Proof of Lemma 1 Form Theorem 1, one obtains eALN. From Assumptions 4, (2), ^ N, :hnf : r Mf and :hng :r M g . (15) and (16), one obtains f^ , gAL From Assumption 2, (11) and (12), one obtains f0, g0ALN. Combining with (18) and ydðnÞ A L1 , one knows that all terms in the right side of (33) are bounded. Thus, one deduces uTALN. According to (32) and Assumption 3, one obtains U

9w9 rðf þ M f Þ þðg U þ M g Þumax , which implies that there must exist a finite constant wsup ¼ U ðf þM f Þ þðg U þM g Þumax A R þ such that 9w9 r wsup , 8x A D. Consequently, one obtains wALN, 8x A D. C. Proof of Theorem 2 Differentiating (41) along (36) and (39) yields T T V_ L ¼ eT ðPA þ AT PÞe=2 þeT Pbh~ f nðxÞ þ eT Pbh~ g nðxÞuT T

T

eT Pbðuh þ wÞge aF e2F þ ge eF h~ f nðxÞ þ ge eF h~ g nðxÞuT Acknowledgments The authors would like to thank reviewers for their insightful suggestions that have greatly improved the quality of this paper. This work is partially supported by the Singapore Agency for Science, Technology and Research (AnSTAR) Science and Engineering Research Council under Grant no.: 1122904016.

T_ T_ þ ge eF ðvwÞ þ h~ f h~ f =gf þ h~ g h~ g =gg :

ðC:1Þ

Applying (18), (19) and (40) to (C.1), and making some transformations, one obtains V_ L ¼ eT Q e=2ge aF e2F þ ðeT PbÞ2 =2r2 eT Pbw T T _ _ þ h~ f ððeT Pb þ ge eF ÞnðxÞ þ h^ f =gf Þ þ h~ g ððeT Pb þ ge eF ÞnðxÞuT þ h^ g =gg Þ:

Y. Pan et al. / Neurocomputing 99 (2013) 15–24

Substituting (42) and (43) into above expression, one obtains V_ L ¼ eT Q e=2ge aF e2F þ ðeT PbÞ2 =2r2 eT Pbw

2 ¼ eT Q e=2ge aF e2F  eT Pb=r þ rw =2þ r2 w2 =2: Thus, one has the following inequality: V_ L r eT Q e=2ge aF e2F þ r2 w2 =2: Noting e~ ¼ ½e; eF  and Q E ¼ diagðQ , ge aF =2Þ, one can change the above expression into T ~ þ r2 w2 =2: V_ L r e~ Q E e=2

ðC:2Þ

According to (C.2) and Lemma 2, one obtains T ~ þ r2 w2 =2: V_ L r e~ Q E e=2

ðC:3Þ

According to the same proof as in [26], one obtains V_ L r kmin V þ kmin V r , where kmin ¼ lmin ðQ E Þ=lmax ðPÞ and V r A R þ is a finite constant. Therefore, one obtains V L ðtÞ rV L ð0Þ þV r ,8t Z0, i.e., VLALN, which implies e, eF, x, h^ f , h^ g ALN. Now, uTALN can be directly obtained by (33). Consequently, the closed-loop system is stable in the sense that all involving signals are UUB. Integrating (C.2) from t ¼0 to t ¼T yields Z Z T 1 T T 1 ~ þ r2 e~ Q E edt VðTÞVð0Þ r  w2 dt, 2 0 2 0 where TA[0,N). After simple manipulation, above expression can be written into (10). Moreover, from (C.3), one obtains 2 V_ L r lmin ðQ Þ:e: =2 þ r2 w2 =2:

which is negative for :e: 4 rwðlmin ðQ ÞÞ1=2 . Thus, e converges to the compact set Oet defined in (44). Similarly, one can obtain eF converges to the compact set Oe defined in (45). References [1] L.X. Wang, Stable adaptive fuzzy control of nonlinear systems, IEEE Trans. Fuzzy Syst. 1 (1993) 146–155. [2] J. Zhou, M.J. Er, J.M. Zurada, Adaptive neural network control of uncertain nonlinear systems with nonsmooth actuator nonlinearities, Neurocomputing 70 (2007) 1062–1070. [3] Y. Pan, M.J. Er, D. Huang, et al., Adaptive fuzzy control with guaranteed convergence of optimal approximation error, IEEE Trans. Fuzzy Syst. 19 (2011) 807–818. [4] Y. Pan, M.J. Er, T. Sun, Composite adaptive fuzzy control for synchronization of generalized Lorenz systems, Chaos 22 (2012) 023144. [5] Y. Pan, M.J. Er, D. Huang, et al., Fire-rule-based direct adaptive type-2 fuzzy HN tracking control, Eng. Appl. Artif. Intell. 24 (2011) 1174–1185. [6] S. Tong, B. Chen, Y. Wang, Fuzzy adaptive output feedback control for MIMO nonlinear systems, Fuzzy Sets Syst. 156 (2005) 285–299. [7] B.S. Chen, C.H. Lee, Y.C. Chang, HN tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach, IEEE Trans. Fuzzy Syst. 4 (1996) 32–43. [8] Y.C. Chang, Adaptive fuzzy-based tracking control for nonlinear SISO systems via VSS and HN approaches, IEEE Trans. Fuzzy Syst. 9 (2001) 278–292. [9] S. Tong, H.X. Li, Direct adaptive fuzzy output tracking control of nonlinear systems, Fuzzy Sets Syst. 128 (2002) 107–115. [10] A. Rubaai, A.R. Ofoli, D. Cobbinah, DSP-based real-time implementation of a hybrid HN adaptive fuzzy tracking controller for servo-motor drives, IEEE Trans. Ind. Appl. 43 (2007) 476–484. [11] C.S. Chen, Quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems, Expert Syst. Appl. 36 (2009) 11827–11835. [12] C.K. Lin, HN reinforcement learning control of robot manipulators using fuzzy wavelet networks, Fuzzy Sets Syst. 160 (2009) 1765–1786. [13] A. Hamzaoui, N. Essounbouli, K. Benmahammed, et al., State observer based robust adaptive fuzzy controller for nonlinear uncertain and perturbed systems, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 34 (2004) 942–950. [14] G.G. Rigatos, Adaptive fuzzy control with output feedback for HN tracking of SISO nonlinear systems, Int. J. Neural Syst. 18 (2008) 305–320. [15] G.G. Rigatos, Adaptive fuzzy control of DC motors using state and output feedback, Electric Power Syst. Res. 79 (2009) 1579–1592.

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[16] W.Y. Wang, M.L. Chan, C.C.J. Hsu, et al., HN tracking-based sliding mode control for uncertain nonlinear systems via an adaptive fuzzy-neural approach, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 32 (2002) 483–492. [17] C.C. Kung, T.H. Chen, HN tracking-based adaptive fuzzy sliding mode controller design for nonlinear systems, IET Control Theory Appl. 1 (2007) 82–89. [18] S. Salehi, M. Shahrokhi, Adaptive fuzzy approach for HN temperature tracking control of continuous stirred tank reactors, Control Eng. Practice 16 (2008) 1101–1108. [19] M. Hojati, S. Gazor, Hybrid adaptive fuzzy identification and control of nonlinear systems, IEEE Trans. Fuzzy Syst. 10 (2002) 198–210. [20] L.X. Wang, Design and analysis of fuzzy identifiers of nonlinear dynamic systems,, IEEE Trans. Autom. Control 40 (1995) 11–23. [21] D. Bellomo, D. Naso, B. Turchiano, et al., Composite adaptive fuzzy control, in: Proc. 16th IFAC World Congress, Prague, Czech Republic, 2005. [22] D. Bellomo, D. Naso, R. Babuˇska, Adaptive fuzzy control of a non-linear servodrive: theory and experimental results, Eng. Appl. Artif. Intell. 21 (2008) 846–857. [23] D. Naso, F. Cupertino, B. Turchiano, Precise position control of tubular linear motors with neural networks and composite learning, Control Eng. Practice 18 (2010) 515–522. [24] H. Du, S.S. Ge, J.K. Liu, Adaptive neural network output feedback control for a class of non-affine non-linear systems with unmodelled dynamics, IET Control Theory Appl. 5 (2011) 465–477. [25] S. Bououden, D. Boutat, G. Zheng, et al., A triangular canonical form for a class of 0-flat nonlinear systems, Int. J. Control 84 (2011) 261–269. [26] P.A. Phan, T. Gale, Two-mode adaptive fuzzy control with approximation error estimator, IEEE Trans. Fuzzy Syst. 15 (2007) 943–955. ˜ ez, et al., Stable Adaptive Control and [27] J.T. Spooner, M. Maggiore, R. Ordo´n Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques, John Wiley & Sons, Inc., New York, NY, 2002. [28] S.S. Ge, C.C. Hang, T.H. Lee, et al., Stable Adaptive Neural Network Control, Kluwer, Boston, MA, 2001. [29] J.A. Farrell, M.M. Polycarpou, Adaptive Approximation based Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches, John Wiley & Sons, Inc., Hoboken, NJ, 2006. [30] C.C. Kung, T.H. Chen, Observer-based indirect adaptive fuzzy sliding mode control with state variable filters for unknown nonlinear dynamical systems, Fuzzy Sets Syst. 155 (2005) 292–308. [31] D. Kim, H. Chung, S. Bhasin, et al., Robust composite adaptive fuzzy identification and control for a class of MIMO nonlinear systems, in: Proc. American Control Conf., San Francisco, CA, 2011, pp. 4947–4952. [32] S.F. Su, J.C. Chang, S.S. Chen, The study on direct adaptive fuzzy controllers, Int. J. Fuzzy Syst. 8 (2006) 150–159. [33] H.N. Nounou, K.M. Passino, Stable auto-tuning of hybrid adaptive fuzzy neural controllers for nonlinear systems, Eng. Appl. Artif. Intell. 18 (2005) 317–334. [34] S.V. Joshi, A.G. Sreenatha, J. Chandrasekhar, Suppression of wing rock of slender delta wings using a single neuron controller, IEEE Trans. Control Syst. Technol. 6 (1998) 671–677. [35] Z.L. Liu, Reinforcement adaptive fuzzy control of wing rock phenomena, IEE Proc. Control Theory Appl. 152 (2005) 615–620.

Yongping Pan received the B.Eng. degree in automation and M.Eng. degree in control theory and control engineering from the Guangdong University of Technology, Guangzhou, China, in 2004 and 2007, respectively, and the Ph.D. degree in control theory and control engineering from the South China University of Technology, Guangzhou, in 2011. From 2007 to 2008, he was an R&D Engineer with the Santak Electronic (Shenzhen) Co., Ltd., Eaton Co. He is currently a Research Fellow of the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has published more than 20 papers in some international journals and conferences such as IEEE Transactions on Fuzzy Systems, Chaos and Neurocomputing. He also serves as the Reviewer for several international journals. His research interests include approximation-based adaptive control, fuzzy systems and neural networks, reinforcement learning, and control system design.

Yu Zhou received his B.Eng. degree in automation from the Henan University, Kaifeng, China in 2004, the M.Eng. degree in detection technology and automatic equipment from the Guangdong University of Technology, Guangzhou, China, in 2007, and the Ph.D. degree in control theory and control engineering from the China University of Mining and Technology (Beijing), Beijing, China, in 2010. Now he is a Lecturer at the School of Electric Power, North China Institute of Water Conservancy and Hydroelectric Power, Zhengzhou, China. His current research interests include neural networks, pattern recognition, intelligence computing, and intelligent control.

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Y. Pan et al. / Neurocomputing 99 (2013) 15–24 Tairen Sun received his M.S. degree in operations research and cybernetics from the Sun Yat-Sen University, Guangzhou, China, in 2008, and the Ph.D. degree in control theory and control engineering from the South China University of Technology, Guangzhou, in 2011. He is currently a Lecturer with the School of Electronic and Information Engineering, Jiangsu University, Zhenjiang, China. His main research interests include robot control, intelligent control, vision-based control, etc.

Meng Joo Er received the B.Eng. and M.Eng. degrees in electrical engineering from the National University of Singapore, Singapore, in 1985 and 1988, respectively, and the Ph.D. degree in systems engineering from the Australian National University, Canberra, Australia, in 1992. He is currently a Full Professor with the School of Electrical and Electronic Engineering, Nanyang Technological University (NTU), Singapore, and the Director of the Renaissance Engineering Program, College of Engineering, NTU. He has authored five books, 16 book chapters, and more than 400 journal and conference papers. His research interests control theory and applications, fuzzy logic and neural networks, computational intelligence, cognitive systems, robotics and automation, sensor networks and biomedical engineering.