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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2016; 26:3921–3936Published online 21 March 2016 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.3541

Composite adaptive dynamic surface control using onlinerecorded data

Yongping Pan1, Tairen Sun2 and Haoyong Yu1,*,†

1Department of Biomedical Engineering, National University of Singapore, Singapore 117583, Singapore2School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China

SUMMARY

This paper presents an online recorded data-based design of composite adaptive dynamic surface control fora class of uncertain parameter strict-feedback nonlinear systems, where both tracking errors and predictionerrors are applied to update parametric estimates. Differing from the traditional composite adaptation thatutilizes identification models and linear filters to generate filtered modeling errors as prediction errors, theproposed composite adaptation integrates closed-loop tracking error equations in a moving time window togenerate modified modeling errors as prediction errors. The time-interval integral operation takes full advan-tage of online recorded data to improve parameter convergence such that the application of both identificationmodels and linear filters is not necessary. Semiglobal practical asymptotic stability of the closed-loop sys-tem is rigorously established by the time-scales separation and Lyapunov synthesis. The major contributionof this study is that composite adaptation based on online recorded data is achieved at the presence of mis-matched uncertainties. Simulation results have been provided to verify the effectiveness and superiority ofthis approach. Copyright © 2016 John Wiley & Sons, Ltd.

Received 9 October 2014; Revised 28 October 2015; Accepted 14 February 2016

KEY WORDS: composite adaptation; dynamic surface control; mismatched uncertainty; strict-feedbacknonlinear system; adaptive control; data-driven control

1. INTRODUCTION

Adaptive control for nonlinear systems with parametric uncertainties had been well established inthe early 1980s [1]. Generally, adaptive control includes two philosophically different schemes,namely, a direct scheme that utilizes tracking errors to update control parameters directly and anindirect scheme that utilizes prediction errors to generate parameter estimations to be used in thecertainty equivalent control law [2]. In the conventional adaptive control, parameter convergencewould be slow even the system satisfies a strong condition named persistent excitation (PE) [1].Composite adaptive control (CAC) is an integrated direct and indirect adaptive control strategythat aims to achieve higher tracking accuracy and better parameter convergence through faster andsmoother parameter adaptation. Seminal works of CAC can be referred to [2–5].

As both tracking errors and prediction errors are applied to update parametric estimates, CACshows an attractive property of obvious performance improvement in many applications, includingrobotic arms [2, 6–15], motor drivers [16–18], aircrafts [19–24], underwater vehicles [25], chemicalprocesses [26, 27], and chaotic systems [28]. Yet, only matched uncertainties are considered inthese results. There exist only a few CAC approaches that consider mismatched uncertainties sofar [29–31]. In [29], a composite adaptive integrator backstepping control (AIBC) approach wasdeveloped for a class of strict-feedback nonlinear systems (SFNSs) with mismatched parametricuncertainties. The major disadvantage of integrator backstepping is the ‘explosion of complexity’

*Correspondence to: Haoyong Yu, Department of Biomedical Engineering, National University of Singapore, Block E4,#04-08, 4 Engineering Drive 3, Singapore 117583, Singapore.

†E-mail: [email protected]

Copyright © 2016 John Wiley & Sons, Ltd.

3922 Y. PAN, T. SUN AND H. YU

resulted from the repeated derivations of virtual control inputs [32–34]. In [30], a robust compositeadaptive dynamics surface control (ADSC) approach was developed to eliminate the drawbacks ofintegrator backstepping, where each virtual control input is passed through a first-order filter at itscorresponding backstepping step so that the derivations of virtual control inputs can be avoided. In[31], a composite ADSC approach based on neural networks was further developed for a class ofSFNSs with mismatched functional uncertainties.

Adaptive control with Q-modification is an emerging alternative CAC approach [35]. The keyissue in the composite adaptation is in how prediction errors are constructed without the derivativesof system states. Differing from the conventional CAC that utilizes identification models and linearfilters to generate filtered modeling errors as prediction errors, the Q-modification-based CAC inte-grates plant dynamics in a moving time window to generate modified modeling errors as predictionerrors. The time-interval integral takes full advantage of online recorded data to improve parameterconvergence. Note that mismatched uncertainties are still not considered in [35]. A Q-modification-based composite AIBC with exact differentiators was developed for a class of SFNSs in [36]. Themajor problem of the approach in [36] lies in the selection difficulty of many parameters in exactdifferentiators. Inspired by the work [35], this paper presents an online recorded data-based designof composite ADSC for a class of SFNSs, where mismatched parametric uncertainties are han-dled by the ADSC technique. Semiglobal practical asymptotic stability of the closed-loop systemis rigourously established by the time-scales separation and Lyapunov synthesis. Compared withthe innovative composite ADSC approaches in [30, 31], the distinctive features of the proposedapproach are shown as follows: (i) composite adaptation is achieved by using online recorded datawithout the usage of both identification models and linear filters and (ii) parametric uncertaintiesand external perturbations can be suppressed by the time-interval integral.

The rest of this paper is organized as follows: The problem under consideration is formulatedin Section 2; the composite ADSC design is provided in Section 3; simulation studies are carriedout in Section 4; conclusions are provided in Section 5. The notations of this paper are relativelystandard, where R, RC, and Rn denote the spaces of real numbers, positive real numbers, and realn-vectors, respectively; k �k denotes the Euclidean-norm; L1 denotes the space of bounded signals;�r WD ¹xjkxk 6 rº denotes the ball of radius r ; min¹�º and max¹�º denote the minimum andmaximum functions, respectively; col.x; z/ WD

�xT ; zT

�T; and Ck represents the space of functions

whose k-order derivatives all exist and are continuous, in which r 2 RC is a constant, x; z 2 Rn

are variable vectors, and n, m, and k are positive integers. For the sake of brevity, in the subsequentsections, the arguments of a function may be omitted while the context is sufficiently explicit.

2. PROBLEM FORMULATION

Consider a class of nth order generalized uncertain parameter SFNSs as follows [37]:8̂<:̂Pxi D fi .xi /C gi .xi /xiC1 CˆTi .xi /�

.i D 1; 2; � � � ; n � 1/

Pxn D fn.x/C gn.x/uCˆTn .x/�

(1)

where u.t/ 2 R and x1.t/ 2 R are the control input and the controlled output, respectively, xi .t/ WDŒx1.t/; x2.t/, � � � ; xi .t/�T 2 Ri .xn.t/ D x.t// are state vectors, fi .xi / W Ri 7! R, gi .xi / W Ri 7! Rand ˆi .xi / W Ri 7! RN are known functions, � 2 �ca is a vector of unknown constant parametersregarded as a mismatched parametric uncertainty [33],N is the number of parameters, ca 2 RC is aknown constant, and i D 1, 2, � � � , n. Let xd .t/ 2 R denote a desired output. The following generalassumptions in [37] are exploited for ADSC synthesis.

Assumption 1There is a constant g0 2 RC so that gi .xi / > g0, 8x 2 Rn for i D 1; 2; � � � ; n.

Assumption 2fi .xi /, gi .xi /, and ˆi .xi / are of C1 for i D 1; 2; � � � ; n.

Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2016; 26:3921–3936DOI: 10.1002/rnc

COMPOSITE ADAPTIVE DYNAMIC SURFACE CONTROL 3923

Assumption 3xd .t/ and Pxd .t/ are continuous and of L1.

Let ˛i .t/ 2 R and ˛ci .t/ 2 R be virtual control inputs and their filtered counterparts, respectively,where i D 1; 2; � � � ; n � 1. Define tracking errors ei .t/ WD xi .t/ � ˛

ci�1.t/ for i D 1, 2, � � � , n,

where ˛c0.t/ D xd .t/. Let e.t/ WD Œe1.t/; e2.t/; � � � , en.t/�T . The control objective of this study isto design an appropriate control law such that the closed-loop system is stable in the sense that thecontrolled output x1 tracks the desired output xd as fast and accurate as possible.

Remark 1This study only considers the class of parameter SFNSs (1) with gi .xi / > 0 for i D 1 to n. How-ever, by exploiting the function approximation property of neural networks or fuzzy systems [22,31, 38–40], it is straightforward to extend the proposed approach to a class of SFNSs with mis-matched functional uncertainties. In addition, as shown in [22, Sec. 2.2], a general class of affinenonlinear systems can be transformed into a chain-integral Brunovsky canonical form under certainconditions. Therefore, the proposed approach is also applicable to a general class of affine nonlin-ear systems. Moreover, according to Reference [40, Remark 1], the following results can be easilymodified to the system (1) with gi .xi / < 0 for i D 1 to n.

3. COMPOSITE ADAPTIVE CONTROL DESIGN

3.1. Dynamic surface control design

Let O� 2 RN be an estimate of � , Q� WD � � O� be an estimation error, ˆ.x/ WDPniD1ˆi .xi / be a

regressor, Q̨ i .t/ WD ˛ci .t/� ˛i .t/ with i D 1 to n be filtering delay errors, and�ce0 ,�cx0 , and�cdbe compact sets for e.0/, x.0/, and xd .t/, respectively, where xd .t/ WD Œxd .t/; Pxd .t/�

T , and ce0,cx0, cd 2 RC are known constants. The derivation of the ADSC law is shown as follows [32].

Step 1: The derivative of e1 is expressed as follows:

Pe1 D f1.x1/C g1.x1/x2 CˆT1 .x1/� � Pxd : (2)

Choose the virtue control input ˛1 to be

˛1 D1

g1.x1/

��k1e1 C Pxd � f1.x1/ �ˆ

T1 .x1/ O�

�(3)

where k1 2 RC is a control gain. Let ˛1 pass through a first-order filter so that

� P̨ c1 D �˛c1 C ˛1; ˛

c1.0/ D ˛1.0/

where � 2 .0; 1/ is a filtering parameter. Subtracting and adding g1˛1C g1˛c1 at the right sideof (2) and applying (3) to the resulting expression, one obtains

Pe1 D �k1e1 C g1.x1/e2 C g1.x1/ Q̨1 CˆT1 .x1/ Q�: (4)

Step 2: The derivative of e2 is expressed as follows:

Pe2 D f2.x2/C g2.x2/x3 CˆT2 .x2/� � P̨c1: (5)

Choose the virtue control input ˛2 to be

˛2 D1

g2.x2/

��k2e2 � g1.x1/e1 C P̨ c1 � f2.x2/ �ˆ

T2 .x2/ O�

�(6)

where k2 2 RC is a control gain. Let ˛2 pass through a first-order filter so that

� P̨ c2 D �˛c2 C ˛2; ˛

c2.0/ D ˛2.0/:



Like Step 1, applying (6) to (5) and making some transformations lead to

Pe2 D �k2e2 C g2.x2/e3 � g1.x1/e1 C g2.x2/ Q̨2 CˆT2 .x2/ Q�: (7)

Step i (2 6 i 6 n � 1): Similarly, the derivative of ei is expressed as follows:

Pei D fi .xi /C gi .xi /xiC1 CˆTi .xi /� � P̨ci�1: (8)

Choose the virtue control input ˛i to be

˛i D1

gi .xi /

��kiei � gi�1.xi�1/ei�1 C P̨ ci�1 � fi .xi / �ˆ

Ti .xi / O�

�(9)

where ki 2 RC is a control gain. Let ˛i pass through a first-order filter so that

� P̨ ci D �˛ci C ˛i ; ˛

ci .0/ D ˛i .0/:

Applying (9) to (8) and making some transformations lead to

Pei D �kiei C gi .xi /eiC1 � gi�1.xi�1/ei�1 C gi .xi / Q̨ i CˆTi .xi / Q�: (10)

Step n: The derivative of en is expressed as follows:

Pen D fn.x/C gn.x/uCˆTn .x/� � P̨cn�1: (11)

Choosing an actual control input u to be

u D1

gn.x/

��knen � gn�1.xn�1/en�1 C P̨ cn�1 � fn.x/ �ˆ

Tn .x/ O�

�(12)

where kn 2 RC is a control gain. Substituting (12) to (11), one obtains

Pen D �knen � gn�1.xn�1/en�1 CˆTn .x/ Q�: (13)

By the collection of (3), (6), (9), and (12), the complete ADSC law is given by8̂̂ˆ̂̂̂<̂ˆ̂̂̂ˆ̂̂:

˛1 D1g1

��k1e1 C Pxd � f1 �ˆ

T1O��

˛i D1gi

��kiei � gi�1ei�1 C P̨

ci�1 � fi �ˆ

TiO��

.i D 2; 3; � � � ; n � 1/

u D 1gn

��knen � gn�1en�1 C P̨

cn�1 � fn �ˆ

TnO��

(14)

where P̨ ci with i D 1 to n � 1 are obtained by

� P̨ ci D �˛ci C ˛i ; ˛

ci .0/ D ˛i .0/: (15)

The following lemmas are introduced to demonstrate two properties of the entire system.

Lemma 1[37]�: Consider the system (1) with Assumptions 1–3 driven by the control law (14) with (15). Givenany x.0/ 2 �cx0 with cx0 2 RC, there exist certain constants cx > cx0 and Tf > 0 such that thesolution x.t/ of (1) satisfies x.t/ 2 �cx , 8t 2 Œ0; Tf /.

Lemma 2[41]: Consider the filter (15) at t 2 Œ0; Tf / with Assumptions 1–3 and the virtual control inputs˛i in (14), where i D 1 to n � 1. For any given � 2 RC, there exists a sufficiently small filteringparameter � in (15) such that j Q̨ i .t/j 6 �, 8t 2 Œ0; Tf /.

†The condition of the control input u being saturated in [37] is not presented here because the boundedness of u givenby (14) is naturally guaranteed by the boundedness of the state vector x.



Remark 2The difference between AIBC and ADSC lies in how the derivatives of ˛1 to ˛n�1 are obtained inthe control law (14). In the AIBC, the analytical expressions of ˛1 to ˛n�1 in (14) are utilized tocalculate P̨1 to P̨n�1, resulting in the repeated derivations of ˛1 to ˛n�1 such that the control lawderived becomes extremely complicated as the increase of the plant order n. On the contrary, in theproposed composite ADSC, the filter (15) is utilized to obtain P̨ c1 to P̨ cn�1, that is, the estimates ofP̨1 to P̨n�1, such that the derivations of ˛1 to ˛n�1 can be greatly simplified. The comprehensivecomparison between AIBC and ADSC can be referred to [32].

3.2. Composite adaptation structure

It is implied from Lemmas 1 and 2 that for a given � 2 RC, the filtering parameter � in (15) can beproperly designed such that the signals ˛ci of the fast system (15) achieves j Q̨ i j 6 �, 8t 2 Œ0; Tf /and all signals of the slow system (1) remain bounded, 8t 2 Œ0; Tf / [37]. By the collection of (4),(7), (10), and (13), the closed-loop tracking error dynamics is expressed as follows:

8̂̂<̂ˆ̂̂:

Pe1 D �k1e1 C g1e2 C g1 Q̨1 CˆT1Q�

Pei D �kiei C gieiC1 � gi�1ei�1 C gi Q̨ i CˆTiQ�

.i D 2; 3; � � � ; n � 1/

Pen D �knen � gn�1en�1 CˆTnQ�

: (16)

For the traditional composite ADSC approaches of [30, 31], n prediction errors "i .t/ are definedas "i .t/ WD xi .t/ � Oxi .t/ with i D 1 to n, where Oxi .t/ 2 R, estimates of xi .t/, are generated by nserial-parallel identification models with linear filters as follows:

8̂<:̂POxi D ˛i"i C fi .xi /C gi .xi /xiC1 CˆTi .xi / O�

.i D 1; 2; � � � ; n � 1/

POxn D ˛n"n C fn.x/C gn.x/uCˆTn .x/ O�

: (17)

Subtracting (1) by (17) leads to the prediction error dynamics as follows:

P"i D �˛i"i CˆTi .xi / Q�; i D 1; 2; � � � ; n (18)

which implies that "i represents a filtered counterpart of the modeling error ˆTi Q� .Differently in this study, the equalities in (16) are integrated in a moving time window to construct

a modified modeling error ".t/ WD qT .t/ Q�.t/ as a prediction error [35], where

q.t/ WD

Z t

t��d

ˆ.x.�//d� (19)

with �d 2 RC being an integral interval. It is worth noting that the definition of q to be an integraltype in (19) is useful for avoiding the use of the immeasurable Pe1 to Pen for the calculation of theprediction error " as shown in the subsequent content.

Let O� WD Œ O�1; O�2; � � � ; O�N �T . The adaptive law of O� is designed as follows:

PO� D �P�Xn

iD1ˆi .xi /ei C kwq"

�(20)

in which � 2 RC is a learning rate, kw 2 RC is a weight factor, and P.�/ D ŒP.�1/, P.�2/, � � � ,P.�N /�T is a projection operator in [42] given by

P.�j / WD

8̂<:̂0 if O�j D �ca and �j < 0

0 if O�j D ca and �j > 0

�j otherwise

:



To derive the computational formula of ", define

ci .t/ WD

Z t

t��d

ˆTi .xi .�//�d� (21)

for i D 1; 2; � � � ; n. Integrating all equalities in (16) at Œt ��d ; t �, one obtains8̂̂ˆ̂̂̂<̂ˆ̂̂̂̂ˆ̂:

c1.t/ D e1.t/ � e1.t � �d /CR tt��d

�k1e1 � g1e2 � g1 Q̨1 Cˆ

T1O��d�

ci .t/ D ei .t/ � ei .t � �d /CR tt��d

�kiei � gieiC1 C gi�1ei�1 � gi Q̨ i Cˆ

TiO��d�

.i D 2; 3; � � � ; n � 1/

cn.t/ D en.t/ � en.t � �d /CR tt��d

�knen C gn�1en�1 Cˆ

TnO��d�

where the time variable � is omitted in the integrals of the foregoing expressions. Thus, notingqT � D

PniD1 ci from (21), the prediction error " in (20) is calculated as follows:

".t/ DXn

iD1ci .t/ � q

T .t/ O�.t/: (22)

Remark 3It is observed that both the prediction errors "1 to "n by the traditional composite ADSC approachesof [30, 31] and the prediction error " by the proposed approach have a similar filtering feature. Thedifferences between the traditional approaches and the proposed approach are as follows: (i) n pre-diction errors "1 to "n are defined for the traditional approaches, whereas only one prediction error" is defined for the proposed approach, and (ii) the identification models (17) are applied to con-struct "1 to "n for the traditional approaches, whereas the integral on the time interval Œt � �d ; t � isimplemented to construct " for the proposed approach. Compared with the traditional approaches,the proposed approach possesses some advantages as follows: (a) the application of both identifica-tion models and linear filters is not necessary, resulting in a simpler control structure, and (b) thetime-interval integral is useful in suppressing parametric uncertainties and external perturbations asshown in [35, Sec. II]. The advantages of the proposed approach are obtained at the costs of addi-tional computational burden and memory storage usage that resulted from the integrals on a movingtime window. Yet, these costs are negligible for contemporary control units.

3.3. Stability and convergence analysis

Define �ce WD ¹ejx 2 �cx ; xd 2 �cd º, ks WD min16i6n¹kiº and Ng WD maxx2�cx ;16i6n�1¹gi .xi /º,where ce 2 RC is a constant. The following theorem shows the major result of this study.

Theorem 1Consider the system (1) with Assumptions 1–3 driven by the control law (14) with (15) and (20).If x.0/ 2 �cx0 and O�.0/ 2 �ca , then there exist suitably large control parameters ks in (14), 1=�in (15), and � in (20) so that the closed-loop system has a semiglobal practical asymptotic stabilityin the sense that (i) the unique solution of the closed-loop system is defined 8t 2 Œ0;1/; (ii) theintegrated squared errors of e and " are bounded by (27) and (28), respectively; and (iii) e convergesto a small neighborhood of zero dominated by � , �, and ks .

ProofIt is derived from Lemmas 1 and 2 that for a given � 2 RC and x.0/ 2 �cx0 , there exists a suitablysmall � in (15) such that j Q̨ i .t/j 6 �, 8t 2 Œ0; Tf / and e.t/ 2 �ce , 8t 2 Œ0; Tf /, where i D 1, 2,� � � , n � 1. Choose a Lyapunov function candidate

V.z/ D eT e=2C Q�T Q�=.2�/ (23)

with z WD col�e; Q�

�2 RnCN for the closed-loop system composed of (16) and (20). Differentiating

V in (23) along (16) and (20) with respect to time t, one obtains



PV D�Xn

iD1kie

2i � kw"

2 CXn�1

iD1gi .xi / Q̨ iei

C Q�T�Xn

iD1ˆi .xi /ei �

PO�=��:

According to the projection modification results in [42], if O�.0/ 2 �ca , then the adaptive law (20)

guarantees that (i) O�.t/ 2 �ca , 8t > 0 and (ii) Q�T�Pn

iD1ˆi .xi /ei �PO�=�

�6 0. Applying the

second result to the foregoing equality and noting j Q̨ i j 6 �, one obtains

PV 6 �Xn

iD1kie

2i � kw"

2 C � NgXn�1

iD1jei j

6 �kskek2=2 � kw"2 � ksXn�1

iD1

�e2i � 2� Ngjei j=ks

�=2:

Applying Young’s inequality 2ab � a2 6 b2 with a; b 2 R to the last term of the foregoing ex-pression and defining .�; ks/ WD .n � 1/.� Ng/2=.2ks/, one obtains

PV 6 �kskek2=2 � kw"2 C .�; ks/ (24)

which is valid on e 2 �ce and t 2 Œ0; Tf /.Applying (23) and O�.t/ 2 �ca to (24), one obtains

PV .t/ 6 �ksV.t/C 2ksc2a=� C .�; ks/6 �ksV.t/=2 � ks.V .t/ � .�; �; ks//=2

(25)

with .�; �; ks/ WD .n � 1/.� Ng=ks/2 C 4c2a=� 2 RC, which implies

PV .t/ 6 �ksV.t/=2; 8V.t/ > ;8t 2 Œ0; Tf /:

The conditions e.t/ 2 �ce and O�.t/ 2 �ca are applied to determine a Lyapunov surface V.z/ D ı

with ı WD c2e =2C 2c2a=� 2 RC. Then, if the selection of � , �, and ks satisfies

ce >p2.n � 1/� Ng=ks C 2ca=

p� (26)

such that ı > , then PV .t/ < 0 at ¹V.t/ D º \�ca . It is clear that for any given ce 2 RC, therecertainly exist suitably large � , 1=�, and ks such that (26) holds. Under the condition (26), the set¹V.t/ 6 º \ �ca is positively invariant such that the trajectory of .e.t/; O�.t// that started from¹V.t/ 6 ıº \�ca enters ¹V.t/ 6 º \�ca in finite time and stays inside ¹V.t/ 6 º \�ca for allfuture time. Therefore, one obtains e.t/ 2 �ce and O�.t/ 2 �ca , 8t 2 Œ0;1/, implying Tf D 1.Because e.t/ 2 �ce and O�.t/ 2 �ca , 8t 2 Œ0;1/, all other closed-loop signals are also bounded,8t 2 Œ0;1/. Therefore, according to [43, Th. 3.3], the unique solution .e.t/; O�.t// of the closed-loop dynamics composed of (16) and (20) is defined for all t 2 Œ0;1/.

Integrating (24) at Œ0; t � and after some transformations, one obtains

Z t

0

ke.�/k2d� 6 1

ks

�V.0/C

Z t

0

.�/d�

; (27)

Z t

0

"2.�/d� 6 1

kw

�V.0/C

Z t

0

.�/d�

: (28)

Under the condition (26), the result of [44, Lemma A.3.2] is applied to (25) to obtain

V.t/ 6 .V .0/ � / exp.�kst /C ;8t > 0:



Applying (23) to the aforementioned result, one also obtains

ke.t/k 6p2V.0/ exp .�kst=2/C

p2;8t > 0

which implies that e converges to ap2-neighborhood of zero. Because ce in (26) can be arbitrarily

enlarged to include all possible e.0/ and can be arbitrary diminished both by the increase of � ,1=�, and ks , the stability result is semiglobally practically asymptotic [45]. �

Figure 1. Simulation trajectories of the two controllers in Example 1 without measurement noise: (a) controltrajectories by the ADSC, (b) adaptation trajectories by the ADSC, (c) control trajectories by the composite

ADSC, and (d) adaptation trajectories by the composite ADSC.

Figure 2. A comparison of system errors for the two controllers in Example 1 without measurement noise:(a) tracking errors and (b) prediction errors.



Remark 4The prediction error " in (24) provides an extra effort for the decrease of the Lyapunov functionV in (23) so that it is possible to obtain higher tracking accuracy related to e and better parameterconvergence related to Q� . Without the term kwq" in the adaptive law (20), parameter convergencewould be slow even if the input signals x make ˆ.x/ satisfy the PE condition. This can also beverified by simulation results in Section 4.2.

Figure 3. Simulation trajectories of the two controllers in Example 1 with measurement noise: (a) controltrajectories by the ADSC, (b) adaptation trajectories by the ADSC, (c) control trajectories by the composite

ADSC, and (d) adaptation trajectories by the composite ADSC.

Figure 4. A comparison of system errors for the two controllers in Example 1 with measurement noise: (a)tracking errors and (b) prediction errors.



Remark 5Because all elements in the adaptive law (20) are updated in real time, the proposed ADSC law(16) is naturally robust against slow parameter variations as the conventional adaptive control. Inaddition, if an external perturbation di .t/ 2 R exists in the ith equation of the system (1) for i D1 to n, then in (25) becomes a positively correlated function of di .t/, resulting in the increasedamplitude of . However, according to the expression of (25), the influence of di .t/ on the systemperformance can be suppressed by the increase of the control parameters k1 to kn and � , and thestability analysis can keep the same as that in the proof of Theorem 1.

Remark 6The parameter selection of our approach can follow the following rules: (i) increasing the controlgains k1 to kn in (14) can improve tracking speed and accuracy, but k1 to kn that are too large maymagnify measurement noises and cause actuator saturation resulting in performance degradation,and consequently, the selection of k1 to kn should be based on the actuator limitation and noiselevel; (ii) increasing the filtering parameter � in (15) can improve tracking accuracy at the cost ofthe increased noise sensitivity for the calculation of P̨ c1 to P̨ cn�1, and consequently, the selection of� should be based on the noise level; (iii) increasing the integral interval �d in (19) can make fulluse of online recorded data to improve control performance at the cost of increased memory storageusage; and (iv) increasing the learning rate � and the weight factor kw in (20) can increase theconvergence rates of both the tracking error e and the prediction error ", but � that is too large maylead to serious oscillations at the control input u.

4. ILLUSTRATIVE EXAMPLES

4.1. Example 1: aircraft wing rock

Consider an aircraft wing rock model with actuator dynamics as follows [20]:8<:Px1 D x2Px2 D b1x3 C �1x1 C �2x2 C �3x

22 C �4x

21x2 C �5x1x

22

Px3 D �x3=b2 C .1=ˇ2/u

(29)

where x1 is the aircraft roll angle (rad), x2 is the roll rate (rad/s), x3 is the actuator output (N), u is theactuator input (V), b1 is an actuator gain, b2 is an aileron time constant, and �1 to �5 are certain coef-ficients. Noting (1), one obtains f1 D f2 D 0, f3 D �x3=b2, g1 D 1, g2 D b1, g3 D 1=b2, � DŒ�1; �2; �3; �4; �5�

T , and ˆ.x/ D ˆ2.x2/ D Œx1; x2; x32 ; x

21x2; x1x

22 �T . For simulations, set x.0/ D

Œ0:4; 0; 0�T , b1 D 1:5, b2 D 1=15, and � D Œ0:2314; 0:6918;�0:6245; 0:0095; 0:0214�T [20]. The

Table I. Performance indexes comparisons for various controllers.

Example Case Controller type ITAE(e1) IAE(e1) ITAE(") IAE(") Ec.u/

1 A (SNRD 0, d D 0) ADSC 2.3541 0.3595 96.610 10.990 10.96CADSC 1.2660 0.2500 0.5078 0.1526 09.95

B (SNRD 50, d D 0) ADSC 2.3400 0.3594 96.580 10.990 11.08CADSC 1.2650 0.2499 0.6293 0.1608 10.07

C (SNRD 50, d ¤ 0) ADSC 2.5790 0.3737 110.20 11.410 11.05CADSC 1.8890 0.2952 1.2990 0.2016 10.03

2 A (SNRD 0, d D 0) ADSC 2.9440 1.1060 40.300 4.7160 0.878CADSC 1.6710 1.1270 0.2452 0.0933 0.660

B (SNRD 50, d D 0) ADSC 2.9010 1.0490 103.40 10.360 1.294CADSC 2.0350 1.1380 0.2154 0.0508 0.600

C (SNRD 50, d ¤ 0) ADSC 3.4150 1.2010 112.10 10.990 1.270CADSC 2.9050 1.1940 0.6566 0.0675 0.621

ADSC, adaptive dynamics surface control; CADSC, composite adaptive dynamics surface control.ITAE(�) is the function of integral time-weighted absolute errors; IAE(�) is the function of integral absolute errors;Ec.u/ is control energy

R t0u2.�/d� .



control objective is to make x1.t/ track xd .t/ generated by filtering xc.t/ D .�=12/sgn.sin 0:2�t/using a linear filter Gc.s/ D 9=.s2 C 6s C 9/ with s being a complex variable.

The parameters selection of the proposed composite ADSC (CADSC) is as follows: Firstly, set thecontrol gains k1 D k2 D k3 D 5 in (14); secondly, set the integral interval �d D 3 s in (19); thirdly,set the filtering parameter � D 0.05 in (15); finally, set the adaptive parameters � D kw D 10 andca D 5 for (20). To demonstrate the superiority of the proposed approach, the conventional ADSC(simply set kw D 0 and other parameters be the same as aforementioned) is selected as a baselinecontroller. Let �r.t/ 2 RC be the minimal singular value of Q.t/ WD

R tt��d

ˆ.x.�//ˆT .x.�//d� attime t . Thus, �r.t/ can be regarded as a PE strength that satisfies Q.t/ > �r.t/I [2]. Simulationsare carried out in MATLAB 2013a (MathWorks, Natick, MA, USA) running on Windows 7, wherethe solver is set to be fixed-step ode 5, the step size is set to be 1 � 10�3 s, and other settings are

Figure 5. Simulation trajectories of the two controllers in Example 2 without measurement noise: (a) controltrajectories by the ADSC, (b) adaptation trajectories by the ADSC, (c) control trajectories by the compositeADSC, (d) adaptation trajectories by the composite ADSC, and (e) comparison of parameter convergence.



kept at their defaults. Let Case A be the noise-free case, Case B be the noisy-measurement case, andCase C be the case with both measurement noises and external perturbations.

Simulation trajectories of the two controllers and a comparison of system errors in Case A aredepicted in Figures 1 and 2, respectively. It is shown that for both the controllers, the estimationerrors Q� are large because the PE strengths of ˆ.x/ are very weak .maxt>0¹�r.t/º 6 3.3 �10�5/[Figure 1(b) and (d)]. Yet, the proposed CADSC still shows much higher tracking and predictionaccuracy [Figure 2] under a similar control input u [Figure 1(a) and (c)] and better parameter con-vergence under the weak PE [Figure 1(b) and (d)]. In addition, consider Case B with 50 dB Gaussianwhite noise in the measured x. Simulation trajectories of the two controllers and a comparison ofsystem errors in Case B are depicted in Figures 3 and 4, respectively. These results are very similar tothose of Case A except that chattering at u occurs because of the noisy measurement, which verifiesrobustness against measurement noises of the proposed approach. Moreover, to verify robustnessagainst external perturbations of the proposed approach, consider Case C with the same measure-ment noise as Case B and external perturbations d1.t/ D 0, d2.t/ D 0.1sin(t), and d3.t/ = 0.1cos(t)[see Remark 5 for the clarification of d1.t/ to d3.t/]. Simulation trajectories in this case are verysimilar to those of Case B so that they are omitted here. Performance index comparisons of the twocontrollers for all cases are provided in Example 1 of Table I, which verify the superior control andadaptation performances of the proposed CADSC under less control energy.

4.2. Example 2: a third-order nonlinear system

To further demonstrate that the online recorded data-based composite adaptation is able to speed upparameter convergence, consider a three-order parameter SFNSs as follows:

8̂<:̂Px1 D x2

Px2 D x21x2 C x3 C �1x1x2 C �2x2 cos.x1/

Px3 D �x1x23 C 5 .2C sin .x2x3// u

: (30)

Thus, one obtains f1 D 0, g1 D g2 D 1, f2 D x21x2, f3 D �x1x23 , g3 D 5.2C sin.x2x3//,ˆ1.x1/D ˆ3.x3/ D 0, ˆ.x/ D ˆ2.x2/ D Œx1x2; x2 cos.x1/�T , and � D Œ�1; �2�

T . For simulations, set �D Œ� 1:5; 2:1�T , x.0/ D Œ�=3; 0; 0�T , and xd .t/ D .�=6/ sin.t/. The procedure of parameter selec-tion, the baseline controller, the simulation cases, and the simulation environment in this exampleare the same as those of Example 1, where the control parameters are set to be k1 D k2 D k3 D 2,�d D 20, � D kw D 30, cw D 5, � D 0.01 for Case A, and � D 0.03 for Cases B and C.

Simulation trajectories of the two controllers and a comparison of system errors in Case A aredirected in 5 and 6, respectively. It is shown that the PE strengths of ˆ.x/ in this example are muchstronger that those of Example 1 for the two controllers .maxt>0¹�.t/º 6 0:23/ [Figure 5(b) and

Figure 6. A comparison of system errors for the two controllers in Example 2 without measurement noise:(a) tracking errors and (b) prediction errors.



(d)]. However, the proposed CADSC spends 8.165 s to achieve Q� 6 0:05, 8t > 8:165 s, whereas theADSC spends 93.75 s, over 11 times more than 8.165 s, to achieve Q� 6 0:05, 8t > 93:75 s [Figure 5(e)]. It is clear that the proposed CADSC is very useful for the speedup of parameter convergence. Itis also observed that the proposed CADSC achieves much higher tracking and prediction accuracy[Figure 6] under a similar control input u [Figure 5(a) and (c)]. In addition, consider Case B with50 dB Gaussian white noise in the measured x. Simulation trajectories of the two controllers anda comparison of system errors in Case B are directed in Figures 7 and 8, respectively, where theseresults are very similar to those of Case A except the chattering at the control input u. Specifically,the proposed CADSC spends 5.819 s to get Q� 6 0:14, 8t > 5:819 s, whereas the ADSC spends75.18 s, over 12 times more than 5.819 s, to get Q� 6 0:14, 8t > 75:18 s [Figure 8 (e)]. Moreover,consider Case C with the same measurement noise as Case B and external perturbations d1.t/ D 0,

Figure 7. Simulation trajectories of the two controllers in Example 2 with measurement noise: (a) controltrajectories by the ADSC, (b) adaptation trajectories by the ADSC, (c) control trajectories by the compositeADSC, (d) adaptation trajectories by the composite ADSC, and (e) comparison of parameter convergence.



Figure 8. A comparison of system errors for the two controllers in Example 2 with measurement noise: (a)tracking errors and (b) prediction errors.

d2.t/ D 0.02sin(t), and d3.t/ = 0.02cos(t). Performance index comparisons of the two controllersfor all cases are given in Example 2 in Table I, and simulation trajectories in Case C are omittedhere because they are very similar to those of Case B. All these results further verify that the pro-posed CADSC is robust against measurement noises and external perturbations and is superior forimproving tracking and prediction accuracy under even less control energy.

5. CONCLUSIONS

This paper has developed an online recorded data-based composite ADSC approach for a class ofparameter SFNSs with mismatched uncertainties. Semiglobal practical asymptotic stability of theclosed-loop system is rigorously established by the time-scales separation and Lyapunov synthe-sis. The proposed approach possesses a distinctive feature that the composite adaptation is achievedthrough data recorded online without the usage of both identification models and linear filters. Sim-ulation results have verified that the proposed approach is robust against measurement noises andexternal perturbations and is superior for improving tracking and prediction accuracy comparedwith the conventional ADSC without composite adaptation. Further work would focus on compositelearning control using data recorded online [46].

ACKNOWLEDGEMENTS

This study is supported by the Defense Innovative Research Programme from MINDEF of Singapore underGrant MINDEF-NUSDIRP/2012/02 and in part by the Biomedical Engineering Programme, Agency forScience, Technology and Research (A*STAR), Singapore under Grant 1421480015.

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