IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 9, SEPTEMBER 2016 2603

Composite Learning From Adaptive Dynamic Surface ControlYongping Pan, Member, IEEE, and Haoyong Yu, Member, IEEE

Abstract—In the conventional adaptive control, a stringent con-dition named persistent excitation (PE) must be satisfied to guaran-tee parameter convergence. This technical note focuses on adaptivedynamic surface control for a class of strict-feedback nonlinearsystems with parametric uncertainties, where a novel techniquecoined composite learning is developed to guarantee parameterconvergence without the PE condition. In the composite learning,online recorded data together with instantaneous data are appliedto generate prediction errors, and both tracking errors and predic-tion errors are utilized to update parametric estimates. The pro-posed approach is also extended to an output-feedback case byusing a nonlinear separation principle. The distinctive feature ofthe composite learning is that parameter convergence can be guar-anteed by an interval-excitation condition which is much weakerthan the PE condition such that the control performance can beimproved from practical asymptotic stability to practical exponen-tial stability. An illustrative example is used for verifying effective-ness of the proposed approach.

Index Terms—Adaptive control, composite learning, interval ex-citation, mismatched uncertainty, parameter convergence.

I. INTRODUCTION

Adaptive integrator backstepping control (AIBC) is effective forhandling mismatched parametric uncertainties in strict-feedback non-linear systems (SFNSs) [1]. Nevertheless, this technique suffers fromthe “explosion of complexity” resulted from repeated derivations ofvirtual control inputs. An adaptive dynamic surface control (ADSC)technique aims to alleviate the limitation of AIBC, where each virtualcontrol input is passed through a first-order filter at its correspondingbackstepping step such that the derivation of virtual control inputs isgreatly simplified [2]. This technique not only prevents the complexityproblem, but also relaxes the smoothness requirement on plant modelsand desired signals [3]. However, ADSC suffers from the “explosionof the dynamic order” of the filters employed and, in the absence ofextreme gain values, from the “explosion of the residual error” (i.e.,the loss of asymptotic tracking) and the “implosion of the region ofattraction” (i.e., the loss of global boundedness).

Composite adaptive control (CAC) is an integrated direct/indirectadaptive control strategy which makes use of both tracking errors andprediction errors to update parametric estimates [4]. The advantageof CAC essentially comes from the smoothness of control responsesresulting in possibility of using high adaptation gain. Hence, smallertracking errors and faster parameter convergence are possible to be

Manuscript received November 10, 2014; revised March 30, 2015 andAugust 19, 2015; accepted October 24, 2015. Date of publication October 27,2015; date of current version August 26, 2016. This work was supported in partby the Defense Innovative Research Programme, MINDEF of Singapore undergrant MINDEF-NUSDIRP/2012/02, and in part by the Biomedical Engineer-ing Programme, Agency for Science, Technology and Research, Singaporeunder grant 1421480015. Recommended by Associate Editor M. de Queiroz.

The authors are with the Department of Biomedical Engineering, NationalUniversity of Singapore, Singapore 117583, Singapore (e-mail: biepany@nus.edu.sg; bieyhy@nus.edu.sg).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2015.2495232

achieved without exciting high-frequency unmodeled dynamics [5].Recent results about CAC can be referred to [6]–[16]. However, onlymatched uncertainties are considered in most CACs [7]–[14]. Thislimitation was eliminated by the integration of AIBC with CAC in [6]and [15]. A composite ADSC was further developed in [16] to alleviatethe drawbacks of AIBC. Although better performances can be obtainedin all these CACs, a stringent persistent excitation (PE) condition stillhas to be satisfied to guarantee parameter convergence.

This work focuses on ADSC for a class of SFNSs with mismatchedparametric uncertainties, where a novel technique coined compositelearning is developed to guarantee parameter convergence without thePE condition. The ability to learn is one of the fundamental featuresof autonomous intelligent behavior [17]. In the composite learning,online recorded data together with instantaneous data are applied togenerate prediction errors, and both tracking errors and predictionerrors are utilized to update parametric estimates. The contributions ofthis study include: 1) a composite learning scheme is proposed undermismatched uncertainties such that parameter convergence can beguaranteed without the PE condition; 2) practical exponential stability[18] of the closed-loop system is rigorously established; 3) an output-feedback extension of the proposed approach is considered, where ahigh-gain observer is utilized to online estimate plant states, and anonlinear separation principle in [19] is exploited to establish stabilityresults of the output-feedback controller. The preliminary idea of thecomposite learning can be found in our works [20], [21].

Throughout this note, N, R, R+, Rn, and Rn×m denote the spaces

of natural numbers, real numbers, positive real numbers, real n-vectorsand realn×m-matrices, respectively,L∞ denotes the space of boundedsignals, ‖x‖ denotes the Euclidean norm of x, σr(A) denotes theminimal singular value of A, sgn(·) denotes the signum function,sat(·) := min{1, | · |}sgn(·) denotes the saturation function, min{·},max{·}, and inf{·} denote the functions of minimum, maximum andinfimum, respectively, Ωc :={x|‖x‖≤c} denotes the ball of radius c,col(x,z) := [xT , zT ]

T , and Ck represents the space of functions forwhich all k-order derivatives exist and are continuous, wherec ∈ R

+, x,z ∈ Rn, A ∈ R

n×m, and n,m, k ∈ N. Define ‖x‖D :=infz∈D ‖x− z‖ with D ⊂ R

n. For the sake of brevity, in the subse-quent sections, the arguments of a function may be omitted while thecontext is sufficiently explicit, and the time variable t of a signal willbe omitted after its first appearance except the dependence of the signalon t is crucial for clear presentation.

II. PROBLEM FORMULATION

Consider the class of nth order generalized SFNSs with parametricuncertainties in [22] as follows:{xi=fi(xi) + gi(xi)xi+1 + ΦT

i (xi)θ(i = 1, 2, . . . , n− 1)

xn=fn(x) + gn(x)u+ ΦTn (x)θ

(1)

with 0 being an equilibrium point, in which u(t) ∈ R and x1(t) ∈ R

are the control input and the system output, respectively, xi(t) :=[x1(t), x2(t), . . . , xi(t)]

T ∈ Ri (x(t) = xn(t)) are the vectors of

system states, fi(xi) : Ri �→ R, gi(xi) : R

i �→ R and Φi(xi) : Ri �→

RN are known functions, θ := [θ1, θ2, . . . , θN ] ∈ Ωca is a vector of

unknown constant parameters, N is the number of parameters, and

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2604 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 9, SEPTEMBER 2016

i = 1, 2, . . . , n. Let xd(t) ∈ R be a desired output. The followingassumptions presented in [22] are introduced for facilitating ADSCsynthesis and analysis, where i = 1, 2, . . . , n.

Assumption 1: There is a constant g0 ∈ R+ such that |gi(xi)| > g0,

∀x ∈ Ωcx . Without loss of generality, let gi(xi) > 0.Assumption 2: fi(xi), gi(xi) and Φi(xi) are of C1.Assumption 3: xd(t) and xd(t) are continuous and of L∞.Let αi(t) ∈ R and αc

i (t) ∈ R denote virtual control inputs and theirfiltered counterparts, respectively, where i=1, 2, . . . , n−1. Then, de-fine tracking errors ei(t) := xi(t)− αc

i−1(t) for i=1, 2, . . . , n withαc0(t):=xd(t).Let e(t):=[e1(t), e2(t), . . . , en(t)]

T ∈Rn andxd(t):=

[xd(t), xd(t)]T ∈ Ωcd . The ADSC law in [22] without control satura-

tion is given in the following form:⎧⎪⎪⎨⎪⎪⎩αi =

1gi

(−kciei + αc

i−1 − gi−1ei−1 − fi −ΦTi θ

)(i = 1, 2, . . . , n− 1)

u = 1gn

(−kcnen + αc

n−1 − gn−1en−1 − fn −ΦTn θ

) (2)

with e0 = g0 = 0 and αc0 = xd, where kc1 to kcn are control gain

parameters, θ := [θ1, θ2, . . . , θN ] ∈ Ωca is an estimate of θ, and αci

with i = 1, 2, . . . , n− 1 are determined by

ωαci = −αc

i + αi, αci (0) = αi(0) (3)

with ω ∈ (0, 1) being a filter parameter. Let θ := θ − θ be a vector ofparameter estimation errors and Φ(x) :=

∑ni=1 Φi(xi) be a vector of

regression functions. The following definitions are introduced for thesubsequent analysis [18], [23].

Definition 1: A bounded signal Φ(t) ∈ RN is of interval excitation

(IE) over t ∈ [Te − τd, Te] with τd ∈ R+ and Te > τd iff there exists

σ ∈ R+ such that

∫ Te

Te−τdΦ(τ )ΦT (τ )dτ ≥ σI .

Definition 2: A bounded signal Φ(t) ∈ RN is of PE iff there are

τd, σ ∈ R+ such that

∫ t

t−τdΦ(τ )ΦT (τ )dτ ≥ σI , ∀ t ≥ 0.

The objective of this study is to design a proper update law of θfor the control law (2) such that the closed-loop system guarantees theconvergence of both e and θ.

III. COMPOSITE LEARNING CONTROL DESIGN

A. Composite Learning Scheme

By the combination of (1) and (2), the closed-loop tracking errordynamics is obtained as follows:⎧⎪⎨

⎪⎩ei = −kciei + giei+1 − gi−1ei−1 + giαi

+ΦTi θ (i = 1, 2, . . . , n− 1)

en = −kcnen − gn−1en−1 +ΦTn θ

(4)

with e0 = g0 = 0 and αi := αci − αi. The following lemmas are in-

troduced for facilitating ADSC synthesis.Lemma 1 [22]:1 Consider the system (1) with Assumptions 1–3

driven by the control law in (2). Given any x(0) ∈ Ωcx0, there exist

cx > cx0 and Ta > 0 such that x(t) ∈ Ωcx , ∀ t ∈ [0, Ta).

1The proof of [22, Lemma 1] uses [24, Lemma 1.1]. However, the controlsaturation in [22] is not presented here since the boundedness of u in (2) isnaturally guaranteed by the boundedness of the state vector x.

Lemma 2 [22]: For the filter (3) with Assumptions 1–3, given anyμ ∈ R

+, there exists a sufficiently small ω in (3) such that |αi(t)| ≤ μ,∀ t ∈ [0, Ta) for i = 1, 2, . . . , n− 1.

In the CAC design, the prediction errors ΦTi θ that only utilize

instantaneous data would be applied together with the tracking errorsei to update θ, where i = 1, 2, . . . , n. It is shown that ΦT

i θ can becalculated by (4) if ei is available. Otherwise, the filtered counterpartsof ΦT

i θ would be employed instead to avoid the usage of ei [5].Although the convergence of both ei and ΦT

i θ can be guaranteedin CAC, the PE condition still has to be satisfied such that accurateparameter estimation, i.e. the convergence of θ, can be guaranteed. Inthe absence of PE, dead-zone modification in the adaptive law of θshould be applied to avoid parameter drift [22].

This section aims to design a composite learning law of θ such thataccurate parameter estimation can be obtained without the PE con-dition and parameter drift can be eliminated without the dead-zonemodification. To this end, assume that Φ(x) in (1) is of IE over t ∈[Te − τd, Te] as in Definition 1, and define a modified modeling errorε(t) := Qeθ(t) as the prediction error, where

Qe :=

{0 for t < Te

Q(Te) for t ≥ Te

(5)

with Te > Ta + τd, σ ∈ R+ and2

Q(t) :=

∫ t

t−τd

Φ(x(τ ))ΦT (x(τ ))dτ. (6)

Then, the composite learning law of θ is designed to be{ϑ(x, θ) = γ

(∑ni=1 Φi(xi)ei + kwε

)˙θ = P

(θ,ϑ(x, θ)

) (7)

in which γ ∈ R+ is a learning rate, kw ∈ R

+ is a weight factor, andP(θ,ϑ) is a projection operator given by [25], [26]

P(θ,ϑ)=

⎧⎨⎩ϑ, if‖θ‖<ca or‖θ‖=ca & θ

Tϑ≤0

ϑ − θθT

‖θ‖2ϑ, otherwise.(8)

For the calculation of the prediction error ε in (7), define

qi(t) :=

∫ t

t−τd

Φi (xi(τ ))ΦTi (xi(τ ))θdτ (9)

with i = 1, 2, . . . , n. Multiplying the ith equality of (4) by Φi(xi) fori = 1, 2, . . . , n and integrating the resulting equalities over [t− τd, t]yields the computational formulas of qi(t) in (10), as shown at thebottom of the page, where the time variable τ is omitted in the integralparts. Hence, the prediction error ε can be calculated by

ε(t) = Qeθ −Qeθ(t) (11)

with Qθ =∑n

i=1 qi, Qe in (5), Q in (6) and q1 to qn in (10).

2The existence of Te is guaranteed by the condition of linear parameteriza-tion as stated in [18, Assumption 3.2].

⎧⎪⎨⎪⎩qi(t) =

∫ t

t−τdΦi(xi)

(ei + kciei − giei+1 + gi−1ei−1 − giαi +ΦT

i (xi)θ)dτ (i = 1, 2, . . . , n− 1, e0 = g0 = 0)

qn(t) =∫ t

t−τdΦn(x)

(en + kcnen + gn−1en−1 +ΦT

n (x)θ)dτ

(10)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 9, SEPTEMBER 2016 2605

Remark 1: It follows from (7) and (11) that the update function ϑof θ in (7) depends on θ via ε, which is similar to robust modificationsof adaptive control such as projection modification P , σ-modificationand ε-modification in [25, Sec. 4.6]. Since θ(0) is prespecified, ε(0)can be computed to update θ together with the projection operator Psuch that the calculation of ε and θ can be performed iteratively. If(11) is substituted into (7), then the update law of θ is nothing buta differential equation of θ with an initial condition θ(0) as thoserobust modifications. Therefore, it is reasonable that the update lawof θ depends on θ as those robust modifications.

Remark 2: The calculation of ε in (11) is as follows: if Q(t) < σ,then Qe is set to be 0 resulting in ε(t) = 0 such that the calculationof q1(t) to qn(t) in (10) is not necessary; else if Q(t) ≥ σ with t =Te, then all data recorded on t ∈ [Te − τd, Te] are applied to calculateq1(Te) to qn(Te) such that ε(t) can be obtained by

ε(t) =n∑

i=1

qi(Te)−Q(Te)θ(t)

for all t ≥ Te. It is noticed that e1(Te) to en(Te) must be available toget q1(Te) to qn(Te) in (10). Since the calculation of q1(Te) to qn(Te)(or Qeθ) does not need to be real-time, fixed-point smoothing can beapplied to the recorded e1(t) to en(t) to obtain e1(Te) to en(Te),where the corresponding algorithm can be referred to [18, Sec. 4].An alternative real-time approach to estimate e1 to en in (10) is toapply linear filters to e1 to en. It is worth noting that differing from theapproach of [18], the integrals in (10) effectively reduce the influenceof measurement noise on the calculation of q1(Te) to qn(Te).

B. Stability and Convergence Analysis

Let ks := min1≤i≤n{kci} and g := maxx∈Ωcx ,1≤i≤n−1{gi(xi)}.The following theorem establishes the stability results of the closed-loop system under the proposed control law.

Theorem 1: Consider the system (1) under x(0) ∈ Ωcx0and As-

sumptions 1–3 driven by the control law (2) with (3), (7) and θ(0) ∈Ωca . If the IE condition Q(Te) ≥ σI in Definition 1 is satisfied forcertain constants σ ∈ R

+ and Te > τd, then there are suitable controlparameters ks, ω and γ so that the closed-loop system achieves:

1) uniformly ultimately bounded (UUB) stability [24] in the sensethat all closed-loop signals are bounded and the unique closed-loop solution is defined for all t ∈ [0,∞);

2) partial practical asymptotic stability [27] during t ∈ [0,∞) inthe sense that e(t) asymptotically converges to a small neigh-borhood of 0 determined by ks, ω and γ;

3) and practical exponential stability [18] on t ∈ [Te,∞) in thesense that both e(t) and θ(t) exponentially converge to smallneighborhoods of 0 determined by ks, ω and γ.

Proof: First, using Lemma 1, one obtainsx(t)∈Ωcx on t∈ [0, Ta)for x(0) ∈ Ωcx0

. In addition, the projection operator in (8) ensurethat θ(t) ∈ Ωca , ∀t ∈ [0,∞) as θ(0) ∈ Ωca [25, Th. 4.6.1]. Choosea Lyapunov function candidate

V (z) = eTe/2 + θTθ/(2γ) (12)

in which z := col(e, θ) ∈ Rn+N . Thus, there exist constants λa :=

min{1/2, 1/(2γ)} and λb := max{1/2, 1/(2γ)} such that

λa‖z‖2 ≤ V (z) ≤ λb‖z‖2. (13)

Differentiating V along (4) and (7) with respect to time t yields

V =−n∑

i=1

kcie2i +

n−1∑i=1

gi(xi)αiei + θT

(n∑

i=1

Φi(xi)ei− ˙θ/γ

).

Using the projection operator result in [25, Th. 4.6.1], one gets

V ≤ −n∑

i=1

kcie2i − kwθ

Tε+

n−1∑i=1

gi(xi)αiei.

From Lemma 2, given any μ ∈ R+, there is a sufficiently small ω in

(3) such that |αi(t)| ≤ μ, ∀ t ∈ [0, Ta) for i = 1, 2, . . . , n− 1. Using

|αi| ≤ μ, ε = Qeθ and the definition of g, one gets

V ≤−n∑

i=1

kcie2i − kwθ

Tε+ μg

n−1∑i=1

|ei|

≤ −ks‖e‖2/2−kw θTQeθ−ks

n−1∑i=1

(e2i −2μg|ei|/ks

)/2.

Applying the Young’s inequality 2ab− a2 ≤ b2 with a, b ∈ R to thelast term of the foregoing expression, one obtains

V ≤ −ks‖e‖2/2 − kwθTQeθ + ρ(ks, ω) (14)

on t ∈ [0, Ta) with ρ(ks, ω) := (n− 1)(μg)2/(2ks) ∈ R+.

Secondly, ignoring the second term at the right side of (14) andnoting (12) and θ(t) ∈ Ωca , ∀ t ∈ [0,∞), one gets

V (t) ≤ −ksV (t) + 2ksc2a/γ + ρ(ks, ω)

= −ksV (t)/2− ks (V (t)− η(ks, ω, γ)) /2

in which η(ks, ω, γ) :=(n− 1)(μg/ks)2 + 4c2a/γ ∈ R

+. Let Ωcz :=Ωcx ∩Ωcd × Ωcw and Ωcz0 := Ωcx0

∩Ωcd × Ωcw such that Ωcz0 ⊂Ωcz . It is implied from the above expression that

V (t) ≤ −ksV (t)/2, ∀V (t) ≥ η (15)

on z(t) ∈ Ωcz and t ∈ [0, Ta). Based on the results presented in (13)and (15), the UUB Theorem [24, Th. 4.5] is invoked to conclude thatif η <

√cz0/λb, then the closed-loop system has UUB stability in

the sense of e(t) ∈ Ωce , θ(t) ∈ Ωca , ∀t ≥ 0 implying Ta = ∞ sothat the solution z(t) is unique, ∀ t ≥ 0 [24, Lemma 3.1]. It is clearfrom the definition of η that given any cx0 ∈ R

+, there certainly existsuitably large ks, 1/ω, and γ such that η <

√cz0/λb holds. Using

e(t) ∈ Ωce , θ(t) ∈ Ωca , ∀ t ≥ 0, one gets x(t), Φ(x(t)), u(t), αi(t),αci (t) ∈ L∞, ∀ t ≥ 0 with i = 1, 2, . . . , n− 1. Thus, all closed-loop

signals are bounded, ∀ t ≥ 0. Besides, one also obtains

‖e(t)‖ ≤√

2(n− 1)

(μg

ks

)+ 2ca

√2

γ

which implies that the closed-loop system achieves partial practicalasymptotic stability for t ≥ 0 in the sense that e(t) can be arbitrarilydiminished by the increase of ks, 1/ω and γ.

Thirdly, since there exist constants σ ∈ R+ and Te > τd such that

Q(Te) > σI , i.e., the bounded signal Φ(x(t)) is if IE over t ∈ [Te −τd, Te], it is obtained from (14) that

V ≤ −ks‖e‖2/2− kwσθTθ + ρ(ks, ω), ∀t ≥ Te.

It follows from the above result and (12) that

V (t) ≤ −keV (t) + ρ(ks, ω), ∀ t ≥ Te

in which ke := min{ks, 2γkwσ} ∈ R+. Solving the above inequality

according to [25, Lemma A.3.2] yields

V (t) ≤ V (Te)e−ket + ρ(ks, ω)/ke, ∀ t ≥ Te

such that z(t) converges to a positively invariant set De := {V ≤ ρ(ks, ω)/ke}. The above result implies that the closed-loop systemachieves practical exponential stability for t ≥ Te in the sense thatboth e(t) and θ(t) converge to

√ρ/ke-neighborhoods of 0 that can

be arbitrarily diminished by the increase of ks, 1/ω and γ. �

2606 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 9, SEPTEMBER 2016

Remark 3: The difference between AIBC and ADSC lies in how thederivatives of α1 to αn−1 are obtained in the control law (2). In theAIBC, the analytical expressions of α1 to αn−1 in (2) are utilized tocalculate α1 to αn−1, whereas in the ADSC, the filter (3) is utilized toobtain αc

1 to αcn−1, i.e., the estimates of α1 to αn−1. The complexity

problem of AIBC and the comparison between AIBC and ADSC areillustrated in detail in [2]. Because the composite learning does notrely on any particular feature of ADSC, it can be easily extended tothe AIBC framework such that the fundamental limitations of ADSCmentioned before can be eliminated.

Remark 4: The only result of composite ADSC can be found in[16]. Compared with the innovative work of [16], the proposed ADSCwith composite learning has some distinctive features as follows:1) online recorded data are utilized to construct the prediction error εsuch that the usage of identification models is avoided; 2) parameterconvergence can be guaranteed by the IE instead of PE condition;3) the control performance can be improved from practical asymptoticstability to practical exponential stability. The items 2–3 also indicatethe distinctive features of the proposed ADSC with composite leaningcompared with the conventional ADSC.

Remark 5: Concurrent learning adaptive control is an emergingtechnique which aims to relax the PE condition using online recordeddata [18]. Yet in this innovative technique, a time-consuming singularvalue maximising algorithm, which makes an exhaustive search overall stored data, must be applied to maximize the singular value ofstored data. This treatment greatly increases computational cost ofthe entire control algorithm. In addition, only matched uncertaintiesare considered in [18]. Although the composite learning also utilizesonline recorded data, it is fundamentally different from the concurrentlearning owing to the facts that in the composite learning, the time-interval integrals are applied to obtain the prediction error, and thesingular value maximization is not necessary. In addition, mismatcheduncertainties are also considered in our approach.

C. Extension to Output Feedback

Consider the case that x1 rather than x is measurable. An output-feedback extension of the above results follows a separation principleof nonlinear control in [19]. Let x = f (x, u) with f : Rn × R �→ R

n

and u = υ(x, θ) with υ : Rn × RN �→ R denote the system (1) and

the controller (2), respectively. The closed-loop system under statefeedback can be written in a compact form

x = f(x, υ(x, θ)

),˙θ = P

(θ,ϑ(x, θ)

). (16)

Let x(t) := [x1(t), x2(t), . . . , xn(t)] ∈ Rn be an estimate of x(t),

x(t) := x(t)− x(t) = [x1(t), x2(t), . . . , xn(t)]T be an observation

error, and [ΦTi (xi)θ]

s := cpisat(ΦTi (xi)θ/cpi), where cpi ∈ R

+ sat-isfy cpi ≥ maxx∈Ωcx

{Φi(xi)}, cx > cx, and i = 1, 2, · · · , n. To es-timate x, a high-gain observer is introduced as follows [24]:

⎧⎪⎪⎨⎪⎪⎩

˙xi = fi(xi) + gi(xi)xi+1 +[ΦT

i (xi)θ]s

+ koix1/εi

(i = 1, 2, . . . , n− 1)

˙xn = fn(x) + gn(x)u+[ΦT

n (x)θ]s

+ konx1/εn

(17)

where ε ∈ R+ is an observer gain parameter, and koi ∈ R

+ with i =1, 2, . . . , n are selected such that the polynomial

sn + ko1sn−1 + · · ·+ ko(n−1)s+ kon (18)

is strictly Hurwitz, where s is a complex variable. The output-feedbackADSC law under the observer (17) is as follows:

˙θ = P

(θ,ϑ(x, θ)

), u = υs(x, θ) (19)

with υs(x, θ) := cusat(υ(x, θ)/cu), in which cu ∈ R+ satisfies cu ≥

maxx∈Ωcx,θ∈Ωca

{|υ(x, θ)|}.Subtracting (17) from (1), one obtains⎧⎪⎨

⎪⎩˙xi = −koix1/ε

i + gi(xi)xi+1 + hi(xi+1, xi, θ)

(i = 1, 2, . . . , n− 1)˙xn = −konx1/ε

n + hn(x, x, θ, u)

(20)

in which hi with i = 1, 2, . . . , n are given by

hi(xi+1, xi, θ) := hi(xi+1, θ)− hi(xi, θ, xi+1)

hi(xi+1, θ) := fi(xi) + gi(xi)xi+1 + ΦTi (xi)θ

hi(xi, θ, xi+1) := fi(xi) + gi(xi)xi+1 +[ΦT

i (xi)θ]s

hn(x, x, θ, u) := h(x, θ, u)− h(x, θ, u)

hn(x,θ, u) := fn(x) + gn(x)u+ ΦTn (x)θ

hn(x, θ, u) := fn(x) + gn(x)u+[ΦT

n (x)θ]s

.

Let χ(t) := Λ−1(ε)x(t) denote a scaled observation error, whereΛ(ε) := diag(εn−1, . . . , ε, 1). Then, combining with u = υs(x, θ),one can rewritten (20) into a compact form

ε ˙χ = Ao(t)χ + εh(x, x, θ, υs(x, θ)

)(21)

where h := [h1, h2, . . . , hn]T and

Ao(t) =

⎡⎢⎢⎢⎢⎢⎢⎣

−ko1 g1 0 . . . 0

−ko2 0 g2. . .

......

.... . .

. . . 0−ko(n−1) 0 . . . 0 gn−1

−kon 0 . . . 0 0

⎤⎥⎥⎥⎥⎥⎥⎦

which is strictly Hurwitz for sufficiently large ko1 to kon [29].By combining (16), (19), (21) with χ = Λ−1(ε)x, the closed-loop

system under output feedback is obtained as follows:

x =f(x, υs(x− Λ(ε)χ, θ)

)(22a)

˙θ =P

(ϑ(x− Λ(ε)χ, θ)

)(22b)

ε ˙χ =Aoχ + εh(x, θ,Λ(ε)χ

). (22c)

The analysis of the system (22) can be regarded as a standard singularperturbation problem, where its reduced system (set χ = 0 in (22a)and (22b)) is the same as its output-feedback counterpart (16), andits boundary-layer system ε ˙χ = Aoχ (set ε = 0 in (22c)) has a uniquesolution χ = 0. Let (z(t, ε), χ(t, ε)), zr(t) and χr(t) be the solutionsof the system (22), its reduced system and its boundary-layer system,respectively. As all conditions in Lemmas 1 and 2 are still satisfied, onegets: 1) x(t) ∈ Ωcx , ∀ t ∈ [0, Ta); 2) θ(t) ∈ Ωca , ∀ t ∈ [0,∞); and3) |αi(t)| ≤ μ with i = 1, 2, . . . , n− 1, ∀ t ∈ [0, Ta). The conditionsfor the application of the separation principle of nonlinear control in[19] are established as follows:

1) According to Theorem 1, the system (16) has partial asymptoticstability and exponential stability with respect to Ds and De fort ∈ [0, Te) and t ∈ [Te,∞), respectively,3 over Ωce ×Ωca ;

2) The functions P(θ,ϑ) and υs in (19) are locally Lipschitz intheir arguments uniformly in t over Ωce ×Ωca , and are globallybounded with respect to x and θ;

3The definitions of asymptotical stability and exponential stability with re-spect to a set can be referred to [28, Definition 3.1].

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 9, SEPTEMBER 2016 2607

Fig. 1. A regulation performance by the conventional state-feedback ADSC.

3) The function f is locally Lipschitz in x and u on Ωce × Ωca ;4) The functions hi are locally Lipschitz in their arguments over

Ωce × Ωca , and their estimated parts [ΦTi (xi)θ]

s are globallybounded with respect to xi and θ, where i = 1, 2, . . . , n.

Accordingly, [19, Th. 1–3] can be invoked to conclude that:4

1) There is ε∗1 ∈ R+ such that, for every ε ∈ (0, ε∗1], the trajectories

(z(t, ε), χ(t, ε)) of the system (22) starting in Ωcx0×Ωca are

bounded for all t ≥ 0, and come arbitrarily close to De × {χ =0} as the time progresses;

2) Given any μ ∈ R+, there is ε∗2(μ) ∈ R

+ and Tb(μ) ≥ Te suchthat ‖z(t, ε)‖De + ‖χ(t, ε)‖ ≤ μ, ∀ t ≥ Tb, ∀ ε ∈ (0, ε∗2];

3) Given any μ ∈ R+, there exists ε∗3 ∈ R

+ such that ‖z(t, ε)−zr(t) ≤ μ, ∀ t ≥ 0, ∀ ε ∈ (0, ε∗3].

IV. AN ILLUSTRATIVE EXAMPLE

Consider a simple system in the form of (1) as follows:⎧⎪⎨⎪⎩x1 = x2

x2 = −x21x2 + x3 + θ1x1x2 + θ2x2 cos(x1)

x3 = x1x3 + 5 (2 + sin(x2x3))u

in which f1 = 0, f2 = −x21x2, f3 = x1x3, g1 = g2 = 1, g3 = 5(2 +

sin(x2x3)), Φ1(x1) = Φ3(x3) = 0, Φ(x) = Φ2(x2)=[x1x2, x2

cos(x1)]T , and θ = [θ1, θ2]

T . For simulations, set θ = [−1.5, 2.1]T

and x(0)=[π/3, 0, 0]T , and add 50 dB additive white Gaussian noise(AWGN) to all states x1 to xn and to the state x1 only for the state-feedback and output-feedback cases, respectively.

The parameters selection of the proposed control law is as follows:First, set kc1 = kc2 = kc3 = 3 in (2); secondly, set ω = 0.05 in (3);thirdly, set τd = 5 s in (6); fourthly, set γ = kw = 10, σ = 0.01 andca = 5 for (7); fifthly, set ε = 0.1 and ko = [15, 75, 125]T in (17);finally, set cu = 20 in (19). To demonstrate superiority of the proposedapproach, the conventional ADSC (simply set τd = 0 s and keep theother parameters the same as above) is selected as a baseline controller.

4The results about asymptotical or exponential stability recovery in [19, Th.4–6] are not presented here since the conditions f = 0 and hi = hi = 0 (i =1, 2, . . . , n) are not satisfied on Ds × {χ = 0} or De × {χ = 0} in thiscase. Please refer to the details of proof in [28, Ch. 3].

Fig. 2. A regulation performance by the proposed state-feedback ADSC.

Fig. 3. A regulation performance by the proposed output-feedback ADSC.

Simulations are carried out in MATLAB R2014a software running onWindows 8.1 operating system, Intel Core i7-4510U CPU and 8 GBRAM memory, in which the measurement noise is generated bySimulink/AGWN Channel. For the Simulink, the solver is set to befixed-step ode 5, the step size is set to be 1× 10−3 s, and the othersettings are kept at their defaults; for the AWGN Channel, the SNR isset as 50 dB, and input signal power is set as 1 watts so that the meanvalue is 0 and the standard deviation is 0.0032.

First, consider a regulation problem where xd is generated by thefollowing reference model:

xd =

[0 1

−25 −10

]xd +

[025

]xc

with xd(0) = [0, 0]T , xc(t) = π/6 for t ∈ [0, 5) and xc(t) = 0 fort ∈ [5,∞). State-feedback regulation performances by the conven-tional ADSC and the proposed ADSC are exhibited in Figs. 1 and 2,respectively, where a vertical dot-dash line is depicted in the lowestsubfigure of Fig. 2 to indicate σr(t) > σ. Note that in practice, thevalue of σ can be increased to σr when σr becomes obviously larger

2608 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 9, SEPTEMBER 2016

Fig. 4. A tracking performance by the proposed state-feedback ADSC.

than the current σ to make full use of excitation information. It isobserved that PE is not satisfied and IE is weak in this case. Yetinterestingly, accurate parameter estimation is shown by the proposedADSC [see Fig. 2], whereas no parameter convergence is shown bythe conventional ADSC [see Fig. 1]. An output-feedback regulationperformance by the proposed ADSC is exhibited in Fig. 3, wherechattering at the control input u is reduced compared with Figs. 1 and 2owing to the filtering property of the observer (17), and parameterestimation accuracy is degraded compared with Fig. 2 because of theadditional error x introduced by the observer (17). It is worth notingthat chattering at u is unavoidable due to the noisy measurement. Yet,it is feasible for a realistic actuator with high bandwidth.

Secondly, consider a tracking problem, where xd is generated by aDuffing oscillator model as follows:

xd =

[0 11.1 −0.4

]xd +

[0

−x3d1 + 1.498 cos(1.8t)

]

which exhibits a chaotic phenomenon for xd(0) = [0.2, 0.3]T . State-feedback and output-feedback tracking performances by the proposedADSC are exhibited in Figs. 4 and 5, respectively. Qualitative analysisof these results are the same as those of the regulation problem exceptthat the exciting strength σ is higher in the this case.

To reveal how performances degrade as the number of stored datand decreases, various values of nd between [0, 10000] are applied insimulations. Since the improved exponential tracking in Theorem 1 isstill not clearly shown in the previous simulations, the conventionalADSC (the case nd = 0) is included here to make comparisons. Theresults of the state-feedback tracking case are depicted in Fig. 6, andthe results of the other cases are not given since they are very similarto Fig. 6, where IAE denotes the function of integral absolute errors.The observations are summarized as follows: 1) both the tracking andlearning performances depend on σ, where the tracking performancereflected by IAE(e) begins degrading as nd < 10, and the learningperformance reflected by IAE(θ) begins degrading as nd < 1000;2) the proposed ADSC (the cases with nd ∈ [1, 10000]) always hasbetter tracking and learning performances than the conventional ADSC(the case nd = 0); 3) the values of nd and IAE(e) is not strictlypositively correlated due to the interplay between e and θ. Therefore,to choose a proper nd (or τd), one can increase nd according to themonitored σ and IAE(e) by trail-and-error till σ or IAE(e) tends to

Fig. 5. A tracking performance by the proposed output-feedback ADSC.

Fig. 6. Relationships between nd and tracking/learning performances.

constant. Note that nd cannot be too large, especially in embeddedapplications, since a large nd leads to large memory cost.

V. CONCLUSION

This work has presented an ADSC strategy for a class of SFNSswith mismatched parametric uncertainties, where a composite learningscheme is proposed to update parametric estimates. It is proven thatparameter convergence can be guaranteed by the IE condition which ismuch weaker than the PE condition. Simulation results under noisymeasurement have demonstrated that the composite learning is notonly efficient for achieving parameter convergence without PE, butalso useful for improving the tracking performance. Further workwould focus on the extension of the composite learning technique towider classes of nonlinear systems.

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