Nonlinear Dyn (2011) 64: 77–87DOI 10.1007/s11071-010-9847-7

O R I G I NA L PA P E R

Pragmatical adaptive synchronization of different orderschaotic systems with all uncertain parameters via nonlinearcontrol

Shih-Yu Li · Zheng-Ming Ge

Received: 28 May 2010 / Accepted: 15 September 2010 / Published online: 21 October 2010© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract A new adaptive synchronization scheme bypragmatical asymptotically stability theorem is pro-posed in this paper. Based on this theorem and non-linear control theory, a new adaptive synchronizationscheme to design controllers can be obtained and espe-cially the constraints for minimum values of feedbackgain K in controllers can be derived. This new strat-egy shows that the constraint values of feedback gainK are related to the error of unknown and estimatedparameters if the goal system is given. Through thisnew strategy, an appropriate feedback gain K can bealways decided easily to obtain controllers achievingadaptive synchronization. Two identical Lorenz sys-tems with different initial conditions and two com-pletely different nonlinear systems with different or-ders, augmented Rössler’s system and Mathieu–vander Pol system, are used for illustrations to demon-strate the efficiency and effectiveness of the new adap-tive scheme in numerical simulation results.

Keywords Constraints of feedback gain · Adaptivesynchronization · Uncertain parameters · Nonlinearcontrol · Pragmatical asymptotically stability theorem

S.-Y. Li · Z.-M. Ge (�)Department of Mechanical Engineering, National ChiaoTung University, 1001 Ta Hsueh Road, Hsinchu 300,Taiwan, Republic of Chinae-mail: zmg@cc.nctu.edu.tw

1 Introduction

Nonlinear dynamics, commonly called the chaos the-ory, changes the scientific way of looking at the dy-namics of natural and social systems, which has beenintensively studied over the past several decades. Thephenomenon of chaos has attracted widespread at-tention amongst mathematicians, physicists and en-gineers. Chaos has also been extensively studied inmany fields, such as chemical reactions, power con-verters, biological systems, information processing,secure communications, etc. [1–9].

Synchronization of chaotic systems is essential invariety of applications, including secure communi-cation, physiology, nonlinear optics and so on. Ac-cordingly, following the initial work of Pecora andCarroll [10] in synchronization of identical chaoticsystems with different initial conditions, many ap-proaches have been proposed for the synchronizationof chaotic and hyperchaotic systems such as linear andnonlinear feedback synchronization methods [11, 12],adaptive synchronization methods [13, 14], backstep-ping design methods [15, 16], and sliding mode con-trol methods [17, 18], etc. However, to our best knowl-edge, most of the methods mentioned above and manyother existing synchronization methods mainly con-cern the synchronization of two identical chaotic orhyperchaotic systems, the methods of synchronizationof two different chaotic or hyperchaotic systems arefar from being straightforward because of their dif-ferent structures and parameter mismatch. Moreover,

78 S.-Y. Li, Z.-M. Ge

most of the methods are used to synchronize only twosystems with exactly known structures and parame-ters. But in practical situations, some or all of the sys-tems’ parameters cannot be exactly known in priori.As a result, more and more applications of chaos syn-chronization in secure communication have made itmuch more important to synchronize two different hy-perchaotic systems with uncertain parameters in recentyears. In this regard, some works on synchronizationof two different hyperchaotic systems with uncertainparameters have been performed [19, 20].

In current scheme of adaptive synchronization,traditional Lyapunov stability theorem and Barbalatlemma are used to prove that the error vector ap-proaches zero as time approaches infinity, but thequestion that why those estimated parameters alsoapproach the uncertain values remains no answer[21–23]. In this article, pragmatical asymptoticallystability theorem and an assumption of equal proba-bility for ergodic initial conditions [24, 25] are usedto prove strictly that those estimated parameters ap-proach the uncertain values. Moreover, traditionaladaptive chaos synchronization in general is limitedfor the same system.

Recently, Meng and Wang [26] proposed a newcontrol law which is designed to achieve the general-ized synchronization of chaotic systems through Bar-balat lemma [27]. In [26], they also derive a synchro-nization condition for controllers, and the generalizedsynchronization between two chaotic systems can bereally attained via these conditions and the controllaw mentioned above. In this paper, we further expandthe innovative idea proposed by [26] to discuss theadaptive synchronization with all uncertain parame-ters in master system by pragmatical asymptoticallystability theorem. Based on this theorem and nonlin-ear control theory, the constraints of feedback gainK in controllers for adaptive synchronization are pro-posed and good effectiveness is shown in simulationresults.

The layout of the rest of the paper is as follows. InSect. 2, adaptive synchronization scheme is presented.In Sect. 3, two simulation cases are illustrated to verifythe new adaptive scheme. In Sect. 4 conclusions aregiven. Pragmatical asymptotically stability theorem isenclosed in Appendix A.

2 Adaptive synchronization scheme

Consider the following master chaotic system:

x = Ax + Bf (x) (2.1)

where x = [x1, x2, . . . , xn]T ∈ Rn denotes a state vec-tor, f is a nonlinear continuous vector function and A

and B are n × n coefficient matrices.The slave system is given by the following equa-

tion:

y = Ay + Bg(y) + u(t) (2.2)

where y = [y1, y2, . . . , yn]T ∈ Rn denotes a state vec-tor, A and B are n × n estimated coefficient matri-ces, g is a nonlinear continuous vector function, andu(t) = [u1(t), u2(t), . . . , un(t)]T ∈ Rn is a control in-put vector.

Function f (x) is globally Lipschitz continuous;i.e., the following condition is satisfied: For functionf (z), there exists constant L > 0, for any two differ-ent z1, z2 ∈ Rn, such that∥∥f (z1) − f (z2)

∥∥ ≤ L‖z1 − z2‖ (2.3)

Property 1 [28] For a matrix A, let ‖A‖ indicate thenorm of A induced by the Euclidean vector norm asfollow:

‖A‖ = (

λmax(

ATA)) 1

2 (2.4)

where λmax(ATA) represents the maximum eigen-

value of matrix (ATA).

Property 2 [11] For a vector,

‖x‖ = (

xTx) 1

2 (2.5)

where xT denotes the transpose of the vector x.

Our goal is to design a controller u(t) so that thestate vector of the chaotic system (2.2) asymptoticallyapproaches the state vector of the master system (2.1).

Theorem 1 If the parametric update laws are chosenas:

˙am = −ame, m = 1 ∼ p (2.6)

where e = x − y are the state errors between the mas-ter and slave systems, p is the number of parameters,

Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters 79

am are the unknown parameters in the master system,am are the estimated parameters in the slave system,and am = am − am are the errors between the unknownand estimated parameters; and the controller u is de-signed as:

u = −yout + K(x − y) + Bf (y)

+ Ay − a2m(e − 1), m = 1 ∼ p (2.7)

where yout is y without controllers, gain K = diag(k1,

. . . , kn) satisfies following constraint:

min(K)

(L‖B‖ + ‖A‖ + max(∑p

i=1 a2i ))

> 1 (2.8)

then master system (2.1) and slave system (2.2) willachieve the adaptive synchronization.

Proof The synchronization errors between master andslave systems are defined

e(t) = x(t) − y(t) (2.9)

controllers in (2.7) are substituted into the error dy-namics system as follows:

e = x − y

= Ae + B(

f (x) − f (y))

− Ke + a2m(e − 1), m = 1 ∼ p (2.10)

We choose a control Lyapunov function as the follow-ing form:

V (t) = 1

2

(

eTe + aT1 a1 + aT

2 a2 + · · · + aTpap

)

= 1

2

(‖e‖2 + a21 + a2

2 + · · · + a2p

)

> 0 (2.11)

The time derivative of V(t) in along any trajectoryof (2.10) is

V (t) = eTAe + eTB(

f (x) − f (y)) − eTKe

+ eTa2m(e − 1) + a2

1e + a22e + · · · + a2

pe

≤ ‖A‖‖e‖2 + ‖e‖‖B‖∥∥(

f (x) − f (y))∥∥

− min(K)‖e‖2 + (

a21 + a2

2 + · · · + a2p

)‖e‖2

≤ ‖A‖‖e‖2 + L‖B‖‖e‖2 − min(K)‖e‖2

+ max(

a21 + a2

2 + · · · + a2p

)‖e‖2

= (‖A‖ + L‖B‖ − min(K)

+ (

a210 + a2

20 + · · · + a2p0

))‖e‖2 (2.12)

where a10, a20, . . . , ap0 are the initial values of a1, a2,

. . . , ap , separately. If min(K) satisfies (2.8), thenV (t) ≤ 0. Let R = min(K) − L‖B‖ − ‖A‖ − (a2

10 +a2

20 + · · · + a2p0), then V ≤ −R‖e‖2, where R > 0.

V is a negative semidefinite function of e and parame-ter differences. In current scheme of adaptive controlof chaotic motion [21–23], traditional Lyapunov sta-bility theorem and Barbalat lemma are used to provethe error vector approaches zero, as time approachesinfinity. But the question, why the estimated or givenparameters also approach to the uncertain or goal pa-rameters, remains no answer. By pragmatical asymp-totical stability theorem, the question can be answeredstrictly. �

3 Numerical simulations

In this section, there are two examples for our newadaptive scheme in numerical simulation. In Case 1,two identical Lorenz systems with different initial con-ditions and parameters are used for master and slavesystems to show the effectiveness of the new scheme.In Case 2, two completely different systems, Rössler’ssystem and Mathieu–van der Pol system, are regardedas master and slave systems separately.

Case 1 Two identical Lorenz systems with differentinitial conditions and parameters.

Master Lorenz system [29]:⎡

⎣

x1

x2

x3

⎤

⎦ =⎡

⎣

a(x2 − x1)

cx1 − x1x3 − x2

x1x2 − bx3

⎤

⎦

=⎡

⎣

−a a 0c −1 00 0 −b

⎤

⎦

⎡

⎣

x1

x2

x3

⎤

⎦

+⎡

⎣

0 0 00 −1 00 0 1

⎤

⎦

⎡

⎣

0x1x3

x1x2

⎤

⎦ (3.1)

where

A =⎡

⎣

−a a 0c −1 00 0 −b

⎤

⎦

80 S.-Y. Li, Z.-M. Ge

B =⎡

⎣

0 0 00 −1 00 0 1

⎤

⎦ and f (x) =⎡

⎣

0x1x3

x1x2

⎤

⎦

x1, x2, x3 are states of master system, when initial con-dition (x10, x20, x30) = (−0.1,0.2,0.3) and parame-ters a = 10, b = 8/3 and c = 28. The chaotic behaviorof (3.1) is shown in Fig. 1.

Slave Lorenz system:

⎡

⎣

y1

y2

y3

⎤

⎦ =⎡

⎣

a(y2 − y1) + u1

cy1 − y1y3 − y2 + u2

y1y2 − by3 + u3

⎤

⎦ (3.2)

y1, y2, y3 are states of slave system, when initial con-dition (y10, y20, y30) = (5,10,5) and initial estimatedparameters a0 = 8, b0 = 3 and c0 = 25. Thus, the er-rors between the estimated and uncertain parameterscan be given as follows:

⎧

⎪⎨

⎪⎩

a0 = a − a0 = 2

b0 = b − b0 = −1/3

c0 = c − c0 = 3

(3.3)

Through (3.1) and (3.3), ‖A‖ = 30.0731, ‖B‖ = 1,and max(a2 + b2 + c2) ∼= 13.111 can be obtained. Ac-cording to our new adaptive scheme, gain K should

satisfy (2.8), i.e.,

min(K)

(L × 1 + 30.0731 + 13.111)> 1 (3.4)

choosing L = 1, the gain matrix can be selected as

follows:

K =⎡

⎣

55 0 00 52 00 0 50

⎤

⎦ (3.5)

By (3.2), we get f (y) = [0 y1y3 y1y2]T, the para-

metric update laws and corresponding controllers canbe decided by (2.6) and (2.7). Then we can obtain thetime derivative of V (t) is semidefinite as follow:

V ≤ (‖A‖ + L‖B‖ − min(K)

+ max(

a20 + b2

0 + c20

))‖e‖2

= (44.1842 − 50)‖e‖2

= (−5.8158)‖e‖2 < 0 (3.6)

which is a negative semidefinite function of e1, e2, e3,

a, b, c. The Lyapunov asymptotical stability theoremis not satisfied. We cannot obtain that common originof error dynamics and parameter dynamics is asymp-totically stable. By pragmatical asymptotically stabil-

Fig. 1 Projections of phaseportrait of chaotic Lorenzsystem with a = 10,b = 8/3, and c = 28

Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters 81

Fig. 2 Time histories oferrors for Case 1

Fig. 3 Time histories ofparametric errors for Case 1

ity theorem (see Appendix A), D is a 6-manifold,n = 6, and the number of error state variables p = 3.When e1 = e2 = e3 = 0 and a, b, c take arbitrary val-ues, V = 0, so X is of 3 dimensions, m = n−p = 6 −3 = 3, m + 1 < n is satisfied. According to the prag-matical asymptotically stability theorem, error vector

e approaches zero and the estimated parameters alsoapproach the uncertain parameters. The equilibriumpoint is pragmatically asymptotically stable. Under theassumption of equal probability, it is actually asymp-totically stable. The simulation results are shown inFigs. 2 and 3.

82 S.-Y. Li, Z.-M. Ge

Fig. 4 Projections of phaseportrait of chaotic Rössler’ssystem with a = 0.2,b = 0.2, and c = 5.7

Case 2 Adaptive synchronization between augment-ed new Rössler’s system as a master system andMathieu–van der Pol system as a slave one.

The augmented new Rössler’s system with four or-ders is:

⎡

⎢⎢⎣

x1

x2

x3

x4

⎤

⎥⎥⎦

=

⎡

⎢⎢⎣

−x2 − x3

x1 + ax2

b + x1x3 − cx3

x1 + x3

⎤

⎥⎥⎦

=

⎡

⎢⎢⎣

0 −1 −1 01 a 0 00 0 −c 01 0 1 0

⎤

⎥⎥⎦

⎡

⎢⎢⎣

x1

x2

x3

x4

⎤

⎥⎥⎦

+

⎡

⎢⎢⎣

0 0 0 00 0 0 00 0 1 00 0 0 0

⎤

⎥⎥⎦

⎡

⎢⎢⎣

00

b + x1x3

0

⎤

⎥⎥⎦

(3.7)

where

A =

⎡

⎢⎢⎣

0 −1 −1 01 a 0 00 0 −c 01 0 1 0

⎤

⎥⎥⎦

,

B =

⎡

⎢⎢⎣

0 0 0 00 0 0 00 0 1 00 0 0 0

⎤

⎥⎥⎦

and f (x) =

⎡

⎢⎢⎣

00

b + x1x3

0

⎤

⎥⎥⎦

x1, x2, x3 and x4 are states of new Rössler’s sys-tem. We choose initial condition (x10, x20, x30, x40) =(0.1,0.5,0.3,0.7) and parameters a = 0.38, b = 0.3,and c = 4.820. Chaos of the new Rössler’s system ap-pears. The chaotic behavior of (3.7) is shown in Fig. 4.

Slave Mathieu–van der Pol system [30] is

⎡

⎢⎢⎣

z1

z2

z3

z4

⎤

⎥⎥⎦

=⎡

⎢⎣

z2 + u1

−(a1 + b1z3)z1 − (a1 + b1z3)z31 − c1z2 + d1z3 + u2

z4 + u3

−e1z3 + f1(1 − z23)z4 + g1z1 + u4

⎤

⎥⎦

(3.8)

z1, z2, z3 and z4 are states of slave system, a1, b1, c1,

d1, e1, f1 and g1 are uncertain parameters. This sys-tem exhibits chaos when the parameters of systemare a1 = 10, b1 = 3, c1 = 0.4, d1 = 70, e1 = 1,f1 = 5, g1 = 0.1 and the initial states of system are(z10, z20, z30, z40) = (0.1,−0.5,0.1,−0.5). The pro-jections of phase portraits are shown in Fig. 5. Wechoose a1 = 3, b1 = 1, c1 = 0.4, d1 = 5, e1 = 1, f1 =1, g1 = 0.1, a = 0, b = 0, c = 0 and the initial states

Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters 83

Fig. 5 Projections of phaseportrait of new chaoticMathieu–van der Polsystem with a1 = 10,b1 = 3, c1 = 0.4, d1 = 70,e1 = 1, f1 = 5, g1 = 0.1

of system are (z10, z20, z30, z40) = (5,7,9,10). The

initial values of a10, b10, c10, d10, e10, f10, g10, a0, b0

and c0 can be decided as: a10 = 3, b10 = 1, c10 = 0.4,

d10 = 5, e10 = 0.3, f10 = 1, g10 = 0.1, a0 = 0, b0 = 0

and c0 = 0. Therefore, the errors of the estimated un-

known parameters can be decided as follow:

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

a10 = a1 − a10 = −3b10 = b1 − b10 = −1c10 = c1 − c10 = −0.4d10 = d1 − d10 = −5e10 = e1 − e10 = −1f10 = f1 − f10 = −1g10 = g1 − g10 = −0.1a0 = a − a0 = 0.2b0 = b − b0 = 0.2c0 = c − c0 = 5.7

(3.9)

Through (3.7) and (3.9), ‖A‖ = 5.8780, ‖B‖ = 1,

and max(a2 + b2 + c2 + a21 +· · ·+ g2

1) = 68.9300 can

be obtained. According to our new adaptive scheme,

gain K must satisfy (2.8), i.e.

min(K)

(L × 1 + 5.8780 + 68.9300)> 1 (3.10)

Choose L = 1. The gain matrix can be selected as fol-low:

K =

⎡

⎢⎢⎣

90 0 0 00 87 0 00 0 83 00 0 0 80

⎤

⎥⎥⎦

(3.11)

In (3.8) we have f (z) = [0 0 b+ z1z3 0]T, the para-metric update laws and corresponding controllers canbe decided by (2.6) and (2.7). Then we can obtain thetime derivative of V (t) is negative semidefinite as fol-low:

V ≤ (‖A‖ + L‖B‖ − min(K)

+ max(

a2 + b2 + c2 + a21 + · · · + g2

1

))‖e‖2

= (75.7080 − 80)‖e‖2

= (−4.2920)‖e‖2 < 0 (3.12)

which is a negative semidefinite function of e1, e2, e3,

e4, a, b, c, a1, b1, c1, d1, e1, f1, g1. The Lyapunov as-ymptotical stability theorem is not satisfied. We can-not obtain that common origin of error dynamics andparameter dynamics is asymptotically stable. By prag-matical asymptotically stability theorem (see Appen-dix A), D is a 14-manifold, n = 14 and the num-ber of error state variables p = 4. When e1 = e2 =e3 = e4 = 0 and a, b, c, a1, b1, c1, d1, e1, f1 and g1

84 S.-Y. Li, Z.-M. Ge

Fig. 6 Time histories oferrors for Case 2

Fig. 7 Time histories ofparametric errors a, c, a1and b1 for Case 2

take arbitrary values, V = 0, so X is of 4 dimen-sions, m = n − p = 14 − 4 = 10, m + 1 < n is sat-isfied. According to the pragmatical asymptoticallystability theorem, error vector e approaches zero andthe estimated parameters also approach the uncer-tain parameters. The equilibrium point is pragmati-cally asymptotically stable. Under the assumption ofequal probability, it is actually asymptotically sta-

ble. The simulation results are shown in Figs. 6, 7and 8.

4 Conclusions

In this paper, a new adaptive synchronization schemewhich is derived by pragmatical asymptotically stabil-

Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters 85

Fig. 8 Time histories ofparametric errorsc1, d1, e1, f1 and g1 forCase 2

ity theorem is proposed to achieve adaptive synchro-nization. We give a simple and useful result in this ar-ticle as: If the master system is given, the minimumvalues in feedback gain K can be calculated by thesum of errors between unknown and estimated para-meters, i.e. our new strategy proves the existence of re-lation between the constraint values of feedback gainK and the sum of the errors of unknown parametersand estimated parameters. By applying this new rela-tion formula, an appropriate feedback gain K can bedecided easily to obtain controllers achieving adaptivesynchronization. Simulation results show that not onlyfor two identical nonlinear systems with all unknownparameters adaptive synchronization can be achieved,but also for two completely different nonlinear sys-tems with different orders and all unknown parame-ters this goal can be attained. Therefore, the new adap-tive scheme is really useful and effective in adaptivesynchronization of various kinds of different nonlin-ear systems.

Acknowledgements This research was supported by the Na-tional Science Council, Republic of China, under Grant NumberNSC 96-2221-E-009-145-MY3.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

Appendix A: Pragmatical asymptotical stabilitytheory

The stability for many problems in real dynamicalsystems is actual asymptotical stability, although maynot be mathematical asymptotical stability. The math-ematical asymptotical stability demands that trajecto-ries from all initial states in the neighborhood of zerosolution must approach the origin as t → ∞. If thereare only a small part or even a few of the initial statesfrom which the trajectories do not approach the originas t → ∞, the zero solution is not mathematically as-ymptotically stable. However, when the probability ofoccurrence of an event is zero, it means the event doesnot occur actually. If the probability of occurrence ofthe event that the trajectories from the initial states arethat they do not approach zero when t → ∞, is zero,the stability of zero solution is actual asymptotical sta-bility though it is not mathematical asymptotical sta-bility. In order to analyze the asymptotical stability ofthe equilibrium point of such systems, the pragmaticalasymptotical stability theorem is used.

Let X and Y be two manifolds of dimensions m andn (m < n), respectively, and ϕ be a differentiable mapfrom X to Y , then ϕ(X) is subset of Lebesque measure0 of Y [31]. For an autonomous system,

dx

dt= f (x1, . . . , xn) (A.1)

86 S.-Y. Li, Z.-M. Ge

where x = [x1, . . . , xn]T is a state vector, the func-tion f = [f1, . . . , fn]T is defined on D ⊂ Rn and‖x‖ ≤ H > 0. Let x = 0 be an equilibrium point forthe system (A.1). Then

f (0) = 0 (A.2)

For a nonautonomous systems,

x = f (x1, . . . , xn+1) (A.3)

where x = [x1, . . . , xn+1]T, the function f =f1, . . . , fn]T is defined on D ⊂ Rn × R+ here t =xn+1 ⊂ R+. The equilibrium point is

f (0, xn+1) = 0 (A.4)

Definition The equilibrium point for the system (A.1)is pragmatically asymptotically stable provided thatwith initial points on C which is a subset of Lebesquemeasure 0 of D, the behaviors of the correspondingtrajectories cannot be determined, while with initialpoints on D − C, the corresponding trajectories be-have as that agree with traditional asymptotical stabil-ity [21–23].

Theorem Let V = [x1, . . . , xn]T : D → R+ be posi-tive definite and analytic on D, where x1, x2, . . . , xn

are all space coordinates such that the derivative of V

through (A.1) or (A.3), V , is negative semidefinite of[x1, x2, . . . , xn]T.

For autonomous system, let X be the m-manifoldconsisted of point set for which ∀x = 0, V (x) = 0 andD is a n-manifold. If m + 1 < n, then the equilib-rium point of the system is pragmatically asymptoti-cally stable.

For nonautonomous system, let X be the m + 1-manifold consisting of point set of which ∀x = 0,V (x1, x2, . . . , xn) = 0 and D is n + 1-manifold. Ifm + 1 + 1 < n + 1, i.e., m + 1 < n then the equi-librium point of the system is pragmatically asymp-totically stable. Therefore, for both autonomous andnonautonomous system, the formula m + 1 < n is uni-versal. So the following proof is only for autonomoussystem. The proof for nonautonomous system is simi-lar.

Proof Since every point of X can be passed by a tra-jectory of (A.1), which is onedimensional, the collec-tion of these trajectories, A, is a (m + 1)-manifold[24, 25].

If m + 1 < n, then the collection C is a subset ofLebesque measure 0 of D. By the above definition,the equilibrium point of the system is pragmaticallyasymptotically stable.

If an initial point is ergodically chosen in D, theprobability of that the initial point falls on the collec-tion C is zero. Here, equal probability is assumed forevery point chosen as an initial point in the neighbor-hood of the equilibrium point. Hence, the event thatthe initial point is chosen from collection C does notoccur actually. Therefore, under the equal probabil-ity assumption, pragmatical asymptotical stability be-comes actual asymptotical stability. When the initialpoint falls on D −C, V (x) < 0, the corresponding tra-jectories behave as they agree with traditional asymp-totical stability because by the existence and unique-ness of the solution of initial-value problem, these tra-jectories never meet C.

In (2.11), V is a positive definite function of n vari-ables, i.e., p error state variables and n − p = m dif-ferences between unknown and estimated parameters,while V = eTCe is a negative semidefinite function ofn variables. Since the number of error state variables isalways more than one, p > 1, m+1 < n is always sat-isfied, by pragmatical asymptotical stability theorem,we have

limt→∞ e = 0 (A.5)

and the estimated parameters approach the uncertainparameters. The pragmatical adaptive control theoremis obtained. Therefore, the equilibrium point of thesystem is pragmatically asymptotically stable. Underthe equal probability assumption, it is actually asymp-totically stable for both error state variables and para-meter variables. �

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