DOI 10.1007/s11071-010-9847-7 O R I G I N A L PA P E R
Pragmatical adaptive synchronization of different orders
chaotic systems with all uncertain parameters via nonlinear
control
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 by pragmatical asymptotically stability theorem is pro-posed in this paper. Based on this theorem and non-linear control theory, a new adaptive synchronization scheme to design controllers can be obtained and espe-cially the constraints for minimum values of feedback gain K in controllers can be derived. This new strat-egy shows that the constraint values of feedback gain K are related to the error of unknown and estimated parameters if the goal system is given. Through this new strategy, an appropriate feedback gain K can be always decided easily to obtain controllers achieving adaptive 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–van der 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· Adaptive synchronization· Uncertain parameters · Nonlinear control· Pragmatical asymptotically stability theorem
S.-Y. Li· Z.-M. Ge (
)Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, Republic of China
e-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 been intensively studied over the past several decades. The phenomenon of chaos has attracted widespread at-tention amongst mathematicians, physicists and en-gineers. Chaos has also been extensively studied in many fields, such as chemical reactions, power con-verters, biological systems, information processing, secure communications, etc. [1–9].
Synchronization of chaotic systems is essential in variety of applications, including secure communi-cation, physiology, nonlinear optics and so on. Ac-cordingly, following the initial work of Pecora and Carroll [10] in synchronization of identical chaotic systems with different initial conditions, many ap-proaches have been proposed for the synchronization of chaotic and hyperchaotic systems such as linear and nonlinear 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 many other existing synchronization methods mainly con-cern the synchronization of two identical chaotic or hyperchaotic systems, the methods of synchronization of two different chaotic or hyperchaotic systems are far from being straightforward because of their dif-ferent structures and parameter mismatch. Moreover,
most of the methods are used to synchronize only two systems 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 it much more important to synchronize two different hy-perchaotic systems with uncertain parameters in recent years. In this regard, some works on synchronization of two different hyperchaotic systems with uncertain parameters have been performed [19,20].
In current scheme of adaptive synchronization, traditional Lyapunov stability theorem and Barbalat lemma are used to prove that the error vector ap-proaches zero as time apap-proaches infinity, but the question that why those estimated parameters also approach the uncertain values remains no answer [21–23]. In this article, pragmatical asymptotically stability theorem and an assumption of equal proba-bility for ergodic initial conditions [24,25] are used to prove strictly that those estimated parameters ap-proach the uncertain values. Moreover, traditional adaptive chaos synchronization in general is limited for the same system.
Recently, Meng and Wang [26] proposed a new control 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 generalized synchronization between two chaotic systems can be really attained via these conditions and the control law mentioned above. In this paper, we further expand the innovative idea proposed by [26] to discuss the adaptive synchronization with all uncertain parame-ters in master system by pragmatical asymptotically stability theorem. Based on this theorem and nonlin-ear control theory, the constraints of feedback gain Kin controllers for adaptive synchronization are pro-posed and good effectiveness is shown in simulation results.
The layout of the rest of the paper is as follows. In Sect.2, adaptive synchronization scheme is presented. In Sect.3, two simulation cases are illustrated to verify the new adaptive scheme. In Sect.4 conclusions are given. Pragmatical asymptotically stability theorem is enclosed in AppendixA.
2 Adaptive synchronization scheme Consider the following master chaotic system:
˙x = Ax + Bf (x) (2.1)
where x= [x1, x2, . . . , xn]T∈ Rndenotes 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 = ˆAy+ ˆBg(y) + u(t) (2.2)
where y= [y1, y2, . . . , yn]T∈ Rndenotes a state vec-tor, ˆA and ˆB are n× n estimated coefficient matri-ces, g is a nonlinear continuous vector function, and u(t )= [u1(t ), u2(t ), . . . , un(t )]T∈ Rnis a control in-put vector.
Function f (x) is globally Lipschitz continuous; i.e., the following condition is satisfied: For function f (z), there exists constant L > 0, for any two differ-ent z1, z2∈ Rn, such that
f (z1)− f (z2) ≤Lz1− z2 (2.3)
Property 1 [28] For a matrix A, letA indicate the norm of A induced by the Euclidean vector norm as follow: 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 xTdenotes the transpose of the vector x. Our goal is to design a controller u(t) so that the state vector of the chaotic system (2.2) asymptotically approaches the state vector of the master system (2.1). Theorem 1 If the parametric update laws are chosen as:
˙ˆam= −˜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 paramemas-ters,
amare the unknown parameters in the master system, ˆam are the estimated parameters in the slave system, and˜am= am− ˆamare the errors between the unknown and estimated parameters; and the controller u is de-signed as:
u= − ˙yout+ K(x − y) + Bf (y) + Ay − ˜a2
m(e− 1), m = 1 ∼ p (2.7)
where ˙youtis ˙y without controllers, gain K = diag(k1, . . . , kn)satisfies following constraint:
min(K)
(LB + A + max(pi=1˜ai2))>1 (2.8)
then master system (2.1) and slave system (2.2) will achieve the adaptive synchronization.
Proof The synchronization errors between master and slave 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 + Bf (x)− f (y) − Ke + ˜a2
m(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+ ˜a2T˜a2+ · · · + ˜apT˜ap =1 2 e2+ ˜a2 1+ ˜a22+ · · · + ˜ap2 >0 (2.11) The time derivative of V(t) in along any trajectory of (2.10) is
˙V (t) = eTAe+ eTBf (x)− f (y)− eTKe + eT˜a2
m(e− 1) + ˜a12e+ ˜a22e+ · · · + ˜ap2e ≤ Ae2+ eBf (x)− f (y)
− min(K)e2+˜a2 1+ ˜a 2 2+ · · · + ˜a 2 p e2 ≤ Ae2+ LBe2− min(K)e2
+ max˜a2 1+ ˜a22+ · · · + ˜ap2 e2 =A + LB − min(K) +˜a2 10+ ˜a202 + · · · + ˜ap20 e2 (2.12)
where˜a10,˜a20, . . . ,˜ap0are the initial values of˜a1,˜a2, . . . ,˜ap, separately. If min(K) satisfies (2.8), then
˙V (t) ≤ 0. Let R = min(K) − LB − A − (˜a2 10+ ˜a2 20 + · · · + ˜a 2 p0), then ˙V ≤ −Re 2, where R > 0. ˙V is a negative semidefinite function of e and parame-ter differences. In current scheme of adaptive control of chaotic motion [21–23], traditional Lyapunov sta-bility theorem and Barbalat lemma are used to prove the error vector approaches zero, as time approaches infinity. But the question, why the estimated or given parameters also approach to the uncertain or goal pa-rameters, remains no answer. By pragmatical asymp-totical stability theorem, the question can be answered
strictly.
3 Numerical simulations
In this section, there are two examples for our new adaptive scheme in numerical simulation. In Case 1, two identical Lorenz systems with different initial con-ditions and parameters are used for master and slave systems to show the effectiveness of the new scheme. In Case 2, two completely different systems, Rössler’s system and Mathieu–van der Pol system, are regarded as master and slave systems separately.
Case 1 Two identical Lorenz systems with different initial conditions and parameters.
Master Lorenz system [29]: ⎡ ⎣˙x˙x12 ˙x3 ⎤ ⎦ = ⎡ ⎣cx1a(x2− x1x3− x1)− x2 x1x2− bx3 ⎤ ⎦ = ⎡ ⎣−ac −1a 00 0 0 −b ⎤ ⎦ ⎡ ⎣xx21 x3 ⎤ ⎦ + ⎡ ⎣00 −1 00 0 0 0 1 ⎤ ⎦ ⎡ ⎣x1x30 x1x2 ⎤ ⎦ (3.1) where A= ⎡ ⎣−ac −1a 00 0 0 −b ⎤ ⎦
B= ⎡ ⎣00 −1 00 0 0 0 1 ⎤ ⎦ and f (x) = ⎡ ⎣x1x30 x1x2 ⎤ ⎦
x1, x2, x3are 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 behavior of (3.1) is shown in Fig.1.
Slave Lorenz system: ⎡ ⎣˙y˙y12 ˙y3 ⎤ ⎦ = ⎡ ⎣ˆcy1ˆa(y− y21− yy31)− y+ u2+ u1 2 y1y2− ˆby3+ u3 ⎤ ⎦ (3.2)
y1, y2, y3are states of slave system, when initial con-dition (y10, y20, y30)= (5, 10, 5) and initial estimated parameters ˆa0= 8, ˆb0= 3 and ˆc0= 25. Thus, the er-rors between the estimated and uncertain parameters can be given as follows:
⎧ ⎪ ⎨ ⎪ ⎩ ˜a0= a − ˆa0= 2 ˜b0= b − ˆb0= −1/3 ˜c0= c − ˆc0= 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= ⎡ ⎣550 520 00 0 0 50 ⎤ ⎦ (3.5)
By (3.2), we get f (y)= [0 y1y3 y1y2]T, the para-metric update laws and corresponding controllers can be decided by (2.6) and (2.7). Then we can obtain the time derivative of V (t) is semidefinite as follow:
˙V ≤A + LB − min(K) + max˜a2 0+ ˜b 2 0+ ˜c 2 0 e2 = (44.1842 − 50)e2 = (−5.8158)e2< 0 (3.6)
which is a negative semidefinite function of e1, e2, e3, ˜a, ˜b, ˜c. The Lyapunov asymptotical stability theorem is not satisfied. We cannot obtain that common origin of error dynamics and parameter dynamics is asymp-totically stable. By pragmatical asympasymp-totically
stabil-Fig. 1 Projections of phase
portrait of chaotic Lorenz system with a= 10, b= 8/3, and c = 28
Fig. 2 Time histories of
errors for Case 1
Fig. 3 Time histories of
parametric 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 ˆa, ˆb, ˆc 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
eapproaches zero and the estimated parameters also approach the uncertain parameters. The equilibrium point is pragmatically asymptotically stable. Under the assumption of equal probability, it is actually asymp-totically stable. The simulation results are shown in Figs.2and3.
Fig. 4 Projections of phase
portrait of chaotic Rössler’s system 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 and Mathieu–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 0 1 a 0 0 0 0 −c 0 1 0 1 0 ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ x1 x2 x3 x4 ⎤ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎣ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ 0 0 b+ x1x3 0 ⎤ ⎥ ⎥ ⎦ (3.7) where A= ⎡ ⎢ ⎢ ⎣ 0 −1 −1 0 1 a 0 0 0 0 −c 0 1 0 1 0 ⎤ ⎥ ⎥ ⎦ , B= ⎡ ⎢ ⎢ ⎣ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎤ ⎥ ⎥ ⎦ and f (x) = ⎡ ⎢ ⎢ ⎣ 0 0 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 −(ˆa1+ ˆb1z3)z1− (ˆa1+ ˆb1z3)z31− ˆc1z2+ ˆd1z3+ u2 z4+ u3 −ˆe1z3+ ˆf1(1− z23)z4+ ˆg1z1+ u4 ⎤ ⎥ ⎦ (3.8) z1, z2, z3and z4are states of slave system, ˆa1, ˆb1,ˆc1, ˆd1,ˆe1, ˆf1 and ˆg1 are uncertain parameters. This sys-tem exhibits chaos when the parameters of syssys-tem are ˆa1 = 10, ˆb1 = 3, ˆc1 = 0.4, ˆd1 = 70, ˆe1 = 1,
ˆ
f1= 5, ˆg1= 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. We chooseˆa1= 3, ˆb1= 1, ˆc1= 0.4, ˆd1= 5, ˆe1= 1, ˆf1= 1, ˆg1= 0.1, ˆa = 0, ˆb = 0, ˆc = 0 and the initial states
Fig. 5 Projections of phase
portrait of new chaotic Mathieu–van der Pol system withˆa1= 10,
ˆb1= 3, ˆc1= 0.4, ˆd1= 70, ˆe1= 1, ˆf1= 5, ˆg1= 0.1
of system are (z10, z20, z30, z40)= (5, 7, 9, 10). The initial values of ˆa10, ˆb10,ˆc10, ˆd10,ˆe10, ˆf10,ˆg10,ˆa0, ˆb0 andˆc0can be decided as:ˆa10= 3, ˆb10= 1, ˆc10= 0.4, ˆd10= 5, ˆe10= 0.3, ˆf10= 1, ˆg10= 0.1, ˆa0= 0, ˆb0= 0 and ˆc0= 0. Therefore, the errors of the estimated un-known parameters can be decided as follow:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˜a10= a1− ˆa10= −3 ˜b10= b1− ˆb10= −1 ˜c10= c1− ˆc10= −0.4 ˜d10= d1− ˆd10= −5 ˜e10= e1− ˆe10= −1 ˜ f10= f1− ˆf10= −1 ˜g10= g1− ˆg10= −0.1 ˜a0= a − ˆa0= 0.2 ˜b0= b − ˆb0= 0.2 ˜c0= c − ˆc0= 5.7 (3.9)
Through (3.7) and (3.9),A = 5.8780, B = 1, and max(˜a2+ ˜b2+ ˜c2+ ˜a21+ · · · + ˜g21)= 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 0 0 87 0 0 0 0 83 0 0 0 0 80 ⎤ ⎥ ⎥ ⎦ (3.11)
In (3.8) we have f (z)= [0 0 b + z1z30]T, the para-metric update laws and corresponding controllers can be decided by (2.6) and (2.7). Then we can obtain the time derivative of V (t) is negative semidefinite as fol-low:
˙V ≤A + LB − min(K) + max˜a2+ ˜b2+ ˜c2+ ˜a2
1+ · · · + ˜g 2 1 e2 = (75.7080 − 80)e2 = (−4.2920)e2< 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 and parameter 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 ˆa, ˆb, ˆc, ˆa1, ˆb1,ˆc1, ˆd1,ˆe1, ˆf1 and ˆg1
Fig. 6 Time histories of
errors for Case 2
Fig. 7 Time histories of
parametric errors˜a, ˜c, ˜a1 and ˜b1for 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 asymptotically stability theorem, error vector e approaches zero and the estimated parameters also approach the uncer-tain parameters. The equilibrium point is pragmati-cally asymptotipragmati-cally stable. Under the assumption of equal probability, it is actually asymptotically
sta-ble. The simulation results are shown in Figs. 6, 7 and8.
4 Conclusions
In this paper, a new adaptive synchronization scheme which is derived by pragmatical asymptotically
stabil-Fig. 8 Time histories of
parametric errors ˜c1, ˜d1,˜e1, ˜f1and ˜g1for Case 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 minimum values in feedback gain K can be calculated by the sum 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 gain K and the sum of the errors of unknown parameters and estimated parameters. By applying this new rela-tion formula, an appropriate feedback gain K can be decided easily to obtain controllers achieving adaptive synchronization. Simulation results show that not only for two identical nonlinear systems with all unknown parameters 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 effecadap-tive in adapadap-tive synchronization of various kinds of different nonlin-ear systems.
Acknowledgements This research was supported by the Na-tional Science Council, Republic of China, under Grant Number NSC 96-2221-E-009-145-MY3.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendix A: Pragmatical asymptotical stability theory
The stability for many problems in real dynamical systems is actual asymptotical stability, although may not be mathematical asymptotical stability. The math-ematical asymptotical stability demands that trajecto-ries from all initial states in the neighborhood of zero solution must approach the origin as t→ ∞. If there are only a small part or even a few of the initial states from which the trajectories do not approach the origin as t→ ∞, the zero solution is not mathematically as-ymptotically stable. However, when the probability of occurrence of an event is zero, it means the event does not occur actually. If the probability of occurrence of the event that the trajectories from the initial states are that they do not approach zero when t→ ∞, is zero, the stability of zero solution is actual asymptotical bility though it is not mathematical asymptotical sta-bility. In order to analyze the asymptotical stability of the equilibrium point of such systems, the pragmatical asymptotical stability theorem is used.
Let X and Y be two manifolds of dimensions m and n(m < n), respectively, and ϕ be a differentiable map from X to Y , then ϕ(X) is subset of Lebesque measure 0 of Y [31]. For an autonomous system,
dx
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 for the 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 that with initial points on C which is a subset of Lebesque measure 0 of D, the behaviors of the corresponding trajectories cannot be determined, while with initial points 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-manifold consisted of point set for which∀x = 0, ˙V (x) = 0 and
D 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. If m+ 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 and nonautonomous system, the formula m+ 1 < n is uni-versal. So the following proof is only for autonomous system. 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 of Lebesque measure 0 of D. By the above definition, the equilibrium point of the system is pragmatically asymptotically stable.
If an initial point is ergodically chosen in D, the probability of that the initial point falls on the collec-tion C is zero. Here, equal probability is assumed for every point chosen as an initial point in the neighbor-hood of the equilibrium point. Hence, the event that the initial point is chosen from collection C does not occur actually. Therefore, under the equal probabil-ity assumption, pragmatical asymptotical stabilprobabil-ity be-comes actual asymptotical stability. When the initial point 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 = eTCeis a negative semidefinite function of nvariables. Since the number of error state variables is always more than one, p > 1, m+ 1 < n is always sat-isfied, by pragmatical asymptotical stability theorem, we have
lim
t→∞e= 0 (A.5)
and the estimated parameters approach the uncertain parameters. The pragmatical adaptive control theorem is obtained. Therefore, the equilibrium point of the system is pragmatically asymptotically stable. Under the equal probability assumption, it is actually asymp-totically stable for both error state variables and
para-meter variables.
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