# Stability Analysis and Control of a New Smooth Chua's System

Authors: Guopeng Zhou; Jinhua Huang; Xiaoxin Liao; Shijie Cheng
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with the stability analysis and control of a new smooth Chua's system. Firstly, the chaotic characteristic of the system is confirmed with the aid of the Lyapunov exponents. Secondly, it is proved that the system has globally exponential attractive set and positive invariant set. For the three unstable equilibrium points of the system, a linear controller is designed to globally exponentially stabilize the equilibrium points. Then, a linear controller and an adaptive controller are, respectively, proposed so that two similar types of smooth Chua's systems are globally synchronized,
and the estimation errors of the uncertain parameters converge to zero ast tends to infinity. Finally, the numerical simulations are also presented.


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## Body

## 1. Introduction

It is well known that Chua’s system is the first analog circuit to realize chaos in experiments. The original Chua’s system is described by the following ordinary differential equations [1]:
(1)x˙=p(y-x-g(x)),y˙=x-y+z,z˙=-qy,
where x,y,z∈R are state variables and g(x)=Gbx+(1/2)(Ga-Gb)(|x-E|-|x-E|), p>0, q>0, and Ga, Gb are constants. Due to the form of a simple circuit, there are a large literature on the dynamical behavior of Chua’s system [2–8]. By changing the parameters or the corresponding functions of Chua’s system, the chaotic phenomenon is very rich, and it is more convenient to study the chaotic mechanism and characteristics [6–8].For the chaotic systems, Lagrange stability, stability of equilibrium points and, synchronization are three important problems which attracted more and more attention (refer to [9–16] and the reference therein). In [11–13], the authors studied the Lagrange stability by applying the attractive set and positive invariant set of the chaotic systems. Moreover, the researchers examined the stabilization of the unstable equilibrium points and the synchronization control for the chaotic systems with linear controllers [7, 14]. Recently, adaptive controllers are used in synchronous control of chaotic systems when the parameters of the systems are uncertain [15–17].Motivated by the previous results, the main purpose of this paper is to construct a new smooth Chua’s system and investigate the stability and control problems. More precisely, we will consider the following smooth Chua’s system:(2)x˙=p(x+y-xln1+x2),y˙=x-y+z,z˙=-qy,
where x,y,z∈R are state variables and p>0, q>0 are constants.We will show that the chaotic characteristics are depended on the parametersp and q and the initial state values of the system (2). All equilibrium points of system (2) are examined to be unstable when p=11 and q=14.87 (see in Section 3). By computing with MATLAB, the maximum Lyapunov exponent of the system (2) is 0.0021, where the embedding dimension m is 3 and the delay time τ is 5. Since the maximum Lyapunov exponent is greater than 0, the Chua’s system is chaotic. It will be of great significance if the solution of (2) is ultimately bounded (Lagrange asymptotically stable). The chaotic phase diagrams of such system is obtained by simulation with MATLAB. Figure 1 shows the phase diagrams of the system (2) with x(0)=-3, y(0)=2, and z(0)=1, the phase diagrams of Chua’s system exhibits chaotic.The phase portraits of Chua’s system (2).
(a)
(b)
(c)The remains of this paper are organized as follows. The existence of globally exponential attractive set and positive invariant set for the system (2) will be discussed in Section 2. The asymptotic stability of the equilibrium points will be studied in Section 3, and the synchronization control for two similar types of the Chua’s systems will be discussed in Section 4. In Section 5, we will give the numerical simulations to demonstrate the correctness of our results, and finally we will give the conclusions in Section 6.


## 2. Existence of Globally Exponential Attractive Set and Positive Invariant Set

The Lagrange stability analysis of the system (2) will be studied in this section. To do so, we first give two definitions [7].Definition 1.
If there exists a radially unbounded, positive definite Lyapunov functionV(X(t)) and positive numbers l>0, α>0 such that for all X0∈R3, when V(X0)>l, the solution of the system (2), X(t)=(x(t),y(t),z(t)) along V(X(t)) satisfies |V(X(t))-l|≤|V(x0)-l|e-α(t-t0), then system (2) has a globally exponential attractive set Ω≜{X∣V(X(t))<l}.Definition 2.
LetΩ⊆R3, if ∀X0∈Ω and for all t≥t0, X(t,t0,X0)⊆Ω, then Ω is called positive invariant set of the system (2).It is easy to prove that the globally exponential attractive set is positive invariant. A system with global attractive set is always called Lagrange globally asymptotically stable system or ultimately bounded dissipative system. For the system (2), we will prove the following the Lagrange stability results.Theorem 3.
The system (2) has the following globally exponential attractive and positive invariant set:
(3)Ω={|x|≤|xb|=min{|xb1|,|xb2|}y2+1qz2-ϵyz≤(5/4η)xb2-τ1μM(G2-)/λM(G2+)|xb1|>e((2-(ϵ/4))2/(2-ϵq-(ϵ/4)))+(ϵ/4)+2-1|xb2|=supΩ-|x|,Ω-≜{xb12ln(1+xb12)-μM(G1-(xb1))/λM(G1+)x∣1px2+y2+1qz2-ϵyz=xb12ln(1+xb12)-μM(G1-(xb1))/λM(G1+)},
where 0<ϵ≪1,
(4)G1+≜[1p0001-ϵ20-ϵ21q],G2+≜[1-ϵ2-ϵ21q],
are symmetric positive definite matrices, and
(5)G1-≜[2(1-ln1+xb12)2-ϵ22ϵq-2ϵ2-ϵ2ϵ2-ϵ],G2-≜[ϵq-2ϵ2ϵ2-ϵ],
are symmetric negative definite matrices. λM(G1+), μM(G1-), λM(G2+), and μM(G2-) are the maximum eigenvalues of G1+, G1-, G2+, and G2-, respectively. ϵ,xb1, and xb2 in (3) are chosen to guarantee that G1+, G2+ are positive definite and G1-, G2- are negative definite. τ1, τ2, and η are chosen such that τ1>0, τ2>0, η>0, τ1+τ2=1, and τ2μM(G2-)+η≤0.Proof.
The proof is divided into three steps.
Step I. The existence of ϵ, xb1 such that G1+ is positive definite and G1- is negative definite.
It is well known thatG1+ is positive definite if and only if all the order principal minors of G1+ are positive [18]. That is,
(6)1p>0,|1p001|>0,|1p0001-ϵ20-ϵ21q|=1p(1q-ϵ24)>0.
Let |ϵ|<2/q; it is easy to see that G1+ is positive definite since p>0, q>0.
Next, let(7)S11=[2(1-ln1+xb12)22ϵq-2],S12=[-ϵ2ϵ2],S21=[-ϵ2,ϵ2],S22=-ϵ.
Then,
(8)G1-=[S11S12S21S22].
From Schur theorem [18], G1- is negative definite if and only if S11<0, S22-S21S11-1S12<0. Let |ϵ|, xb1 satisfy
(9)0<ϵ≪1,|xb1|>e((2-(ϵ/4))2/(2-ϵq-(ϵ/4)))+(ϵ/4)+2-1.
It is easy to verify that G1- is negative definite.
Step II. Existence of globally exponential attractive set of x.
Letλm(G1+) and λm(G1-) be the minimum eigenvalues of G1+ and G1-, respectively. By constructing a radially unbounded Lyapunov function as follows:
(10)V=1px2+y2+1qz2-ϵyz=[xyz]TG1+[xyz],
then, one can obtain that(11)λm(G1+)(x2+y2+z2)≤V≤λM(G1+)(x2+y2+z2).
The time derivative ofV along the system (2) is given by
(12)dVdt|(2)=2pxx˙+2yy˙+2qzz˙-ϵy˙z-ϵyz˙=2pxp(x+y-xln1+x2)+2y(x-y+z)+2q(-qy)-ϵ(x-y+z)z-ϵy(-qy)=[xyz]T[2(1-ln1+x2)2-ϵ22ϵq-2ϵ2-ϵ2ϵ2-ϵ][xyz]=[xyz]TG1-[xyz]-2(ln1+x2-ln1+xb12)x2.
If|x|>|xb1|, one has
(13)dVdt|(2)≤[xyz]TG1-[xyz]≤μM(G1-)λM(G1+)V.
Then, the following inequality holds
(14)V(X(t,t0,X0))≤V(X0)e(μM(G1-)/λM(G1+))(t-t0).
Thus, the trajectory of the system (2) will exponentially decay into the area U={x∣|x|≤|xb1|} if |x|>|xb1|.
When the trajectory of the system (2) is in the area U, it holds that
(15)dVdt|(2)=[xyz]TG1-[xyz]+2(ln1+xb12-ln1+x2)x2≤μM(G1-)λM(G1+)V+2(ln1+xb12-ln1+x2)x2≤μM(G1-)λM(G1+)V+xb12ln(1+xb12)=μM(G1-)λM(G1+)(V-xb12ln(1+xb12)-μM(G1-)/λM(G1+)).
LetVl≜(xb12ln(1+xb12))/(-μM(G1-)/λM(G1+)). If V(X0)>Vl and V(X(t,t0,X0))>Vl, it holds that
(16)V(X(t,t0,X0))-Vl≤(V(X0)-Vl)e(μM(G1-)/λM(G1+))(t-t0).
LetΩ-≜{X∣V(X)=Vl}, |xb2|≜supx∈Ω-|x|. Then, V(X(t,t0,X0))-Vl exponentially decays when |x|≥|xb|≜min{|xb1|,|xb2|}. Thus, Ωx≜{x∣|x|≤|xb|} is the globally exponential attractive set of x.
Step III. Existence of globally exponential attractive set of y and z.
Let(17)G2+≜[1-ϵ2-ϵ21q],G2-≜[ϵq-2ϵ2ϵ2-ϵ],
then, G2+ is a positive matrix if ϵ∈(0,2/q), and G2- is a negative matrix if ϵ∈(0,8/(4q+1)).
At the same time, it is easy to examine thatG2+ is positive and G2- is negative if let ϵ∈(0,8/(4q+1)), since q>0 and 8/(4q+1)=2/(q+1/4)≤2/q.
Letλm(G2+), λM(G2+) be, respectively, minimum and maximum eigenvalues of G2+, and let μm(G2-) and μM(G2-) be the minimum and maximum eigenvalues of G2-, respectively. Since |x|≤|xb|, a radially unbounded and positive definite Lyapunov function about y and z is constructed as follows:
(18)W=y2+1qz2-ϵyz=[yz]TG2+[yz].
Then, the time derivative of W along the system (2) yields
(19)dWdt|(2)=2yy˙+2qzz˙-ϵy˙z-ϵyz˙=[yz]TG2-[yz]+2xy-ϵxz≤μM(G2-)(y2+z2)+2|xb||y|+ϵ|xb||z|≤τ1μM(G2-)(y2+z2)+τ2μM(G2-)(y2+z2)+ηy2+xb2η+ηz2+xb24η≤τ1μM(G2-)λM(G2+)W+(τ2μM(G2-)+η)(y2+z2)+54ηxb2≤τ1μM(G2-)λM(G2+)(W-(5/4η)xb2-τ1μM(G2-)/λM(G2+)),
where τ1,τ2, and η are chosen such that τ1>0, τ2>0, η>0, τ1+τ2=1, and τ2μM(G2-)+η≤0.
LetWl≜(5/4η)xb2/(-τ1μM(G2-)/λM(G2+)); if W(X0)>Wl and W(X(t,t0,X0))>Wl, it holds that
(20)W(X(t,t0,X0))-Wl≤(W(X0)-Wl)e(τ1μM(G2-)/λM(G2+))(t-t0).
Similarly, ifW(X0)≤Wl, the state trajectory will stay in the area such that W(X(t,t0,X0))≤Wl holds. Hence, y and z are exponentially decreased and ultimately enter into the attractive region W(X(t,t0,X0))≤Wl, that is,
(21)y2+1qz2-ϵyz≤(5/4η)xb2-τ1μM(G2-)/λM(G2+).
Combining Steps I, II, and III, Theorem3 is obtained and the proof is completed.Remark 4.
In this section, a constructive method is proposed to prove the main results of existence of globally exponential attractive set and positive invariant set for the Chua’s systems. By constructing the matricesG1+, G1-, G2+, G2-, and Lyapunov function candidate V and W, the problem is solved ingeniously.


## 3. Global Linear Stabilization of the Equilibrium Points

In this section, the stability of the equilibrium points for the system (2) will be discussed with the aid of a linear controller.Firstly, it is easy to examine that the system (2) has three equilibrium points:
(22)S0=(0,0,0),S+=(e2-1,0,-e2-1),S-=(-e2-1,0,e2-1).
Moreover, the equilibrium points of the system is independent of parameters p and q. It should be mentioned that, however, the stability of the equilibrium points is depended on p and q. In the following, we will design a linear controller to stabilize the unstable equilibrium points.Letp=11, q=14.87, and (x*,y*,z*) be any equilibrium point of the system (2), the corresponding Jacobian matrix is given by
(23)J0=[p(-ln1+x*2+11+x*2)p01-110-q0].
Then the characteristic equation of the corresponding local linearization system of system (2) is as follows:
(24)a0λ3+a1λ2+a2λ+a3=0,
where a0=1, a1=1-pw, a2=-q-p-pw, a3=pqw, and w=-ln1+x*2+(1/(1+x*2)).(i)
Ifx*=0, one has a0=1>0, a1=-10<0. According to Hurwitz stability criterion [19], the necessary condition of stable equilibrium point is the same sign of the coefficients of the characteristic equation. Consequently, the equilibrium S0 is unstable.(ii)
Ifx*=±e2-1, one has a0=1, a1=10.5113>0, and a2=-16.3587<0. Similarly, one can obtain that S+ and S- are the unstable equilibrium points.Now, we will discuss how to design a linear feedback controller such that the unstable equilibrium points are exponentially stable. For this purpose, we add the control terms to the system (2):
(25)x˙=p(x+y-xln1+x2)+u1,y˙=x-y+z+u2,z˙=-qy+u3.LetX*=(x*,y*,z*) be any of the three unstable equilibrium points, let X(t)=(x(t),y(t),z(t)) be the solution of the system (25), and X~(t)=(x~,y~,z~)=X(t)-X*, then the error system is given by
(26)x~˙=p(x~+y~-xln1+x2+x*ln1+x*2)+u1,y~˙=x~-y~+z~+u2,z~˙=-qy~+u3.Definition 5.
∀
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3, if ui(i=1,2,3) is appropriately selected such that
(27)|X~(t,t0,X0)|≤|X(0)-X*|e-α(t-t0)
holds (α>0). Then, the control input ui(i=1,2,3) can globally exponentially stabilize the equilibrium point X*.Theorem 6.
If the following linear controller is added to the error system (26),
(28)u1=-pσxx~,u2=u3=0,
where σx is any parameter given beforehand such that σx>2; then the equilibrium point X* is globally exponentially stable.Proof.
The proof is divided into two steps.(1)
We will find the existence ofϵ>0 such that G3+ is positive definite and G3- is negative definite, where
(29)G3+≜[1p0001-ϵ20-ϵ21q],G3-≜[2(1-σx)2-ϵ22ϵq-2ϵ2-ϵ2ϵ2-ϵ].It is easy to obtain that G3+ is positive definite if |ϵ|<2/q. Now we focus on choosing ϵ>0 such that G3- is negative definite. By the Schur theorem [18], G3- is negative definite if and only if
(30)ϵ>0,ϵq+ϵ4-2<0,(2(σx-1)-ϵ4)(2-ϵq-ϵ4)>(2-ϵ4)2,
that is
(31)ϵ>0,ϵ<84q+1,σx>64-16ϵ+ϵ264-32ϵq-8ϵ+1+ϵ8.
Obviously, G3- is negative definite if 0<ϵ≪1, σx>2.(2)
We construct a positive definite and radially unbounded Lyapunov function to prove the stability of closed-loop systems (26) with controller (28) which is written as
(32)V1(X~(t,t0,X~0))=1px~2+y~2+1qz~2-ϵy~z~=[x~y~z~]TG3+[x~y~z~].
Supposeλm(G3+) and λM(G3+) are minimum and maximum eigenvalues of the positive definite matrix G3+, respectively. Then, we have
(33)λm(G3+)(x~2+y~2+z~2)≤V1≤λM(G3+)(x~2+y~2+z~2).
Letf(x~)=xln1+x2-x*ln1+x*2. Obviously, xln1+x2 is a monotonically increasing odd function. Then, (i)
ifx~≥0, then f(x~)≥0 and
(34)0≤x~f(x~)≤+∞,(ii)
ifx~≤0, then f(x~)≤0 and
(35)0≤x~f(x~)≤+∞.Thus
(36)0≤x~f(x~)≤+∞,∀x~∈R.
DifferentiatingV1 with respect to time yields
(37)dV1dt|(26)=2px~x~˙+2y~y~˙+2qz~z~˙-ϵy~˙z~-ϵy~z~˙=[x~y~z~]TG3-[x~y~z~]-2x~f(x~)≤μM(G3-)(x~2+y~2+z~2)=μM(G3-)λM(G3+)λM(G3+)(x~2+y~2+z~2)≤μM(G3-)λM(G3+)V1,
where μM(G3-) is the maximum eigenvalues of the negative definite matrix G3-. Then,
(38)V1(X~(t,t0,X~0))≤V1(X~0)e(μM(G3-)/λM(G3+))(t-t0),x~2(t)+y~2(t)+z~2(t)≤V1(X~(t,t0,X~0))λm(G3+)≤V1(X~0)λm(G3+)e(μM(G3-)/λM(G3+))(t-t0).
Hence,x~2(t), y~2(t), and z~2(t) converge to zero exponentially. According to Definition 5, the equilibrium point X* is globally exponentially stable. The proof is complete.Remark 7.
A constructive method to stabilize the unstable equilibrium points is proposed in this section, matricesG3+, G3-, and Lyapunov function candidate V1 are given. Then, a linear controller is obtained to solve the problem. Comparing with nonlinear controller, linear controller is easy to implement in reality.


## 4. Globally Exponential Synchronization of Two Chua’s Systems

In this section, the globally exponential synchronization of two Chua’s systems will be discussed. The drive system is given by(39)x˙d=p(xd+yd-xdln1+xd2),y˙d=xd-yd+zd,z˙d=-qyd,
and the response system is described as follows:
(40)x˙r=p(xr+yr-xrln1+xr2)+u1,y˙r=xr-yr+zr+u2,z˙r=-qyr+u3,
where the subscripts d and r denote the drive and response systems and ui(i=1,2,3) is feedback control input which satisfies ui(0,0,0)=0.Letex=xr-xd, ey=yr-yd, and ez=zr-zd; one obtains that
(41)e˙x=p(ex+ey-xrln1+xr2+xdln1+xd2)+u1,e˙y=ex-ey+ez+u2,e˙z=-qey+u3.Theorem 8.
If the following controller is added to the error system (41),
(42)u1=-pδxx~,u2=u3=0,
where δx is any parameter given beforehand with δx>2; then the zero solution of (41) is globally exponentially stable and the systems (39) and (40) are globally exponentially synchronized.Proof.
Since the proof of this theorem is parallel to that of Theorem6, we omit it here.If the parametersp and q in the drive system are uncertain, one can construct the following controlled response system:
(43)x˙r=p^(xr+yr-xrln1+xr2)+u1,y˙r=xr-yr+zr+u2,z˙r=-q^yr+u3,
where p^ and q^ are the estimates of the uncertain parameters p and q, respectively. Let ex=xr-xd, ey=yr-yd, ez=zr-zd, p~=p^-p, and q~=q^-q, then, the error system of (39) and (43) is given by
(44)e˙x=p^(ex+ey-xrln1+xr2+xdln1+xd2)+p~(xd+yd-xdln1+xd2)+u1,e˙y=ex-ey+ez+u2,e˙z=-q^ey-q~yd+u3.Theorem 9.
If the following adaptive controller is added to the error system (44),
(45)u1=-p^(ex+ey-xrln1+xr2+xdln1+xd2)-ex,u2=-ex-ez,u3=q^ey-ez,p^˙=p~˙=-ex(xd+yd-xdln1+xd2),q^˙=q~˙=ezyd,
then, we have the following.(1)
The equilibrium points (p~=0, q~=0, ex=0, ey=0, ez=0) of the system (44) with adaptive control law (45) are globally stable; in addition, limt→∞ex=0, limt→∞ey=0, and limt→∞ez=0. And thus, the two systems (39) and (44) are globally synchronized.(2)
The parameter estimatesp^ and q^ will, respectively, converge to p and q as t tends to infinity.Proof.
The proof contains two steps.
Firstly, a Lyapunov function candidate is constructed as follows:(46)V2=12(ex2+ey2+ez2+p~2+q~2).
Then, one has(47)dV2dt|(44)=exe˙x+eye˙y+eze˙z+p~p~˙+q~q~˙=ex(p^(ex+ey-xrln1+xr2+xdln1+xd2)+u1p^(ex+ey-xrln1+xr2)+ey(ex-ey+ez+u2)+ez(-q^ey+u3)-q~(ezyd-q~˙)+p~(ex(xd+yd-xdln1+xd2)+p~˙)=-ex2-ey2-ez2≤0.
By LaSalle-Yoshizawa theorem [20], all the equilibrium points of the closed-loop systems are globally stable. Additionally, limt→∞ex=0, limt→∞ey=0, and limt→∞ez=0. And thus, the two systems (39) and (43) are globally synchronized.
By referring to Lemma  4.1 in [21], limt→∞p~=0, limt→∞q~=0 if there exist two functions f1(t), f2(t) which satisfy persistency of excitation condition, such that
(48)limt→∞p~˙=0,limt→∞q~˙=0,limt→∞p~f1(t)=0,limt→∞q~f2(t)=0.
Sincelimt→∞ex=0, limt→∞ez=0, it is easy to obtain from (45) that
(49)limt→∞p~˙=limt→∞-ex(xd+yd-xdln1+xd2)=0,limt→∞q~˙=limt→∞ezyd=0.
Add u1 and u3 from (45) to (44), we have
(50)limt→∞e˙x=limt→∞-ex+p~(xd+yd-xdln1+xd2)=limt→∞p~(xd+yd-xdln1+xd2)=0,limt→∞e˙z=limt→∞-ez-q~yd=limt→∞-q~yd=0.
Let f1(t)≜xd+yd-xdln1+xd2 and f2(t)≜yd; it is easy to see that f1(t) and f2(t) satisfy persistency of excitation condition [21]; thus, we have
(51)limt→∞p~=0,limt→∞q~=0.
The proof is complete.Remark 10.
Synchronization control methods for two Chua’s systems are proposed in this section. A linear controller is given when parametersp and q are known. In the case of uncertain parameters p and q, an adaptive controller is proposed to solve the problem. Comparing with the results in reference [22, 23], the output tracking error only converges to a small neighborhood of the origin, yet the synchronization errors of Chua’s systems can exponentially approach zero. On the other hand, compared with [23], a novel exponentially convergent method is used to solve the convergence of the estimation error of the uncertain parameters.


## 5. Numerical Simulations

In this section, several examples of numerical simulations are proposed to illustrate the theoretical results obtained in the previous sections. A fourth-order Runge-Kutta method is used to obtain the simulation results with MATLAB.Chua’s system (2) and the error systems (26), (41), and (44) are considered in this section for the numerical simulations. Let p=11, q=14.87, the initial state x(0)=-3, y(0)=2, and z(0)=1. Figure 2 shows the state trajectories of the closed-loop system (26) with linear control input (28); it is easy to see that the equilibrium point S0 is asymptotically stable. Similarly, Figure 3 shows that, with the corresponding control input (28), the equilibrium point S- is asymptotically stable. Figure 4 shows the synchronous errors of the system (39) and (40) with linear controller (42); it is easy to see that the two systems are globally asymptotically synchronized. When parameters p and q are uncertain, by using adaptive controller (45), the synchronous errors are asymptotically convergent to ex=0, ey=0, ez=0 (see Figure 5), and the estimate values of the uncertain parameters p and q asymptotically converge to the real values p=11, q=14.87 (see Figure 6).The asymptotic stability ofS0 with controller (28).
(a)
(b)
(c)The asymptotic stability ofS- with controller (28).
(a)
(b)
(c)The synchronization errors of the system (39) and (40) with linear controller (42).
(a)
(b)
(c)The synchronization errors of the system (39) and (43) with adaptive controller (45).
(a)
(b)
(c)The convergence of parametersp^ and q^.
(a)
(b)


## 6. Conclusions

A new smooth Chua’s system is constructed, and the chaotic characteristics is confirmed by computing the Lyapunov exponents of the system. A Constructive method is used to prove the existence of globally exponential attractive set and positive invariant set. For the three unstable equilibrium points of the system, a linear controller is designed to achieve globally exponential stability of the equilibrium points. Then, a linear controller and an adaptive controller are, respectively, proposed so that two similar types of smooth Chua’s systems are globally synchronized, and the estimate errors of the uncertain parameters converge to zero ast tends to infinity.



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