A New Robust Sliding-Mode Observer Design for ... - Semantic Scholar

Rafael Martıńez-Guerra e-mail.-rguerra兵apoznyak其@ctrl.cinvestav.mx Departamento de Control Automa´tico, CINVESTAV-IPN, Apdo. Postal 14-740, C.P. 07360, Mexico, D.F. Mexico

Ricardo Aguilar [email protected] Departamento de Energıá Universidad Autońoma Metropolitana— Azcapotzalco, Apartado Postal 75-338, C.P. 07300, Me´xico, D.F. Mexico

Alexander Poznyak Departamento de Control Automa´tico, CINVESTAV-IPN, Apdo. Postal 14-740, C.P. 07360, Mexico, D.F. Mexico

1

A New Robust Sliding-Mode Observer Design for Monitoring in Chemical Reactors The robust observer design for the online estimation of heat in continuous stirred tank reactors, containing nonstructured uncertainties within its model description as well as noisy temperature measurements, is addressed. The proposed observer contains a slidingmode term and is designed based on Differential Algebraic technique. The concept of the algebraic observability for a given class of model uncertainty is introduced. It is applied to the uncertainty estimation from noisy temperature measurements providing a simple observer structure which turns out to be robust against output (sensors) noises as well as sustained disturbances. The performance of this observer is shown to be calculated numerically. The obtained results look promising for possible industrial applications. 关DOI: 10.1115/1.1789534兴

Introduction

Currently, due to the necessity of online estimates of unknown terms related to mathematical models used for identification and control purposes, the state and parameter estimation approach is actually one of the most attractive research fields 共see, for example, 关1兴兲. To obtain an effective operation management using monitoring and control of chemical processes, the on-line estimation of unknown variables is a task which spontaneously arises 关2兴. A particular example widely used in the chemical industry is the Continuous Stirred Tank Reactor 共CSTR兲. This kind of reactors is involved, for example, in polymerization, petrochemical, pharmaceutical and biochemical industries 关3兴. Usually, the evaluation of temperature variations, due to heat generation by an exothermic chemical reaction, is a difficult task because of the complexity of the related physic and chemical phenomena. The estimators or observers for states and uncertainties can play a key role during the early detection of hazardous and unsafe operating conditions. Following this spirit, several researches have been focused in the proposition of methodologies for states and uncertainties estimation to be applied in chemical industry to eliminate unpredictable effects during a control process. For example, Schuler and Schmidt 关4兴 use the uncertainty estimation based on the calorimetric balances to infer heats of reaction. This methodology may be successfully applied in steady state operation, but if the system measurements are too noisy it may lead to instabilities during transient operation. Another approach is related to the construction of observers 共basically, asymptotic兲 for nonlinear processes using Differential Geometric methods 关5兴. The main idea is to find a state transformation to represent the system as a linear equation plus a nonlinear term, which is a function of the system output. However, finding a nonlinear transformation, that places a system of order n into observer form, requires the integration of n-coupled partial differential equations. Furthermore, this approach needs accurate knowledge of the nonlinear dynamics of the system and, hence, turns out to be inapplicable in the case of an uncertainty presence. The extended Kalman filters are widely used in the process industry because of their easy implementation and capabilities to deal with errors in the modeling and the measurements in its Contributed by the Dynamic Systems, Measurement, and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the ASME Dynamic Systems and Control Division November 19, 2003. Associate Editor: E. Misawa.

structure. Nevertheless, this design is based on the linearizing approaches providing difficulties in the robustness and convergence analysis 关3– 6兴. Another kind of methodology is related to Differential Algebra approach which has been recently shown to be a very effective tool for understanding some basic questions related to the process under consideration such as the input-output inversions and the observer realizations 关7–18兴. Algebraic Differential methodologies have been employed to construct asymptotic and even exponential observers 共further details can be found in 关10–12兴 and therein兲. However, up to date, these designs only include fullorder observers for state estimation and do not include uncertainty estimation. The early papers dealing with sliding mode observers considering measurement noise were considered by Utkin 关19兴 and Drakunov, 关20兴. Drakunov and Utkin discussed the problem of designing observers for state estimation using sliding modes 关21兴. De Carlo et al. discussed the variable structure control as a highspeed switched feedback control resulting in a sliding mode 关22兴. Anulova 关23兴 treated an analysis of systems with sliding mode in presence of noises. Slotine et al. 关13兴, successfully designed, so named, Sliding-Mode approach to construct observers which are highly robust with respect to noises in the input of the system. But, it turns out that the corresponding stability analysis can not be directly applied in the situations with the output noise 共or, mixed uncertainty兲 presence 关18兴. So, it is still a challenge for the control system society to suggest a workable technique to analyze the stability of identification error generated by Sliding-mode 共discontinuous nonlinearity兲 type observers. Currently, online monitoring of chemical reactors is still very important because the operation is related to safety, quality and productivity. However their direct measurement of chemical reaction evolution is often expensive or even impossible considering the current sensor technology. Additionally, heat generation by chemical reaction is an uncertain term in the system, because of the fact that thermal and kinetic phenomena related are complex and difficult to measure, experimentally. The main contribution of this paper consists in the designing of an uncertainty observer, applied to Continuous Stirred Tank Reactors 共CSTR’s兲, in order to infer online the heat of reaction from noisy temperature measurements and exclude their negative effect to the following control process. The concept of the algebraic observability for a given class of a model uncertainty is introduced. The main feature of this concept is that the corresponding algebraic observability condition of the uncertainty-state pair can be easily obtained. Using these conditions, the original model description is transformed to a generalized observability canonical

Journal of Dynamic Systems, Measurement, and Control Copyright © 2004 by ASME

SEPTEMBER 2004, Vol. 126 Õ 473

form where the implementation of the proposed sliding-mode uncertainty observers seems to be a very simple task. Such observers provide a high robustness property against of external noises compared to a standard high-gain observer. The Lyapunov-like technique is suggested in this paper to analyze the asymptotic stability property of the corresponding state-estimation error. The numerical experiments are also presented here to justify a high effectiveness of the suggested approach.

2

Problem Statement

Consider the following nonlinear dynamic plant with the mathematical model of a CSTR given by 关12兴:

So, to estimate X 2 it is sufficient to obtain an estimation of ␩ 2 ⫽y˙ Remark 1. From Definition 2 and Definition 3 it follows that the pair (X 1 ,X 2 ) is universally observable in the Diop-Fliess sense 关17兴. It should be noted that a partial change of coordinate enables us to estimate ␩ 1 ⫽y and ␩˙ 1 ⫽y˙ ⫽ ␩ 2 共or, equivalently, x 1 and x˙ 1 ). Then, the estimate of X 2 can be obtained from the Eq. 共2兲 or 共6兲 without the knowledge of its dynamic given by the Eq. 共3兲共or 共8兲 in the new coordinate change兲. The corresponding Input-Output representation of Eqs. 共2兲 and 共3兲 can be rewritten in new coordinates as follows:

␩ i⫽

• Mass Balance: X˙o⫽␪共Xoe⫺Xo兲⫺KX2o • Energy Balance: X˙1⫽␪共X1e⫺X1兲⫹X2⫹␥共u⫺X1兲

(1)

(7)

␩˙ 2 ⫽⌽ 共 ␩ 1 , ␩ 2 ,u˙ 兲

(8)

Y ⫽␩1 (3)

that is, X 2 ⫽X 2 (X 0 ,Y ,t). • System output:

where ⌽共.兲 is treated as an unmodelled dynamic. 3.2 Output Noise Effect. Now, considering the noise case presence, we consider

Y⫽X1

Y ⫽ ␩ 1⫹ ␦

(4)

Here K Kinetic constant 共Arrhenius model兲. u Given system input 共temperature of the cooling jacket兲. X 0 Concentration of the reactive chemical. X 0in Reactive input concentration. X 1 Reactor temperature. X 1in Input temperature. X 2 Uncertain term related to the heat generation by chemical reaction. • Y System measurable output. • ␥ Global heat transfer coefficients. • ␪ Inverse of the residence time. • • • • • • •

The main issue of the paper is to design and analyze an uncertainty Sliding-Mode type observer, to infer online the heat of reaction X 2 from noisy temperature measurements, despite the presence of sustained disturbances.

3

共 i⫽1,2兲

␩˙ 1 ⫽ ␩ 2

(2)

• Uncertainty dynamics: X˙2⫽f 共X0 ,X1 ,X2兲

d i⫺1 Y dt i⫺1

where ␦ is an additive bounded noise. Our aim is to design an observer to obtain ␩ 2 共the uncertainty term in the transformed space兲. However, as it can be seen from the nature of the system given by Eq. 共8兲, a standard structure of an observer, based on a copy of the system plus measurement error correction is not realizable in this case since the term ⌽ is a priory unknown.

4

Sliding-Mode Observer

4.1 Observer Structure. Proposition 1. The following dynamic system is a Sliding-mode asymptotic type observer of the system 共Eqs. 共7兲–共8兲 to estimate the variables ␩ 1 and ␩ 2 , respectively:

␩ˆ˙ 1 ⫽ ␩ˆ 2 ⫹m ␶ ⫺1 sign 共 Y ⫺Yˆ 兲 ,

m⬎0,

␩ˆ˙ 2 ⫽m 2 ␶ ⫺2 sign 共 Y ⫺Yˆ 兲 ,

(9) (10)

where 兩

Preliminaries

3.1 Basic Definitions. In order to introduce a background previous to the estimation methodology proposed, the following definitions are considered below 共see 关10–12,15–17兴兲. Definition 1.-A dynamic is supposed to be a finitely generate differential algebraic extension I/k 具 u 典 . This means that any element of I satisfies a differential algebraic equation with coefficients given by rational functions over k in the components of u and a finite number of their time derivatives. Definition 2.-Let 兵 u,y 其 be a subset of I in a dynamic I/k 具 u 典 . An element in I is said to be observable with respect to 兵 u,y 其 if it is algebraic over k 具 u,y 典 . Therefore, a state x is observable if it is observable with respect to 兵 u,y 其 . Definition 3.-A dynamic I/k 具 u 典 with output y is said to be observable if any state is observable. Following 关14兴, we say that a given class of uncertainty X u in I satisfies the algebraic observability condition, if X u satisfies Definition 2. So, any algebraic differential dynamic system is an extension verifying Definition 1. In view of this, the following differentialalgebraic equations can be obtained: X 1 ⫺Y ⫽0

(5)

Y˙ ⫹ 共 ␪ ⫹ ␥ 兲 Y ⫺ ␪ X 1e ⫺ ␥ u⫺X 2 共 X 0 ,Y ,t 兲 ⫽0

(6)

474 Õ Vol. 126, SEPTEMBER 2004

Yˆ ⫽ ␩ˆ 1 ⫹ ␦ and

sign 共 Y ⫺Yˆ 兲 ª

再

1 ⫺1

(11)

i f 共 Y ⫺Yˆ 兲 ⬎0 i f 共 Y ⫺Yˆ 兲 ⬍0

unde f ined

i f 共 Y ⫺Yˆ 兲 ⫽0

Now, returning back to the original state space, in view of 共6兲, the heat of reaction can be evaluated as Xˆ 2 ⫽ ␩ˆ 2 ⫺ ␪ 共 X 1e ⫺ ␩ˆ 1 兲 ⫺ ␥ 共 u⫺ ␩ˆ 1 兲

(12)

According to the variable change given by 共7兲, the variable ␩ 1 is the thermodynamic reactor temperature 共system output兲. From Eqs. 共12兲, if temperature measurements are noisy, the noise would be transmitted to the estimation of the heat of reaction that may lead to poor performance in the estimation procedure. That is because it is necessary to filter the Temperature measurements. This is the main reason why the structure of the proposed observer 共9兲-共10兲 makes sense. Furthermore, note that the equation 共10兲 contains a sliding-mode term which, as it will be shown below, is added in order to provide the robustness against an inaccurate modeling of measurements and output noises. Transactions of the ASME

T ⫹eTP⌳⫺1 f Pe⫹共⌬f 兲 ⌳ f ⌬f

4.2 Error Dynamics. Now, let us define the following estimation errors: e 1 ⫽ ␩ 1 ⫺ ␩ˆ 1 e 2⫽

⭐eT关 PA␮⫹A␮T P⫹P⌳ f P⫹Q兴e⫺eTQe

(13)

␩ 2 ⫺ ␩ˆ 2

2 ⫹共L1f ⫹储A␮储兲2储e储2兲2储 ⌳ f 储 ⫺2ke T C T sign 共 Ce⫹ ␦ 兲 ⫹共L0f

(14)

m

By 共7兲–共10兲, it follows that the estimation errors e⫽(e 1 ,e 2 ) T verify the following ordinary differential equation: e˙ ⫽A ␮ e⫺Ksign 共 Ce⫹ ␦ 兲 ⫹⌬ f

(15)

⫽e T 共 PA ␮ ⫹A ␮T P⫹ PR P⫹Q 兲 e⫺e T Qe⫹2 储 ⌳ f 储 L 02 f ⫺2k 共 Ce 兲 T sign 共 Ce⫹ ␦ 兲

The main ‘‘trick’’ consists in the implementation of the following inequalities valid for each component: x T sign 共 x⫹z 兲 ⫽ 共 x⫹z 兲 T sign 共 x⫹z 兲 ⫺z T sign 共 x⫹z 兲

where A ␮ ⫽ 关 ⫺0 ␮⫺m␮ 兴 , ␮ ⬎0 is a regularizing parameter, K ␮e1 1 ⫽m ␶ ⫺1 关 m ␶ ⫺1 兴 , C⫽ 关 1,0兴 and ⌬ f ⫽ 关 1/m⌽⫹ ␮ e 2 兴 is an uncertainty term 共or unmodelled dynamics term兲.

n

⭓

兺兩共 x⫹z 兲兩 ⫺ 兺兩 z 兩 n

兺

i⫽1

⭓

兩 x i 兩 ⫺2

兺兩z 兩 i

i⫽1

兺兩 x 兩 ⫺2 冑n 储 z 储

Here, we have used that: 兩共 x⫹z 兲 i 兩 ⭓ 兩 x i 兩 ⫺ 兩 z i 兩

(19)

and n

兺兩 z 兩 ⭐ 冑n 储 z 储

(20)

i

i⫽1

The last is resulted from the Cauchy-Bounyakowski-Schwarz inequality:

with ,

冑冑兺

n

has a positive definite solution P⫽ P ⬎0. A4. The gain matrix K is selected as K⫽k P ⫺1 C T where K is a positive constant. Comment 1. The algebraic Riccati equation in A3 has a positive definite solution if the matrix A ␮ is stable 共that is valid for any positive ␮兲 and the following matrix inequality is fulfilled 共see 关24兴, Appendix A, Lemma 12.3兲: T

i⫽1

n

i i

a i2

i⫽1

i⫽1

b i2

(21)

for a i ªn ⫺1 , b i ª 兩 z i 兩 . Applying 共17兲 to 共18兲 and in view of A3, it follows: V˙ 共 e 兲 ⫽⫺e T Qe⫹2 储 ⌳ f 储 L 02 f ⫺2k 共 Ce 兲 T sign 共 Ce⫹ ␦ 兲

冉兺 n

⭐⫺e T Qe⫹2 储 ⌳ f 储 L 02 f ⫺2k

1 A ␮T R ⫺1 A ␮ ⫺Q⬎ 关 A ␮T R ⫺1 ⫺R ⫺1 A ␮ 兴 R 关 A ␮T R ⫺1 ⫺R ⫺1 A ␮ 兴 T 4 In our case this inequality may be transformed to the following one:

n

兺 ab⭐ 兺

Q⫽Q 0 ⫹2 共 L 1 f ⫹ 储 A ␮ 储兲 I 2

i⫽1

兩共 Ce 兲 i 兩 ⫺2 冑n 储 ␦ 储

冊

n

⭐⫺e T Qe⫺2k

兺兩共 Ce 兲兩 ⫹ ␳ 共 k 兲

(22)

i

i⫽1

where

␳ 共 k 兲 ª2 储 ⌳ f 储 L 02 f ⫹4k 冑n 储 ⌳ ⫺1 储 ␦ ⫹

1 T ⫺1 关A R 4 ␮

(23)

since 2 储 ␦ 储 ⫽ 冑␦ T ⌳ 1/2⌳ ⫺1 ⌳ 1/2␦ ⭐ 冑储 ⌳ ⫺1 储 ␦ T ⌳ ␦ ⫽ 冑储 ⌳ ⫺1 储储 ␦ 储 ⌳

⫺R ⫺1 A ␮ 兴 R 关 A ␮T R ⫺1 ⫺R ⫺1 A ␮ 兴 T ⬎Q 0 Hence, the matrix Q 0 providing the existence of the solution to the Riccati equation, always exists if G ␮ ⬎0.

and

冉兺 n

4.4 Lyapunov-Like Analysis. Let us define the Lyapunov function candidate V(e) as 0⬍ P⫽ P T 苸Rnxn

(18)

i

i⫽1

PA ␮ ⫹A ␮T P⫹ PR P⫹Q⫽0

V 共 e 兲 ⫽ 储 e 储 2P ªe T Pe,

i

i⫽1

n

2 A2. The output noise is assumed to be bounded as 储 ␦ 储 ⌳ ª ␦ T ⌳ ␦ ⭐( ␦ ⫹ ) 2 ⬍⬁ where ⌳ is a symmetric definite positive matrix playing role of a normalizing matrix 共since different components of the output measurements may have a different physical nature兲. A3. There exits a positive definite matrix Q 0 ⫽Q T0 ⬎0 such that the following matrix Riccati equation

G ␮ ªA ␮T R ⫺1 A ␮ ⫺2 共 L 1 f ⫹ 储 A ␮ 储兲 2 I⫺

i

n

⭓

储 ⌬ f 储 ⭐L 0 f ⫹ 共 L 1 f ⫹ 储 A ␮ 储兲储 e 储

0⬍⌳ f ⫽⌳ Tf

n

i⫽1

4.3 Main Assumptions. A1. There exist nonnegative constants L 0 f , L 1 f such that for any e the following generalized quasi-Lipschitz 共strip-bound兲 condition holds:

Rª⌳ ⫺1 f ⫹2 储 ⌳ f 储 L 1 f I,

(17)

i⫽1

兩共 Ce 兲 i 兩

冊

with ␣ P ª␭ min(P

X T Y ⫹Y T X⭐X T ⌳ f X⫹Y T ⌳ ⫺1 f Y

V˙ 共 e 兲 ⫽

valid for any X, Y 苸R n⫻m , 0⬍⌳ f ⫽⌳ Tf , it follows: V˙ 共 e 兲 ⫽2e T Pe˙ ⫽2e T P 关 A ␮ e⫺Ksign 共 Ce⫹ ␦ 兲 ⫹⌬ f 兴 ⫽2e T PA ␮ e⫺2ke T C T sign 共 Ce⫹ ␦ 兲 ⫹2e T P⌬ f ⭐e T 共 PA ␮ ⫹A ␮T P 兲 e⫺2ke T C T sign 共 Ce⫹ ␦ 兲 Journal of Dynamic Systems, Measurement, and Control

n

⭓

兺关兩共 Ce 兲兩兴 ⫽ 储 Ce 储 2

2

i⫽1

⫽ 储 C P ⫺1/2P 1/2e 储 2 ⭓ ␣ P e T Qe

(16)

In view of 共15兲 and using the matrix inequality

2

⫺1/2

T

C CP

⫺1/2

(24)

)⭓0. Then 共22兲 implies

d 2 ⫺2k ␣ P 储 e 储 P ⫹ ␳ 共 k 兲共储 e 储 2P 兲 ⭐⫺ 储 e 储 Q dt

⭐⫺ ␣ Q V 共 e 兲 ⫺ ␽ 冑V 共 e 兲 ⫹ ␤

(25)

where

␣ Q ª␭ min共 P ⫺1/2Q T Q P ⫺1/2兲 ⬎0, ␽ ⫽2k ␣ p ,

␤⫽␳共 k 兲.


At this point we are ready to formulate the main result. 4.5 Main Result. Theorem 1 If the assumptions A1–A3 are satisfied then

冋册 1⫺

where

␮ ˜ ⫽␮ ˜ 共 k 兲ª

冉

␮ ˜ V

→0

(26)

⫹

␳共 k 兲

冑共 k ␣ P 兲 2 ⫹ ␳ 共 k 兲 ␣ Q ⫹k ␣ P

冊

2

and the function 关䊉兴 ⫹ is defined as 关Z兴⫹ ª

再

Z

i f Z⭓0

0

i f Z⬍0

⽧ The proof of this result is given in Appendix. Remark 2. The theorem 1 actually states that the weighted estimation error V⫽e T Pe converges to the zone ␮ ˜ asymptotically, that is, it is ultimately bounded.

5

Numerical Simulation and Discussions

In order to show the improved features of the observer proposed, some numerical simulations were performed. The heat of reaction generated in a CSTR is estimated via temperature measurements, note that the unmodeled dynamic 共heat of reaction兲 is considered as Arrhenius model which is Lipchitz respect to thermodynamic temperature, this characteristic satisfies assumption A1. These measurements are corrupted by a bounded white noise ( ␦ ⫹ ⫽2 K) around the actual temperature value, which is in accordance with assumption A2. Additionally, there is a sustained 共bounded兲 disturbance in the reactor temperature inlet X 1e ⫽X 1e0 ⫹2 sin(␲t). For the results given at Figures 1–2 the value of the observer gain is m⫽0.00001 and of the sampling-time is ␶ ⫽1.0. To calculate the solution to the Riccati equation the following parameters have been selected:

Fig. 1 On-line estimation of the heat of reaction with the sustained disturbances „mÄ0.00001….

6

Concluding Remarks

A new type of a sliding-mode observer, intended to infer the heat of reaction in CSTRs via temperature measurements, is designed using the differential algebraic tools. The concept of algebraic observability condition for a given class of uncertainty is introduced to estimate the uncertain term from the output selected, which is easily obtained from this approach. Moreover, the implementation of this observer seems to be very simple following the

⌳ f ⫽␭ f I, L 1 f ⫽ ␮ , R⫽ 共 ␭ ⫺1 f ⫹2␭ f ␮ 兲 I, Q 0 ⫽q 0 I,

Q⫽ 共 q 0 ⫹8 ␮ 2 兲 I

with ␭ f ⫽20, ␮ ⫽0.0001, q 0 ⫽ ␮ 2 . With these parameters we obtain P⫽10⫺3

冋

3.16099

⫺0.22096

⫺0.22096

3.16099

册

⬎0.

The observer filters adequately the noised temperature measurements, as one can see in Fig. 1. This temperature estimate is used in the estimation of the reaction heat using the correspondent differential algebraic structure. The observer is able to infer the heat of reaction with good precision, as it can be seen from Fig. 2. Figures 3 and 4 show the effect of the observer parameter gain m. The value m⫽0.001, which is considered less than the temperature sampling time, is selected in order to show the effect of a small gain. As one can see, the observer can not provide the quick convergence to the values of the corresponding terms, temperature and heat of reaction respectively. Figure 5 shows the effect of the parameter m⫽0.1, which is larger than the temperature sampling time, the temperature filtered tends to reconstruct the temperature, however it also reconstruct the corruption by noisy. Figure 6 shows the performance of the uncertainty observer for this case. Consequently, the uncertainty estimated presents peaking phenomena in the start-up of the estimation procedure and besides, the noisy is transmitted to the uncertainty estimated. 476 Õ Vol. 126, SEPTEMBER 2004

Fig. 2 Filtering process of the reactor temperature measurements with the sustained disturbances „mÄ0.00001….

Transactions of the ASME

Fig. 5 Filtering process of the measurements of reactor temperature „mÄ0.1…. Fig. 3 Filtering process of the measurements of reactor temperature „mÄ0.001….

transformation proposed. Despite of noisy temperature measurements and sustained disturbances, the performance of the observer developed looks satisfactory, considering the restrictions mentioned. It is important to notice that the same observer presents bad performance when using very high and small gains of the observer, relative to the sampling time. The reaction network of the second order considered in this paper is very common in chemical industry. So, the suggested approach has practically an

Fig. 4 On-line estimation of the heat of reaction „mÄ0.001….

Journal of Dynamic Systems, Measurement, and Control

unlimited scope of application and it can be easily applied to a more general reaction network dealing with more complex 共including feedbacks兲 chemical processes.

Appendix 1 Proof of theorem 1 Consider the Lyapunov function V(e) verifying the differential equation: V˙ ⫽⫺ ␣ V⫺ ␽ 冑V⫹ ␤ The equilibrium point V * of this equation, satisfying

Fig. 6 On-line estimation of the heat of reaction „mÄ0.1….


⫺ ␣ V⫺ ␽ 冑V⫹ ␤ ⫽0 is as follows:

V *⫽

冉冑冉冊 ␽

2␣

So, it follows that G * ⫽0, that is equivalent to the fact:

冉冊冊冉冑冉冊 ␤

␤

2

␽

⫹ ⫺ ␣ 2␣

2

冋册

2

1⫺

␣

⫽

␽

2

␤

␽

⫹ ⫺ ␣ 2␣

2␣

冊

2

˙ ⫽2 共 V⫺V * 兲 V˙ ⭐2 共 V⫺V * 兲共 ⫺ ␣ V⫺ ␽ 冑V⫹ ␤ 兲 ⌬ ⫽2 共 V⫺V * 兲共 ⫺ ␣ V⫺ ␽ 冑V⫹ ␤ ⫹ 共 ␣ V * ⫹ ␽ 冑V * ⫺ ␤ 兲兲 ⫻2 共 V⫺V * 兲共 ⫺ ␣ 共 V⫺V * 兲 ⫺ ␽ 共冑V⫺ 冑V * 兲兲 ⫽⫺2 ␣ 共 V⫺V * 兲 2 ⫺2 ␽ 共冑V⫹ 冑V * 兲共冑V⫺ 冑V * 兲 2 ⭐0 for any V⫽V * , that implies: lim V→V * . For t→⬁

冋册

2 Gª 关 V⫺ ␮ ˜ 兴⫹ ⫽V 2 1⫺

冋册

冋册冋册

⭐2V 1⫺

␮ ˜ V

⫽⫺2V 1⫺

␮ ˜ V

␮ ˜ V

2 ⫹

V˙ ⫹

共 ⫺ ␣ V⫺ ␽ 冑V⫹ ␤ 兲

⫹

␮ ˜ V

关 ␣ 共 V⫺V * 兲 ⫹ ␽ 共冑V⫺ 冑V * 兲兴 ⫹

⭐0 The last inequality implies that G t converges, that is, G t →G * ⬍⬁ The integration of the last inequality from 0 to t yields

冕冋册冋 t

G t ⫺G 0 ⭐⫺2

V 1⫺

0

␮ ˜ V

⫹

册

␣ 共 V⫺V * 兲 ⫹ d␶ ␽ 共冑V⫺ 冑V * 兲

that leads to the following inequality

冕冋册 t

2

V 1⫺

0

␮ ˜ V

关 ␣ 共 V⫺V * 兲 ⫹ ␽ 共冑V⫺ 冑V * 兲兴 d ␶ ⭐G 0 ⫺G t ⭐G 0 ⫹

Dividing by t and taking the upper limits of both sides, we obtain: 1 t→⬁ t ¯ lim

冕冋册冋 t

V 1⫺

0

␮ ˜ V

⫹

冋册冋 ␮ ˜ V tk

⫹

册

␣ 共 V⫺V * 兲 ⫹ d ␶ ⭐0 ␽ 共冑V⫺ 冑V * 兲

and there exists a subsequence t k such that: V t k 1⫺

Theorem is proven.

References

Defining ⌬⫽(V⫺V * ) 2 , we derive:

˙ t ª2 关 V⫺ ␮ G ˜ 兴 ⫹ V˙ ⫽2V 1⫺

→0 ⫹

⫽␮ ˜

⭓0

we obtain:

␮ ˜ Vt

␣ 共 V t k ⫺V * 兲 ⫹

␽ 共冑V t k ⫺ 冑V * 兲

G t k ——→ 0 k→⬁

478 Õ Vol. 126, SEPTEMBER 2004

册

→0

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