Automatica, Vol. 28, No. 1, pp. 27-33, 1992 Printed in Great Britain.

0005-1098/92 $5.00 + 0.00 Pergamon Press pie

~) 1991 International Federation of Automatic Control

Singularity-free Adaptive Pole-placement without Resorting to Persistency of Excitation:

Detailed Analysis for First Order Systems*

ROGELIO L OZ ANOt

Globally convergent adaptive pole placement of possibly nonminimum phase systems can be obtained by an appropriate modification of the Least Squares parameter estimate that ensures controllability of the estimated plant model without resorting to exciting probing signals.

Key Words--Adaptive control; convergence analysis; discrete-time systems; least-squares estimation; nonminimum-phased systems; parameter estimation; pole placement.

Abstract--This paper presents a novel approach to avoid singularities in adaptive pole-placement control algorithms. The control scheme is applicable to either minimum or nonminimum phase plants and the closed-loop stability is guaranteed in spite of bounded disturbances. The algorithm uses a Least Squares identification algorithm with a particular dead zone that allows the definition of modified estimate that retain the convergence properties of the original estimates. Explicit modification to the parameters estimates is given that secures well-conditioned matrices in the adaptive control law. Detailed analysis is given for one-pole one-zero systems.

1. INTRODUCTION THE DEVELOPMENT of adaptive control algorithms for possibly nonminimum phase systems has been hampered by singularities that may arise in the control law. The stumbling block has always been the possibility of obtaining plant para- meters estimates for which the control input cannot be computed. This singularity explicitly occurs when the estimates are such that if they were taken as the true plant parameters it would imply that the plant were not controllable.

In order to avoid any singularities in the control law we must somehow guarantee that the estimated plant model preserves the control- lability properties of the original system. To solve this problem, two research streams have emerged; one securing convergence of the estimates to their true values by inducing

* Received 18 September 1989; revised 13 February 1990; revised 20 November 1990; received in final form 15 April 1991. This paper was presented at the 29th IEEE Conference on Decision and Control, Honolulu, Hawaii, December, 1990. This paper was recommended for publication in revised form by Associate Editor I. M. Y. Mareels under the direction of Editor P. C. Parks.

~" Universit6 de Technologie de Compi6gne, HEUDIASYC, B.P. 649-60206 Compi6gne cedex, France.

27

excitation in the plant signals, and another in which the estimates are adequately modified to meet the controllability requirements, without resorting to external dither signals.

The technique relying on excitatiofi has been developed (Kreisselmeier and Smith, 1986; Giri et al., 1989; Goodwin and Sin, 1984; Cristi, 1987a; Polderman, 1989). In Kreisselmeier and Smith (1986) and Giri et al. (1989), the excitation vanishes asymptotically, Goodwin and Sin (1984) and Cristi (1987a) use persisent excitation while Polderman (1989) requires only excitation at a finite number of times. However in the presence of noise, the excitation signal should be large enough to predominate over the noise. Therefore, since it is not always feasible or desirable to introduce additional disturbances of such a size into the process, adaptive control not resorting to persistency of excitation arises as an appealing alternative approach.

In the design of adaptive controllers not resorting to excitation arguments, one is confronted with the problem of finding an appropriate modification for the estimates. The parameters estimates should be modified in such a way that they meet the controllability requirements without losing their convergence properties. The first works in this direction not requiring excitation of the plant input and output were presented in De Larminat (1984) and Lozano and Goodwin (1985) where different modifications were proposed using the properties of the covariance matrix in least squares-type identification algorithms. An extension of these algorithms has been presented in Cristi (1987b). A different interesting perspective to the problem was given in Praly (1984) where a

28 R. LOZANO

solution was shown to exist by identifying a system that is bilinear in the parameters. More recently, a different approach was proposed in Lozano (1989) securing a positive lower bound for the determinant of the controllability matrix estimate. The proposed control scheme performs only output regulation and requires to carry out a matrix polar decomposition on-line.

This paper presents an adaptive pole place- ment control algorithm for discrete-time systems that is robust with respect to bounded disturbances. A uniform positive lower bound for the determinant of the Sylvester matrix is obtained. An explicit modification to the estimates is given to meet the controllability requirements for the case of first-order systems. The study presented in this paper is restricted to first-order systems. Extension of the proposed technique for high-order systems is not straight- forward and its complexity increases with the order of the system.

2. IDENTIFICATION OF SYSTEMS SUBJECT TO B O U N D E D DISTURBANCES

Consider a discrete-time linear system

A(q-~)y t = B(q -1 )u t + v, (2.1)

with A ( q -~) = 1 + a~q -1 (2.2)

n ( q ) -~ = b~q -1 + b~q -2 (2.3)

where Yt and u, are the plant output and input. Equation (2.1) can be rewritten as

Yt = o*Tdpt-1 ÷ Vt (2.4) with

0 *r = [b~, b~, a~] (2.5)

dptr_l = [u,_~, u,_2, -Yt-~]. (2.6)

We will consider the class of disturbances satisfying

Iv, I-< r /÷/~ II~t-lll (2.7) where

r/_>0, u_>0 (2.8)

are positive constants. Therefore, we will consider bounded disturbances with known upper bound "r/" and a class of disturbances due to unmodelled dynamics bounded by /~ [Itp~_xl[ for /~ small. The stability of the controller proposed in the paper will hold for /~ smaller than a certain threshold.

The following an a priori knowledge of the plant will be required.

Assumpt ion 1. r/ and /~ in equations (2.7) are known.

Description o f the parameter estimation algorithm Normalized variables

Xt_x = q~,-l/(1 + 119,-,11) (2.9)

y, =y , / (1 + II~,-dl) (2.10)

6, =/~ + r//(1 + II0,-dl). (2.11)

Normalized prediction error

e, = )7 , - O~_,x,_,. (2.12)

Augmented error

w, (e~ 2 - r ~ 2 -1,2 (2.13) = "-I-Xt_ll~t_lXt_l)

Least squares with dead zone

~., = if w, - 6~ (2.14) otherwise

where 6,--2_ o~(62t + cO,) (2.14a)

ct = 1 ÷ tr (F0) (2.14b)

and E > 0 is chosen arbitrarily.

~,,F,_ ~ x ,_ ~x ,r_ ~F,_ ~ r (2.15) F~ = F~_~ - 1 + )~x,_iF~_~x,_~

Ot = Ot-~ + ~,tFtXt-le,. (2.16)

The main convergence properties of the estimation algorithm described above are given in the next theorem.

Theorem 1. The parameter estimation algorithm described in equations (2.1)-(2.16) has the following properties:

F~ and 0, converge (2.17)

lim sup (w~ 2 - 3t 2) --< 0

where w, and 3~ are given in (2.13) and (2.14a), respectively.

IIFTa(O * - 0311-<ho (2.18)

where

ho--IIFff~0011 + (O~FffaOo + tr Fo)/e. (2.18a)

Proof. The proof follows the ideas in Lozano (1989) but in this case the results hold also for /~ = 0 and for ~ /=0 and ho in (2.18a) is not a function of/~. The proof is given in Appendix A.

Equation (2.17) establishes that 6, is a bound in the limit, not only for the normalized error et in (2.12), but also for IIF,-~x,-lll [see also (2.13)]. This property is a consequence of using the augmented error wt in (2.13) as a scheduling variable to stop the algorithm.

In order to understand the utility of equation (2.18) let us define the vector fl* = FTI(O * - 0~)

Singularity-free adaptive pole-placement 29

which is bounded in view of (2.18). This equation can also be written as 0 " = 0, + F~fl~* which means that we can modify 0, to obtain the true parameters 0* by simply adding Ftflt* to the current estimates Or. On the other hand, knowledge of fl,* implies knowledge of the true parameters 0". Nevertheless, the above equa- tions motivate the modification to the current estimates proposed in the following section.

3. A D A P T I V E P O L E - P L A C E M E N T

The main contribution of the paper is to propose an adaptive pole placement algorithm that converges globally and presents no sin- gularities. By singularity we mean that Ot could assume values for which the corresponding Bezout equation might not have a solution or in other words, that Ot might correspond to a non controllable model.

In this section we will use the convergence properties of the estimation algorithm presented in the previous section to design a singularity free adaptive pole-placement that requires no persistency of excitation in the plant input and output. Therefore, we must first procure estimates that will prevent any singularities in the control law. For that purpose let us define the following modified parameter estimates.

Ot = Ot .~L Ftflt (3.1)

fit in the above equations should satisfy the following conditions: (C1) fl, should be bounded and converge; and (C2) fl, should be such that the absolute value of the determinant of the Sylvester matrix associated with ~ is uniformly bounded from below.

Finding a fl, that satisfies the above require- ments is the main contribution of the present paper. This problem will be solved for first-order systems in the next section.

However, in order to show how such a fl~ solves our control problem, let us proceed with the controller design.

Define the following polynomials:

f l t (q -~) = 1 + a~(t)q -~ (3.2)

Bt(q -1) =/~l(t)q -1 +/~2(/)q -2 (3.3)

where t~ and/~ are the elements of ~, as follows [see also (2.5)]

Ot = [/~(t),/~2(t), al(t)]. (3.4)

We can now write the expression for the estimated model of the plant as follows:

A, (q -~ )y , = B , (q -~ )u , + e~(t). (3.5)

From the above and (2.6) it becomes clear that e~(t) in (3.5) is the a posteriori identification

error using the modified parameter estimates, i.e.

e~(t) = ,~,(q-1)y, _ B , ( q - l ) u t

= Yt - 0fCt-l . (3.6)

A bound for e~(t), which will be required later, is given in the following lemma.

L e m m a 1. ea(t) in (3.6) satisfies the following inequality

limsup {~2(t)- -2 2 3t~,flmax } -< 0 (3.7) where

e~(t) ~a( t ) ---- (3.8)

1+11¢,-111

and timex is such that

max (11/~,11, 1) -< flmax. (3.9)

Proof. Introducing (3.1) into (3.6)

ea(t) = y, - 0,r¢,_1 - fl~rF/p,_l

= (~, - Ofx,_a - fltrFtX,_l)(1 + I1¢,-111)

[using (2.9) and (2.10)]

= (e, - [~TtFt_lXt_ 1 -]- Zt)(1 + I1¢,-111)

[using (2.12)] (3.10)

where

Z, = ( 0 t _ 1 -- o t ) T X t _ l "~ ~tT(Ft_l -- F t ) x , _ 1. ( 3 . 1 1 )

Combining (3.8) and (3.10) it follows:

~a(t) = e, - f l f F t - l x , _ , + z,. (3.12)

From (2.13), (3.9), (3.12) and the fact that (a + b + c) 2 -< 3(a 2 + b 2 + c 2) for any a, b and c, we have

2 T 2 ~(t) -< 3[e~ + flrnaxXt_lFt_lXt_l -]- Z2t] 2 2

<-- 3 f lmaxW t d- 3zEt. ( 3 . 1 3 )

Note also that zt in (3.11) converges to zero because 0t and Ft converge, and x~, fit are bounded. (3.13) can also be written as

- 2 2 2 -- 3flraaxOt <--3flmax(Wt -- ~ ) + 3 Z t z- ( 3 . 1 4 )

Therefore, introducing (2.17) into (3.14) and the fact that z,--~ 0 we finally obtain (3.7). V

We can now design an adaptive pole- placement algorithm for the plant estimated model in (3.5), which is asymptotically time- invariant, where e , ( t ) plays the role of an unmodelled dynamics. Thus the pole-placement control algorithm can be expressed as:

S t ( q - 1 ) u t = C(q - ~ ) - y ~ g , ( q - 1 ) y t (3.15) with

C ( q -1 ) -- 1 + c l q -1 d- ¢2q -2 (3.15a)

30 R. LOZANO

where the zeros of C(z-~)z 2 represent the desired closed-loop poles, y f is a reference signal and St and Rt satisfy the following polynomial identity

,~t(;~)S,(;t) +/~t(;t)R,(Z) = C(;t). (3.16)

Combining (3.5), (3.15) and (3.16) we obtain the following closed-loop model:

[ C + A~(t)A~(/) ] [u , ] A t

A3 (t) C + A4(t ) l l_y ,J=[B,] Cy*~

(3.17)

We can now state the following lemma.

Lemma 2. Consider the adaptive pole- placement control defined in (3.15) and (3.16) where ,~/(q-1) and /)t(q -~) are obtained by modifying the current estimates ~ as indicated in (3.1) with fl, satisfying conditions C1 and C2. Then, there exists an upperbound ~ small enough but different from zero such that the plant input and output remain bounded for the class of disturbances (2.7) for any ~-> 0 smaller than/~.

Proof. The proof follows closely the ideas given in Goodwin and Sin (1984) and Lozano (1989) and thus we will only give an outline of the proof.

Let us study equation (3.17) which describes the closed-loop system behaviour. A~(t) i = 1 . . . . . 4 in (3.17) depend on the parameters estimates increments O ( t ) - O ( t - 1 ) and there- fore converge to zero [see (2.17)]. All the polynomials in (3.17) converge, C is asymptoti- cally stable and y* is bounded. Considering a time arbitrarily large but finite so that the estimates are close to their final values, we can interpret (3.17) as the description of an asymptotically stable system having [u,, Yt] as the output and e~(t) as the input. Therefore, we conclude that, in the limit, the sequences u, and Yt cannot grow faster than linearly with e,(t). Thus, if we assume that II~t-~ll grows un- bounded, ~o(t) in (3.8) will have, in the limit a nonzero lower bound because in view of (3.17), ut, Yt and e,(t) should diverge at the same rate. The lower bound on ~( t ) will essentially depend on C and Rt and St in (3.17) and so, is not a function of /~ [see (2.18) (2.18a) and condition C.2]. On the other hand, if II~t-~ll grows unbounded, 6, in (2.14) will converge to /~ and thus tSt in (2.14) will converge to cr(~t~+ e/u). Hence, there exists a threshold /~ small enough such that for /~-</~ the lowerbound on i~(t) is greater than 3fl2ma.O:(/~+ e~u) and so (3.7) will not hold. Since this is a contradiction one

concludes that II@t-~ll should remain bounded for #-</Z V

In the next section we will present an expression for fl, in (3.1) satisfying conditions C1 and C2.

4. MODIFICATION OF THE ESTIMATES FOR FIRST-ORDER SYSTEMS

In this section we will present an expression for fit in (3.1) satisfying conditions (C1) and (C2) for the following system:

(1 + a~q-~)yt = (b~ + b~q-1)q-~uv (4.1)

If we consider S(q -~) = So + s~q -1 and R(q -1) = r0, equation (3.16) can be expressed as:

M(0*)/Sl/= c, (4.2) L roJ c2

where c~ and c2 are given in (3.15a) and

M(O*) = a~ 1 b . (4.3)

0 a~" b 2 ._]

The determinant of the Sylvester matrix in (4.3) is

det M(O*) = b~ - brat . (4.4)

We will require the following a priori knowledge on a measure of the controllability of the plant.

Assumption 2, A lower bound ~0 is known such that Idet M(O*)I ~ ~o.

We then have the following result.

Lemma 3. Consider the system in (4.1) subject to assumption 2. The following expression for fit satisfies conditions C1 and C2 of Section 3. [For simplicity of notation we have ommited the ar_gument (t). ]

fl = $wl Sw2 i1-~ sgh (det M(O)) + ~,2x

\11~11

where

k =f2 - blf3 - all1 (4.6)

g = 2 + 4 Ilkll/(d - 2a ) (4.7)

d = eo/4h z (4.8)

h = max (1, h0) (4.9)

Sw, = ½[1 - sgh (Idet M ( 0 ) I - d)] (4.10)

Sw2 = ½[1 - sgh (llf~ll I[f3ll - d)] (4.11)

•2 = 1 - S.~2 (4.12)

Singularity-free adaptive pole-placement 31

I -A

sgh(g)

-1

]~

FIG. 1. Signum function with hysteresis.

where sgh(-) is the signum function with hysteresis depicted in Fig. 1, {A} is the hysteresis width and is assumed to be much smaller than d(A-<d/10) , h0 is given in (2.18) and 60 is given in assumption 2, {f~} is the i th row of the matrix Ft.

fl, in (4.5) ensures the following property on the modified estimates

- d Idet M(0)I >- ~ . (4.13)

Remark 1. Equation (4.5) was obtained by studying all the different possible situations that may occur in the expression for det M(O). Therefore fit was selected to exploit the term in that expression that dominated over the others for every particular case.

Remark 2. The signum function with hysteresis sgh (.) was used in (4.5), instead of simply the sign function, to guarantee that fl, converge. Note for example that sgn (e -t sin(t)) does not converge while sgh (e -t sin (t)) does.

Proof. Since O, and Ft are bounded and converge, it then follows from (4.5) to (4.12) that fit converges, which is condition C1. Let us now prove that condition C2 is also satisfied.

For the case of the system in (4.1), expression (3.1) can be written as:

[bl(O l [ T(t)l /~2(t)/= b2(t)/+ (4.14) t~l(/) 1 al(t) 1 mf~'(t)_]

where fir(t) represents the ith row of Ft. To simplify the notation we will omit the argument (t).

The determinant of the Sylvester matrix associated with the modified estimates 0 is obtained from (4.4) and (4.14) as follows:

det M(O) = 6 2 - - ~lt~l = b~ + f f ~ - (b, +frlfl)(a , +f~[3)

= det M(O) + f l rk - (frlfl)(f~fl) (4.15)

where k is given in (4.6).

From (2.18) and (3.1), there exists a fl* with Ilfl*ll <-ho, such that 0 = 0". Introducing such a value for fl* in (4.15) and considering assump- tion 2 we have:

e0 [detM(0*)[ IdetM(0(fl*)[ - - < _ _

h 2 - h 2 h 2

= Idet M(O) + krfl * _ ¢ T ~ * ¢ T t ~ * l / h 2

J l l "~ J 3 l ~ I

-< Idet M(0)I + Ilkll ÷ IIf~llll/311

(4.16)

where h is given in (4.9). Depending on the values of S,~1 and Sw2 in

(4.10) and (4.11), we can have different situations. We will prove next that (4.13) holds in all possible cases which are (Swl, Sw2)= (a) (0, 0); (b) (0, 1); (c) (1, 0) and (d) (1, 1). In the first two cases S~I = 0 and therefore, from (4.10), fl = 0. It follows that Idet M(0)I >- d - A (see also Fig. 1). Since A < d/lO, (4.13) holds.

(c) Let us now study the case Swl= 1 and Sw2 = 0. Since S~1 = 1, from (4.10) we have

[det M(0)I -< d + A (4.17)

Since S,~z = 0, from (4.11) and Fig. 1 we have

Ilfll111/311->d - A. (4.18)

Let us now find a different expression for the term (f~fl)(f~fl) in (4.15). In this case, fl in (4.5) reduces to.

f l = g ( ~ + s g h ( f ~ / 3 ) ~ ) . (4.19)

Therefore

/ , ~ f~/3 ~ ( f ~ ) ( f ~ ) -- ge~ llfd] ÷ sgh ~ , J 3 J 3 1

× (f~/3+ sgh (f~/3)11/311) \IIAII

= g2 sgh (fT/3)

× (llflll + sgh (fr~/3)f~/3~ IIf311/

× (11/311 + sgh (flT/3) f~r/3~ IIf llJ

= g 2 s g h (fT/3)/llfdl 11/311

+ 2 sgh (f~r/3)f~'/3 + (f~r/3)z ~ Ilfll111/311]

_ g2 sgh (f~r/3)

Ilfd111/311

× (llfll111/311 ÷ sgh (f~/3)f~f3) 2 (4.20)

32 R. LOZANO

Note from Fig. 1 that:

sgh (g)g -> - A . (4.21)

Introducing (4.21) into (4.20) for g2=f(f3 it follows

g2

I(flrfl)(f~fl)l-> - - (llflll Ill311 - A) 2 IIAII IIf~ll

=g= IIAII IIAII (1 ,ii i A 2

(using (4.18)) = g ~ ( d - 2 ~ ) ~

(d ? A ) (since < d ~g~ A - - ~ ) .

Introducing (4.17) and (4.22) into (4.15) and taking into account that ~ in (4.19) is such that ~1~11 ~ 2g ~ we have

Idet M(0)I ~ - Idet M(0)I - I~rk~ + I I ~ l

( ( d - 2 A ) 2 1 1 k l O _ ( d + ~ ) e g g 2

=[ 2 + 411kll ]'d~jt - 2 ~ ) - ( d + ~ )

(using 4.7))

- > 2 ( d - 2 A ) - (d + A) = d - 5A

-> - since A ~ (4.23) 2

(d) Let us finally study the case &,t = 1 and &,2 = 1. (4.17) also holds in this case i.e.

Idet M(0)I <- d + A. (4.24)

Since &,l = 1, from (4.11) and Fig. 1 it follows that

Ilflll [If3ll ~ d + A. (4.25)

In this case fl in (4.5) reduces to

k h fl = ~ sg (det M(O)). (4.26)

Introducing (4.26) into (4.15) and noting that

II/~11 = 1 we get

[det M(0)I ~ Idet M(O) + Ilkll sgh (det M(0))I

- II/~11 z IIf~ll Ill311

-> Isgh [det M(0)] det M(O) + Ilkll I

- (d + A) (using (4.25))

-> Ilkll - A - (d + A) (using (4.21))

- > 4 d - I d e t M ( 0 ) I - IIf~ll I I A I I - d - 2 A

(using (4.16) and (4.8))

-> 4d - 2(d + A) - d - 2A

(using (4.24) and (4.25))

= d - 4A - ~ since A -< ]-~

which concludes the proof.

(4.27)

V

CONCLUSION

We have presented a way to circumvent singularities in adaptive pole placement control algorithms. The analysis has been detailed only for the first-order systems. In this case, the modifications to the estimates have been explicitly given.

Extensions of the proposed technique for higher-order systems is not straightforward and its complexity increases with the order of the system.

REFERENCES

Cristi, R. (1987a). Internal persistency of excitation in indirect adaptive control. IEEE Trans. Aut. Control, AC-32, 1101-1103.

Cristi, R. (1987b). A note on indirect adaptive control with stabilizable plant estimates. IEEE Trans. Aut. Control, AC-32, 1039-1040.

De Larminat, P. (1984). On the stabilization condition in indirect adaptive control. Automatica, 20, 793-795.

Giri, F., J. M. Dion, L. Dugard and M. M'Saad (1989). Robust pole placement direct adaptive control. 1EEE Trans. Aut. Control, AC-34, 353-356.

Goodwin, G. C. and K. S. Sin (1984). Adaptive Filtering, Prediction and Control. Prentice-Hall, Englewood Cliffs, NJ.

Knopp, K. (1956). Infinite Sequences and Series. Dover, New York.

Kreisselmeier, G. and M. C. Smith (1986). Stable adaptive regulation or arbitrary n-th order plants. IEEE Trans. Aut. Control, AC-31, 299-305.

Lozano, R. (1989). Robust adaptive regulation without persistent excitation. IEEE Trans. Aut. Control, AC-34, 1260-1267.

Lozano, R. and G. C. Goodwin (1985). A globally convergent pole placement algorithm without persistency of excitation requirement. IEEE Trans. Aut. Control, AC.30, 795-797.

Poiderman, J. W. (1989). A state space approach to the problem ot adaptive pole assignment. 2, 71-94.

Praly, L. (1984). Towards a globally stable direct adaptive control scheme for not necessarily minimum phase systems. IEEE Trans. Aut. Control, AC-29, 946-948.

Samson, C. (1980). Commande adaptative des syst~mes ~ minimum de phase ou non. Ph.D. Thesis, Universit6 de Rennes, France.

Singulari ty-free adapt ive po le -p lacement 33

A P P E N D I X A

Proof of Theorem 1 From (2.4) and the definitions in (2.9) and (2.10) we have

~, = O*rx,_~ + O, (A.1) where

O, = v,/(1 + II¢,-dl). (A.2)

In view of (2.7) and (2.11)

I~tl -< 6,. (A.3)

Introducing (A.1) into (2.12)

" T e, = -O,_~x,_~ + O, (A.4) where

0, = 0, - 0". (A.5)

From (2.16) and (A.5) we o b t ~ n

~ = Ot-~ + A,F~t_~et (A6)

From (2.15) and the matrix inve~ion lemma [3] we have

F~ ~ = F;j~ + ~ , _ ~x,~ ~. (A.7)

C o m b i ~ n g (A.6) and (A.7)

#Y~7'~, = ~ Y - , ~ # , - , + ~,(#y_~x,_, + e,) ~ + ~ ~ r re, (A~,_ ,F~,_ , - I) (A.8)

On the other hand from (2.15) it f o l l o ~ that

- 1 T ~ - ~ F ~ t - ~ - 1 =- r . (A.9) 1 + ~ t _ ~ _ ~ x t _ ~

In t roduong (A.4) and (A.9) into (A.8) we obtain

. . . . 2 ~,e~ V, - V~_~ + ~,v, - 1 + ~ _ ~ _ ~ x ~ _ ~ (A.10)

with v~ = #~v; '# , .

From (2.15) we ~ have

~ T F2 ~ - ~ t-~x~-~ (A.11) T • tr ~ = tr ~_~ - 1 + ~ - ~ - ~ x t - t

Adding (A.10) and (A.11) and using (2.13) we get

~tw~ ~ (A.12) ~ = E_~ + X,0~ - 1 + ~ , _ ~ E _ ~ x , _ ~

with E = V~ + tr E = O ~ F ? ~ , + tr E. (A.13)

In view of (2.9), (2.14), (2.14b) and (2.15) we have

1 + ~.~xtr_ iFt_ lxt_ 1 < 1 + tr Fo = tr. (A.14)

Introducing (A.3) and (A.14) into (A.12)

V~-< V~_I + ~ (t~6~ 2 - w~2). (A.15)

Using (2.14) and (2.14a) it follows:

-~ ~ 2 ~ 6 , ) - < - ~ A , 6 , • (~a, ~ - C) -- (~,~- w, -

(A.16)

In view of the above and since V t is positive, it follows that V t converges and At(o~6~ 2 - wt 2) converges to zero. Fur thermore , A t = 0 or 0~6~ 2 - wt 2 = 0 imply both that w~ 2 - ~2 _~ 0 [see also (2.14)] and therefore (2.17) follows.

Introducing (A.16) into (A.15) and iterating, we obtain

E -< E-~ - ~A,~, -< Vo - ~ ~ Ai~i. i = 1

Thus, we have

~ ~,~, _<v0. (A.17) i = l ~-

Combining (A.6) and (A.7) and using (A.4) we obtain

F~-~O, = F,-J~O,_~ + Aert_~xr,_~Ot_~

+ ;~,x,_~(-OL~xt_~ + ~,)

= FT-'~O,-~ + A:~,_~o,

= F~Oo + ~ ~.~x~_~f: i. (A.18) i = l

Note that in view of (2.9), (A.3) and (A.17) we have

-- ~ 4,6, --< V°. (A.19) i = l ~"

Introducing (A.13) and (A.19) into (A.18) we obtain (2.18)

and (2.18a). In view of (A.19), ~ Aixi_~Oi converges [see i ~ l

Knopp (1956)] and so does F~-lO, [see (A.18)]. Since F~ in (2.15) ~ n v e r g e s ( ~ z a n o , 1989; Samson, 1980) it follows that 0~ converges which concludes the proof. V