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Design of a Prescribed Convergence TimeUniform Robust Exact Observer in the Presence

of Measurement NoiseLiset Fraguela∗, Marco Tulio Angulo, Jaime Alberto Moreno and Leonid Fridman

Abstract—In practice, any observer is expected to have smallconvergence time and high precision despite the presence ofdisturbances. Decreasing the convergence time usually requiresto increase the overall gain of the observer. However, thisalso amplifies the effect of the noise and deteriorates theprecision. This trade-off is studied in this paper for the UniformRobust Exact Observer for mechanical systems: an observerwith uniform convergence with respect to the initial condition.In addition, a method to compute its gains is presentedguaranteeing a prescribed convergence time while keeping thebest possible precision under measurement noise. A simulationexample comparing the performance of the uniform observerwith respect to a linear one is presented.

I. INTRODUCTION

The Uniform Robust Exact Observer (UREO) and Differ-entiator were recently introduced in [1] and [2], respectively.Both are based on a modification of the Super-Twisting (ST)observer [3] and differentiator [4], by including high-degreeinjections. These new injections provide for the so-called“uniform convergence with respect to the initial condition”.In other words, the convergence time of the UREO for anyinitial observation error is uniformly bounded by a constant.In particular, its convergence time can be prescribed: it canbe set in advance by choosing adequately its gains. In thisform, the UREO significantly improves the convergence timeof the original ST observer [3], while keeping its exactness.

In [4], A. Levant shows that any exact differentiator overa class of signals with bounded second derivative (by someconstant L) has an accuracy no better1 than

√Lδ in the

L. Fraguela, M.T. Angulo and L.Fridman are with Facultad de Ingenierıa,Division de Ingenierıa Electrica, Departamento de Ingenierıa de Control yRobotica, Universidad Nacional Autonoma de Mexico, 04510, Mexico, D.F.Corresponding author e-mail: lisetfrc@siu.buap.mx

J. A. Moreno is with Computacion y Electronica, Instituto de Ingenierıa,Universidad Nacional Autonoma de Mexico, 04510, Mexico, D.F.

1A more careful reading shows that a better accuracy is 2√2√Lδ. This

constant is ”the same” as the one obtained by Kolmogorov in [5]. We write”the same” because Kolmogorov considers a signal defined in R and obtainsa constant with an uncertainty half as much as the one mentioned before.Levant only considers the signal in the positive semi-axis, thus obtainingtwice the uncertainty

presence of measurement noise uniformly bounded by δ.He then presents the ST differentiator [4], a discontinuousalgorithm which, in absence of noise, computes exactly thefirst order derivative. In addition, in the presence of smallmeasurement noise, the ST differentiator has an accuracy ofKs

√Lδ, for some Ks > 0.

Prescribed time convergence is useful for the state esti-mation of hybrid systems with strictly positive dwell-time.It allows splitting the state estimation problem into severaltime intervals in which each mode of the system is active.When the observer is not uniform, it is possible for the initialcondition for some mode of the system to be so large thatthe observer does not converge before the next switchingor impulse time takes place. In such a case, this can bealleviated only by making the gain of the observer largerand larger. However, in presence of measurement noise,this strategy also amplifies the noise obtaining larger andlarger errors. Therefore, in general, there exists a trade-offbetween decreasing the convergence time of the observerand increasing its errors due to the presence of noise.

Our paper explores this trade-off and whether the uniformconvergence property can offer some advantage. We take ascase of study the UREO and propose a method to optimallydesign its gains. The method guarantees that the observerconverges in a desired prescribed time and that its precisionwith respect to noise and perturbation is the best possible(based on an adaptation of the method presented in [6]).In a simulation example, the performance of the UREOis compared with respect to a linear optimally designedHigh Gain Observer (HGO) in terms of both convergencetime and precision [6]. For the first order HGO the optimalgain provides a precision of Kl

√Lδ, for some Kl > 0

explicitly calculated in [6]). Additionally, an analytic methodto compute the precision of the UREO adapted from [7] isintroduced to support our analysis.

The remainder of this paper is organized as follows:Section II introduces the Problem Statement, in Section IIIwe present a method to optimally design the UREO. InSection IV the simulation study is presented and Section

51st IEEE Conference on Decision and ControlDecember 10-13, 2012. Maui, Hawaii, USA

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V gives some conclusions. Finally, the Appendix presentsa summary of the method to compute the observation errorbased on [7].

II. PROBLEM STATEMENT

Consider the mechanical system

x1 = x2x2 = f(t, x1, x2) + ξ(t, x1, x2),

(1)

where x1 ∈ R is the position and x2 ∈ R is the velocity.For the sake of simplicity, we shall only consider the scalarcase. In the vector case, one independent observer can beconstructed for each position. The function f is assumed tobe known and the function ξ is unknown and represents aperturbation. Only one assumption is imposed on the system:

|f(t, x1, x2)− f(t, x1, x2)− ξ(t, x1, x2)| ≤ L > 0,

for all x2 ∈ R.The aim is to design a velocity observer for x2 based on

the noisy measurement of the position y = x1 + η. Theonly assumption on the measurement noise η(·) is that it isuniformly bounded by a known constant δ.

To solve the problem, the UREO of [1] may be used. Itcan takes the following form

ΣU :

{˙x1 = −α1

ε φ1(x1 − y) + x2, x1(0) = y(0),˙x2 = −α2

ε2 φ2(x1 − y) + f(t, x1, x2),(2)

where αi > 0, i = 1, 2 are fixed constants, 1/ε > 0 is thegain of the observer and the functions φi, i = 1, 2, are givenby

φ1(θ) = |θ|1/2 sign(θ) + µ|θ|3/2 sign(θ),φ2(θ) = 1

2 sign(θ) + 2µθ + 32µ

2|θ|2 sign(θ),

with µ ≥ 0 a design parameter. In this form, the UREO hastwo parameters (µ, ε). Note that when µ = 0, the UREO isreduced to the ST observer of [3].

We will compare the performance in convergence timeand precision of the UREO of equation (2), with the possibleperformance using the linear High-Gain observer (HGO)

Σl :

{˙x1 = −α1

ε (x1 − y) + x2˙x2 = −α2

ε2 (x1 − y) + f(t, x1, x2),(3)

optimally designed in the sense of measurement noise andperturbation [6]. For the case of a linear observer, we willassume that the variables of the system are restricted to acompact set, so there exists a maximum convergence timefor the observer with respect to the initial condition. Dueto the uniform convergence property, this assumption isunnecessary for the UREO [2].

A. Formulation of the Problem.

We address the following question: how to optimallydesign the parameters (µ, ε) of the UREO to provide adesired convergence time TC while maintaining the bestpossible precision with respect to noise and perturbation?

The convergence time of all the observers will be mea-sured in the absence of noise and perturbation.

For this, we propose a method to optimally design theparameters of the UREO. To compute the asymptotic error ofthe observer due to the disturbances (i.e. the ultimate boundfor x2−x2), we will use results adapted from our companionpaper [7], summarized in the Appendix. Additionally, acomparative example of the performance of the optimalUREO with respect to the performance of a linear optimallydesigned HGO is presented.

III. A METHOD TO DESIGN THE UREO PARAMETERS

This section describes a method to optimally select theparameters (µ∗, ε∗U ) of the UREO of equation (2). The aimis to guarantee a desired convergence time Tc, together withthe best possible precision for a given perturbation and noise.

We start by picking µ as follows. Set ε = 1 and pickµ∗ such that the convergence time T0 of the observerwithout disturbances is smaller or equal than Tc. A boundfor the convergence time of the UREO is given in [2].However, this is a very conservative estimate and is nothelpful in finding the parameter µ for a desired convergencetime. An alternative method to solve this problem is toconstruct a graph of the convergence time T0 as a functionof µ by running a simulation with a very (very!) largeinitial condition. Note that due to the uniform convergenceproperty of the UREO, this graph is independent of the initialcondition. This graph needs to be constructed only once fora given set of parameters αi. For α1 = 1.5, α2 = 1.1, theresults of Figure 1 were obtained. The blue circles in Figure1 are experimental results obtained by simulation.

Interestingly enough, the behavior of Figure 1 can beeffectively described by the expression

T0(µ) = m(α1, α2)1√µ, (4)

with m(1.5, 1.1) = 4.01. This particular value for theconstant m was obtained by adjusting equation (4) usinga single measurement. Surprisingly, it can predict the othermeasurements with high accuracy. In this form, it is possibleto directly compute the value for µ∗ as follows

µ∗ =

(m(α1, α2)

Tc

)2

. (5)

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0 100 200 300 400 5000

0.1

0.2

0.3

0.4

0.5

0.6

µ

T0

Fig. 1. Convergence time T0 for x2 as a function of µ, for ε =1 and α1 = 1.5, α2 = 1.1 The circles show convergence timesmeasured by simulation. The interpolation using formula (4) isshown in solid line.

The next step is to choose ε keeping the convergencetime smaller than Tc and providing the best precision. Firstnote that once µ has been fixed as µ∗, any ε < 1 willproduce a convergence time smaller than Tc. Indeed, from(2), the parameter ε only re-scales time by a factor 1/ε inthe absence of disturbances. Thus, the problem is reducedto pick an ε ≤ 1 that yields the best precision with respectto the perturbation and noise.

The precision of the UREO is the ultimate bound x2,ubof the observation error x2− x2. In other words, the largestasymptotic velocity error that the observer will make due tothe disturbances. For a given set of parameters αi, µ, ε, Land δ, the precision of the UREO can be computed usingTheorem 1 presented in the Appendix. It only requires to findthe solution of equation (10) and then to substitute it in theexpression (11). Equation (10) can be solved numerically, forinstance using the MATLAB fzero function, or graphically,by finding the intersection of the two graphs. In this form, itis possible to construct a graph of the ultimate bound x2,ubas a function of ε.

The function x2,ub is convex, so it has a unique minimumat ε = ε∗. If it turns out that ε∗ < 1, then the optimalselection for ε in the UREO, providing convergence timeand precision, is given by ε∗U = ε∗. Otherwise, i.e. ε∗ ≥ 1,the optimal selection is ε∗U = 1.

IV. SIMULATION EXAMPLE

For the simulation study, let us consider a pendulum withunknown Coulomb friction and an external perturbation. Its

dynamics are described by system (1) with

f(t, x1, x2) =1

Jnu− g

lnsin(x1)− Vs,n

Jnx2 −

Ps,nJn

sign(x2),

(6)ξ(t, x1, x2) =

[1J −

1Jn

]u−

[gl −

gln

]sin(x1)−

−[Vs

J −Vs,n

Jn

]x2 −

[Ps

J −Ps,n

Jn

]sign(x2) + v,

with v(t) = 0.5 sin(2t) + 0.5 cos(5t).Let u = u(t) be the twisting controller

u = −30sign(y − xd1)− 15sign(x2 − xd2)

and xd1 = sin(t), xd2 = cos(t) the reference signals.The real parameters of the system are: gravity acceleration

g = 9.815 m/s2, pendulum longitude l = 0.9 m, massm = 1.1 g, inertial moment J = ml2 = 0.891 gm2, viscousand dry friction coefficients Vs = 0.18 gm2/s, Ps = 0.45gm2/s2.

Let the nominal parameters be α1 = 1.5, α2 = 1.1 arefixed and ln = 1 m, mn = 1 g, Jn = 1 gm2, Vsn = 0.2gm2/s, Psn = 0.5 gm2/s2, that represents a 10% differencefrom the actual parameters of the system. Thus, the proposedvelocity observer has the form

˙x1 = x2 −α1

εφ1(x1 − y)

˙x2 =1

Jnu−

g

lnsin(y)−

Vs,n

Jnx2 −

Ps,n

Jnsign(x2)−

−α2

ε2φ2(x1 − y),

(7)

An easy calculation shows that the given controller isbounded, with |u| ≤ 45, and the inequality |x2| ≤ 70is ensured, when nominal values of parameters and theirmaximal possible deviations are taken into account. Taking|x2| ≤ 70 and assuming that |x2| ≤ 140, we obtain that

|ρ| = |f(t, x1, x2)− f(t, x1, x2)− ξ(t)| < 6 = L.

The measurement noise η was created using MAT-LAB/Simulinks block Uniform Random Number with asample time of 0.05 and initial values x1(0) = x1(0) = 0,x2(0) = 0 and x2(0) = 140.

Following the method in Section III for a desired pre-scribed convergence time Tc = 0.2 s, using formula (5) weobtain µ∗ = 402. The set of parameters required for theUREO and HGO in order to construct the graph of x2,ub asa function of ε is (α1, α2, L, δ) = (1.5, 1.1, 402, 6, 0.001)and for the UREO also µ∗ = 402. In Figure 2 (a) and(b) are depicted those graphs for the UREO and HGO,respectively. Notice in Figure 2 (a) that εoptU = 0.6 < 1,then the performance of the UREO can be improved settingε∗U = 0.6 instead of its prefixed value 1. This way, we obtain

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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

ε

xtild

e 2,ub

0.3615

εUopt=ε

U*

0.39

(a) UREO

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xtild

e 2,ub

ε

εlopt=0.019ε

l*=0.005

0.705

0.3504

(b) HGO

Fig. 2. Graph of x2,ub as a function of ε (L = 6, δ = 0.001 andµ∗ = 402)

the minimum x2,ub(0.6) = 0.3605 that is computed usingthe methodology presented in the Appendix. At the sametime, the convergence time will be improve due to fact thatthe optimal gain εoptU is smaller than 1.

Meanwhile, for the HGO with the maximum initial ob-servation error equal to 100000, the prescribed convergencetime Tc yields ε∗l = 0.005. In [6] an expression to computethe ultimate bound of x2 as a function of ε is presented andhas the following form:

x2,ub = P2Lε+δ

ε

(P2

P1Q12 +Q21

), (8)

where the constants Pk, Qik (i, k = 1, 2) are computed asdescribed in the Appendix, yielding x2,ub(ε∗l ) = 0.705. Onthe other hand, equation (8) allows to obtain an optimal

0 2 4 6 8 10−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

time (s)

xhat

1−y

HGOUREO

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−3

−2

−1

0

1

2

3

x 10−3

time (s)

xhat

1−y

HGOUREO

Fig. 3. Comparison of the estimation error x1−y using UREO with εoptU =0.6 and HGO with ε∗l = 0.005, with respect to the desired trajectory xd1(L = 6, δ = 0.001 and µ∗ = 402)

gain that guarantees the minimum x2,ub(εoptl ) = 0.3504

for an optimal gain εoptl = 0.019. It is important to notice(see Figure 2 (b)) that the gain that guarantees the desiredprescribed convergence time is smaller than the optimalone, i.e., ε∗l < εoptl . Therefore, it is impossible to improvex2,ub(ε

∗l ) = 0.705 with ε∗l = 0.005 for the HGO to ensure

convergence in the desired prescribed convergence time.We can then conclude that the UREO will over-perform

the HGO with respect to the ultimate bound of the velocityestimation error in the presence of the considered measure-ment noise and perturbation. Figures 3 and 4 support thisstatement.

Figure 3 shows the estimated positions error x1 − y asfunctions of time using UREO and HGO . Figure 4 displaysthe behavior of the velocity estimation error as a functionof time for the UREO and HGO. It is important to observethat the ultimate bound of x2 of the UREO is considerablysmaller than the one of the HGO, as claimed before. Theachievement of the prescribed convergence time Tc = 0.2s of both observers, as was intended in the first place, is

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0 2 4 6 8 10−20

0

20

40

60

80

100

120

140

time (s)

xtild

e 2

HGOUREO

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.2

−0.1

0

0.1

0.2

0.3

time (s)

xtild

e 2

HGOUREO

Fig. 4. Comparison of the velocity estimation error x2 (and a detail) usingUREO with εoptU = 0.6 and HGO with ε∗l = 0.005 (L = 6, δ = 0.001and µ∗ = 402)

reflected in the two figures.

V. CONCLUSIONS

We have presented a method to design the UREO pro-viding a desired convergence time together with the bestpossible precision with respect to noise and perturbation.The main property of the UREO is its uniform convergence.The method was illustrated in an example showing thatthe UREO can over-perform a linear HG observer in bothconvergence time and precision.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial sup-port form: (i) CONACyT (Consejo Nacional de Ciencia yTecnologıa), grants 51244, 132125 and CVUs 160335 and229959, (ii) Programa de Apoyo a Proyectos de Investi-gacion e Innovacion Tecnologica (PAPIIT) UNAM, grants117610 and 117211 and (iii) Fondo de Colaboracion del II-FI, UNAM, IISGBAS-165-2011.

REFERENCES

[1] E. Cruz-Zavala, J. Moreno, and L. Fridman, “Uniform second-ordersliding mode observer for mechanical systems,” in Variable StructureSystems (VSS), 2010 11th International Workshop on, June 2010, pp.14 –19.

[2] ——, “Uniform robust exact differentiator,” Automatic Control, IEEETransactions on, vol. 56, no. 11, pp. 2727 –2733, Nov. 2011.

[3] J. Davila, L. Fridman, and A. Levant, “Second-order sliding-mode ob-server for mechanical systems,” Automatic Control, IEEE Transactionson, vol. 50, no. 11, pp. 1785 – 1789, Nov. 2005.

[4] A. Levant, “Robust exact differentiation via sliding mode technique,”Automatica, vol. 34, no. 3, pp. 379–384, 1998.

[5] A. N. Kolmogorov, “On inequalities between upper bounds of consecu-tive derivatives of an arbitrary function defined on an infinite interval,”Amer. Math. Soc. Transl, vol. 2, no. 9, pp. 233–242, 1962.

[6] L. K. Vasiljevic and H. K. Khalil, “Error bounds in differentiationof noisy signals by high-gain observers,” Systems & Control Letters,vol. 57, no. 10, pp. 856 – 862, 2008.

[7] M. Angulo, J. Moreno, and L. Fridman, “Optimal gain for the super-twisting differentiator in the presence of measurement noise,” in 2012American Control Conference, 2012.

[8] D. Bernstein and W. So, “Some explicit formulas for the matrixexponential,” Automatic Control, IEEE Transactions on, vol. 38, no. 8,pp. 1228 –1232, Aug 1993.

[9] A. Filippov, Differential equations with discontinuous right-hand side.London: Kruwler, 1960.

APPENDIX

This section summarizes a method to compute the ultimatebound of the observation error for the UREO adaptedfrom our results in [7] for the Generalized Super-twistingAlgorithm.

The dynamics of the observation error x = x− x for theUREO is described by

˙x1 = x2 − α1

ε φ1(x1) + α1

ε w1

˙x2 = ρ− α2

ε2 φ2(x1) + α2

ε2 w2,(9)

where ρ = ρ(t, x1, x2, x2) = f(t, x1, x2, u)− f(t, x1, x2)−g(t, x1, x2) and

wi = φi(x1)− φi(x1 − η); i = 1, 2.

The aim is to obtain an upper bound for the maximumultimate bound of the estimation error x2 due to the presenceof the uncertainties η and ρ. Due to the assumptions givenin Section II, both |ρ(t)| ≤ L = C1 + C2 and |η(t)| ≤ δ.

The ultimate bound of the estimation error x2 is obtainedin terms of the ultimate bound for x1. This value is foundby obtaining the largest solution of an algebraic equation.These considerations are proved in the following theorem.

Theorem 1: The ultimate bound for x1, denoted as x1,ub,is the largest solution of the equation

x1/2 + µx3/2 = Γ(x)[P1Lε

2 + 2µQ12δ]

+Q11 [∆1(x)+

(10)

+µ∆3(x)] +1

2Q12Γ(c)

[∆0(x) + 3µ2∆2(x)

],

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in the unknown x. Once this value is computed, the ultimatebound for x2 is given by

x2,ub = P2LΓ(x)ε+1

εΞ(x, δ) (11)

where,

Ξ(x, δ) =P2

P1Q12Γ(x)

[2µδ +

1

2∆0(x) +

3

2µ2∆2(x)

]+

+Q21 [∆1(x) + µ∆3(x)] ,

and the functions ∆i (i = 0, 1, 2, 3) and Γ are given by

∆0(x) = 1− sign(|x| − δ),∆1(x) = |x|1/2 − ||x| − δ|1/2 sign(|x| − δ),∆2(x) = |x|2 − ||x| − δ|2 sign(|x| − δ),∆3(x) = |x|3/2 − ||x| − δ|3/2 sign(|x| − δ),Γ(x) = 2|x|1/2

1+3µ|x| .

(12)

The constants Pk =∫∞0

∣∣{eAtB}k

∣∣ dt and Qkj =∫∞0

∣∣{eAtDj

}k

∣∣ dt (k, j = 1, 2). The notation {·}k de-notes the k-th element of a two-dimensional vector. Theexplicit formulas from [8], Corollary 2.4, for the case whenα21 − 4α2 < 0, are applied to compute the values of this

constants. The matrices A, B, D1 and D2 are defined as

A =

(−α1 1−α2 0

), B =

(01

), D1 =

(α1

0

), D2 =

(0α2

).

(13)Proof: Using Fillipov’s argument presented in [9, Chap-

ter 2, p. 99], each solution of the system

dζ

dτ= Aζ +Bε2ρ+D1w1 +D2w1 (14)

with ρ = ρ/φ′1, w1 = w1 and w2 = w2/φ′1, can be

transformed into a solution of (9) by using (x1, x2) =φ−1(ζ) =

(φ−11 (ζ1), 1εζ2

)as a formal change of coordinates.

One just need to restrict x1 to lie into a compact, as largeas required, and be careful of what happens when x1 isidentically zero.

The transformed disturbances ρ and wi can be shown tobe bounded as

|w1| ≤ ∆1(x) + µ∆3(x), |ρ| ≤ LΓ(x),|w2| ≤ Γ(x)

(12∆0(x) + 2µδ + 3

2µ2∆2(x)

),

(15)By assuming that x1 lies in a compact, then the trans-

formed disturbances are uniformly bounded. Using the trans-formed system (14) and the uniform bounds for the trans-formed uncertainties |ρ| ≤ ρ0, |w1| ≤ w0

1 and |w2| ≤ w02 ,

the ultimate bound for ζ1 can be computed using [6, Lemma1] takes the form

x1/21,ub + µx

3/21,ub = ζ1,ub = P1ε

2ρ0 +Q11w01 +Q12w

02,

and for ζ2

ζ2,ub = P2ε2ρ0 +Q21w

01 +Q22w

02,

and by consequence

x2,ub =1

εζ2,ub =

1

ε

[P2

P1ζ1,ub +

(Q21 −

P2

P1Q11

)w0

1

].

However, this result is a very crude approximation of theobservation error, since the transformed disturbances dependon the state.

To consider the dependance of the disturbances on thestate, the following recursive algorithm is proposed:

Step 0: initialize the disturbances at their maximum.Step 1: compute the corresponding ultimate bound for x1

(here denoted x0) as the unique solution to

x1/20 + µx

3/20 = P1ε

2ρ0 +Q11w01 +Q12w

02

Step 2: update the bounds of the disturbances using theobtained value of the ultimate bound

ρ1 = LΓ(x0), w11 = ∆1(x0) + µ∆3(x0),

w12 = Γ(x0)

(1

2∆0(x0) + 2µδ +

3

2µ2∆2(x0)

)Step 3: go back to Step 1 and repeat!The proof follows then by induction: the first step of

the algorithm obviously corresponds to an upper bound ofthe ultimate bound of the error. If at step n the ultimatebound is xn and the disturbances are (ρn, wn1 , w

n2 ), then the

disturbances are indeed bounded by (ρn+1, wn+11 , wn+1

2 ) sothe ultimate bound is in fact xn+1. This shows that xn, n ≥ 0are upper-bounds for the ultimate bound of x1. Then thelimit of the algorithm (fixed point) is also an upper boundfor the ultimate bound of x1. The fixed point is found bysolving (10).

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