Fuzzy iterative learning control applied in a biological ...

Applied Mathematics and Computation 246 (2014) 608–618

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Fuzzy iterative learning control applied in a biological reactor using a reduced number of measures M.A. Márquez-Vera a,⇑, L.E. Ramos-Velasco b, J. Suárez-Cansino b, C.A. Márquez-Vera c a

Universidad Politécnica de Pachuca, C. Pachuca-Cd. Sahagún Km 20, C.P. 43830 Zempoala, Hgo, Mexico Universidad Autónoma del Estado de Hidalgo, C. Pachuca-Tulancingo Km 4.5, 42090 Mineral de la Reforma, Hgo, Mexico c Universidad Veracruzana, Prolongación Venustiano Carranza S/N, Col. Revolución, C.P. 93390 Poza Rica, Veracruz, Mexico b

a r t i c l e

i n f o

Keywords: Iterative learning control Fuzzy logic Batch reactor

a b s t r a c t There exist some processes difficult to control as the chemical ones, a common problem takes place when the output cannot be measured on-line, and so, closed-loop control cannot be implemented. In this work an iterative learning control type proportional-derivative is analyzed and theoretical results are shown, this control is applied to a biological reactor to degrade phenol by working in discontinuous batch state, as the measures of the substrata concentrations are taken by hand, it was proposed to have a sample time of one hour. To guarantee convergence and to improve the control, cubic splines were used to interpolate the measures. Fuzzy logic was used to compute the control gains used to build the control signal. Simulation results are shown and the control signals are presented through iterations, here it is possible to see that the error is smaller using fuzzy logic to compute the control signal when iterations run. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction There exists a lot of chemicals and biochemical systems where it is not possible to measure all state variables that exists in the process [1]. In this type of processes, the control just to be made in open-loop, and the input to the process, usually the xenobiotic substance, is the inlet to the reactor [2]; this input is a load profile computed by an expert in the process. In this case the xenobiotic component is a toxic substance that should not to be present in the water, for example phenol derivations [3]. To biodegrade the phenol, a sequencing batch reactor (SBR) can be used, it is a process designed to work in unstable conditions, this kind of processes has some phases that are iterated with a particular time period, depending on the kind of treatment and the physiologic state of process [4]. In this work, the SBR is a recipient that contains waste water, where activated sludge works to consume a xenobiotic substance as the phenol, which is present in water deposits due textile factories and the wood treatment. The use of granular reactors can improve the results but with these reactors the main problem is their construction, it is difficult to build them in huge scale, and the actuators required consume a lot of energy. One problem to control a SBR is that there does not exists on-line sensors for some substrates, so closed-loop control cannot be implemented, it is possible to use states observers to estimate the concentrations and to make a closed-loop [5–8], this implies to measure all possible state variables for which there exists sensors. The biotechnological processes are ⇑ Corresponding author. E-mail addresses: [email protected] (M.A. Márquez-Vera), [email protected] (L.E. Ramos-Velasco), [email protected] (J. Suárez-Cansino), [email protected] (C.A. Márquez-Vera). http://dx.doi.org/10.1016/j.amc.2014.08.072 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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nonlinear and not well known systems to easily design the observers required, in this way, the fuzzy logic can be used to have an approached model and to used it for the observers design [9–11]. Another control technique for these processes is the iterative learning control (ILC), as in the processes the substrata concentrations are obtained after some treatment in the measures as spectrography; it is not possible to have the substrata concentrations in real time. Fortunately, it is possible to use the information from previous iterations of the SBR to build the next control signal (the load profile); usually, to tune the controller, the model must to be considered [12], but by using fuzzy logic, the control gains can be adapted [13,14]. The ILC was developed initially for robotics, where a robot must to make some action many times [15], each time the tracking error from the previous iteration is used to build the control signal for the next iteration. As the SBR works in iterations, this idea is used in the reactor because some substances concentrations are unknown after certain time, there exists some results for ILC in nonlinear systems [16], also its fuzzy version to control processes [13], the main problem is that the analysis for this controllers is for continuous time systems or in fact, when the sample time is small enough. In this work, the measures are supposed to be made by hand, so, it is not possible to guarantee a small sample time. The main idea was to interpolate measures to have a sample time of one hour to reduce cost and to help the technics whom work in the laboratory, and to use fuzzy logic to compute the control gains. Some theoretical results are presented in this work and there was made a comparison between simulation results by using a proportional-derivative (PD) ILC with a sample time of 1.2 min; and the simulation where the measures were taken each hour and interpolated with cubic splines to have a virtual sample time of 1.2 min, and the control gains were computed using fuzzy logic. This work is organized as follows: first the bioprocess description is shown in Section 2, and the subSection 2.1 shows the model used for simulation. In Section 3 it is presented the iterative learning control with some assumptions used in subSection 3.1 for an analytic study to determine the ILC convergence. The Section 4 has some concepts about fuzzy logic and its use to implement a fuzzy ILC type PD. The comparative results are in Section 5, and a discussion about errors is in Section 6. Finally, conclusions are in Section 7. Appendix A shows the proof for the theoretic results given in subSection 3.1. 2. Bioprocess description The phenol is a chemical product used to produce resins, explosives, perfume, disinfectants, tints and it is present in the petrol and gas industrial residues; it is difficult to treat this substance due it is an inhibitor for the microbial grow-up [3]. However, this substance cans arrive to water deposits and it is toxic for the people, so water for consumption must to be treated in a special way. There are some techniques to purify wastewater, for example the biofiltration, it uses microorganisms to remove pollution, another purification process is the granular biomass, this technique has the bigger rate of purification [17], the main problem for bioreactors with granular biomass is that it is required a lot of energy to purify important quantities of water; finally, the activated sludge is the common treatment applied for wastewater, here, the biomass has the appearance of sludge, this technique was used in this work. The principal bacteria in the microbial consortium in this case is the pseudomonas putida, furthermore, it has a big activity of digestion for organic material. This living cells are called the biomass. The aim in this reactor is to see the xenobiotic substance as food for the microorganisms, basically, this reactor is a tank where biological reactions take place in a liquid medium. To make that bacteria consume the toxic elements, they must to be acclimated to the phenol. The SBR process operates in five phases, first the reactor is filled with wastewater containing the toxic elements, it is made by using some filling profile, if huge quantities of phenol ingress, the bacteria can die due pollution and with a low inlet, there will be not enough food for microorganisms. The next step is the reaction, here, some actuators are turnedon, as the aeration and the agitation, it is made to propitiate the reaction and to guaranty that the concentrations are the same in the entire reactor. Then, all actuators are turned-off and the biomass is deposited at the bottom. After the sedimentation, the clear water can be retired from the top of reactor where the water has not pollution; finally, a dead time is used to relax the biomass in this phase, and to have similar initial conditions to the last one, before to continue with the next cycle. 2.1. Biodegradation model To get the simulation results, a model for the bioprocess must to be used. The experimentation to have an approached model was made in the Autonomous University of Hidalgo State, the mass balance differential equations are (1)

1 0 1 1 0 _ XðtÞ XðtÞ lðtÞXðtÞ   C C B_ C B B S1 ðtÞ C B qS1 ðtÞXðtÞ Sin  S1 ðtÞ C C B C¼B CDðtÞ; B B 1 þ C B C C B_ B A @ S2 ðtÞ A @ v S2 ðtÞXðtÞ  qS2 ðtÞXðtÞ A @ S2 ðtÞ _ 0 VðtÞ 1 0

ð1Þ

where XðtÞ it the biomass concentration, S1 ðtÞ is the phenol concentration, S2 ðtÞ is the metabolic intermediate concentration which is also toxic and VðtÞ is the volume of liquid inside the reactor; DðtÞ is the dilution rate and it is the input of the system.

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qS1 is the phenol consumption, qS2 is the intermediate consumption and v S2 is the intermediate production rate, lðtÞ is the microbian grow, it is defined by modified Haldane and Monod equations [18] and it is defined by (2) and determined as follows

lðtÞ ¼ l1 ðtÞ þ l2 ðtÞ;

ð2Þ

lmax1 S1 ðtÞ K2 ; K S1 þ S1 ðtÞ þ S21 ðtÞ=K i K 2 þ S2 ðtÞ l S2 ðtÞ K1 l2 ðtÞ ¼ max2 : K S2 þ S2 ðtÞ K 1 þ S1 ðtÞ l1 ðtÞ ¼

The consumption and production for substrates are defined by the equations set (3)

qS1 ðtÞ

l1 ðtÞ=Y 1 ;

¼

qS2 ðtÞ ¼ l2 ðtÞ=Y 2 ; v S2 ðtÞ ¼ al2 ðtÞ:

ð3Þ

The values used for simulation are shown in Table 1 which were verified in [3] using this kind of process, these parameters must to be tuned if the process has a reset after clean the reactor or if the biomass is eliminated and a new one is acclimated. 3. Iterative learning control When measures cannot be obtained on-line, the ILC is an alternative; its difference with other control schemes is that it works with information from the last operation; as the SBR works in iterations, the information from concentrations in the last iteration is used to build the next control signal. The original idea was to use the dynamic history from previous movements of a robot, which made the same activity each time, and the result was improved when iterations run [16]. In this case, the closed-loop is in iterations, not in time, a diagram for this kind of control is shown in Fig. 1; for the process in this research, the input is the inlet profile and the output is the phenol concentration. The process works 15 h to degrade phenol, and a dead time of 10 h is used to know the concentrations to build the next control signal and to arrive to some similar initial conditions between iterations. The iterative learning is made by building a signal control for the next iteration from the error, the derivative of error and the previous control signal which was known from the previous iteration [19]. In order to apply the ILC to the SBR, it was considered the system (4):

x_ i ðtÞ ¼ f ðxi ðtÞÞ þ bðxi ðtÞÞui ðtÞ þ xi ðtÞ;

ð4Þ

yi ðtÞ ¼ gðxi ðtÞÞ for i ¼ 1; 2; 3; . . .

where xi ðtÞ; xi ðtÞ 2 Rn and ui ðtÞ; yi ðtÞ 2 Rr represent the state, the disturbance, the input and the output in the i-th iteration, respectively. Furthermore, f : Rn ! Rn ; b : Rn ! Matnr ðRÞ ¼ Rnr , and g : Rn ! Rr are non-linear smooth mappings. Consider the desired system:

x_ d ðtÞ ¼ f ðxd ðtÞÞ þ bðxd ðtÞÞud ðtÞ; yd ðtÞ ¼ gðxd ðtÞÞ;

ð5Þ

xd ð0Þ ¼ x0 : Let U be a convex subset of Rn , a set including all possible states. We are going to work under the following assumptions:

Table 1 Parameter values in phenol biodegradation process. Symbol

lmax1 lmax2

Value

Unit

0.39

h

1

0.028

h

1

K S1

30

mgl

1

K S2

350

mgl

1

Ki

170

mgl

1

K1

66

mgl

1

K2

160

Y1 Y2

a

0.57 0.67 1.6

1

mgl mg=mg mg=mg mgl

1

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Fig. 1. ILC diagram.

S1. The Jabobian matrix of f ; g and b are bounded on U. S2. The mapping b is bounded on U. S3. We know the desired output yd . S4. The Hessian matrix [20] of each g s is bounded on the convex set U; s ¼ 1; . . . r, where g ¼ ðg 1 ; . . . ; g r Þ. The condition S1 implies that f ; g and b are Lipschitz maps [21] on U. Therefore the time-dependent vector field defined on J  Rn ; Fðt; xÞ ¼ f ðxðtÞÞ þ bðxðtÞÞud ðtÞ, satisfies a Lipschitz condition on Rn uniformly with respect to I, where I is an open interval containing 0. So there exists a unique integral curve xd defined on a maximal interval I0 containing 0 such that xð0Þ ¼ x0 . We suppose that ½0; T  I0 . Let Dt > 0 be the sampling time and let m be the bigger positive integer such that mDt 6 T. Through this paper, for n ¼ 0; . . . m, we are going to denote the number nDt by n, the number nDt þ Dt by n þ 1. Finally, we shall consider the following: S5. For every n ¼ 0; . . . ; m  1,

ui ðtÞ ¼ ui ðnÞ;

8t 2 ½n; n þ 1Þ:

S6. We know an initial error bound e1 and an input disturbance bounds e2 . So, for all index i we have, jx0  xi ð0Þj < e1 and maxt2½0;T jxi ðtÞj < e2 . S7. Let ei ðtÞ ¼ yd ðtÞ  yi ðtÞ. For every n ¼ 0; . . . ; m  1, we shall consider the following PD Learning Law:

  1 uiþ1 ðnÞ ¼ ui ðnÞ þ C ðei ðn þ 1Þ  ei ðnÞÞ  Uei ðnÞ ; Dt

ð6Þ

were C ¼ Ci ðnÞ and U ¼ Ci ðnÞ are r  r matrices corresponding to the proportional and the derivative gains. Both may depend on i and n. Finally, we are going to use the following notation: dui ðnÞ :¼ ud ðnÞ  ui ðnÞ and dxi ðnÞ :¼ xd ðnÞ  xi ðnÞ. 3.1. Convergence results For every index i and n ¼ 0; . . . ; m  1, there exist matrices M 1 ; M2 ; M3 and a vector a, all depending on n and i, such that, for all n ¼ 0; . . . ; m  1 and all i:

1 ðei ðn þ 1Þ  ei ðnÞÞ ¼ M 1 dui ðnÞ þ M 2 dxi ðnÞ þ a; Dt ei ðnÞ: ¼ M3 dxi ðnÞ

ð7Þ

Furthermore, kM 1 k; kM2 k; kM 3 k have a bound independent of n and i and also jaj 6 c0 e2 þ oðDtÞ for some constant c0 . As consequence, for every index i and every n ¼ 1; . . . ; m:

jduiþ1 ðnÞj 6 q jdui ðnÞj þ 1 max jdui ðkÞj þ e: k¼0;...;n1

ð8Þ

where:

q ¼ sup max kI  Ci ðnÞM1 k; i

n¼0;...m

ð9Þ

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1 ¼ sup max kCi ðnÞðUi ðnÞM3  M2 Þkc1 ; i

ð10Þ

n¼0;...m

e 6 c2 e1 þ c3 e2 þ oðDtÞ:

ð11Þ r

for some constants c1 ; c2 and c3 . The symbol j  j using above lemma represents the ‘1 -norm in R and the symbol k  k represents the matrix norm induced by the ‘1 -norm. If the system (4) were linear, i.e.:

x_ i ðtÞ ¼ Axi ðtÞ þ Bui ðtÞ þ xi ðtÞ;

ð12Þ

yi ðtÞ ¼ Cxi ðtÞ: then M 1 ¼ CB; M 2 ¼ CA y M3 ¼ C. Consequently, C and U are constant matrices and we have:

q ¼ kI  CCBk; 1 ¼ kCðUC  CAÞkc1 :

ð13Þ ð14Þ

for some constant c1 . Let .; 1 and e the constants described in (9) and (10). For every index i and n ¼ 1; . . . ; m, we establish: minðn;iÞ X 

Pði; nÞ ¼



max qis 1sk¼0;...;ns jdu0 ðkÞj;

s

s¼1

  i

Xi

Q ði; nÞ ¼

Rði; nÞ ¼

i

s¼minðn;iÞ

minðn;iÞ1 X

.is 1s ;

s

ð16Þ

1spði;sÞ; where

s¼0

pði; sÞ ¼

ð17Þ

 i1  X sþj

j

q:

j

j¼0

ð15Þ

Theorem 1. For every index i and n ¼ 1; . . . ; m:

jdui ðnÞj

6

.i jdu0 ðnÞj þ Pði; nÞ þ jdu0 ð0ÞjQði; nÞ þ eRði; nÞ:

ð18Þ

  i .iminðn;iÞ ; s¼1;...;minðn;iÞ s

ð19Þ

And:

Pði; nÞ 6 Cu0 C1

max

Pði; nÞ þ jdu0 ð0ÞjQ ði; nÞ 6 Cu0 ð. þ 1Þi ; Rði; nÞ 6



1

qþ11

1

minðn;iÞ

1.

ð20Þ !

1 :

ð21Þ

where C1 ¼ maxf1; 1minðn;iÞ g and

Cu0 ¼ max jdu0 ðkÞj: k¼0;...;n1

Therefore, if n ¼ 1; . . . ; m is fixed and

qþ1 10:

ð28Þ

The experimental validated model (1), with the parameters values shown in Table 1 was simulated with the following initial conditions Xð0Þ ¼ 400 mg/l, S1 ð0Þ ¼ 0 mg/l, S2 ð0Þ ¼ 0 mg/l y Vð0Þ ¼ 1 l. Each iteration has as initial conditions the final values of the state variables from the last iteration. The biomass concentration, in practice, use to increase with this type of process, so a purge can be made and then the biomass excess is incinerated; in this case, a dead time was considered to have some similar initial conditions and adding a limited random value 10 mg/l to simulate different initial conditions for each iteration. The variation in the initial conditions are considered in e1 , and the noise signal and external disturbance signals are considered in e2 as denoted in the assumption S6 in Section 3. To make the first iteration the input flow was of zero ml, so the error signal is the reference profile, this information makes the first training to build the control signal for the next iteration. 5. Results To have convergence, it is useful to use a small sampling time, in biotechnology the time reaction use to be in order of hours in most cases, so a sampling period of some minutes can be applied, in the next figures it is shown the biodegradation simulation by using a sample time of 1.2 min and by using fuzzy logic to build the control signal with a sample time of one hour. To propose the control gains for the classical ILC, it is possible to use (22), where it was used that q ¼ 0:99 and f ¼ 0:009, in this way q þ f < 1 as it is shown in subSection 3.1; the disturbance and the initial conditions variation are considered in e ¼ 0:1, so jdui ðnÞj  1, because if ui ðnÞ is constant, the volume inside the reactor increase with time and it cans exceed the reactor volume, so the variation for the inlet flow was saturated to 1 ml each 1.2 min. In this manner, using (9) and (10) it is possible to select some control gains. From (7) it is required to have some bounds for M 1 ; M 2 and M 3 , here it was defined that dxi ðnÞ ¼ 30 because in the first iteration, the maximum variation required was xi ðnÞ ¼ 60, as the reference (28) shows, and its medium value was taken dxi ðnÞ ¼ 30. To have a higher variation for the input, it was proposed that D1t ðei ðn þ 1Þ  ei ðnÞÞ ¼ 35. So, using (7) it was assumed M1 ¼ 5; M 2 ¼ 1 and M 3 ¼ 60. With the matrix and the constants q and f, the control gains proposed for the classical ILC were,

q1 ¼ sup max kI  Ci ðnÞM1 k; i

n¼0;...m

ð29Þ

where it was computed C ¼ 0:0018 < Ci ¼ :002 and from

11 ¼ sup max kCi ðnÞðUi ðnÞM3  M2 Þkc1 ; i

n¼0;...m

ð30Þ

it was obtained U ¼ 0:08 < Ui ¼ :1. In this way the gains used for the classical ILC type PD were U ¼ 0:08 and C ¼ 0:0018. As a first simulation, there were made seven iterations for the classical ILC, in the first one the input is a feeding profile of zero ml/s, the sampling time was 1.2 min and a zero order hold is used to interpolate measures. Results are in Fig. 4. Due the sampling time cannot be of 1.2 min due to technical restrictions, to simplify the real experiments and to make cheaper the procedure; it was proposed to have measures each hour, and between measures, they were used cubic splines as interpolation having in this manner a virtual sampling time of 1.2 min. The final error increase as it was supposed, but in simulation it is possible to see that in iteration seven the error is smaller than with 1.2 min of sampling period with the classical ILC, as shown in Fig. 5; finally, fuzzy gains were used to improve convergence. The initial consequents for each rule were h ¼[1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1], after the iterations, the values change in order to (25), according to the gradient, they change each instant time and the final vector found was, for example to obtain the U gain, hU ¼[1.0062 0.9778 0.6000 0.4000 0.2000 0.0000 0.2320 0.4569 0.6777 0.8000 8.1760], this vector was used to compute the control gain using (24); it was forced the central rule to have zero as consequent, this rule is activated with major frequency, so chattering can appear if this value changes continuously. 6. Discussion In Table 3 they are shown the integral of square error (ISE) (31) and the integral of absolute error (IAE) (32) [29] to have an idea about the performance of fuzzy ILC, these errors criteria show a better performance in the seventh iteration for the fuzzy ILC, the ISE is 21.82% lower than by using the classical ILC, also, an over shoot of 41% was presented using the classical ILC in the third iteration, whereas by using the fuzzy logic, the output concentration does not exceed the reference signal because the fuzzy logic saturates the changing in the control gains.

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Fig. 4. ILC with a sample time of 1.2 min.

Fig. 5. Cubic splines ILC with a sample time of 60 min.

Table 3 Comparison between classical ILC and the use of fuzzy logic with interpolation. Method

Sampling time

Iteration

ISE

IAE

1.2 min

3 5 7

20349.5133 204.7893 105.2601

382.9890 37.9309 27.1221

60 min

3 5 7

12316.5573 219.4160 82.2973

276.0031 128.4609 24.2729

ILC type PD

Fuzzy ILC

Fig. 6. Errors by using the cubic splines.

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Fig. 7. Input signal through iterations.

ISE ¼

Z

T

e2 ðtÞdt;

ð31Þ

jeðtÞjdt:

ð32Þ

0

IAE ¼

Z

T

0

In Fig. 6 they are shown the errors for different iterations when the sampling period is 1 h and an interpolation with cubic splines is made for measures, so there exists a virtual sampling time of 1.2 min. The input signal to the process is presented in Fig. 7, here it is possible to see the variation for the feeding profile, the maximum value was around of 0.3 l/h, it is lower than the supposed to compute the control gains for the classical ILC shown in Section 5. This kind of control can be seen as a PD controller in open-loop for time due it uses the error and its derivative approached by a difference equation, whereas it can be seen as a PI controller in closed-loop for iterations because in (26) it is added the last value of ui ðnÞ, so the error tends to cero when iterations run. 7. Conclusions It was shown the convergence analysis for an ILC type PD using the infinity norm, the control gains are fuzzy to improve the control when model is not well known, and this fuzzy system saturates the gains values to guarantee convergence. It was assumed the system as a continuous one due the reaction velocity, it was used a sample time of 1.2 min in simulation to make the classical ILC, but it is difficult to make it in practice, so it was proposed to take measures each hour and to interpolate samples having a virtual time of 1.2 min. The control gains can be determined for each instant time, but the advantage of not having a model dependence is important for not well understood processes, and the fuzzy logic can work with this kind of systems, this was the reason of its implementation. The differential control gain C is important for the control convergence, the proportional one U determines the error in steady state according to Theorem 1, and it has an effect for the interpolation error; the disturbance and initial conditions variation (e1 ; e2 ) were used in the theoretic results. Unfortunately, for the considerations taken, it was not possible to guarantee perfect tracking, although it is like a PI controller for the iterations. The use of fuzzy logic accelerates the convergence, in simulations it was possible to see that an over shoot was presented in classical ILC, whereas by using fuzzy logic to compute the control gains the errors criteria were lower than for classical ILC in the seventh iteration, unfortunately, fuzzy logic did not reduces the final error. As future work it is proposed to have a variable sample time for not having to use a sample time smaller than 30 min and to use less measures than the used in the present work. Appendix A. Proof for Theorem 1

Proof. We are going to prove the theorem by induction on i, with n fixed. Let i ¼ 1. If minðn; 1Þ ¼ n, then Eq. (18) holds trivially because Pð1; 1Þ ¼ qjdu0 ð0Þj; Q ð1; 1Þ ¼ 0 and Rð1; 1Þ ¼ 1. The case minðn; 1Þ ¼ 1 is analogous. Suppose now that Eq. (18) holds for i, we shall verify that also holds for i þ 1. We shall assume that minðn; iÞ ¼ n, the case minðn; iÞ ¼ i is analogous.  þ e, for some k  ¼ 0; . . . ; n  1. Because, jdu ð0Þj 6 jdu ðkÞj,  it is not hard to see that Then, jduiþ1 ðnÞj 6 qjdui ðnÞj þ 1jdui ðkÞj 0 0

 þ Pði; kÞ  þ jdu0 ð0ÞjQ ði; kÞ  þ eRði; kÞ  qi jdu0 ðkÞj is less than equal to

qi Cu0 þ Pði; n  1Þ þ jdu0 ð0ÞjQ ði; n  1Þ þ eRði; n  1Þ; where Cu0 ¼ maxk¼0;...;n1 jdu0 ðkÞj. Therefore,

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jduiþ1 ðnÞj 6 qiþ1 jdu0 ðnÞj þ p þ jdu0 ð0Þjq þ er; where

p ¼ .Pði; nÞ þ .i 1Cu0 þ 1Pði; n  1Þ; q ¼ .Q ði; nÞ þ 1Q ði; n  1Þ; r ¼ .Rði; nÞ þ 1Rði; n  1Þ þ 1: Since



r¼

i s1

n1 X



þ

    i iþ1 ¼ then p ¼ Pði þ 1; nÞ; q ¼ Q ði þ 1; nÞ and s s

1s ðqpði; sÞ þ pði; s þ 1ÞÞ 6

s¼0

Therefore, Eq. (18) holds for i þ 1.

n1 X

1s pði þ 1; sÞ ¼ Rði þ 1; nÞ:

s¼0

h

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