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176 B. Zhang, W. Zhang / ISA Transactions 45, (2006) 175–183͑PPD͒ is used to linearize the nonlinear systems Assumption 2 (A2). The system is generalizedonline. This idea was also used to develop an Lipschitz, that is, satisfying ͉⌬y P͑k + 1͉͒adaptive-predictive PI controller ͓6͔. ഛ C͉⌬u͑k͉͒, for "k and ⌬u͑k͒ 0, where ⌬y P͑k Model predictive control has been applied to + 1͒ = y P͑k + 1͒ − y P͑k͒, ⌬u͑k͒ = u͑k͒ − u͑k − 1͒ andnonlinear processes ͓7–10͔, resulting in predictive C is a constant.functional control ͑PFC͒ ͓11–14͔͒, which is a most The Lipschitz constant C is often required to bepromising model predictive control algorithm. The known for the control design purpose. Based onPFC algorithm achieves computational simplicity the above assumption, the following result can beby using simpler but more intuitive design guide- obtained.lines ͓13͔ and in the past decade has been success- Theorem 1. ͓5͔ For the nonlinear system ͑1͒, wefully used in industrial applications. The advan- assume that Assumptions ͑A1͒ and ͑A2͒ hold.tages of fewer online calculations, a simpler Then there must exist G͑k͒, called PPD, whenalgorithm and higher control precision are at- ⌬u͑k͒ 0,tributes of the PFC, which contribute to its indus-trial use. ⌬y P͑k + 1͒ = G͑k͒⌬u͑k͒ , ͑2͒ Motivated by the work of Hou and Huang ͓5͔and Tan et al. ͓6͔, this paper presents an extension whereof PFC to nonlinear system control in which thePPD concept is used to dynamically linearize the ͉G͑k͉͒ ഛ C ͑3͒nonlinear system. In other words, the PPD re-linearizes the nonlinear model as the plant moves With Theorem 1, Eq. ͑2͒ can be used as an internalfrom one operating point to another, and uses the model to predict future process outputslatest linear model as the internal model at eachstep, resulting in solving a quadratic performance ˆ͑QP͒. Aggregation as part of the algorithm ͓15͔ is y ͑k + 1͒ = y ͑k͒ + G͑k͒⌬u͑k͒ , ͑4͒used to predict future values of the PPD. Then wherePFC is used to design the nonlinear adaptive con-trol algorithm, in which only two coincidencepoints are selected to calculate the manipulated ˆ ͉G͑k͉͒ ഛ C, ͑5͒variable. The resultant controller has a simplestructure, hence tuning is not a problem. The pro- ˆ y͑k͒ is the model output, and G͑k͒ is an estimateposed algorithm can also provide bounded input/ of G͑k͒.output sequences and track the setpoint withoutsteady-state error. Last, the paper discusses the pa-rameter tuning, and uses simulations for a long 2.2. Predictive output and structured controltime delay plant and an experimental setup for the variablesmeasurement of the acidity or alkalinity in a Using Eq. ͑4͒, at sampling time k + Hi inside thechemical reaction process to show that the pro- optimization horizon, the future output can be pre-posed algorithm has higher precision and better dicted byrobustness to parameter perturbation than a PI orPID controller. Hi y ͑k + Hi͒ = y ͑k͒ + ͚ G͑k + j − 1͒⌬u͑k + j − 1͒ . ˆ j=12. Nonlinear adaptive predictive functional ͑6͒control algorithm (NPFCA) The PFC algorithm is different from other2.1. The dynamic linearized internal model model predictive controls. Instead of calculating For a nonlinear system ͑1͒, the following two control signal with no restrictions, which may re-assumptions are necessary: sult in an unstable control signal, PFC uses struc- Assumption 1 (A1). The partial derivative of tured future manipulated variables, that is, the fu-f͑·͒ with respect to the control input u͑k͒ is con- ture manipulated variables are parameterized bytinuous. nB base functions uBj:
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B. Zhang, W. Zhang / ISA Transactions 45, (2006) 175–183 177 nB inputs are calculated by minimizing the sum of the u͑k + n͒ = ͚ j͑k͒uBj͑n͒ n = 0,1, . . . ,Hi − 1, quadratic difference between the predicted process j=1 output and the reference trajectory at all coinci- ͑7͒ dence points. The criterion takes the following form:where j͑k͒ ͑j ͓1 , nB͔͒ are unknown coefﬁ- H2cients. There are no restrictions for selecting these base min J P = ͚ ͓͑y ref͑k + n͒ − y ͑k + n͒ − e͑k + n͔͒͒2 n=H1functions, and the selection of base functions hasno inﬂuence on the dynamic response or robust- Mness and stability of the closed-loop system ͓12͔. + r ͚ ⌬u2͑k + n − 1͒ , ͑11͒The base function can be selected as polynomial, n=1sine, or exponential format. For many applications where r is a weighting efﬁcient, ͓H1 , H2͔ ͑H2it is sufﬁcient to describe the process input using a Ͼ H1͒ is a coincidence horizon, M ͑M ഛ H2͒ is aform such as u͑k + n͒ = 1͑k͒ + 2͑k͒n ͑n control horizon, and e͑k + n͒ is the prediction error= 0 , 1 , ¯ , Hi − 1͒, which results in compensation which is given by e͑k + n͒ = e͑k͒ u ͑ k ͒ = 1͑ k ͒ , = y P͑k͒ − y͑k͒. Substituting Eqs. ͑8͒–͑10͒ into Eq. ͑11͒, the cal-⌬u͑k + Hi − 1͒ = ⌬u͑k + Hi − 2͒ = ¯ = ⌬u͑k + 1͒ culation of the process input u͑k͒ is straightfor- ˆ ward provided G͑k + j͒ is known. Note, Eq. ͑5͒ = 2͑ k ͒ . ͑8͒ ˆ states that for "k , ͉G͑k͉͒ Ͻ C, require that futureTherefore, the determination of control input u͑k ˆ predicted PPD G͑k + j͒ be bounded. To deal with+ n͒ ͑n = 0 , 1 , ¯ , Hi − 1͒ means to ﬁnd coefﬁ- this restriction the idea of aggregation is adoptedcients 1͑k͒ and 2͑k͒ at each instant time k, and ˆ to predict future PPD at G͑k + j͒.only u͑k͒ = 1͑k͒ is applied to the process. Substituting Eq. ͑8͒ into Eq. ͑6͒ results in Let the aggregated variable be the current PPD Gˆ ͑k͒, then future predicted values of the PPD Hi y ͑ k + H i͒ = y ͑ k ͒ + ͚ G ͑ k + j − 1 ͒ 2͑ k ͒ ˆ ˆ ˆ ˆ ˆ ͕G͑k + 1͒ , G͑k + 2͒ , . . . , G͑k + j͒ , . . . , G͑k + Hi − 1͖͒ j=2 can be described as the amplitude decaying se- quence related to the aggregated variable, namely: ˆ + G͑k͓͒1͑k͒ − u͑k − 1͔͒ ͑9͒ ˆ ˆ G͑k + j ͒ = G͑k͒ j ͑0 Ͻ Ͻ 1, j = 1, ¯ Hi − 1͒2.3. Optimization and control law equation ͑12͒ The PFC algorithm computes future process in- where is an unknown decaying coefﬁcient.put so that the predicted process output can follow ˆa reference trajectory. In the PFC algorithm, the Then, G͑k + j͒ can automatically meet the con-reference trajectory is used to specify the desired straint requirement ͑5͒. Moreover, to facilitate thefuture process behavior. For many applications a ˆ tuning of the controller, this paper sets G͑k͒ = ,ﬁrst-order exponential reference trajectory is suf- then Eq. ͑12͒ can be modiﬁed toﬁcient: ˆ ˆ G͑k + j͒ = j+1 = G͑k͒ j+1 ͑0 Ͻ Ͻ 1, j y ref͑k + Hi͒ = w͑k + Hi͒ − Hi͑w͑k͒ − y P͑k͒͒ , ͑10͒ = 1, . . . ,Hi − 1͒ . ͑13͒where = exp͑−Ts / Tref͒, Tref is the desired re- ˆ However, the constraint for PPD G͑k͒ may causesponse time of the closed loop system, w is the an uncontrolled overshoot of ⌬u͑k͒, thus a weightsetpoint, and for constant value setpoint tracking for the control input ⌬u͑k͒ is introduced in thew͑k + Hi͒ = w͑k͒. performance index Eq. ͑11͒. Since the control in- The PFC algorithm requires an online optimiz- put ͑8͒ is used and the future predicted output ising method. When a QP index is used, the process given by Eq. ͑9͒, here we set M = H2.
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178 B. Zhang, W. Zhang / ISA Transactions 45, (2006) 175–183 For the assumption, 1͑k͒ and 2͑k͒ are un- e ͑ k + H 1͒ = e ͑ k + H 2͒ = y P͑ k ͒ − y ͑ k ͒ . ͑15͒known coefﬁcients in Eq. ͑11͒, which requires thatat least two coincidence points H1Ts and H2Tsshould be selected. Using Eq. ͑8͒, Eq. ͑11͒ is re- Substituting Eqs. ͑8͒–͑10͒, ͑13͒, and ͑15͒ into Eq.written as ͑14͒, lettingmin J P = ͓y ref͑k + H1͒ − y ͑k + H1͒ − e͑k + H1͔͒2 + ͓y ref͑k + H2͒ − y ͑k + H2͒ − e͑k + H2͔͒2 ץJ P ץJ P M = 0, = 0. ͑16͒ ͑1 ץk ͒ ͑2 ץk ͒ + r͓1͑k͒ − u͑k͔͒ + r ͚ 2 2͑ k ͒ , 2 ͑14͒ n=2where The manipulated variable is given by ˆ ˆ ˆ ˆ ͓− A1G͑k͒ − A2G͑k͒ − ru͑k − 1͔͓͒S2 + S2 + r͑ M − 1͔͒ − ͑− A1S1 − A2S2͓͒S1G͑k͒ + S2G͑k͔͒ 1 2u ͑ k ͒ = 1͑ k ͒ = , ˆ ˆ ˆ ͓S G͑k͒ + S G͑k͔͒2 − ͓2G2͑k͒ + r͔͓S2 + S2 + r͑ M − 1͔͒ 1 2 1 2 ͑17͒where Ai = w͑k + Hi͒ + Hi͑w͑k͒ − y P͑k͒͒ + G͑k͒u͑k− 1͒ − y P͑k͒ ͑i = 1 , 2͒, ˆ ˆ ͉G͑k͉͒ = ͯ ␥ ␥ + ⌬u2͑k − 1͒ ˆ G͑k − 1͒ Hi + ⌬u͑k − 1͒ ⌬y ͑k͒ ␥ + ⌬u2͑k − 1͒ P ͯ Si = ͚ G͑k͒ j, ͯ ͯ ˆ ͑i = 1,2͒, M = H2 . ␥ ˆ j=2 ഛ ͉G͑k − 1͉͒ ␥ + ⌬u2͑k − 1͒ ͯ ͯ ˆ Subject to Eq. ͑4͒, online searching for G͑k͒ isrequired, however in Eq. ͑2͒, many algorithms to ⌬u2͑k − 1͒ + ͉G͑k − 1͉͒ ˆestimate G͑k͒ can be used. We adopted the adap- ␥ + ⌬u2͑k − 1͒ ˆtive learning algorithm ͓5͔ for G͑k͒: ␥ ⌬u2͑k − 1͒ ഛ C+ C = C. ␥ + ⌬u ͑k − 1͒ 2 ␥ + ⌬u2͑k − 1͒ ͑19͒ ˆ ˆ ⌬u͑k − 1͒ ˆ G͑k͒ = G͑k − 1͒ + ͑⌬y P͑k͒ − G͑k This inequality implies that the adaptive learn- ␥ + ⌬u2͑k − 1͒ ˆ ing algorithm ͑18͒ for G͑k͒ also meets the con- − 1͒⌬u͑k − 1͒͒ , ͑18͒ straint requirement ͑5͒. Note that explicit control input constraints are ˆ ˆ not addressed in this paper, however when inputwhere ␥ Ͼ 0, and the initial value G͑0͒ of G͑k͒ and/or state-related constraints need be consid-are in the range of 0–1. ered, the technique proposed by Abu el Ata-Doss ˆ Since ͉G͑k − 1͉͒ ഛ C and ͉G͑k − 1͉͒ ഛ C, it is ͓16͔ is usable.easy to obtain the following relation with Eq. ͑18͒:
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B. Zhang, W. Zhang / ISA Transactions 45, (2006) 175–183 1793. Performance analysis of the closed loop Now we summarize the design procedure of thecontrol system and the algorithm proposed NPFAC as follows:implementation Step 1: ͑Initialization͒ At time k = 0, take the ini- ˆ ˆ tial value G͑0͒ of PPD G͑k͒ are in the range of In order to assure the convergence of the closed 0–1, and set ␥ Ͼ 0.loop system, the following assumption is made. Step 2: Find appropriate coincidence points Assumption 3 (A3): The PPD satisﬁes G͑k͒ H1 , H2, and the closed-loop response time Tref byϾ 0 for "k. using the criterion ͑A4͒ and ͑21͒. Further, for a Based on the previous assumption, the stability time delay system both the two coincidence pointsof the closed loop system is guaranteed in the fol- H1 and H2 should be selected larger than the dead-lowing theorem. time in samples. Set the input horizon M = H2 and Theorem 2: Subject to Assumptions ͑A1͒–͑A3͒, the weighting coefﬁcient r to verify inequalitythe algorithm Eq. ͑17͒ for the nonlinear system ͑A6a͒.Eq. ͑1͒ is used to track the setpoint w, then coin- Step 3: At time k ജ 1, collect the process input/cidence points Hi͑i = 1 , 2͒, the weighting coefﬁ- ˆ output data, and ﬁnd G͑k͒ by using adaptive learn-cient r Ͼ 0, and the control horizon M = H2 Ͼ 1 ex- ˆist such that ing algorithm ͑18͒. Substitute G͑k͒ into ͑A4͒, ͑A6a͒, and ͑21͒, if these three conditions fail, re- lim ͉y P͑k + 1͒ − w͉ = 0 ͑20͒ turn to Step 2. Otherwise, go to Step 4. k→ϱ Step 4: Use the control law equation ͑17͒ to cal-and ͕y P͑k͖͒ , ͕u͑k͖͒ are bounded sequences. culate the process control input and apply it to the Proof: See the Appendix. process. Note, the G͑k͒ in Eq. ͑A6b͒ is unknown at cur- Step 5: At the next point, repeat Steps 3 and 4.rent time k and G͑k͒ Ͻ C. If we take the followinginequality as a criterion for selecting controller pa- 4. Illustrative examplesrameters, then the condition ͑A6b͒ always holds The following long time delay plant and pH G͑k͒2 Ͼ CG͑k͒͑1 − Hi͒ ͑i = 1,2͒ . ˆ ˆ ͑21͒ measurement of acidity or alkalinity process are Theorem 2 actually provides a criterion to select used to show the effectiveness of the proposed al-controller parameters. That is, existing parameters gorithm.r Ͼ 0, Hi͑i = 1 , 2͒, and M = H2 Ͼ 1 insure that Eqs. Example 1: Consider a plant described by͑A4͒, ͑A6a͒, and ͑21͒ are applicable. Further, from KEq. ͑A3͒ increasing r leads to decreasing, and P͑s͒ = e −s ͑22͒ ͑s + 1͒3Eq. ͑A2͒ shows that increasing r results in aslower tracking of the setpoint. Theorem 2 also in which K = 1 and = 15. Our goal is to use theshows that Tref has no inﬂuence on stability. The proposed NPFCA to control such a large time-rapidity of Tref will inﬂuence the dynamic re- delay plant and to show that the proposed controlsponse and robustness of the closed-loop system. method gives a better performance than a PI orThe shorter Tref, the more active the controller will PID controller. A comparison to the methods ofbe with larger amplitude variations ͓12͔. Astrom-Hagglund’s PI tuning ͑A-H PI͒ ͓17͔, Sig- ˆ urd Skogestad’s IMC-PI and IMC-PID tuning Note that parameter estimation for G͑k͒ is con- ͑SIMC-PI, SIMC-PID͒ ͓18͔ is given. Using thevergent and the coincidence points H1 , H2, the in- above-mentioned tuning methods, the recom-put horizon M = H2, the weighting coefﬁcient r, mended controller parameters are as follows:and the reference trajectory time Tref are easilyselected using Eqs. ͑A4͒, ͑A6a͒, and ͑21͒. Since A-H PI: Gc͑s͒ = 0.2115 + 0.0286/sthe developed method has no plant structural re-quirement, the closed-loop control scheme is very SIMC-PI: Gc͑s͒ = 0.0455͑1 + 1/1.5s͒ ˆrobust. In practice, H1 , H2, the initial value G͑0͒, SIMC-PID: Gc͑s͒ = 0.0322͑1 + 1/s͒͑1 + 1.5s͒and Tref can be ﬁxed, and r is used to tune so thatan optimal compromise between performance and For the proposed NPFCA, we choose the sam-robustness can be reached. pling time Ts = 1 s. Follow the design procedure in
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180 B. Zhang, W. Zhang / ISA Transactions 45, (2006) 175–183Fig. 1. Step responses of the nominal closed loop system. Fig. 2. Step responses of the perturbed closed loop system. ˆSection 3, we take the initial value of PPD G͑0͒ and disturbance rejection. Further, it is easy to= 0.9, and ␥ = 0.9. In Step 2, since the plant con- verify that at each sampling time for the proposedsidered contains a large time delay, the coinci- controller parameters the conditions ͑A4͒, ͑A6a͒,dence points H1 and H2 should be selected larger and ͑21͒ always hold.than the dead-time. Then by Eqs. ͑A4͒, ͑A6a͒, and Fig. 2 shows the responses to process parameter͑21͒, the controller tuning parameters used here perturbation: K = 1.3 and = 16. Note that theare assigned at coincidence points H1 = 20, H2 SIMC-PI and SIMC-PID methods are very oscil-= 22, the input horizon M = H2 = 22, the desired latory, and A-H method provides poor disturbanceclosed-loop response time Tref = 1 s, and the rejection. The proposed method provides a smoothweighting coefﬁcient r = 75. setpoint response and an acceptable disturbance For a step change in the unit setpoint and a load rejection. The improvement on performance is duedisturbance at t = 0 s and t = 185 s, respectively, to that the proposed algorithm has no plant struc-Fig. 1 presents these step responses of the closed- tural requirement and adopts an adaptive learningloop control system. One can see that for setpoint algorithm to on-line estimate PPD G͑k͒.changes, as well as disturbance inputs the A-H PI Example 2: A pH neutralization process can bemethod, SIMC-PI, and SIMC-PID methods do not modeled by the following equation ͓14͔:reach the ﬁnal set point at about time t = 300 s, x͑k͒ = f 1͑u͑k͒͒ = u͑k͒ − ͑1.207 + r1͒u2͑k͒whereas the newly developed method providessigniﬁcant improvement in both setpoint response + 1.15u3͑k͒ , ͑23a͒ y ͑k͒ ͑0.0185 + r2͒z−2 + ͑0.0173 + r3͒z−3 + 0.00248z−4 = , ͑23b͒ x͑k͒ 1 − ͑1.558 + r4͒z−1 + 0.597z−2where r1, r2, r3, and r4 are time-varying param- It can be found that the proposed method doeseters of the process in which the initial values are achieve excellent results. Fig. 4 shows the re-set to zero. Selecting the initial value of the PPD sponse to a unit output disturbance at time t ˆto G͑0͒ = 0.98, ␥ = 0.9, coincidence points H 1 = 250 s with no change in controller parameters.= 17, H2 = 25, the input horizon M = 25, the sam- Again the proposed controller performs very well.pling time Ts = 1 s, the desired closed loop re- The proposed control system can track setpointsponse time Tref = 1 s, and r = 300, the step re- without steady-state error although there is an ex-sponses for setpoint tracking are shown in Fig. 3. isting external disturbance. In addition, for the
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B. Zhang, W. Zhang / ISA Transactions 45, (2006) 175–183 181Fig. 3. Output ͑top͒ and input ͑bottom͒ of the closed loop Fig. 5. Step response of the closed loop system with time-step responses for setpoint tracking. varying parameters.case of process parameter perturbation occurring structure of the plant or any further external forc-at different times t = 200 s ͑r1 = 0.1, r2 = 0.01͒ and ing for purposes of model development. Note, thet = 400 s ͑r3 = 0.001, r4 = −0.008͒, the step re- results presented in this paper can be extended tosponse is shown in Fig. 5. One can see that the MIMO nonliner processes, and the proposedproposed algorithm has excellent robustness. scheme can be easily implemented.5. Conclusions Appendix: Proof of theorem 2 The PPD was used to dynamically linearize anonlinear process, and aggregation was used to For constant value setpoint tracking, the errorpredict the PPD, resulting in an adaptive predic- between the output y P and the constant setpoint wtive functional control algorithm ͑APFCA͒ for can be written as follows:nonlinear processes. The proposed algorithm was E ͑ k + 1 ͒ = ͉ y P͑ k + 1 ͒ − w ͉ = ͉ y P͑ k ͒ − wtested on two processes and was shown to clearlyoutperform existing algorithms. + G͑k͒⌬u͑k͉͒ = ͉y P͑k͒ − w + G͑k͒͑u͑k͒ A theorem, which illustrates that the designedcontrol system can track the setpoint with zero er- − u͑k − 1͉͒͒ . ͑A1͒ror and input/output sequences are bounded, was Substituting Eq. ͑17͒ into Eq. ͑A1͒ results in Eq.derived in this paper. A merit of the proposed con- ͑A2͒.troller algorithm is that it does not require the E͑k + 1͒ ഛ ͉1 − ͉E͑k͒ , ͑A2͒ where = = G͑k͒G͑k͒͑S2 − S1͓͒͑1 − H1͒S2 − ͑1 − H2͒S1͔ ˆ + G͑k͒G͑k͒r͑ M − 1͒͑2 − H1 − H2͒ , ˆ ˆ ˆ = G2͑k͒͑S2 + S2 − 2S1S2͒ + ͓2G2͑k͒ + r͔r͑ M − 1͒ 1 2 + rS2 + rS2 . 1 2 ͑A3͒Fig. 4. Step response of the closed-loop system with output From Assumption 3 and Eq. ͑18͒, it is easy todisturbance. ˆ know that G͑k͒ Ͼ 0. Note, S2 Ͼ S1, Assumption 3
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182 B. Zhang, W. Zhang / ISA Transactions 45, (2006) 175–183 ˆstates G͑k͒G͑k͒ Ͼ 0, and Eq. ͑10͒ shows that 0 G͑k͒2 Ͼ G͑k͒G͑k͒͑1 − Hi͒͑i = 1,2͒ . ˆ ˆഛ ͑1 − ͒ Ͻ 1͑i = 1 , 2͒, therefore, if we set r Ͼ 0, HiM = H2 ജ 1, and Hi͑i = 1 , 2͒ such that ͑A6b͒ Then Ͼ Ͼ 0, and is = / Ͻ 1, and 0 Ͻ 1 − ͑ 1 − H1͒ S 2 Ͼ ͑ 1 − H2͒ S 1 ͑A4͒ Ͻ 1. By Eq. ͑A2͒, the following inequality holds:results in Ͼ 0. E ͑ k + 1 ͒ ഛ ͉ 1 − ͉ E ͑ k ͒ ഛ ͉ 1 − ͉ 2E ͑ k − 1 ͒ ഛ ¯ Furthermore, from Eq. ͑A3͒ one can derive Eq.͑A5͒. ഛ ͉1 − ͉k+1E͑0͒ . ͑A7͒ Then − = S2͓G͑k͒2 − G͑k͒G͑k͒͑1 − H2͔͒ + S2͓G͑k͒2 1 ˆ ˆ 2 ˆ lim ͉y P͑k + 1͒ − w͉ = lim E͑k + 1͒ 2 k→ϱ k→ϱ − G͑k͒G͑k͒͑1 − H1͔͒ + ͚ ͓G͑k͒2 ˆ ˆ i=1 = lim ͑1 − ͒k+1E͑0͒ k→ϱ − G͑k͒G͑k͒͑1 − Hi͔͓͒r͑ M − 1͒ − S1S2͔ ˆ ͑A8͒ + r 2͑ M − 1 ͒ + r ͑ S 2 + S 2͒ . 1 2 ͑A5͒ Since 0 Ͻ 1 − Ͻ 1 and E͑0͒ = ͉y P͑0͒ − w͉ = ͉w͉, it is easy to know by ͑A8͒ that Eq. ͑20͒ holds andIf r Ͼ 0, M = H2 Ͼ 1, and Hi͑i = 1 , 2͒ are selected the control system track setpoint with no steadysuch that state error. Moreover, we use the following form for 1: r ͑ M − 1 ͒ Ͼ S 1S 2 ͑A6a͒ = G͑k͒1 , ͑A9͒and where G͑k͒͑S2 − S1͓͒͑1 − H1͒S2 − ͑1 − H2͒S1͔ + G͑k͒r͑ M − 1͒͑2 − H1 − H2͒ ˆ ˆ 1 = ˆ ˆ G2͑k͒͑S2 + S2 − 2S S ͒ + ͓2G2͑k͒ + r͔r͑ M − 1͒ + rS2 + rS2 1 2 1 2 1 2and a bound for and G͑k͒ exist, 1 will be ͉u͑k͉͒ ഛ ͉u͑k͒ − u͑k − 1͉͒ + ͉u͑k − 1͉͒bounded. Combing Eqs. ͑17͒ and ͑A3͒ resulting in ഛ ͉⌬u͑k͉͒ + ͉u͑k − 1͒ − u͑k − 2͉͒ + ͉u͑k − 2͉͒ ഛ ¯ ഛ ͉⌬u͑k͉͒ + ͉⌬u͑k − 1͉͒ + ¯ ⌬u͑k͒ = u͑k͒ − u͑k − 1͒ = 1͓w − y P͑k͔͒ + ͉⌬u͑2͉͒ + ͉u͑1͉͒ ͑A12͒ ͑A10͒ Thus by Eq. ͑A11͒ ͕y P͑k͖͒, ͕u͑k͖͒ are bounded sequences.and ᮀ ͉⌬u͑k͉͒ ഛ 1maxE͑k͒ ͑A11͒ Acknowledgmentswhere 1max is the upper bound of 1. Using the The authors would like to thank the Nationalabsolute triangle inequality property to Eq. ͑A10͒, Natural Science Foundation of China ͑60274032͒,it results, Specialized Research Fund for the Doctoral Pro-
9.
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