外文翻譯--非線性跟蹤-微分器的分析與改進【PDF+WORD】【中文2700字】【中英文文獻譯文】,PDF+WORD,中文2700字,中英文文獻譯文,外文,翻譯,非線性,跟蹤,微分,分析,改進,改良,pdf,word,中文,中英文,文獻,譯文 Analysis and Improvement of theNonlinearTracking-Differentiator Zhu Faguo and Chen Xueyun (Department of ElectricalEngineering,Harbin Institute of Technology#Harbin, 150001, P.R.China) Abstract: This paper proposesan extensiveanalysisof thenonlineartracking-differentiator on theworkingprocess and pa- rameters tuning, based on which the source of theovershoot and the causethat coupling to tune are cleared. Thus, an improved switching function is put forwardtothedeveloped nonlineartracking-differentiator, which exhibitsan agreeableperformance and owns a flexible, uncoupletuning technique. This amelioration renders this unit more applicable to practice than ever. Keywords: nonlinear system; tracking-differentiator; switching function dL??6-±s ￥sD?é ??S ?D (W:^y 0), the property of the nonlinear tracking-differentiator can be analyzed as follows approximatively: Manuscript received May 29,1998, revised Aug. 4,1999. ?16 ?6 ù1999 M12 e? ? ?D?¨CONTROL THEORY AND APPLICATIONS Vol.16,No.6Dec.,1999 Response of v ( t) is shown in Fig.1. Defining a switch- ing function of the unit as s = x1- c + | x2 | # x22R . (2) Assuming the first time s = 0 at time t1, during t I [0, t1] the unit satisfies x1( t) = R # t2/2, x2( t) = R # t, t [ t1, s = R # t2- c, (3) where t1 = c/ R, x1( t1) = c2 , x2( t1) = R # c . Assuming the unit arrives at steady state x1( t2) = c at time t2, during t1 < t [ t2 the unit satisfies s = 0, x1( t) = c2 + Q t2 t1 R # c - R # ( t - t1) dt, x2( t) = R # c - R # ( t - t1), t1 t2, the unit will end thetracking process and be kept in the steady state. 2. 2 Nonlinear tracking-differentiator with lin- ear area To diminish the chatter in the steady state of the u- nit, thefunction sgn ( s) is substituted with a linearsatu- ration function sat ( s, D) in [ 1] and Equation ( 1) turned to be ¤x 1 = x2, ¤x 2 = - R #sat x1- v( t) + | x2 | # x22R , D , (5) where sat( s, D) = sgn( s), | s | \ D,s/ D, | s | 0) and we obtain that ¤s = x2 + x2R #¤x 2. (6) From (5) and (6), we can obtain that ¤s = x2 # 1- sD . (7) Assuming $x2 is the increment of x2 during t0 to t1, ¤s must vary in(2x20, x20+ $x2). Because Dand $t = t1- t0 are both very small, $x2 must be much less thanx20. Considering¤s as a constant and $x2 U 0, thus ¤s U (2x20+ x20+ $x2)/ 2= 1. 5x20. (8) At time t1 st1 U st0+ s## $t = - D+ ¤s # $t = 0. (9) So $t = 1. 5D/ x20. ( 10) Thus, the equations below can be obtained: x2t 1 = x20+ Q t1 t0¤x2dt U x20+ R #$t/2, Qt1t0(- s)dt U D/2 , ( 11) x1t1= x10+Q t1 t0 x2dt U x10+ x20 # $t+ R # $t 2/4. Simplify (11), then x2t 1 = R #( c - D) + D3 Rc - D, x1t1 = c2 + D 2 16( c - D) . ( 12) In the same way, when s = + Dat time t2c, we can ob- tain that x2tc2 = R #( c - D) , x1tc 2 = c - D2 + 43 D- D 2 8( c - D) . ( 13) No.6 Analysis and Improvement of the Nonlinear Tracking-Differentiator 899 After t > t2c, it is easy to find that the switching function will track along with the inside of s = + Duntil it enters the steady state. The curve is similar to that of ideal nonlinear tracking-differentiator during ( t1, t2) . In the analysis above, it is found that x2 varied more slowly when it is closer to zero because of the lin- ear area. This feature is conducive to diminishing the chatter, and also shifting the decreasing curve to right. When x2 reaches zero, overshoot yields because x1 must be greaterthan the valueof v ( t) = c. In Fig.2, S1 and S2 are respectively defined as the area between x2 and time axis, t0 and t2cof the unit with linear area and the ideal unit, which are shadowed with right ramp line and left ramp line. The overshoot is equal to the difference between S1 and S2. Because S 1 can be expressed as S1 = 43 D- D 2 8( c - D), (14) and S2 can be expressed as S 2 = + D, (15) the overshoot is nearly equal to $S = S1- S 2 = D3 - D 2 8( c - D). (16) The results of the simulation shows that the real overshoot is 1. 5 to 3 times of ( 16) because the lin- earization above is conservative and in fact x2 has de- layed to decrease since s tracks inside the boarder of+ D after t > t2. 2.2.2 Steady property The steady state of the nonlinear tracking-differen- tiator with linear area is indicated by x2 decreasing to zero. In the state x1fluctuates nearby x1 = c. Assuming at time a it satisfies $x1a = x1a - c, x2a = 0 and $x1 = x1- c. Simplifying the analysis with consider- ing | x2 | as a constant and substitute ( b) with ( a) in the following equations $¤x1 = x2, (17a) ¤x2 = - RD $x1+ | x2 |# x22R . (17b) Transfer it with a Laplace operator p: p (p $x1(p) - $x1a ) = - RD $x1(p ) + p $x1( p) - $x1a2a #| x2 | . (18) Defining T = D/ R, (18) can be described as $x1( p) = $x1a #| x2 | 2D + p $x1a p + | x2 |4D 2 + 1T - | x2 | 2 16D2 . ( 19) Because | x2 | n R # D, (19) can be further simplified to $x1( p) = $x1a #| x2 | 2D + p $x1a p + | x2 |4D 2 + 1T . ( 20) Inversely transfer (20) with a Laplace operator, thus $x1( t)= a1 #e- B( t)#t #sinXt+ $x1a #e- B(t)#t #cosXt, ( 21) where a1 = $x1a #| x2 |2D # T = $x1a # | x2 |2 R # D n $x1a , B( t) = | x2 |4D , X= 1T = RD. Ignoring the first term of right side of (21), then $x1( t) U $x1a # e- B( t)#t #cosXt. ( 22) It means that if any disturbance $x1a to x1 in the steady state happens, the error of tracking signal will fluctuate at frequency X = R/ D, initial scope $x1a and decreasing rate B( t) = | x2 |4D . 3 Developed nonlinear tracking-differen- tiator When s approaches zero at the first time in which process | x2| has a large value, the overshoot of x1yields for the delayed decrease of x2 by the action of linear area. To weaken the effect of the linear area in that pro- cess but not influence the property of the steady state, a penalty function e| x2| is multiplied to the switching func- tion s and the advanced nonlinear tracking-differentiator should be expressed as ¤x 1 = x2, ¤x 2 = - RD #sat( s, D), s = e| x2| # x1- v( t) + | x2 | #x22R . ( 23) It is easy to understand that the new nonlinear 900 CONTROL THEORY AND APPLICATIONS Vol.16 tracking-differentiator has both the tracking property of an ideal unit and the steady property of theone with lin- ear area. In all the processes, the tracking signal and differential signal have an admirable quality. 4 Parameters tuning In the developed nonlinear tracking-differentiator, it becomes smooth to uncouple tuning of R and Dfor the weak relation with each other and clear meanings. De- fine Ttd = 2 c/ R as the time inertia of the nonlinear tracking-differentiator, which is the time needed for the output tracking signal reaching the value of the step in- put. If Ttd can be confirmed at first, R can be tuned as R = 4c/ T2td, where c is the amplification of the input. If the input can be expressed as a standard sinuous v( t ) = c #sin 2P# tT 0 , T tdshould satisfy Ttd 64c T 20 , where T0 is the period of the sinuous input. If the input is a non-sinuous signal, FFT should be needed to calculate T0that should bethe period of the highest order harmonic. A great value of R is beneficial to a fast tracking to the variation of the input but it also increases the sensitivity of x2 to the disturbance. So R should not be too huge if it is great enough to satisfy the tracking property. Though the linear erea is useful to get rid of the chatter, the frequency of the decreasing fluctuation X= R/ Ddeclines with the augment of D. This feature is nuisance to the steady property obviously. So Dshould be tuned to a relatively small value if it is great enough to reject the disturbance. 5 Digitalsimulation Three examples aregiven to make the simulation in step h = 0. 001s of the 4th order Lunge-kutta method. In Fig.3 to Fig.5, the solid line represents the result of simulation of a developed nonlinear tracking-differentia- tor, dashed line represents that of a unit with a linear area and dotted line represents the input ( overlapped with the solid line sometimes) . Example1 v( t) = 5( t \0), R = 100, D= 0. 5, disturbance of + 5% is attached to the input v( t) at t = 1s. The result of simulation is shown in Fig.3. Example 2 v( t) = 3 sin12. 56 t, R = 300, D= 0. 5, disturbance of + 5% was attached to the in- put v( t) at t = 0. 8s. The result of simulation is shown in Fig.4. Example3 v( t) = 3sin 9. 42t + sin(12. 56t + 0. 1), R = 300, D= 0. 5, disturbanceof + 5% was at- tached to the input v ( t) at t = 0. 8s. The result of sim- ulation is shown in Fig.5. 6 Conclusion Nonlineartracking-differentiator isa significant unit in the control field, but it is far from being widely used because of the difficulty of tuning. This paper has pro- posed an improved switching function and got a new structure of the unit whose parameters can be uncoupled No.6 Analysis and Improvement of the Nonlinear Tracking-Differentiator 901 to tune. The digital simulation has proved that the new unit has not only an agreeable tracking property but also a good property of rejecting disturbance and chatter. References 1 Han Jingqing and Wang Wei. Nonlinear tracking-differentiator. System Science and Mathematics, 1994, 14(2):177- 183 2 Han Jingqing. NonlinearPIDcontroller. JournalofAutomatic, 1994, 20 (4):487- 491 3 Wang Shunhuang, LiXiaotian and Zhen Qiubao. Nonlinear PID and its application in distrbuted control system of electric furnace. Journal of Automatic, 1995, 21(6): 676- 680 4 Nan Laishun, Liu Hongjie, Wang Zhibao and Wang Xiufeng. Nonlin- ear PID regulator and its application in DCS. Automatic Instrument, 1997, 12(6):15- 18 5 Wang Fan, Li Junyi and XiYugeng. Analysis and improvement of a d-i rected control method of nonliear system. Control Theory and Applica- tion, 1993,10(6): 621- 631 本文作者簡介 ??S 1972 M 3.p V.?1V Yè ?"ddL? e?Z ? ￥ù?. ?D 1934 M 3.? q,p V 3? =,?Sèy?D? ? Y.?1V Yè ?"d×?s# e?Z ?￥ù?. (Continued from page 897) 16 KhalilH K. Nonlinear Systems. Second Edition,Prentice Hall,1996 本文作者簡介 ￠Z C 1965 M 3.? 2 ?yvD1? e?y?"?? q,p V.?1ù?Z_1:dL? e?,  | e?? *ü? ? e?. · 3 1989 M_E DevDp V-a ? 5 =))) ? HvD  |ù? ?ú)ù? SDE,11997 Mà?P??óvD ?D1??"? q