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本科畢業(yè)設(shè)計(jì)論文
英文翻譯
專業(yè)名稱 自動(dòng)化
學(xué)生姓名 王偉博
指導(dǎo)教師 王 佩
畢業(yè)時(shí)間 2014.06
非線性跟蹤-微分器的分析與改進(jìn)
朱發(fā)國(guó) 陳學(xué)允
( 哈爾濱工業(yè)大學(xué)電氣工程系#哈爾濱, 150001)
摘要: 本文對(duì)非線性跟蹤-微分器的工作機(jī)理和參數(shù)意義進(jìn)行了深入分析, 發(fā)現(xiàn)了超調(diào)產(chǎn)生的根源及參數(shù)相互制約的原因, 通過(guò)對(duì)其開(kāi)關(guān)平面函數(shù)的改進(jìn), 改善了信號(hào)的品質(zhì), 得到了靈活解耦的參數(shù)整定方法, 使非線性跟蹤微分器更具有工程實(shí)用意義.
關(guān)鍵詞: 非線性系統(tǒng); 跟蹤-微分器; 開(kāi)關(guān)平面函數(shù)
1引言
連續(xù)信號(hào)和它們的差距都在控制工程中普遍使用。他們的質(zhì)量，特別是該差異是極大地影響了整個(gè)控制系統(tǒng)。因此，對(duì)一個(gè)結(jié)構(gòu)非線性跟蹤微分器。它通過(guò)積分產(chǎn)生的跟蹤信號(hào)及其鑒別方法，這是對(duì)那些非持續(xù)還是很有用的非差分信號(hào)，例如，對(duì)干擾信號(hào)。它經(jīng)常被用在非線性控制系統(tǒng)并導(dǎo)致性能。但其作用機(jī)制目前尚不清楚，其調(diào)整是困難的，尤其是條件R/ D=常數(shù)，導(dǎo)致新的沖突是快速跟蹤和顫振抑制。因此其應(yīng)用受到嚴(yán)格限制。理論分析和仿真該單位是由在本文和開(kāi)發(fā)結(jié)構(gòu)，驗(yàn)證時(shí)，這被證明是有點(diǎn)過(guò)分超調(diào)，靈活和不耦合調(diào)整。
2非線性跟蹤微分和分析
2.1理想的非線性跟蹤微分器
非線性跟蹤微分是這樣的結(jié)構(gòu)：對(duì)一個(gè)輸入信號(hào)、和產(chǎn)生，在這里， 在的跟蹤輸入信號(hào)和=的值，換句話說(shuō)是是擴(kuò)展不同差分的。
一個(gè)理想的二階非線性跟蹤微分可以表示為
（1）
其中R是實(shí)數(shù)大于零。
設(shè)為階躍信號(hào)使得= C，（T> 0），非線性跟蹤微分器的屬性可以按如下方式近似分析：
圖1 理想的跟蹤微分特性
的響應(yīng)示于圖1.1，定義該單元的開(kāi)關(guān)功能
（2）
假設(shè)在第一時(shí)間s = 0時(shí)，在時(shí)刻t1， [0，]的單元滿足
（3）
得到
，，
假設(shè)單元到達(dá)穩(wěn)定狀態(tài)= C的時(shí)間，單元滿足時(shí)，
（4）
得到
，
經(jīng)過(guò)t> t2時(shí)，本機(jī)將結(jié)束跟蹤過(guò)程和被保持在穩(wěn)定狀態(tài)。
2.2非線性跟蹤微分線性區(qū)
以減少在該單元的穩(wěn)態(tài)顫，函數(shù)SGN（s）是取代的直鏈飽和函數(shù)在[1]的和方程（1）變成為
（5）
得到，
2.2.1跟蹤屬性
在圖如圖2所示，實(shí)線表示的屬性，一個(gè)理想的非線性跟蹤微分和虛線指那些具有線性區(qū)域的屬性。 假設(shè)它是t0當(dāng)單位滿足，我們得到
，，
兩側(cè)區(qū)分方程（2）兩側(cè)，（X2> 0）和我們得到的
（6）
圖2 跟蹤微分性能
從（5）和（6），我們可以得到該
（7）
假設(shè)是在期間的增益，必須在區(qū)間內(nèi)變化，因?yàn)楹投挤浅Ｐ?，必須要比少的多，考慮到，從而
（8）
在時(shí)間
（9）
所以
（10）
因此，下面的公式可以得到：
（11）
簡(jiǎn)化（11），然后
（12）
以同樣的方式，當(dāng)?shù)臅r(shí)間時(shí)，我們可以得到
（13）
后，很容易地發(fā)現(xiàn)，在開(kāi)關(guān) 功能將跟蹤隨著第內(nèi)側(cè)在進(jìn)入穩(wěn)定狀態(tài)。該曲線是類似的期間（t1，t2）的理想非線性跟蹤微分器。
在上面的分析中，發(fā)現(xiàn)使得多樣化速度比較慢，它更接近，因?yàn)榫€性區(qū)域?yàn)榱?。此功能有利于削弱了抖顫，并且還減少曲線轉(zhuǎn)移到右側(cè)。當(dāng)達(dá)到零時(shí)，收益率超調(diào)因?yàn)楸仨毚笥赩（T）= c的值。在圖2，S1和S2被分別定義為和時(shí)間軸，t0和t2 單元具有線性區(qū)域和理想的單元，其陰影與右斜行和左斜線之間的區(qū)域。過(guò)沖等于S1和S2之間的差額。由于S1可以表示為
（14）
和S2可以表示為
（15）
過(guò)沖幾乎等于
（16）
模擬的結(jié)果表明，實(shí)際的過(guò)沖是1.5?3中的（16）倍，因?yàn)樯鲜龅木€性化是保守的，而事實(shí)上具有延遲的自+在t> t2邊內(nèi)的足跡降低。
2.2.2穩(wěn)定性能
(17a)
非線性跟蹤微分線性區(qū)域的穩(wěn)定狀態(tài)是由降至表示為零。在該點(diǎn)附近波動(dòng)X1= C。假設(shè)在時(shí)間它滿足，和簡(jiǎn)化了分析與考慮| X2|為一個(gè)常數(shù)，并以（b）與（a）以下面的公式
（17b）
用拉普拉斯算子轉(zhuǎn)換：
（18）
定義：可以被描述為
（19）
因?yàn)?
（19）可進(jìn)一步簡(jiǎn)化為
(20)
逆轉(zhuǎn)換（20）與一個(gè)拉普拉斯算子，從而
(21)
得到
忽略右邊的（21）中的第一項(xiàng)，然后
(22)
這意味著，如果任何干擾到在穩(wěn)定狀態(tài)下發(fā)生的，跟蹤信號(hào)的誤差會(huì)在波動(dòng)頻率，初步范圍和和減小速率。
3 擴(kuò)展的非線性跟蹤微分器
當(dāng)S趨近于零在第一時(shí)間在此過(guò)程|X2|有一個(gè)較大的值，X1產(chǎn)量為X2的延遲減少線性的作用的過(guò)沖區(qū)域。削弱在這一過(guò)程中線性區(qū)域的影響，但不影響穩(wěn)態(tài)的性能，乘以于開(kāi)關(guān)函數(shù)s和先進(jìn)的非線性跟蹤微分應(yīng)表示為
(23)
這是很容易理解的新的非線性跟蹤微分有一個(gè)理想的單元的兩個(gè)跟蹤屬性和一個(gè)與線性區(qū)域的穩(wěn)定性能。在所有的進(jìn)程，跟蹤信號(hào)和差分信號(hào)有一個(gè)較優(yōu)的品質(zhì)。
4 參數(shù)整定
在擴(kuò)展的非線性跟蹤微分器，它變得光滑脫鉤研發(fā)部門(mén)的調(diào)整為弱關(guān)系，彼此明確的含義。
定義= 2C/ R作為非線性跟蹤微分，這是所需的輸出跟蹤信號(hào)到達(dá)步驟輸入的值的時(shí)間的時(shí)間慣性。如果可在第一證實(shí)，R可以被調(diào)整為：R =4C/，其中c是輸入的放大。如果輸入可以表示為一個(gè)標(biāo)準(zhǔn),應(yīng)滿足,一個(gè)良好的跟蹤性能和R應(yīng)調(diào)整為。其中T0是漸進(jìn)的輸入的期間。如果輸入是一個(gè)非彎曲的信號(hào)，F(xiàn)FT應(yīng)該需要 計(jì)算T0，應(yīng)該是最高階的時(shí)期諧波，為R的極大值，是一個(gè)有益的快追蹤到的輸入的變化，但它也增加了X2與擾動(dòng)的靈敏度。因此R不應(yīng)過(guò)于龐大，如果它足夠大，可以滿足跟蹤屬性。被調(diào)整到一個(gè)較小的值，如果它是足夠大的，將拒絕干擾。
5 數(shù)字仿真
這三個(gè)實(shí)例中。在圖3至圖5中，實(shí)線表示仿真開(kāi)發(fā)非線性跟蹤分化的結(jié)果，虛線表示與線性單元的區(qū)域和虛線表示輸入（重疊與實(shí)線有時(shí)）。
實(shí)例1，，+5％干擾被附加在輸入v（t），在t =1秒的模擬的結(jié)果如圖3。
實(shí)例2，， +5％擾動(dòng)被附加在輸入V（T），在t=0。8秒的模擬的結(jié)果如圖 4。
實(shí)例3，，5％的干擾被附加在輸入v（t），在t =0.8秒的模擬的結(jié)果如圖3。
圖3 實(shí)例1結(jié)果
圖4 實(shí)例2結(jié)果
圖5 實(shí)例3結(jié)果
6 結(jié)論
非線性跟蹤微分ISA顯著單元在控制字段，因?yàn)檎{(diào)整的難度它遠(yuǎn)遠(yuǎn)沒(méi)有廣泛使用。本文提出了一種改進(jìn)切換功能，并得到了一個(gè)新的的模型，其參數(shù)的結(jié)構(gòu)可脫開(kāi)調(diào)。數(shù)字仿真證明，新系統(tǒng)不僅具有一個(gè)良好的跟蹤性能，也有拒絕干擾和顫振的良好性能。
本文作者簡(jiǎn)介
朱發(fā)國(guó) ,1972年生.博士.主要從事電力系統(tǒng)非線性控制方面的研究.
陳學(xué)允 ,193年生.教授,博士生導(dǎo)師,中國(guó)電機(jī)工程學(xué)會(huì)理事.主要從事電力系統(tǒng)穩(wěn)定分析及控制方面的研究.
裴海龍,1965年生.華南理工大學(xué)自動(dòng)控制工程系副教授,博士.主要研究方向?yàn)? 非線性控制,機(jī)器人控制和神經(jīng)網(wǎng)絡(luò)控制.
徐楊生，1989年獲賓夕法尼亞大學(xué)博士之后任卡內(nèi)基-梅隆大學(xué)機(jī)器人研究所高級(jí)研究科學(xué)家, 自 1997年至今香港中文大學(xué)機(jī)械與自動(dòng)化系教授、系主任, 中國(guó)國(guó)家高技術(shù)遙科學(xué)領(lǐng)域首席顧問(wèn).主要研究領(lǐng)域?yàn)? 空間機(jī)器人設(shè)計(jì)與控制, 實(shí)時(shí)技能獲取與建模,高性能機(jī)電系統(tǒng)研究, 在國(guó)際著名刊物發(fā)表有關(guān)論文 40 余篇.
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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(jiǎn)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(jiǎn)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