液壓類外文翻譯-體積模量對(duì)液壓傳動(dòng)控制系統(tǒng)的影響【中文4650字】【PDF+中文WORD】

液壓類外文翻譯-體積模量對(duì)液壓傳動(dòng)控制系統(tǒng)的影響【中文4650字】【PDF+中文WORD】,中文4650字,PDF+中文WORD,液壓,外文,翻譯,體積,傳動(dòng),控制系統(tǒng),影響,中文,4650,PDF,WORD 附錄2【中文4650字】 體積模量對(duì)液壓傳動(dòng)控制系統(tǒng)的影響 Sadhana Vol.31, Part 5, October2006, pp. 543–556.(C)Printed in India Yildiz Technical University Mechanical Engineering Department,，34349 Besiktas,Istanbul,Turkey e-mail:aakkaya@yildiz.edu.tr MS received 9 September 2005;revised 20 February 2006 摘要. 這篇研究報(bào)告，我們主要通過PID（比例積分微分）控制方式檢測(cè)液壓控制系統(tǒng)對(duì)角速度控制的Matlab仿真。有一個(gè)地方很值得關(guān)注，包括對(duì)體積模量控制分析系統(tǒng)。仿真結(jié)果表明，體積模量通過變參數(shù)可以獲得更實(shí)用的模型。此外，PID控制器的不足之處在于對(duì)變體積模量角速度的控制，而模糊控制能夠?qū)崿F(xiàn)較好的控制。 關(guān)鍵詞 液壓傳動(dòng)；體積模量；PID(比例積分微分)；模糊控制 1.引言 液壓傳動(dòng)系統(tǒng)是種輸出可實(shí)現(xiàn)無級(jí)調(diào)速的理想動(dòng)力傳遞方式，這樣在工程中得到了廣泛的應(yīng)用，特別是在制造領(lǐng)域，自動(dòng)化和重型車輛。它能夠提供快速的響應(yīng)，在變負(fù)載情況下能保持精確的傳動(dòng)速度，可以改善能量的利用效率和變功率傳動(dòng)。液壓傳動(dòng)的基礎(chǔ)為液壓系統(tǒng)。一般來講，它包括由異步電動(dòng)機(jī)驅(qū)動(dòng)的變量泵，定量或變量馬達(dá)，所有要求控制的都在一個(gè)簡(jiǎn)單的控制柜中。通過調(diào)節(jié)泵或者馬達(dá)的排量，實(shí)現(xiàn)無級(jí)調(diào)速。 制造廠商和研究人員不斷的改進(jìn)性能和降低液壓傳動(dòng)系統(tǒng)成本。尤其是近十年，體積模量在液壓傳動(dòng)和控制系統(tǒng)的研究中引起了人們的關(guān)注。一些這方面的研究專題在學(xué)術(shù)期刊中可以找到。Lennevi和Palmberg、Lee和Wu運(yùn)用各種轉(zhuǎn)速控制算法求液壓系統(tǒng)的液壓力得到了很好的發(fā)展和應(yīng)用。所有這些設(shè)計(jì)用的體積模量都是固定值，適用的壓力范圍廣。但是，實(shí)際上體積模量是液壓系統(tǒng)中必須考慮的因素。因溫度變化和大氣壓，體積模量可在運(yùn)行過程中求出液壓系統(tǒng)的液壓力。一點(diǎn)空隙足以大副減少體積模量。此外，系統(tǒng)壓力起著重要的作用在體積模量值上。非線性影響了體積模量的不穩(wěn)定，例如：壓力振動(dòng)導(dǎo)致的壓力波會(huì)對(duì)液壓系統(tǒng)的運(yùn)行不利，還有可能會(huì)因磨損而導(dǎo)致部件的使用壽命縮短，干擾控制系統(tǒng)，降低了效率和增加了噪音。盡管有這些不良的影響，但在液壓傳動(dòng)系統(tǒng)中很少有關(guān)于體積模量的研究。1994年Yu等人開發(fā)了一個(gè)參數(shù)辯識(shí)的方法，通過長(zhǎng)的管子來測(cè)量壓力波在液壓傳動(dòng)系統(tǒng)中對(duì)液壓油體積模量的影響。Marning (1997)發(fā)現(xiàn)了液壓油體積模量與液壓系統(tǒng)壓力之間的線性關(guān)系。但是，迄今為止，在液壓傳動(dòng)控制的設(shè)計(jì)過程中，還沒有文獻(xiàn)將體積模量考慮進(jìn)液壓傳動(dòng)控制系統(tǒng)的動(dòng)態(tài)模型中。事實(shí)上，典型的液壓傳動(dòng)系統(tǒng)變體模量比普通的液壓傳動(dòng)系統(tǒng)有更復(fù)雜的動(dòng)態(tài)過程。因此,伺服控制系統(tǒng)的穩(wěn)態(tài)、 動(dòng)態(tài)狀況對(duì)體積模變得更為重要,因?yàn)殚]環(huán)系統(tǒng)本身不會(huì)引起穩(wěn)定性問題。體積模量無法直接確定,這樣須要估計(jì)?；谶@一估計(jì), 在液壓控制系統(tǒng)中可能要采用修正的方法。體積模量復(fù)雜的動(dòng)態(tài)相互作用和控制方式是用仿真建模和分析軟件來監(jiān)測(cè)的。做一個(gè)真正的模型系統(tǒng)是非常復(fù)雜和費(fèi)時(shí)的，模擬仿真測(cè)試是非常有利的。伺服液壓傳控制系統(tǒng)是解決這個(gè)問題的好辦法。靜態(tài)和動(dòng)態(tài)模的仿真試驗(yàn)不需要昂貴的模型。仿真還能縮短產(chǎn)品的設(shè)計(jì)周期。 這項(xiàng)研究的重點(diǎn)是一個(gè)典型的液壓傳動(dòng)控制系統(tǒng)。非線性系統(tǒng)模型是通過MATLAB的仿真軟件來研究的。該系統(tǒng)模型是由泵、閥、液壓馬達(dá)、液壓管等組件組合而成。另外，變體積模量將描述出影響系統(tǒng)動(dòng)力學(xué)的現(xiàn)象與控制算法。為此，兩個(gè)不同的液壓軟管仿真模型被分別接入系統(tǒng)模型中。另外，利用模型來設(shè)計(jì)控制的過程。液壓馬達(dá)角速度的控制是通過PID(比例積分微分)和 模糊控制器來完成的。在第一個(gè)模型中，液壓系統(tǒng)的體積模量和角速度假設(shè)為一個(gè)定值，并由典型的PID(比例積分微分)和模糊控制器來控制。第二個(gè)模型,體積模量被定義為可變參數(shù)，這個(gè)參數(shù)取決于大氣壓和系統(tǒng)的壓力。在應(yīng)用同一PID控制參數(shù)的情況下，這種新模式適用于液壓系統(tǒng)的速度控制。接下來，模糊控制器應(yīng)用于這一新模式中，可以判斷體積模量的非線性關(guān)系。兩種控制辦法的仿真結(jié)果被用來對(duì)比分析體積模量在液壓系統(tǒng)中的不同情況。 2．?dāng)?shù)學(xué)模型 液壓系統(tǒng)的物理模型如圖1所示。變量泵由異步電動(dòng)機(jī)驅(qū)動(dòng)，提供液壓能給傳動(dòng)系統(tǒng)來產(chǎn)生固定的體積模量效應(yīng)，變量馬達(dá)驅(qū)動(dòng)負(fù)載。為了不讓系統(tǒng)產(chǎn)生過高的壓力，使用減壓閥來解決。 圖1. 液壓傳動(dòng)系統(tǒng) 從客觀的角度來看這個(gè)研究，系統(tǒng)的數(shù)學(xué)模型應(yīng)該越簡(jiǎn)單越好。與此同時(shí)，它必須包括重要的實(shí)際特征。了解單獨(dú)組件的目的是為了更好的了解系統(tǒng)模型。利用物理基礎(chǔ)知識(shí)，目前可以得到平衡和連續(xù)性方程。模型反映出了每個(gè)組件動(dòng)態(tài)狀態(tài)時(shí)的情況。通過了解每個(gè)組件，將所有組件聯(lián)系起來可以了解整個(gè)系統(tǒng)，從而得到整個(gè)系統(tǒng)模型。本文中，利用各組件來開發(fā)液壓系統(tǒng)模型是早期所用到的方法。 2.1 變量泵 假設(shè)原動(dòng)機(jī)（異步電動(dòng)機(jī)）的角速度是個(gè)常數(shù)。因此，聯(lián)結(jié)泵的軸的角速度也是個(gè)恒定的值。泵的流量可以通過變量泵的斜盤角度和位移得到如下關(guān)系： Qp = αkpηvp, (1) 式中，Qp表示泵的流量(m3/s)，α表示斜盤的傾斜角度(?)，kp表示泵的系數(shù)，ηvp表示泵的容積效率，假設(shè)這個(gè)參數(shù)與泵自轉(zhuǎn)角度沒有關(guān)系。 2.2 減壓閥 為了簡(jiǎn)化，減壓閥不考慮動(dòng)態(tài)因素的影響，這樣，可以得到減壓閥在開啟和關(guān)閉時(shí)的兩個(gè)流量方程。 Qv = kv(P ? Pv), 如果P 大于Pv, (2) Qv = 0, 如果 P 小于等于 Pv, (3) 式中，kv表示閥的靜態(tài)特性，P表示系統(tǒng)的壓力（帕），Pv表示開啟壓力（帕）。 2.3 液壓管 作為傳統(tǒng)模型，高壓管用于連接泵和馬達(dá)，在這里體積模量是個(gè)固定值。變體積模量在接下來的章節(jié)中討論。 流體的可壓縮性關(guān)系如下式（4）所示。等式（5）提出了在給定流量時(shí)壓力值的求法。假設(shè)液壓管對(duì)系統(tǒng)的壓降忽略不計(jì)。 Qc = (V /β)(dP/dt), (4) (dP/dt) = (β/V )Qc, (5) 式中，Qc表示經(jīng)過壓縮后的流量(m3/s), V表示流體經(jīng)過壓縮后的體積(m3)，β表示流體的固定體積模量，在液壓系統(tǒng)和動(dòng)能傳動(dòng)中它是一個(gè)重要的參數(shù)，因而它將影響動(dòng)力系統(tǒng)和控制系統(tǒng)的狀況。非氣液壓油的體積模量取決于溫度和壓力，礦物油根據(jù)添加劑數(shù)量不同，體積模量為1200~2000Mpa。但是，系統(tǒng)壓力和融合空氣，將影響體積模量的值。如果采用液壓膠管而非鋼管，體積模量在這里就回大大降低。由于這些參數(shù)影響體積模量，液壓傳動(dòng)系統(tǒng)模型必須具有更準(zhǔn)確的動(dòng)力系統(tǒng)。 流體和空氣的混合體在液壓管中的變體積模量可以如下所示： (6) 式中，下標(biāo)α、f和h分別指空氣、流體和液壓管。假設(shè)初始總體積為=+，還有 >>。這樣體積模量會(huì)比任何, , 和 Vt/Va都要小。積模量中流體的來自于生產(chǎn)廠家體的數(shù)據(jù)。(Cp/Cv)P = 1.4P主要用于絕熱狀態(tài)下空氣的體積模量。（6）式還可以改寫如下： (7) 式中：s表示融入空氣的總體積(s = Va/Vt )。 2.4 液壓馬達(dá)和負(fù)載 液壓馬達(dá)的流量(m3/s)可以用公式表示如下： Qm = kmω/ηvm, (8) 式中：km表示液壓馬達(dá)的系數(shù)，ω表示液壓馬達(dá)的角速度，ηvm表示液壓馬達(dá)的容積效率。假設(shè)液壓馬達(dá)的效率不受轉(zhuǎn)動(dòng)軸的影響。液壓馬達(dá)的扭矩可有公式表示如下： Mm = kmt_Pηmm, (9) 式中：kmt表示液壓馬達(dá)的扭矩系數(shù)，P表示液壓馬達(dá)的壓降，ηmm表示液壓馬達(dá)的機(jī)械效率。液壓馬達(dá)所產(chǎn)生的扭矩等于瞬間馬達(dá)負(fù)載的總和，可由公式表示如下： Mm = MI +MB +Mo, (10) 式中，MI、MB和Mo表示瞬間形成的負(fù)載慣性，摩擦力伴隨機(jī)械運(yùn)行而生，這樣可以描述為： Mm = Im(dω/dt) + Bω +Mo, (11) 式中，Im表示液壓馬達(dá)軸的轉(zhuǎn)動(dòng)慣量，B表示馬達(dá)和軸的摩擦系數(shù)，ω表示馬達(dá)軸的角速度。等式（11）用于確定液壓馬達(dá)軸的角速度。從新定義角速度公式如下： dω/dt = (Mm ? Bω ?Mo)/Im. (12) 2.5 液壓傳動(dòng)系統(tǒng) 通過基本數(shù)學(xué)模型，結(jié)合液壓系統(tǒng)中各組件和發(fā)生的現(xiàn)象，從而方便獲得總體液壓傳動(dòng)系統(tǒng)模型。由此，液壓系統(tǒng)是根據(jù)模型仿照而成的系統(tǒng)。在開發(fā)動(dòng)態(tài)模型系統(tǒng)時(shí)，假設(shè)傳動(dòng)的靜態(tài)和動(dòng)態(tài)特性不取決于液壓馬達(dá)的旋轉(zhuǎn)方向，傳動(dòng)處于平衡狀態(tài)。不考慮模型中液壓泵和馬達(dá)的泄露量。通過數(shù)學(xué)模型可以得到液壓傳動(dòng)系統(tǒng)的兩個(gè)等式如下： 流量方程： Qp = Qm + Qc + Qv, (13) 瞬時(shí)： Mm = MI +MB +Mo. (14) 聯(lián)合等式（5）和（12），可以得到如下公式： dP/dt = (β/V )(Qp ? Qm ? Qv), (15) dω/dt = (Mm ? Bω ?Mo)/Im. (16) Matlab仿真一個(gè)常用的模擬仿真方式，它主要用于求解非線性方程。仿真模型是基于圖2中所示的液壓傳動(dòng)系統(tǒng)的數(shù)學(xué)模型。系統(tǒng)模型中的組件可以很容易在規(guī)定要求內(nèi)變換。據(jù)此，改變液壓組件中的液壓管，通過等式（7）可以得到第二種模型。 3.控制應(yīng)用 許多相關(guān)的刊物記載出版了液壓傳動(dòng)系統(tǒng)中馬達(dá)與相連負(fù)載的速度控制方法。為了完成這個(gè)目標(biāo)，設(shè)計(jì)中采用了不同的閉環(huán)控制。但是，1996年Lee和Wu通過調(diào)節(jié)泵的位移來調(diào)節(jié)負(fù)載的速度，這種測(cè)試方法是最有用的。此外，1996年Re等人解決了用改變泵的排量來控制負(fù)載的速度，改變泵和馬達(dá)的流量是最高效的，在任何時(shí)候應(yīng)該盡可能首選這種控制方法。為此，正在研究液壓傳動(dòng)系統(tǒng)的這一問題，輸出角速度通過液壓馬達(dá)提供的流量來控制，通過調(diào)節(jié)變量泵斜盤的角度來調(diào)節(jié)流量。為了研究的方便，在應(yīng)用中不考慮斜盤的動(dòng)力學(xué)影響。此外，斜盤控制系統(tǒng)動(dòng)態(tài)速度通常比其它系統(tǒng)要快，因此忽略動(dòng)力學(xué)影響是有理由的。液壓傳動(dòng)控制系統(tǒng)中液壓馬達(dá)的角速度通過精確控制得到，因而事先必須設(shè)計(jì)好控制器。在工業(yè)中，經(jīng)典的控制方法有PI、PID，它們被用于液壓傳動(dòng)系統(tǒng)中的速度控制。關(guān)鍵是要確定控制參數(shù)，因?yàn)镻ID控制方法具有線性的特性。特別是在控制器中應(yīng)該把體積模量當(dāng)作一個(gè)非線性的。由于有可變范圍，這樣控制器的性能要非常的穩(wěn)定。以理論知識(shí)為基礎(chǔ)的控制越來越多，特別是在模糊控制領(lǐng)域。不像經(jīng)典控制方法，模糊控制結(jié)合非線性來設(shè)計(jì)控制思路。因此，這種控制方法的應(yīng)用可以用于判斷對(duì)減少體積模量影響的控制能力。 3.1 PID控制 液壓傳動(dòng)系統(tǒng)對(duì)角速度控制的算法在公式（17）、（18）中已經(jīng)給出。用Ziegler-Nichols法校正控制參數(shù)，例如比例增益(Kp),響應(yīng)時(shí)間常數(shù)(τd )，積分時(shí)間常數(shù)(τi)。通過參考角速度來確定最優(yōu)的控制參數(shù)。圖3表示液壓傳動(dòng)系統(tǒng) 仿真模型。 uv(t) = Kp·e(t) + Kp·τd·de(t)/dt +Kp/τi·dt, (17) e(t) = ωr ? ω. (18) 4、結(jié)論 利用系統(tǒng)模型和仿真技術(shù)分析了體積模量非線性對(duì)液壓傳動(dòng)系統(tǒng)的影響。通過這個(gè)研究表明，如果忽略了液壓傳動(dòng)系統(tǒng)體積模量的動(dòng)態(tài)影響，對(duì)系統(tǒng)的響應(yīng)和安全運(yùn)行將帶來很大的錯(cuò)誤。因此，應(yīng)該把體積模量作為變參數(shù)考慮，這樣可以得到實(shí)際的整體模型和確定更精確的PID控制器參數(shù)。迄今為止，還沒有分析液壓系統(tǒng)模型體積模量的同時(shí)描述模型的設(shè)計(jì)特點(diǎn)的文獻(xiàn)。于是，對(duì)于當(dāng)時(shí)最早的設(shè)計(jì)，PID控制器應(yīng)用于液壓傳動(dòng)控制系統(tǒng)可能是有用的。這樣可以清楚的看到模糊控制器消除變體積模量的不良影響。這樣有利于控制設(shè)計(jì)開發(fā)更好的控制器。今后的研究發(fā)展的方向，將包括模型斜盤的動(dòng)力學(xué)問題、閥的動(dòng)力學(xué)問題、液壓馬達(dá)和泵的流動(dòng)復(fù)雜和轉(zhuǎn)矩問題。這樣，一個(gè)合適的控制方法將被應(yīng)用于調(diào)速和變負(fù)載的情況。 參考文獻(xiàn) Dasgupta K 2000 Analysis of a hydrostatic transmission system using low speed high torque motor. Mech. Mach. Theory 35: 1481–1499 Dasgupta K, Chattapadhyay A, Mondal S K 2005 Selection of fire-resistant hydraulic fluids through system modelling and simulation. Simul. Model. Pract. Theory 13: 1–20 Eryilmaz B,Wilson B H 2001 Improved tracking control of hydraulic systems. Trans. ASME: J. 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Control 8: 338–353 7 S adhan a Vol.31,Part 5,October 2006,pp.543556.Printed in IndiaEffect of bulk modulus on performance of a hydrostatictransmission control systemALI VOLKAN AKKAYAYildiz Technical University,Mechanical Engineering Department,34349,Besiktas,Istanbul,Turkeye-mail:aakkayayildiz.edu.trMS received 9 September 2005;revised 20 February 2006Abstract.In this paper,we examine the performance of PID(proportionalintegral derivative)and fuzzy controllers on the angular velocity of a hydrostatictransmission system by means of Matlab-Simulink.A very novel aspect is that itincludes the analysis of the effect of bulk modulus on system control.Simulationresultsdemonstratesthatbulkmodulusshouldbeconsideredasavariableparameterto obtain a more realistic model.Additionally,a PID controller is insufficient inpresence of variable bulk modulus,whereas a fuzzy controller provides robustangular velocity control.Keywords.Hydrostatic transmission;bulk modulus;PID(proportional integralderivative);fuzzy controller.1.IntroductionHydrostatic transmission(HST)systems are widely recognized as an excellent means ofpower transmission when variable output velocity is required in engineering applications,especially in field of manufacturing,automation and heavy duty vehicles.They offer fastresponse,maintainprecisevelocityundervaryingloadsandallowimprovedenergyefficiencyand power variability(Dasgupta 2000;Kugi et al 2000).A basic hydrostatic transmission isan entire hydraulic system.Generally,it contains a variable-displacement pump driven byan induction motor,a fixed or variable displacement motor,and all required controls in onesimple package.By regulating the displacement of the pump and/or motor,a continuouslyvariable velocity can be achieved(Wu et al 2004).Manufacturers and researchers continue to improve the performance and reduce the costof hydrostatic systems.Especially,modelling and control studies of hydrostatic transmissionsystemshaveattractedconsiderableattentioninrecentdecades.Somestudiesonthistopiccanbe found in the literature(Huhtala 1996;Manring&Luecke 1998;Dasgupta 2000;Kugi et al2000;Dasguptaetal2005).Variousrotationalvelocitycontrolalgorithmsforhydrostaticsys-temsaredevelopedandappliedbyLennevi&Palmberg(1995),Lee&Wu(1996),Piotrowska(2003).All these designs use the bulk modulus as a fixed value through a wide pressurerange.However,in practice,the bulk modulus is an essential part of dynamic behaviours of543544Ali Volkan Akkayathe hydraulic systems(McCloy&Martin 1980;Watton 1989).Due to temperature variationsand air entrapment,the bulk modulus may vary during the operation of the hydraulic sys-tems(Eryilmaz&Wilson 2001).A little entrapped air is enough to reduce the bulk modulussignificantly(Merrit 1967;Tan&Sepehri 2002).Moreover,system pressure plays an impor-tant role on the bulk modulus value(Wu et al 2004).Some effects of instabilities induced bybulk modulus nonlinearities such as pressure oscillations in the form of pressure waves canbe detrimental to operation of hydraulic systems and may result in reduced component life,loss of performance,disturbance in control systems,reduced efficiency and increased acous-tic noise.In spite of these adverse effects,there are few studies about bulk modulus withinhydrostatic transmission systems.Yu et al(1994)developed an on-line parameter identifica-tion method,determining the effective oil bulk modulus within an actual hydraulic system bymeasuring the propagation of a pressure wave through a long pipe.Marning(1997)devel-oped a linear relation between oil bulk modulus and pressure for a HST system.However,todate,nothing has appeared in the literature that addresses the effect of bulk modulus dynam-ics incorporated into a hydrostatic transmission model on control design process of the HSTsystem.In fact,models of hydrostatic transmission systems with variable bulk modulus havemore complex dynamic behaviour than normal.Moreover,having servo control of the sys-tem,dynamics of bulk modulus becomes more important because the closed-loop systemitself raises the issue of stability.Bulk modulus cannot be determined directly and hence needs to be estimated.Based onthis estimation,corrective actions may be taken in control applications for HST systems.Thecomplex dynamic interactions between variable bulk modulus and the control action is inves-tigated using modelling and simulation analysis.Simulation tests are particularly beneficialwhen preparing a model of a real system is complicated and time-consuming.A servo hydro-statictransmissioncontrolsystemisagoodexampleforthisissue.Thedeterminationofstaticand dynamic behaviours using simulation tests is possible without expensive prototypes.Thesimulation also makes a shorter product-designing cycle possible.This study focuses on control performance of a typical HST system.A nonlinear modelof the system is studied by means of Matlab-Simulink software.The system model is acombination of each individual component model consisting of pump,valve,hydraulic hoseand hydraulic motor.In addition,the variable bulk modulus is presented to describe theeffects of this phenomenon on system dynamics and control algorithm.For this purpose,twodifferent hydraulic hose Simulink models are incorporated separately into the system model.In addition,the models are utilized in the control design process.The control of the angularvelocity of the hydraulic motor coupled with load is achieved by PID(proportional integralderivative)and fuzzy types of controller.In the first model,bulk modulus is assumed to havea fixed value and angular velocity control of the HST system is carried out with the classicalPID control algorithm.In the second model,bulk modulus is defined as a variable parameterdependingonentrappedairandsystempressure.Thisnewmodelisappliedonvelocitycontrolof the HST system under the same PID control parameters.In the following,fuzzy controllerisimplementedinthisnewmodelinordertojudgeitscapabilityagainstvariablebulkmodulusnonlinearity.The simulation results of two control approaches are then compared to analysethe differences in the performance of the HST system in terms of bulk modulus dynamics.2.Mathematical modelThe physical model of the HST system considered for this study is shown in figure 1.Thevariable displacement pump driven by an induction motor supplies hydraulic power to a fixedEffect of bulk modulus on performance of a transmission control system545Figure 1.Hydrostatic transmission system.displacementhydraulicmotorfordrivingload.Toprotectthesystemfromexcessivepressure,a pressure relief valve is used.From a research objective point of view,the descriptions of a system mathematical modelshould be as simple as possible.At the same time,it must include important characteristics ofthe real event.One way to understand the system is to separate the system into componentsfor the purpose of modelling.Using a fundamental knowledge of physics,for instance themomentequilibriumandcontinuityequation,amodelthatrepresentsthedynamicsbehaviourofeachcomponentcanbederivedatthecomponentlevels.Havingunderstoodeachindividualcomponent,wecanunderstandtheoverallsystembyinterconnectingthecomponentstogetherto obtain an overall system model(Prasetiawan 2001).In this paper,the model of eachcomponent used for the HST system is developed using earlier methods(Jedrzykiewicz et al1997,1998).2.1 Variable-displacement pumpIt is assumed that the angular velocity of the prime mover(induction motor)is constant.Therefore,angular velocity of the pump shaft is constant.Pump flow rate can be adjustedwith variable displacement via the swashplate displacement angle and can be given asQp=kpvp,(1)where,Qpis pump flow rate(m3/s),is displacement angle of swashplate(),kpis pumpcoefficient(m3/s),vpis pump volumetric efficiency()which is assumed not to depend onpump rotation angle.2.2 Pressure relief valveTo simplify,pressure relief valve dynamics is not taken into consideration.Therefore,twoequation as below are given for passing flow rate through pressure relief valve(m3/s)in thestate of opening and closing.Qv=kv(P Pv),if P Pv,(2)Qv=0,if P Pv,(3)546Ali Volkan Akkayawhere,kvis slope coefficient of valve static characteristic(m5/Ns),P is system pressure(Pa)and Pvis valve opening pressure(Pa).2.3 Hydraulic hoseAs in traditional modelling,the pressurized hose that connects the pump to the motors ismodelled as volume with a fixed bulk modulus in this section.Variable bulk modulus arediscussed in the following subsection.The fluid compressibility relation can be given as in(4).Equation(5)provides the pressurevaluefromagivenflowrate.Itisassumedthatpressuredropinthehydraulichoseisnegligible.Qc=(V/)(dP/dt),(4)(dP/dt)=(/V)Qc,(5)where,Qcis flow rate deal with fluid compressibility(m3/s),V is the fluid volume(m3)subjected to pressure effect,is fixed bulk modulus(Pa).2.3a VariablebulkmodulusFluidisanimportantelementofhydrostaticsystemsandenablespower transmission,hence it can influence the dynamic behaviours of the system and thecontrol system.The bulk modulus of non-aerated hydraulic oil depends on temperature andpressure,for mineral oils with additives its value ranges from 1200 to 2000MPa.Moreover,system pressure and entrapped air affect the bulk modulus value.If a hydraulic hose is usedratherthanasteelpipe,thebulkmodulusofthissectionmaybeconsiderablyreduced.Owingto these reasons,the parameters influencing bulk modulus value must be included in the HSTmodel for more accurate system dynamics.The equation which gives the variable bulk modulus of fluid-air mixture in a flexiblecontainer is as follows(McCloy&Martin 1980):1v=1f+1h+VaVt1a,(6)where,the subcripts a,f and h refer to air,fluid,and hose respectively.It is assumed that theinitial total volume Vt=Vf+Va,and that f?a.Thus bulk modulus will be less than anyf,h,or Vt/Vaa.The bulk modulus of the fluid fis obtained from the manufacturersdata.The adiabatic bulk modulus used for air is(Cp/Cv)P=14P.With these assumptions,(6)can be rewritten as in,1v=1f+1h+s14 P,(7)where,s is entrapped air percent in the total volume(s=Va/Vt).2.4 Hydraulic motor and loadFlow rate used in the hydraulic motor(m3/s)can be written as inQm=km/vm,(8)where,kmis hydraulic motor coefficient(m3),is angular velocity of hydraulic motor(1/s)andvmisvolumetricefficiencyofthemotor().Itisassumedthathydraulicmotorefficiencydoes not depend on its shaft rotation angle.Hydraulic motor torque(Nm)can be written as,Mm=kmt?Pmm,(9)Effect of bulk modulus on performance of a transmission control system547where,kmtis motor torque coefficient(m3),?P is pressure drop in hydraulic motor(Pa)and mmis mechanical efficiency of hydraulic the motor().The torque produced in thehydraulic motor(Nm)is equal to the sum of the moments from the motor loads and can begiven as,Mm=MI+MB+Mo,(10)where,MI,MBandMoarethemomentsresultingfromloadinertia,frictionforceandmachineoperation respectively.These moments can be denoted asMm=Im(d/dt)+B+Mo,(11)where,Imis the inertia of the hydraulic motor shaft(Nms2),B is viscous friction coefficientof motor and its shaft(Ns/m),and is angular velocity of motor shaft(1/s).Equation(11)can be used to determine the angular velocity of the hydraulic motor shaft.This equation isrearranged for angular velocity asd/dt=(Mm B Mo)/Im.(12)2.5 Hydrostatic transmission systemThe fundamental mathematical models of the system components and phenomena occurringin hydrostatic systems are conveniently combined to obtain the overall HST system model.Accordingly,a hydrostatic transmission is modelled as a lumped system.In the developmentof the dynamic model of the system,it is assumed that static and dynamic features of thetransmissiondonotdependuponthedirectionofhydraulicmotorrotationandthetransmissionis a state of thermal balance.Leakage flows in pump and motor are not taken into accountduring the modelling.The mathematical model of the HST system consists of two equations as below:equality of flow rate:Qp=Qm+Qc+Qv,(13)moment:Mm=MI+MB+Mo.(14)Using(5)and(12),the following are then obtained,dP/dt=(/V)(Qp Qm Qv),(15)d/dt=(Mm B Mo)/Im.(16)A commonly available general purpose simulation package Matlab/Simulink is used tosolve the nonlinear equations.The Simulink model based on the component mathematicalmodels of HST system is given in figure 2.The component models can be easily modifiedin accordance width specific constructions.Accordingly,when bulk modulus is rebuilt in thehydraulic hose component with regard to(7),the second model can be generated.548Ali Volkan AkkayaFigure 2.Simulink model of hydrostatic transmission system.3.Control applicationsMost publications related to the HST control are related to the speed control of the hydraulicmotorconnectedtotheload.Inordertoachievethisgoal,differentclosed-loopcontroldesignstrategiescanbeused.However,Lee&Wu(1996)showedthatusingonlypumpdisplacementto regulate load speed is the most effective of all the methods they tested.In addition,Re et al(1996)concluded that the sole use of pump displacement actuation to control one load speedof a system with variable-displacement pump and motor is the most efficient,and should bealways preferred whenever possible.For this reason,in the HST systems being considered inthis study,the output angular velocity is controlled by the flow rate supplied to the hydraulicmotor,and this flowrate is adjusted by the swashplate angle of the variable-displacementpump.Swashplate dynamics are not taken into consideration in the control application inthis study for the sake of simplicity.In addition,the swashplate control system usually hasfaster dynamics than the rest of the system,and therefore neglecting its dynamics is justified(Watton 1989).Topreciselycontroltheangularvelocityofthehydraulicmotorinhydrostatictransmissioncontrol systems,an appropriate controller must be designed in advance.In industrial appli-cations,classical control methods such as PI,PID are being used for velocity control of HSTsystems.ItiscrucialtodeterminecontrollerparametersaccuratelybecausePIDcontrolmeth-ods have linear characteristics.They are sometimes insufficient to overcome nonlinearitieswhich exist in the nature of the HST systems for high precision applications(Tikkanen et al1995;Prasetiawan 2001).In particular,the bulk modulus ought to be regarded as a source ofsignificant nonlinearity for this type of controller.Thus,the controller has to be very robustto account for such wide variation.Use of knowledge-based systems in process control isincreasing,especially in the fields of fuzzy control(Tanaka 1996).Unlike classical controlmethods,the fuzzy controller is designed with linguistic terms to cope with the nonlineari-ties.Therefore,this control method is also applied to judge its capacity to reduce the adverseeffect of variable bulk modulus.3.1 PID controlThe structure of the PID control algorithm used for the angular velocity control of HSTsystem is given in(17)and(18)below.Ziegler-Nichols method is implemented for tuningcontrol parameters,such as proportional gain(Kp),derivative time constant(d)and integraltime constant(i)(Ogata 1990).After fine adjustments,the optimal control parameters areEffect of bulk modulus on performance of a transmission control system549Figure 3.Simulink model of HST system for PID control.determined for the reference angular velocity.Figure 3 shows the Simulink model of thePID-controlled HST system.uv(t)=Kpe(t)+Kpdde(t)dt+Kpi?e(t)dt,(17)e(t)=r.(18)3.2 Fuzzy controlFuzzy logic has come a long way since it was first presented to technical society,whenZadeh(1965)first published his seminal work.Since then,the subject has been the focusof many independent research investigations.The attention currently being paid to fuzzylogic is most likely the result of present popular consumer products employing fuzzy logic.The advantages of this method are its applicability to nonlinear systems,simplicity,goodperformance and robust character.These days,this method is being applied to engineer-ing control systems such as robot control,flight control,motor control and power systemssuccessfully.In fuzzy control,linguistic descriptions of human expertise in controlling a process arerepresented as fuzzy rules or relations.This knowledge base is used by an inference mecha-nism,in conjunction with some knowledge of the states of the process in order to determinecontrol actions.Unlike the conventional controller,there are three procedures involved in theimplementation of a fuzzy controller:fuzzification of inputs,and fuzzy inference based onthe knowledge and the defuzzification of the rule-based control signal.The structure of thefuzzy controller is seen in figure 4.An applied fuzzy controller needs two input signals.These signals are error(e)and deriva-tive of error(de)respectively.The usual overlapped triangular fuzzy membership functionsare used for two input signals(e,de/dt)and the output signal(u).Figure 5 shows the struc-ture of the membership functions of input and output signals.Input signals are transformedat intervals of 1,1 by scaling factors which are Ge and Gde.In the fuzzification process,all input signals are expressed as linguistic values which are:NB negative big,NM negative medium,NS-negative small,ZE-zero,PS-positive small,PM-positive medium,PB-positive big.After input signals are converted to fuzzy linguisticvariables,these variables are sent to the inference mechanism to create output signals.550Ali Volkan AkkayaFigure 4.Structure of a fuzzy controller.Theinferenceprocessconsistsoffortyninerulesdrivenbythelinguisticvaluesoftheinputsignals.These fuzzy rules written as a rule base are shown given in table 1.The rule base isdeveloped by heuristics with error in motor angular velocity and derivation of error in thisvelocity.For instance,one of the possible rules is:IF e=PS and de=NB THAN u=NM.Thisrulecanbeexplainedasinthefollowing:Iftheerrorissmall,angularvelocityofhydraulicmotor is around the reference velocity.Significantly big negative value of derivation of errorshows that the motor velocity is rapidly approaching the reference position.Consequently,controller output should be negative middle to prevent overshoot and to create a brake effect.Asarule-inferencemethod,theMamdaniMethodisselectedbecauseofitsgeneralacceptance(Tanaka 1996).Defuzzification transforms the control linguistic variables into the exact control output.Indefuzzification,the method of centre of gravity is implemented(Tanaka 1996),asu=n?i=1WiBi/n?i=1Wi(19)Figure 5.Triangular fuzzy member-shipfunctions,(a)einputsignal,(b)deinput signal,(c)u output signals.Effect of bulk modulus on performance of a transmission control system551Table 1.Rule base for fuzzy control.deeNBNMNSZEPSPMPBNBNBNBNBNMNMNSZENMNBNBNMNSNSZEPSNSNBNMNSNSZEPSPMZENMNSNSZEPSPSPMPSNMNSZEPSPSPMPBPMNSZEPSPSPMPBPBPBZEPSPMPMPBPBPBwhere,u is the output signal of the fuzzy controller,Wiis the degree of the firing of the ithrule,Biisthecentroidoftheconsequentfuzzysubsetofithrule.Realvaluesofcontroloutputsignal(uv)are determined by the scaling factor of Guv.As a result,the fuzzy controllerbuilt-in Fuzzy Logic Toolbox of the Matlab program has been added to the Simulink modelof hydrostatic transmission system for simulation analysis(figure 6).4.Simulation results and discussionThe validity of the influence of bulk modulus dynamics on HST control system has beentestedincomputersimulations.Inordertocarryoutsimulation,somephysicalandsimulationparameters corresponding to HST system are taken from work of McCloy&Martin(1980)and Jedrzykiewicz et al(1997,1998),and other control parameters are as given in table 2.OpenlooppressureandangularvelocityresponsesoftheHSTsystemaregiveninfigures 7aand b respectively,under fixed bulk modulus and variable bulk modulus.Comparing the sim-ulation results shows that the model including variable bulk modulus shows flexible dynam-ics and decreasing system stiffness(figure 7a).Moreover,a degree of aeration less than 1%brings about considerable changes of velocity and pressure responses because the aeration ofthe work