机械类外文翻译【FY001】体积模量对液压传动控制系统的影响【PDF+WORD】
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体积模量对液压传动控制系统的影响
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@.tr
MS received 9 September 2005;revised 20 February 2006
摘要. 这篇研究报告，我们主要通过PID（比例积分微分）控制方式检测液压控制系统对角速度控制的Matlab仿真。有一个地方很值得关注，包括对体积模量控制分析系统。仿真结果表明，体积模量通过变参数可以获得更实用的模型。此外，PID控制器的不足之处在于对变体积模量角速度的控制，而模糊控制能够实现较好的控制。
关键词 液压传动；体积模量；PID(比例积分微分)；模糊控制
1.引言
液压传动系统是种输出可实现无级调速的理想动力传递方式，这样在工程中得到了广泛的应用，特别是在制造领域，自动化和重型车辆。它能够提供快速的响应，在变负载情况下能保持精确的传动速度，可以改善能量的利用效率和变功率传动。液压传动的基础为液压系统。一般来讲，它包括由异步电动机驱动的变量泵，定量或变量马达，所有要求控制的都在一个简单的控制柜中。通过调节泵或者马达的排量，实现无级调速。
制造厂商和研究人员不断的改进性能和降低液压传动系统成本。尤其是近十年，体积模量在液压传动和控制系统的研究中引起了人们的关注。一些这方面的研究专题在学术期刊中可以找到。Lennevi和Palmberg、Lee和Wu运用各种转速控制算法求液压系统的液压力得到了很好的发展和应用。所有这些设计用的体积模量都是固定值，适用的压力范围广。但是，实际上体积模量是液压系统中必须考虑的因素。因温度变化和大气压，体积模量可在运行过程中求出液压系统的液压力。一点空隙足以大副减少体积模量。此外，系统压力起着重要的作用在体积模量值上。非线性影响了体积模量的不稳定，例如：压力振动导致的压力波会对液压系统的运行不利，还有可能会因磨损而导致部件的使用寿命缩短，干扰控制系统，降低了效率和增加了噪音。尽管有这些不良的影响，但在液压传动系统中很少有关于体积模量的研究。1994年Yu等人开发了一个参数辩识的方法，通过长的管子来测量压力波在液压传动系统中对液压油体积模量的影响。Marning (1997)发现了液压油体积模量与液压系统压力之间的线性关系。但是，迄今为止，在液压传动控制的设计过程中，还没有文献将体积模量考虑进液压传动控制系统的动态模型中。事实上，典型的液压传动系统变体模量比普通的液压传动系统有更复杂的动态过程。因此,伺服控制系统的稳态、 动态状况对体积模变得更为重要,因为闭环系统本身不会引起稳定性问题。体积模量无法直接确定,这样须要估计。基于这一估计, 在液压控制系统中可能要采用修正的方法。体积模量复杂的动态相互作用和控制方式是用仿真建模和分析软件来监测的。做一个真正的模型系统是非常复杂和费时的，模拟仿真测试是非常有利的。伺服液压传控制系统是解决这个问题的好办法。静态和动态模的仿真试验不需要昂贵的模型。仿真还能缩短产品的设计周期。
这项研究的重点是一个典型的液压传动控制系统。非线性系统模型是通过MATLAB的仿真软件来研究的。该系统模型是由泵、阀、液压马达、液压管等组件组合而成。另外，变体积模量将描述出影响系统动力学的现象与控制算法。为此，两个不同的液压软管仿真模型被分别接入系统模型中。另外，利用模型来设计控制的过程。液压马达角速度的控制是通过PID(比例积分微分)和 模糊控制器来完成的。在第一个模型中，液压系统的体积模量和角速度假设为一个定值，并由典型的PID(比例积分微分)和模糊控制器来控制。第二个模型,体积模量被定义为可变参数，这个参数取决于大气压和系统的压力。在应用同一PID控制参数的情况下，这种新模式适用于液压系统的速度控制。接下来，模糊控制器应用于这一新模式中，可以判断体积模量的非线性关系。两种控制办法的仿真结果被用来对比分析体积模量在液压系统中的不同情况。
2．数学模型
液压系统的物理模型如图1所示。变量泵由异步电动机驱动，提供液压能给传动系统来产生固定的体积模量效应，变量马达驱动负载。为了不让系统产生过高的压力，使用减压阀来解决。
图1. 液压传动系统
从客观的角度来看这个研究，系统的数学模型应该越简单越好。与此同时，它必须包括重要的实际特征。了解单独组件的目的是为了更好的了解系统模型。利用物理基础知识，目前可以得到平衡和连续性方程。模型反映出了每个组件动态状态时的情况。通过了解每个组件，将所有组件联系起来可以了解整个系统，从而得到整个系统模型。本文中，利用各组件来开发液压系统模型是早期所用到的方法。
2.1 变量泵
假设原动机（异步电动机）的角速度是个常数。因此，联结泵的轴的角速度也是个恒定的值。泵的流量可以通过变量泵的斜盘角度和位移得到如下关系：
Qp = αkpηvp, (1)
式中，Qp表示泵的流量(m3/s)，α表示斜盘的倾斜角度(?)，kp表示泵的系数，ηvp表示泵的容积效率，假设这个参数与泵自转角度没有关系。
2.2 减压阀
为了简化，减压阀不考虑动态因素的影响，这样，可以得到减压阀在开启和关闭时的两个流量方程。
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表示阀的静态特性，P表示系统的压力（帕），Pv表示开启压力（帕）。
2.3 液压管
作为传统模型，高压管用于连接泵和马达，在这里体积模量是个固定值。变体积模量在接下来的章节中讨论。
流体的可压缩性关系如下式（4）所示。等式（5）提出了在给定流量时压力值的求法。假设液压管对系统的压降忽略不计。
Qc = (V /β)(dP/dt), (4)
(dP/dt) = (β/V )Qc, (5)
式中，Qc表示经过压缩后的流量(m3/s), V表示流体经过压缩后的体积(m3)，β表示流体的固定体积模量，在液压系统和动能传动中它是一个重要的参数，因而它将影响动力系统和控制系统的状况。非气液压油的体积模量取决于温度和压力，矿物油根据添加剂数量不同，体积模量为1200~2000Mpa。但是，系统压力和融合空气，将影响体积模量的值。如果采用液压胶管而非钢管，体积模量在这里就回大大降低。由于这些参数影响体积模量，液压传动系统模型必须具有更准确的动力系统。
流体和空气的混合体在液压管中的变体积模量可以如下所示：
(6)
式中，下标α、f和h分别指空气、流体和液压管。假设初始总体积为=+，还有 >>。这样体积模量会比任何, , 和 Vt/Va都要小。积模量中流体的来自于生产厂家体的数据。(Cp/Cv)P = 1.4P主要用于绝热状态下空气的体积模量。（6）式还可以改写如下：
(7)
式中：s表示融入空气的总体积(s = Va/Vt )。
2.4 液压马达和负载
液压马达的流量(m3/s)可以用公式表示如下：
Qm = kmω/ηvm, (8)
式中：km表示液压马达的系数，ω表示液压马达的角速度，ηvm表示液压马达的容积效率。假设液压马达的效率不受转动轴的影响。液压马达的扭矩可有公式表示如下：
Mm = kmt_Pηmm, (9)
式中：kmt表示液压马达的扭矩系数，P表示液压马达的压降，ηmm表示液压马达的机械效率。液压马达所产生的扭矩等于瞬间马达负载的总和，可由公式表示如下：
Mm = MI +MB +Mo, (10)
式中，MI、MB和Mo表示瞬间形成的负载惯性，摩擦力伴随机械运行而生，这样可以描述为：
Mm = Im(dω/dt) + Bω +Mo, (11)
式中，Im表示液压马达轴的转动惯量，B表示马达和轴的摩擦系数，ω表示马达轴的角速度。等式（11）用于确定液压马达轴的角速度。从新定义角速度公式如下：
dω/dt = (Mm ? Bω ?Mo)/Im. (12)
2.5 液压传动系统
通过基本数学模型，结合液压系统中各组件和发生的现象，从而方便获得总体液压传动系统模型。由此，液压系统是根据模型仿照而成的系统。在开发动态模型系统时，假设传动的静态和动态特性不取决于液压马达的旋转方向，传动处于平衡状态。不考虑模型中液压泵和马达的泄露量。通过数学模型可以得到液压传动系统的两个等式如下：
流量方程：
Qp = Qm + Qc + Qv, (13)
瞬时：
Mm = MI +MB +Mo. (14)
联合等式（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仿真一个常用的模拟仿真方式，它主要用于求解非线性方程。仿真模型是基于图2中所示的液压传动系统的数学模型。系统模型中的组件可以很容易在规定要求内变换。据此，改变液压组件中的液压管，通过等式（7）可以得到第二种模型。
3.控制应用
许多相关的刊物记载出版了液压传动系统中马达与相连负载的速度控制方法。为了完成这个目标，设计中采用了不同的闭环控制。但是，1996年Lee和Wu通过调节泵的位移来调节负载的速度，这种测试方法是最有用的。此外，1996年Re等人解决了用改变泵的排量来控制负载的速度，改变泵和马达的流量是最高效的，在任何时候应该尽可能首选这种控制方法。为此，正在研究液压传动系统的这一问题，输出角速度通过液压马达提供的流量来控制，通过调节变量泵斜盘的角度来调节流量。为了研究的方便，在应用中不考虑斜盘的动力学影响。此外，斜盘控制系统动态速度通常比其它系统要快，因此忽略动力学影响是有理由的。液压传动控制系统中液压马达的角速度通过精确控制得到，因而事先必须设计好控制器。在工业中，经典的控制方法有PI、PID，它们被用于液压传动系统中的速度控制。关键是要确定控制参数，因为PID控制方法具有线性的特性。特别是在控制器中应该把体积模量当作一个非线性的。由于有可变范围，这样控制器的性能要非常的稳定。以理论知识为基础的控制越来越多，特别是在模糊控制领域。不像经典控制方法，模糊控制结合非线性来设计控制思路。因此，这种控制方法的应用可以用于判断对减少体积模量影响的控制能力。
3.1 PID控制
液压传动系统对角速度控制的算法在公式（17）、（18）中已经给出。用Ziegler-Nichols法校正控制参数，例如比例增益(Kp),响应时间常数(τd )，积分时间常数(τi)。通过参考角速度来确定最优的控制参数。图3表示液压传动系统
仿真模型。
uv(t) = Kp·e(t) + Kp·τd·de(t)/dt +Kp/τi·dt, (17)
e(t) = ωr ? ω. (18)
4、结论
利用系统模型和仿真技术分析了体积模量非线性对液压传动系统的影响。通过这个研究表明，如果忽略了液压传动系统体积模量的动态影响，对系统的响应和安全运行将带来很大的错误。因此，应该把体积模量作为变参数考虑，这样可以得到实际的整体模型和确定更精确的PID控制器参数。迄今为止，还没有分析液压系统模型体积模量的同时描述模型的设计特点的文献。于是，对于当时最早的设计，PID控制器应用于液压传动控制系统可能是有用的。这样可以清楚的看到模糊控制器消除变体积模量的不良影响。这样有利于控制设计开发更好的控制器。今后的研究发展的方向，将包括模型斜盘的动力学问题、阀的动力学问题、液压马达和泵的流动复杂和转矩问题。这样，一个合适的控制方法将被应用于调速和变负载的情况。
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QQ 29467473
- 内容简介：
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Sadhana 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: aakkaya.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. Simulationresults demonstrates that bulk modulus should be considered as a variable parameterto 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, maintain precise velocity under varying loads and allow improved energy efficiencyand 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 transmissionsystems have attracted considerable attention in recent decades. Some studies on this topic canbe found in the literature (Huhtala 1996; Manring & Luecke 1998; Dasgupta 2000; Kugi et al2000; Dasgupta et al 2005). Various rotational velocity control algorithms for hydrostatic sys-tems are developed and applied by Lennevi & 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 of543nts544 Ali 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-static transmission control system is a good example for this issue. The determination of staticand 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 parameterdepending on entrapped air and system pressure. This new model is applied on velocity controlof the HST system under the same PID control parameters. In the following, fuzzy controlleris implemented in this new model in order to judge its capability against variable bulk modulusnonlinearity. 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 fixedntsEffect of bulk modulus on performance of a transmission control system 545Figure 1. Hydrostatic transmission system.displacement hydraulic motor for driving load. To protect the system from excessive pressure,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 themoment equilibrium and continuity equation, a model that represents the dynamics behaviourof each component can be derived at the component levels. Having understood each individualcomponent, we can understand the overall system by interconnecting the components togetherto 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 PPv, (2)Qv= 0, if P Pv, (3)nts546 Ali 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 pressurevalue from a given flow rate. It is assumed that pressure drop in the hydraulic hose is negligible.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 Variable bulk modulus Fluid is an important element of hydrostatic systems and enablespower 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 2000 MPa. Moreover,system pressure and entrapped air affect the bulk modulus value. If a hydraulic hose is usedrather than a steel pipe, the bulk modulus of this section may be considerably reduced. 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 fgreatermuch a. Thus bulk modulus will be less than anyf, h,orVt/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)and vmis volumetric efficiency of the motor (). It is assumed that hydraulic motor efficiencydoes not depend on its shaft rotation angle. Hydraulic motor torque (Nm) can be written as,Mm= kmtDelta1Pmm, (9)ntsEffect of bulk modulus on performance of a transmission control system 547where, kmtis motor torque coefficient (m3), Delta1P 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, MBand Moare the moments resulting from load inertia, friction force and machineoperation 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 thetransmission do not depend upon the direction of hydraulic motor rotation and the transmissionis 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.nts548 Ali 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 hydraulicmotor connected to the load. In order to achieve this goal, different closed-loop control designstrategies can be used. However, Lee & Wu (1996) showed that using only pump displacementto 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).To precisely control the angular velocity of the hydraulic motor in hydrostatic transmissioncontrol 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. It is crucial to determine controller parameters accurately because PID control meth-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 arentsEffect of bulk modulus on performance of a transmission control system 549Figure 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) = Kp e(t) + Kp dde(t)dt+Kpiintegraldisplaye(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,
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