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A Systems Approach for Modelling Mechatronics Systems
Baam A.Huein Production and Quality Engineering Department, The Norwegian University of Science and
Technology, NTNU, Trondheim, Norway
Received on December 3,1997
ABSTRACT
This paper presents a unified approach based on utilizing multidimensional arrays in order to model the physical and logical properties of mechatronics systems.A mechatronics system model consists of two interacting submodels.A submodel that describes aspects related to energy flow in the physical system, and another submodel that describes aspects related to information flow in the control system.The multidimensional array based approach of modelling provides us with the poibility to use one terminology and the same formalism for modelling both subsystems.The consequence of using the same formalism is that simulation of the mechatronics system can be performed using only one simulation environment
Keywords: Mechatronics, System, Modelling
1.INTRODUCTION
Mechatronics system is defined as the synergetic integration of mechanical engineering with electronics, and intelligent computer control in the design and manufacturing of industrial products and procees [5].The components of mechatronics systems must be designed concurrently, that is, the constraints imposed on the system by each discipline must be considered at the very early stages.Therefore, proper system design will depend heavily on the use of modelling and simulation throughout the design and prototyping stages.The integration within a mechatronics system is performed through the combination of the hardware components resulting in a physical system and through the integration of the information proceing system resulting in an intelligent control system [7].The mechatronics system then, is the result of applying computer based control systems to physical systems.The control system is designed to execute commands in real time in order to select, enhance, and supervise the behavior of the physical system.The only poible way to guarantee that these control functions will keep the behavior of the whole system within certain boundaries before we actually build it, is to create a model of the real system that takes into account all the imposed constraints by both the hardware and software components.This implies that a model of the real system must be powerful enough to capture all the properties of mechatronics system.That includes;the dynamic, static, discrete event, logic, as well as cost related properties of the real system, a task we believe, defies any fragmented approach of modelling.In this paper we present a unified approach for modelling mechatronics systems.This unified approach utilizes geometric objects or multidimensional arrays to formulate models of mechatronics systems.The multidimensional array based approach of modelling provides us with the poibility to use the same formalism for a large variety of systems [2,3,4,9].The consequence of using the same formalism is that simulation of mechatronics systems can be performed using only one simulation environment.2.MODEL STRUCTURE
Intuitively speaking, a model that describes the dynamic behavior of a given system can not be used to investigate the static behavior of the very same system.Therefore, in order to capture all aspects, we need a variety of models, each one of them encapsulates some aspects of the real system.We will consider the mechatronics system model as a set of connected submodels, each submodel corresponds to some realizable aspects.In this regard, the term connected was used to emphasize the dependency between the variables in these submodels.Throughout the proce of modelling, we shall distinguish between the following concepts, see Figure 1.Decomposition: in order to handle the complexity of mechatronics systems, they should be decomposed into subsystems.This decomposition is carried out on a multilevel fashion until we reach the basic elements that constitute the total system.The primitive system model: is a description of the system in the disconnected state.It exprees the relation between the variables in the individual elements when the bonds between these elements are removed.By this model we isolate a specific behavior;static, dynamic, etc., in each element.A pair of local variables defines the behavior of a given element locally.The Connected system model: is a description of the same system after taking the internal constraints into account.The internal constraints within the system are given by the way the local variables are connected or related directly as well as indirectly by the variables of the connected system.The connected system model resembles the actual structure of the real system.The applied sources are generated due to interaction between the system and its environment.They could be seen as the external constraints imposed on the system or even inherent constraints in the form of stored energy in system elements.3.APPLICATION EXAMPLE Consider, the manufacturing system shown in Figure 2.The system consists of a boring spindle powered by a direct current motor.The feed forward motion of the boring spindle is carried out by means of a hydraulic linear actuator.The hydraulic actuator is powered by a constant preure hydraulic pump.The volumetric flow in the hydraulic circuit is controlled by a servo valve [8].The above manufacturing system has the following specifications: The positions of boring spindle are sensed by three micro breakers.Breaker(B)which indicates that the boring spindle is at the rear position.At the rear position the rapid phase valve(I)will be switched on in order to allow a rapid forward motion(F)and the signal(S)will switch on the spindle motor.Breaker(M)indicates that the boring spindle has reached the feeding position.At this position the rapid phase valve will be switched off in order to start a controlled feed forward motion.This motion is regulated by the servo valve(St).Breaker(€)which indicates that the boring spindle has reached its final position, at this position and the backward motion(R)will begin, simultaneously the rapid phase valve(I)will be switched on in order to allow a rapid backward motion.It is also specified that the rotating speed of the spindle motor should be kept at 3000 rpm.during boring the work piece and the feed forward speed must be kept at 2cm/sec under all loading conditions.Our objective is to set up a complete model of the given system using multidimensional arrays and to carry out neceary experiments on the model to verify that specifications are satisfied.3.1 Physical System Modelling
When modelling physical systems, we are concerned with modelling the evolution of the physical variables that lives within this system.The decomposition of the physical system is shown in Figure 3.The groups of basic physical elements are claified into three categories: Generalized resistor: examples of this category are;electric resistor, mechanical damper, and hydraulic resistor.Generalized capacitor: examples of this category are;electric capacitor, mechanical spring, and hydraulic reservoir.Generalized inductor examples of this category are;electric inductor, mechanical ma, and hydraulic inductor.Breaking down the physical system into subsystems and further into basic elements will provide us with a sharp insight about the evolution of the physical quantities within each subsystem, yielding to better understanding of the modes and the states that each subsystem would attain.The advantages of having such insight will become visible during the design phase of a local control system.Modelling can be considered as the opposite procedure of decomposition.The difference is that, in decomposition, we divide the system into independent physical entities, while in modelling we reconnect the models of these physical entities.Therefore, modelling can be seen as the procedure of connection.In modelling, we start at the bottom level of this hierarchy and move upwards.At each level, we propagate from a primitive system model to a connected system model.In the succeeding level, the primitive system model would then be established by aggregating the connected system models from the former level as shown in Figure 4.At the bottom level of each subsystem, the primitive system model will be established by utilizing the governing equation or the fundamental law of each individual element.That fundamental law, such as Newton's law or Ohm's law, describes the local behavior of that element.Direct and indirect connections that resemble the internal constraints within the boundaries of each subsystem define the transformation from the primitive system model to the connected system model.For systems with linear connections such as direct current servomotor, the internal constraints are given by one connection object, the velocity object(V).The velocity object is a 2-dimensional array, the rows in that array correspond to the variables in the primitive system(local variables)and the columns correspond to the variables in the connected system(global variables).Thus, the velocity object is a transformation from the global variables in the connected system model to local variables in the primitive system model.The model of the physical system is set up by aggregating diagonally the connected system models of the hydraulic subsystem and the boring spindle.Modelling the physical system resulted in a set of different all algebraic equations [7].In a state space form, the behavior of the physical system is given by: y = ~(A , x , u , ~)Where(x)is the set of initial state variables,(u)is the set of input sources,(A)is the state transition matrix for the physical specific control function of truth or falsehood(1 0).3.2 Control System Modelling Before a control algorithm can be designed and implemented we need a description of its required properties or behavior.A precise and comprehensive mathematical model of the properties of the control system could be expreed by employing logic notation.This mathematical model provides us with means to reveal the inconsistency and conflicts in the control system and to verify that the control system meets design specifications.In order to carry out all control functions outlined in problem description, the control system should be decomposed into three subsystems.A proce controller subsystem, which will be responsible to iue start and stop commands for the different physical entities and two continuos controllers.One controller for the servo valve in the hydraulic subsystem in order to regulate the feed forward motion of the hydraulic actuator.The second controller is for the servomotor in order to regulate the angular speed of the spindle motor.The decomposition is shown in Figure 5.The functions of each subsystem are described by a set of logical arguments or rules.Each of these logical arguments could be considered as a subsystem that can be decomposed further into a number of factious logical elements.These elements could be literally anything that could carry a logical variable that aumes either the stateof truth or falsehood(1 0).These elements represent the primitive system model of aspecific control function.The procedure of modelling the control system will also move upward along the hierarchy until a total model is obtained as shown in Figure 6.In the primitive system model, the connections between the logical variables are defined by three connection objects.In claical logic, they are referred to as basic logical connectives.The group of basic logical connectives includes;conjunction(AND),disjunction(OR), and negation(NOT).We propagate to the connected system model by aggregating the logical variables in the primitive system using the above logical connectives.A connected subsystem is nothing else but the truth table of a logical argument expreed in a multi-dimensional array form.The number of axes in that array should be equal to the number of variables, therefore all repeated axes must be fused together by the method of colligation.The connected system exprees all the poible states of the system after imposing the internal constraints on the structure by connecting its individual elements.The behavior of the control system could be represented in the following form s = f(p , , i , n).Where,(0)is a set of input variables that is external constraints due to interaction with the environment.(P,)is the state transition matrix of the control system expreed in multidimensional array format.(s)is a set of output variables.The index(n)is analogues to a time index in that it specifies the order of a given state.3.3 Model of The Total System
Since both systems utilize different types of signals internally, then intuitively speaking, the only poible interface between the physical and the control system model will take place externally, through the environment by means of the impreed sources.In the above manufacturing system, we can distinguish between two ways of interface between the physical and the control system.Discrete interface: takes place in the proce controller when the purpose of the control system is to coordinate asynchronous tasks to satisfy system requirements.For example, when an event command “start the spindle motor” is iued by the proce controller, the spindle motor starts rotating.The proce of rotation itself is controlled by the lower level controller(continuos controller).Continuous interface: takes place locally on lower level control schemes when the purpose of the control system is to keep the behavior of the physical system within given boundaries such as implementing speed control.The resultant system model in this case is said to be a hybrid system model.The identifying characteristics of hybrid systems are that they incorporate both continuos dynamic behavior, i.e., the evolution of physical quantities governed by differential and algebraic equations(y = f(A , x , u , r)), and discrete event dynamic behavior governed by logic equations:(s = f(p , , i , n)).A total model can be obtained by generating a simple interface between the physical system model and the control system model.The interface will be consisting of two simple memoryle mapping functions(a)and(p)[l].The first map(a)converts the controller output(s)into a constant incremental input to the physical system as follows: u(i)= a(s n)The second map(p)converts the physical system output into a set of input logic variables to the control system as follows: i= p(y(r)), as shown in Figure 7.What we have gained so far is establishing a consistent and complete mathematical description of mechatronics system model by using arrays to identify the properties of the whole system.The interface between the submodels is kept as simple as poible by employing simple mapping functions.4.SIMULATION
Considering that the whole system is at rest and the boring spindle is at the rear position and the user has just preed start button.The combination of input signal from the breakers and from the interface with the physical system will cause the control system to attain a new state and consequently a new set of output logical variables will be generated.This combination of output signals will cause the boring spindle to start moving forward in a rapid phase motion(uncontrolled motion).At the same time the spindle motor will be switched on and start rotating.However, since the spindle motor has not yet reached the feeding position, this rotation speed will remain unaffected by the servo motor control algorithm.Simulation for the angular velocity of the spindle motor is shown in Figure 8.It is shown from Figure 8 that the spindle motor will attain a constant rotation speed of 3173 rpm.after a transient period of about 5 seconds.The spindle motor was simulated auming zero load torque on the spindle that is because the boring spindle has not yet reached feeding position.The objective of the control system will be to keep spindle motor within 3000 r.p.m.under all loading conditions.Simulation of the linear speed and the differential preure of the hydraulic actuator is shown in Figure 9.I t shows that the rapid phase velocity of the actuator is about 6cm/sec.The system will continue to operate within the boundaries shown in Figure 8 and Figure 9 until i t receives a new set of input sources.That set will be initiated when the boring spindle reaches position M.Due to the signals generated from the interface with physical system, which is no longer at rest, combined with a new set of signals from the micro breakers.The control system will attain a new state and generate another set of output signals to be interpreted by the mapping function and converted into new input physical signals.In this case, the boring spindle will go from rapid phase motion(6cmkec)to a controlled feed forward motion in such way that the feed forward motion will be kept at 2cm/sec, and the rotating speed of the spindle motor should be reduced from 3173 r.p.m.to be within 3000 r.p.m.under all loading conditions.The actuator linear velocity will be controlled by the servo valve controller algorithm.And the boring spindle motor will be controlled by the servo motor controller algorithm.Auming that the servomotor is subjected to cosine load torque given by(q = 2 x cost)and the hydraulic cylinder is subjected to load force given by(F ~
= 0.0s x c o s t).Simulation results are shown in Figure 10 and Figure 1 1.The simulation shows that the output speed of both the spindle motor and the actuator cylinder are kept within the boundaries specified by control algorithm.5.CONCLUSIONS
A systems approach that utilizes multidimensional arrays for modelling mechatronics systems has been proposed and presented in this paper.The array approach provided us with a powerful mathematical representation of the real system.By utilizing multidimensional arrays we set up two submodels embodying the physical and the logical properties of mechatronics system.The interface between these two submodels is kept as simple as poible by employing a simple mapping functions.6.REFERENCES
Antsaklis, et.al.1993, Hybrid system modelling and autonomous control systems.Hybrid systems workshop, Technical university of Denmark.Bjrarke, 0., 1989, Manufacturing systems theory A geometric approach to connection.Tapir-Trondheim
Franksen, Ole I., 1992, The geometry of logic, from truth tables to nested arrays.The 4th international symposium on systems analysis and simulations.Berlin
Harashima, et.al.1996, Mechatronics-What is it, why, and how?.IEEWASME Transactions on Mechatronics, Vol.1, No.Huein, B.A., 1997, On modelling mechatronics systems.NTNU report, Trondheim, Norway Isermann, R., 1996, Modelling and design methodology for mechatronics system.IEEE/ASME Transactions on Mechatronics, Vol.1, No.Mprller, G.L, 1995, On the technology of array based logic.Ph.D.diertation Electric power engineering department, Technical University of Denmark
建立机电一体化模型的系统方法
Baam A.Huein 生产和质量 工程部，挪威科技大学，师大，挪威特隆赫姆
1997年11月3日
摘要
本文介绍了一个统一的基于 利用多维阵列以模拟 物理和逻辑 机电系统的性能。机电系统模型由两个相互作用的子模型描述与物理系统的能量流方面，和另一个子模型，介绍了有关控制系统中的信息流方面。多维数组的建模方法，为我们提供的可能性，使用的术语和相同的两个子系统建模形式主义。使用相同的形式主义的后果是，机电系统的仿真，可以只使用一个仿真环境
关键词 ： 机电一体化，系统建模
1.引言
机电一体化系统就是在工业产品设计和制造的过程中综合了电子和计算机控制的机械产品设计的协同作用的协同整合[5]。机电一体化系统在最初级阶段就必须考虑到各个学科对于本系统的约束。因此，适当的系统设计将在很大程度上依赖于整个设计和原型阶段使用的建模与仿真。集成了机电一体化的系统通过硬件的组合来影响物理系统还有信息处理系统来影响智能控制系统来执行它的功能[7]。
机电系统是基于计算机控制系统的应用物理系统的结果。控制系统设计实时执行的命令，以选择、增强和监督物理系统的正常运行。唯一可能验证这些控制功能可以将这整个系统的动作在我们建立边界之前就能明显的区分开来的途径，就是建立一个同时包括硬件和软件部分的包括所有约束条件的全真的系统模型。这表明了一个真正的系统模型必须强大到足以捕获所有的机电一体化系统的性能。这包括动态、静态的、离散事件，以及逻辑的上的与真正的系统的成本以及相关的属性和任务，不忽视其中任何零散的建模方法。在本文中，我们提出了一个统一的机电一体化系统的建模方法。这种统一的方法是利用几何对象或多维数组来制定机电一体化系统的型号。多维数组的建模方法，为我们提供了可能使用相同的形式主义，种类繁多的系统[2,3,4,9]。使用相同的形式主义的后果是，机电一体化系统的仿真可以只使用一个仿真环境中进行。
2． 模型结构
直观地说，一个用来描述给定系统的动态行为的模型并不能用来研究一个完全相同的系统的静态行为。因此，为了捕获所有相关信息，我们需要用到多种型号的模型，其中每一个模型封装真实系统的某些方面。我们会考虑的机电一体化系统模型作为一套综合在一起的子模型集，每个子模型对应真实系统的一些方面。所以从这个角度讲，长期的关联被用来强调这些子模型的变量之间的依赖关系。整个建模过程中，我们应当区分以下概念，见图1。
分解：为了处理复杂的机电一体化系统，他们应该被分解成子系统。这样进行了一个多层次的分解，直到我们达到构成整个系统的基本要素。原始系统模型应该是对一个断开状态下的系统的描述。它表示在这些元素之间互不影响时一个独立的元素的变量之间的关系。通过这个模型，我们 隔离每个元素特定的行为，比如静态，动态等。对局部变量再义一个给定的元素的行为。一个联合的系统模型在描述了相同的系统的同时还要考虑到系统内部的制约。局部变量的间接相关或直接相关以及连接系统变量间的连接的方式给系统以内部约束。相互关联的系统模型就类似于真正的系统的实际结构。应用的来源是由系统和环境之间的相互作用产生的。他们可以被看作是系统或 形式存储在系统要素的能源，甚至是在固有的限制下施加外部约束。
3.应用实例
图2所示的是制造系统。
该系统包含一个由直流电动机驱动的镗杆。镗主轴向前运动的进给运动采用液压线性驱动的方法。液压执行机构是一个恒定的压力液压泵，由伺服阀控制液压回路中的体积流量[8]。上述制造系统具有以下规格：镗轴的位置由三个微型断路器来感应，断路器（B）表明镗主轴是在后方的位置。在后方位置的快速阶段（I）将切换阀，以便快速向前运动（F），信号（S）将切换主轴电机。断路器（M）用来表明镗主轴是否已达进给位置。在这个位置上的快速阶段阀将关闭以开始控制进给运动。这项动作是由伺服阀（ST）监管。断路器（€）用来表明镗主轴已达到其最后的位置，在这个位置，反向运动（R）将开始，同时阶段阀（I）将被迅速的打开，以便快速向后运动。同时还规定主轴电机转速应保持3000r/min，同时在所有负载条件下必须保持进给速度2cm/sec。我们的目标是建立一个完整的模型和系统，使多维数组在实验模型验证中均能达到我们的要求。
3.1物理系统建模
物理系统建模时，我们需要关注的是融合在这个系统的物理变量的演化建模。物理系统的分解如图3所示。
基本物理元素组分为三类：广义电阻：如普通电阻，机械阻尼器，液压电阻。广义电容：如电力电容器，机械弹簧，液压水库。广义电感：如电动机械电感，质量，和液压电感。基本要素分解成子系统，物理系统内各子系统的物理量的演变进一步的分解，以让每个子系统的功能得以实现。这种可以随时感知的优势，将在每个控制系统的设计阶段成为现实。建立模型可以被视为分解系统的相反过程。所不同的是，在分解的过程中我们是将系统划分成独立的物理实体的子系统，而在建模过程中我们是重新对这些物理实体组合建成模型。因此，建立模型可以看作为将这些子系统重新组合的过程。
在建模时我们刚开始在这些所有层次的最底层，在每个级别上，我们从原始系统模型进而到全部关联起来的系统模型。在随后的层面，原始的系统模型，然后建立由前一级连接的系统模型如图4所示。
在各子系统的最底层，原始系统模型将利用方程或每个元素的基本规则来建立。这些基本规则如牛顿定律或欧姆定律，用来说明该元素的原本动作。直接和间接的关联就类似于各子系统的转变，从原始的系统模型定义关联之后的系统模型的边界内的内部制约。如直流伺服电机的线性连接的系 统，一个连接对象，速度对象（V）内部制约。速度对象是一个二维数组，该数组中的行对应在原始系统中的变量（局部变量），列对应的变量在连接系统（全局变量）。因此，速度的目的是在连接的系统模型，以实现原始的系统模型中的局部变量、全局变量的转变。
通过聚合对角线的连接的液压子系统和镗主轴系统模型的物理系统的模型建立，模型的物理系统中[7]所有不同的代数方程组。在状态矢量空间，物理系统的运动方程是：Y =（，X，U，〜），其中（x）是初始状态变量，（U）是输入源，（A）是为控制具体的物理功能的状态是真或假的（1 0）转移矩阵。
3.2在控制算法之前控制系统建模
在控制算法之前控制系统建模，可以设计和实施我们需要描述的属性或行为。一个准确和全面的数学模型控制系统的性能可以用逻辑符号表示。这个数学模型，为我们提供了揭示在控制系统中的互相冲突的方法，同时可以验证该控制系统是否满足我们的设计规范，这样就可以开展问题说明我们列出的所有控制功能。控制系统应分解成三个子系统，一个过程控制器子系统，这将是负责发出启动和停止不同的物理实体和两个连续的控制器命令。一个是控制器伺服阀的液压子系统，以约束进给液压执行机构的运动。另一个是为伺服电机，主轴电机调节角速度的控制器。如图5所示的分解。
每个子系统的功能是由一组逻辑来约束。这些元素可以携带一个假定的真理或谬误（1 0）的逻辑变量。这些元素代表原始的具有普通的控制功能的系统模型。而模拟控制系统的程序也将沿着层次结构向上，直到总的模型，得到如图6所示。
在原始系统模型，逻辑变量之间的关联是指由三个对象的连接。在经典逻辑，他们被称为基本逻辑连接词，包括基本的逻辑连接词，如结合（AND），分离（OR），否定（NOT）。我们传送到所连接的系统模型，通过聚集在原始系统中使用上述逻辑连接词的逻辑变量。
连接子系统的不是别的，而是在一个多维数组的形式上表达逻辑论证的真值表。该数组中的轴数应该等于变量的数目，因此，所有重复的轴必须绑扎方法融合在一起。连接系统在连接各个元素的结构后对系统的所有可能的状态应该实行内部制约。控制系统的行为，可以在以下形式表示小号= F（P，I，N）。其中，（0）是输入变量的集合，是由于外部约束与环境的相互作用（P）是多维数组，表示控制系统的状态转移矩阵（S）是一组输出变量。该指数（n）是时间指数，它指定一个同样条件情况下给定的状态。
3.3总系统模型
由于这两个系统内部使用不同类型的信号，直观地说，物理和控制系统模型之间唯一可能的接口采取的是通过外部环境来联通。在上面的制造系统，我们可以将物理和控制系统区分为两种接口方式：离散界面：在这个过程中的控制器中的控制系统的目的是协调异步任务，以满足系统的要求。例如，当由程序控制器发出一个事件命令“启动主轴电机”时，主轴电机开始转动。在旋转过程本身是由下级控制器（连续控制器）控制。连续界面：处于较低的水平控制级别的地方，目的是为了保持物理系统的运动，如使其执行速度处于控制界限内。在这种情况下产生的系统模型被认为是一个混合动力系统模型。混合动力系统的识别特征是它们包含两个连续的动态行为，即，微分代数方程（Y = F（A，X，U，R））和离散事件动态行为受约束的物理量的演变逻辑方程：（S = F（P，I，N））。一个总的模型可以由一个简单的接口连接的物理系统模型和控制系统模型来组成。该接口将包括两个简单的记忆映射功能（a）和（p）[1]。第一个（a）转换成一个常数的增量输入，物理系统的控制器输出如下：（S）：U（I）=（SN）第二个（P）转换成物理系统输出的输入，逻辑变量的一套控制系统如下：I= P（Y（R）），如图7所示。
通过使用数组来确定整个系统的性能，这是我们迄今取得是建立一致和完整的机电一体化系统模型的数学描述。子模型之间的接口应采用简单的映射功能，并且保持尽可能简单。
4.模拟
我们可以看到整个控制系统是在主轴后方的位置，用户只需按下启动按钮。断路器和物理系统的接口输入信号的组合将使控制系统达到一个新的状态，因此将会产生一套新的输出逻辑变量。这种输出信号的组合将导致镗轴开始运动（不受控制的约束）在快速阶段向前迈进。同时将开启主轴电机开始旋转。然而，由于主轴电机尚未达到进给位置，将保持这个由伺服电机控制算法的影响的转速。主轴电机角速度仿真如图8所示。
从图8所示，主轴电机将达到3173转的恒定转速后约5秒钟的短暂。主轴电机是模拟假设零负载转矩主轴，是因为主轴尚未达到进给的位置上，将所有负载条件下保持在3000转主轴电机控制系统的目的。如图9所示的线性速度和液压执行器压差模拟。我T显示，快速的执行机构相速度是6cm/sec。
该系统将在图8和图9所示的边界范围内操作，直到它接收到新的输入源。该集将启动时，镗主轴到达M处。控制系统将达到一个新的状态，并产生另一种映射功能并转换成新的输入输出信号集。在这种情况下，镗主轴将进入快速运动阶段（6cmkec），进给运动将保持在2cm/sec的速度，主轴电机这在这样负载条件下转速应从3173转减少到3000转以内。执行器的线性速度将由伺服阀控制算法来控制。镗主轴的电机由伺服电机控制算法来控制。现在 假设收到余弦负载转矩伺服电机（q = 2 x cost）和液压缸受到加载（F=0.0sxcost）共同作用的负载，其 模拟结果如图10和图11。
仿真结果表明，主轴电机和油缸的输出速度控制算法由保持在指定的边界内。
5.结论 现在已经提出了一个系统的方法，利用建模，机电系统的多维数组和本文提出的 阵列的方法提供了强大的实时系统的数学表示。利用多维数组，我们设立了两个子模型，体现了机电一体化系统的物理和逻辑特性。这两个子模型之间的接口保持尽可能简单，并采用一个简单的映射功能。
6．参考文献
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Franksen, Ole I., 1992年，几何逻辑，从真值表嵌套数组，第四届国际研讨会上系统的分析和模拟，柏林
Harashima, et.al.1996年，机电一体化是什么，为什么，怎么样？ 机电一体化，第一卷 Huein, B.A., 1997年，机电一体化系统的建模，师大的报告，挪威特隆赫姆
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