Fuzzy Sliding Mode Control and Observation

Hydraulic LibraryRev 9 - November 2009Copyright © LMS IMAGINE S.A. 1995-2009AMESim® is the registered trademark of LMS IMAGINE S.A.AMESet® is the registered trademark of LMS IMAGINE S.A.AMERun® is the registered trademark of LMS IMAGINE S.A.AMECustom® is the registered trademark of LMS IMAGINE S.A.LMS b is a registered trademark of LMS International N.V.LMS b Motion is a registered trademark of LMS International N.V.ADAMS® is a registered United States trademark of MSC.Software Corporation. MATLAB and SIMULINK are registered trademarks of the Math Works, Inc. Modelica is a registered trademark of the Modelica Association.UNIX is a registered trademark in the United States and other countries exclusively licensed by X / Open Company Ltd.Python is a registered trademark of the Python Software Foundation.Windows and Visual C++ are registered trademarks of the Microsoft Corporation. The GNU Compiler Collection (GCC) is a product of the Free Software Foundation. See the GNU General Public License terms and conditions for copying, distribution and modification in the license file.All other product names are trademarks or registered trademarks of their respective companies.Hydraulic Library Rev 9 Table of contentsChapter 1:Tutorial examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2Example 1: A simple hydraulic system . . . . . . . . . . . . . . . . . . . . . . 2Cavitation and air release. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3Example 2: Using more complex hydraulic properties . . . . . . . . . 11Using one of the special fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4Example 3: Using more complex line submodels . . . . . . . . . . . . . 171.5Example 4: Valves with duty cycles. . . . . . . . . . . . . . . . . . . . . . . . 221.6Example 5: Position control for a hydraulic actuator. . . . . . . . . . . 271.7Example 6: Simple design exercise for a hydraulic suspension . . 33Chapter 2:Theory of fluid properties. . . . . . . . . . . . . . . . . . . . . .412.1Density and compressibility coefficient . . . . . . . . . . . . . . . . . . . . . 41Entrapped air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Dissolved air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.2Air release and cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Viscosity influence on the flow. . . . . . . . . . . . . . . . . . . . . . . . . . 48Flow through orifices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Frictional drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Chapter 3:AMESim Fluid Properties . . . . . . . . . . . . . . . . . . . . . .553.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55FP04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2Tutorial example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Chapter 4:Hydraulic Line modeling. . . . . . . . . . . . . . . . . . . . . . .614.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Zero-dimensional line submodels. . . . . . . . . . . . . . . . . . . . . . . . 61“Lumped” and “Lumped distributive” line submodels. . . . . . . . . 62Lax-Wendroff “CFD 1D3” line models . . . . . . . . . . . . . . . . . . . . 63Choosing between Lumped/Distributive and CFD 1D Lax-Wendroffmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2Line submodel selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641Table of contents24.3Three important quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65Aspect ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65Dissipation number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66Communication interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 4.4The selection process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68Hydraulic Library Rev 9 Chapter 1: Tutorial examples1.1IntroductionThe AMESim Hydraulic library consists of:• A collection of commonly used hydraulic components such as pumps, motors, orifices, etc. including special valves.•Submodels of pipes and hoses.•Sources of pressure and flow rate.•Sensors of pressure and flow rate.• A collection of fluid properties.Hydraulic systems in isolation serve no purpose. It is necessary to do somethingwith the fluid and also to control the process. This means that the library must becompatible with other AMESim libraries. The following libraries are frequently usedwith the Hydraulic library:Mechanical libraryUsed in fluid power application when hydraulic power is translated into mechanicalpower.Signal, Control and Observer libraryUsed to control the hydraulic system.Hydraulic component design libraryUsed to build specialist components from very basic hydraulic and mechanicalelements.Hydraulic resistance libraryThis is a collection of submodels of bends, tee-junctions, elbows etc. It is usedtypically in low pressure applications such as cooling and lubrication systems.1Chapter 1Tutorial examples2Chapter 1 of the manual consists of a collection of tutorial examples. We strongly recommend that you do these tutorial examples. They assume you have a basic level of experience using AMESim . As an absolute minimum you should have done the examples in Chapter 3 of the AMESim manual and the first example of Chapter 5 which describes how to do a batch run.1.2Example 1: A simple hydraulic systemObjectives•Construct a very simple hydraulic system •Introduce the simplest pipe/hose submodels •Interpret the results with a special reference to air release and cavitationFigure 1.1: A very simple hydraulic system In this exercise you will construct the system shown in Figure 1.1. This is perhaps the simplest possible meaningful hydraulic system. It is built partly fromcomponents from the Hydraulic category (which are normally blue) and partly from the Mechanical category.The hydraulic section is built up from standard symbols used for hydraulic systems. The prime mover supplies power to the pump, which draws hydraulic fluid from a tank. This fluid is supplied under pressure to a hydraulic motor, which drives a rotaryNote:•You can use more than one fluid in the Hydraulic library. This is important because you can model combined cooling and lubrication systems of a library.•The hydraulic library assumes a uniform temperature throughout the system. If thermal effects are considered to be important, you should use the Thermal Hydraulic and Thermal Hydraulic Component Design libraries.•There are models of cavitation and air release in the hydrauliclibrary. Note also there is a special two-phase flow library. Atypical application for this is air conditioning systems.Hydraulic Library Rev 9load. A relief valve opens when the pressure reaches a certain value. The output from the motor and the relief valve returns to the tank. The diagram shows three tanks but it is quite likely that a single tank is employed.The first category contains general hydraulic components. The second contains special valves.The hydraulic components used in the model you will build can all be found in the first of these Hydraulic categories. If you click on this category icon, the dialog box shown in Figure 1.2 opens.First look at the components available in this library. Display the title of components by moving the pointer over the icons:Figure 1.2: The components in the first hydraulic category.Close the Library before continuing.3Chapter 1Tutorial examples4Step 1:Use File X New... to produce the following dialog box.Figure 1.3: The hydraulic starter system.Step 2:Construct the rest of the system and assignsubmodels1.Construct the systemwith the components as shown in Figure 1.1.2.Save it ashydraulic1.3.Go to Submodel mode.Notice that the drop, prime mover, node and pipes do not appear the same as they usually do. This is because they do not have sub models associated with them.The easiest way to proceed is as follows:4.Click on the Premier submodel button in the menu bar.Select the hydraulic starter circuit libhydr.amt and then click onOK. A new system with a fluid properties icon in the top leftcorner of the sketch is created.You could also have clicked on the New icon in the tool bar butif you do this you will have to add the fluid properties iconyourself.Hydraulic Library Rev 95Figure 1.4: The line submodels.You get something like Figure 1.4. It is possible that your system may have HL000associated with one of the other line runs. These minor variations are dependent on the order in which you constructed the lines. They will not influence the simulation results.An important feature to note is that a line run has a special submodel (HL000) which is not a direct connection. To emphasize this point the line run has a special appearance:Remember the submodel DIRECT does nothing at all. It is as if the ports at each end of the line were connected directly together.In contrast, HL000 computes the net flow into the pipe and uses this to determine the time derivative of pressure. If the net flow into the pipe is positive, pressure increases with time. If it is negative, it decreases with time. The pressure created by HL000 is conveyed to the relief valve inlet. The motor inlet is conveyed by the node and submodel DIRECT .5.Click the mouse right-button.6.Select Show line labels in the label menu.Chapter 1Tutorial examples6Step 3:Set parameters1.Change to Parameter mode.2.Set the following parameters and leave the others at their default values:Figure 1.5: Setting the line submodel HL000 parameters.3.To display the parameters of a line submodel click the left mouse button with thepointer on the appropriate line run.Part of the dialog box for HL000 is shown in Figure 1.5. The compressibility of the oil and the expansion of the pipe or hose with pressure are taken into ac-count together with the pipe volume. HL000 normally requires the bulk modulus of the hydraulic fluid and pipe wall thickness together with the Young’s modulus of the wall material. From these values an effective bulk modulus of the com-bined fluid and pipe walls can be calculated. The effective bulk modulus of a hose is normally very much less than that of a rigid steel pipe.4.Click on the fluid icon FP04 in the sketch.A new dialog box as shown in Figure 1.6 is displayed. This shows you the prop-erties of the hydraulic fluid. Currently they are at their default values and the ab-solute viscosity, bulk modulus, air/gas content and temperature are given in common units.Submodel Title ValueHL000pipe length [m]4RL00coefficient of viscous friction [Nm/(rev/min)]0.02Figure 1.6: Parameter for fluid properties submodel FP04.5.Click on Close .Step 4:Run a simulation1.Go toSimulation mode and do a simulation run.The default values in the Run Parametersdialog box are suitable for this exam-ple.2.Click on the Start a simulation button.3.Click on the pump component to produce the dialog box shown in Figure 1.7.Some variables such as a pressure have no direction associated with them. ANote that the first item in the list is an enumeration integerparameter. A collection of properties of varying complexityare available but for this exercise elementary is satisfacto-ry.Tutorial examples(gauge) pressure of -0.1 bar indicates that the pressure is below atmospheric.In contrast other variables, such as flow rate, do have a direction associatedwith them. A flow rate of -6 L/min indicates that the flow is in the opposite direc-tion to some agreed standard direction.Figure 1.7: The Variable List for PU001.Note that you can use the Replay facility to give you a global picture of the re-sults. Figure 1.8 also shows the flow rates in L/min at a time of 10 seconds.Figure 1.8: Flow rates displayed in replay.4.To plot a variable associated with a line submodel, click on the correspondingline run.5.Plot pressure at port 1 for HL000.Figure 1.9: The pressure in the hydraulic pipe.Notice how the pressure goes up to just over the relief valve setting (150 bar).During this time the load speeds up rapidly and actually 'over-speeds'. At thispoint the motor is demanding more hydraulic flow than the pump can supply.The result is that the pressure must drop and the relief valve closes. The pres-sure continues to drop and falls below zero bar gauge. However, pressure is notlike voltage or force. We cannot have a pressure of -100 bar. The absolute zeroof pressure is about -1.013 bar gauge. It is time to introduce two terms. Cavitation and air releaseWhen pressure falls to very low levels, two things can happen:•Air previously dissolved in the fluid begins to form air bubbles.•The pressure reaches the saturated vapor pressure of the liquid andbubbles of vapor appear.These phenomena are known respectively as air release and cavitation. They cancause serious damage. Using the Zoom facility, the graph gives a better view of thelower pressure values:Figure 1.10: Low pressure in the hydraulic pipeAll AMESim submodels have hydraulic pressure in bar gauge. The low pressureshown in Figure 1.10 : Low pressure in the hydraulic pipe is caused by the loadTutorial examplesspeed exceeding its steady state or equilibrium value. This is highly undesirablebehavior as it can result in damage to the real system.In reality the starting values we have given for the pipe pressure and load speed arenot very realistic and the prime mover would start from rest or a valve would be usedto regulate the flow to the motor. However, hydrostatic transmission systems likethis often do suffer badly from cavitation and air release problems.Note that all AMESim submodels display hydraulic volumetric flow rate in L/min.There are two possible interpretations of this flow rate:•The flow rate is measured at the local current hydraulic pressure, or•The flow rate is measured at a reference pressure.AMESim adopts the second alternative with a reference pressure of 0 bar gauge.This means that the volumetric flow rate is always directly proportional to the massflow rate. In most situations the difference between the two flow rates is negligible.However, there are three situations when there is a significant difference:1.There is a very large air content; the pressure drops below the satu-ration pressure for air in the liquid and air bubbles are formed in theliquid.2.The pressure drops to the level of the saturated vapor pressure of theliquid and cavities of vapor form.3.Extremely high variations in pressure occur such as in certain typesof fuel injection systems.The first situation is called air release and the second cavitation. If there is cavitationor significant air release at the inlet to a pump, the flow rate according to the firstdefinition will not be reduced. However, with the AMESim approach (measuringflow rate at a reference pressure) it is significantly reduced.The properties of hydraulic fluids vary a great deal. Modeling them is a veryspecialist process and the model can be extremely simple or highly complex. Therun times are greatly influenced by this level of complexity.1.3Example 2: Using more complex hydraulicpropertiesObjectives:•Use more complex models of fluid properties.•See how air content changes the performance of the system.In the Hydraulic category two special components can modify the fluid properties:This is an example of a component without ports. We cannot connect this icon to any other.There are two important points to note aboutFP04.1.It has an integer parameter index of hydraulic fluid that is in the range 0 to 100inclusive. This arrangement means that it is possible to have more than one fluidin an AMESim system.•simplest This has a constant absolute viscosity. The bulk modulus isconstant above the gas saturation pressure and is 1/1000 of this valuebelow the gas saturation pressure. This model is very old but is still usedby some AMESim users.It is likely to give the fastest runs.•elementary This is the default and features a constant liquid bulk mod-ulus with absolute viscosity. The treatment of fluid properties under airrelease and cavitation is done.•advanced This gives you access to some cavitation parameters not ac-General Hydraulic PropertiesIn AMESim always use this fluid properties icon. It is associated withone submodel: FP04. This is a collection of simple and complex fluidproperties.Drop Hydraulic PropertiesThis is a special model, only used to ensure backward compatibilitybetween 4.0 models and earlier. Do not use this model.2.The characteristics of the fluid properties are de-termined by its parameters. These are set in thetype of fluid properties list. There are 7 possi-bilities:Tutorial examplescessible in the elementary properties.•advanced using tables This is like the advanced option but you install tables of data to give variation of bulk modulus and absolute viscositywith pressure and temperature.•Robert Bosch adiabatic diesel These properties are provided by Rob-ert Bosch GmbH and comprise a number of common types of diesel fuel.•elementary with calculated viscosity•advanced with calculated viscosityUsing one of the special fluidsStep 1:Use the Advanced fluid properties.1.Return to the first example of this manual, add another fluid properties icon.e Premier submodel and go to Parameter mode. Your sketch should look likethis.Figure 1.11: The sketch with two instances of FP04.3.Look at parameters of FP04-2. Change the enumeration integer parameter toadvanced. The Change Parameters list should now look like this:Figure 1.12: The advanced fluid properties.Change the index of hydraulic fluid in FP04-2 to 1. This is a number in the range0 to 100. If you look at the other hydraulic components in the system you willfind they have index 0 and hence they will still use the fluid properties of FP04-1. We could go into every hydraulic component using this second fluid and setthe parameter index of hydraulic fluid to 1. This would be extremely tedious witha big system and there is always the possibility of missing one.Instead we can set all fluid indices to the same value of 1.Step 2:Set all fluid indices to the same value of 1The best way to do this is to use the common parameters facility.e Edit X Select all.All the system components will be selected, unselect FP04-1 by holding the SHIFT key and clicking on the component.Figure 1.13 Select componentse Settings X Common parameters.Tutorial examplesFigure 1.14 shows the Common parameters dialog box. This is a list of commonparameters for selected objects. They occur at least twice. Since there are 3 hy-draulic tanks and they all have pressures of 0 bar, this value is displayed. Thereare a number of submodels that have a parameter index of hydraulic fluid. InFP02 the index of hydraulic fluid is set to 1 whereas in other submodels its valueis 0. The value is displayed as ???. Similarly the prime mover and rotary loadboth have a parameter (strictly speaking a variable) with title shaft speed. Sincethe two values are different, ??? is displayed.Figure 1.14: Different values for common parameters3.Set the parameter index of hydraulic fluid to 1. This will change all the parame-ters in the system except FP01 (remember we used Select all and deselectedFP01).Step 3:Run a simulation and plot some variablesYou will probably find the results very much the same as in example 1.Step 4:Organize a batch run to vary the air content1.In Parameter mode use Settings X Batch parameters.2.Drag and drop the air gas content from FP04-2 to the Batch control parametersetup dialog box.3.Set up the batch parameters as in Figure 1.15 so that the air content goes from0% to 10% in steps of 2%.Figure 1.15: Setting up a batch run varying air content4.Specify a batch run in the Run parameters dialog box and initiate the run.5.Plot several graphs of the batch run to compare results with various air contents:Tools X Batch Plot .Figure 1.16: Pressure in pipe.By zooming in on the curve in regions where the pressure is below 0 bar, you will probably find some variation in the results, but not to a significant degree.6.Change the saturation pressure in FP04-2 to 400 bar.7.Repeat the batch run and update your plot.Tutorial examplesFigure 1.17: Pressure in pipe with saturation pressure 400 bar.The variation between the runs is now very pronounced. The dynamiccharacteristics of the system are completely transformed. A few words ofexplanation are necessary.Normally the air content of a hydraulic oil is well below 1%. A typical value is 0.1%.It is normally considered good practice to keep the value as low as reasonablypossible. However, in a few applications, such as lubrication oil in gearboxes, theoil and air are well mixed. In this case, 2.5% is a typical value, and up to about 10%is possible.A reasonable quantity of air, given time, will completely or partially dissolve in thehydraulic fluid. The lowest pressure at which all the air is dissolved is called thesaturation pressure. For very slow systems all the air is dissolved above thesaturation pressure and partially dissolved below this pressure. Henry’s law givesa reasonable approximation for the fraction of air that is dissolved in equilibrium.Some systems are slow enough to stay very close to this equilibrium position(Figure 1.16). Often classic fluid power systems behave like this. The originalsaturation pressure is better for the current example.However, it does take time for the air to dissolve and this time will not be availablein fast acting systems. Fuel injection systems are a good example of this. Hencewith such systems it may be appropriate to set the saturation pressure artificiallyhigh to allow for significant quantities of air to be undissolved at all pressures.1.4Example 3: Using more complex line sub-modelsObjectives:•Use more complex line submodels.•Understand the need for a variety of line submodels.•To understand the importance of setting an appropriate line submodel.The system for this example is the same as for example 2 (Figure 1.11). We will describe the modification of the system to use more complex line submodels and the experiments performed. Finally we present a little of the theory behind the submodels.Step 1:Change submodelsAll the submodels in the current system were selected automatically. We willchange some of them manually.1.Go to Submodel mode.You will now change some line submodels.Before continuing note the following points:•The corners in the pipe runs are not physical but diagrammatic.•There are three hydraulic pipes and they meet at a point which physically will be a tee-junction.•It is necessary to have a large number of hydraulic pipe submodels.•In the present system three submodels are set: DIRECT , DIRECT andHL000.Figure 1.18: The current line submodels.None of these line submodels takes friction into account. We will suppose that the relief valve is close to the node but the pump and the motor are at such distances•This tee-junction in the sketch is described as a 3-portnode and it has the submodel H3NODE1. This modelsthe junction has a common pressure with flow rates thatgive conservation of mass.Tutorial examplesfrom the node that the pressure drop along the pipes cannot be ignored. We need to select new pipe submodels that take friction into account for the pipe runs:•from the pump to the node •from the node to the motor.2.Click on the line run attached to the pump and select HL03 in the Submodel list .Figure 1.19: The hydraulic line submodels available.Why did we not choose a more complex submodel that also included inertia? We answer this question later in this exercise.3.For the line from the node to the motor, select the submodel HL01.4.For the line between the node and the relief valve, the submodel DIRECT is al-ready selected and this is exactly what we want.Step 2:Set parameters and run a simulation1.Go to Parameter mode and set parameters for HL01 and HL03 so that both pipelengths are 5 m and pipe diameters are 10 mm .This can be done one at a time. However, we can do it another way. Press the Shift key on click on the HL03 and HL01 line runs so that they are selected. Use Settings X Common parameters . Figure 1.20 shows the Commonparameters dialog box.Note the brief description of each line submodel. In these de-scriptions C stands for compressibility, R for resistance (pipefriction) and I for inertia (fluid momentum). HL000 which weused before takes into account compressibility only. HL03takes into account compressibility and friction. It is modeled liketwo hydraulic compressible volumes with a resistance betweenthem.Figure 1.20: The common parameters of the two line submodels.Note that ??? indicates that different values are set in the line submodels. Set the index of hydraulic fluid to 1, diameter of pipe to 10 and pipe length to 5.2.In FP04-2 reset the saturation pressure (for dissolved air/gas) to 0 bar.3.Run a simulation with the default run parameters. Do not forget to reset RunType to Single run if you have previously run a Batch.4.Plot the two pressures in HL03.Figure 1.21: Pressures at the ends of pipe joining pump to node.Note that there is a large pressure drop along the line. This could be regarded as a sizing problem but in addition it would be bad practice to site the relief valve so far from the high pressure point.Tutorial examplesStep 3:We now investigate other line submodels.1.Return to Sketch mode and Copy-Paste part of the system as shown:Figure 1.22: Part of the system is duplicated.2.In Submodel mode change the lower two line submodels as follows:Figure 1.23: New line submodels.This system will enable you to make direct comparisons between results.3.Go to Run mode and do a simulation. Plot the pressure at the pump outlet (pres-sure at port 2).Figure 1.24: Pressure at pump outlet.We note that the curves are virtually the same. (Try zooming.) There is abso-lutely no advantage to using HL07 and HL09 instead of HL01 and HL03. If we separated the two systems and ran them independently we would find run times for the more complex submodels were higher.4.C hange the communication interval in the Run Parameters dialog box to 0.001sand rerun the simulation.If you have a look at the Warnings/Errors tab of the Simulation run dialog box, you will find that some checks are performed by the line submodels (see Figure1.25). A similar message is issued for HL03.Figure 1.25: Messages under the Warning tab.It is suggested that:•HL01 should be replaced by HL07 and•HL03 should be replaced by HL09.In other words with this communication interval the lower subsystem is better than the upper. If you replot the pressures at the pump outlets, there are clearly differences. This is what happens if you zoom.Figure 1.26: Zoomed pressures at pump outlet.The violent (and unrealistic) start up has created this oscillation in the pressure of about 56 Hz. It is damped out by 0.1 seconds. Why did we get no warning message in the previous run? The answer is that a lot of checks are applied to your submodel choices when the run starts. These take into account the fluid properties, the pipe dimensions and the communication interval.。

摘要汽车电子液压制动系统特性测试与非线性控制研究环境污染、能源危机等问题迫使传统汽车向新能源汽车转型，而新能源汽车对制动系统的结构、功能和性能均提出了更高的要求：（1）新能源汽车取消了传统发动机的使用或者发动机排量较小，无法产生足够的真空助力，需要使用新的设备以产生足够的真空度或寻找新的助力方式；（2）新能源汽车需要具有制动能量回收功能；（3）新能源汽车采用多种复合制动模式，需要解决制动主缸和轮缸的解耦问题。

电子液压制动系统适应了新能源汽车的发展要求，它采用电力作为制动压力的能量来源，解决了助力问题，同时也实现了制动主缸与轮缸的解耦，并且系统更易于集成能量回收装置。

但是，电子液压制动系统是机、电、液混合系统，存在较强的非线性特性，为了精确控制动系统轮缸压力，需要充分考虑系统的非线性特性。

因此，本文首先对电子液压制动系统特性进行试验测试，并建立电子液压制动系统精确的数学模型，然后设计系统非线性控制策略。

本文的主要研究内容包括：（1）电子液压制动系统特性试验测试。

首先搭建电子液压制动系统硬件在环试验台，该试验台包括电子液压制动系统、dSPACE实时平台和液压测试模块三个部分。

然后利用搭建的硬件在环试验台对电子液压制动系统特性进行试验测试。

（2）电子液压制动系统一体化建模。

采用功率键合图建模方法建立系统模型，所建立的制动系统-车轮一体化模型包括压力源、增压阀、减压阀、制动管路、制动钳和车轮等元件，考虑了制动管路的液容、液阻和液感效应，电磁阀的阻尼效应，制动液的可压缩特性等系统特性。

（3）非线性控制策略设计及验证分析。

采用两种方法设计控制策略：滑模变结构控制和非线性反步法控制。

滑模变结构控制策略是从底盘系统一体化控制的目的出发，将制动系统与车轮作为一个整体设计系统控制算法，使用MATLAB/Simulink和AMESim联合仿真的方式验证其可行性。

非线性反步法控制策略是从实时性的目的出发，将制动系统和车轮分为两个部分，分别设计控制算法，最后采用软件仿真和硬件在环试验验证控制方法的有效性。

Automatica38(2002)2159–2167/locate/automaticaBrief PaperNon-singular terminal sliding mode control of rigid manipulatorsYong Feng a,Xinghuo Yu b;∗,Zhihong Man ca Department of Electrical Engineering,Harbin Institute of Technology,Harbin150006,People’s Republic of Chinab School of Electrical and Computer Engineering,Royal Melbourne Institute of Technology University,GPO Box2476V Melbourne,Vic.3001,Australiac School of Computer Engineering,Nanyang Technological University,SingaporeReceived26June2001;received in revised form16June2002;accepted9July2002AbstractThis paper presents a global non-singular terminal sliding mode controller for rigid manipulators.A new terminal sliding mode manifold isÿrst proposed for the second-order system to enable the elimination of the singularity problem associated with conventional terminal sliding mode control.The time taken to reach the equilibrium point from any initial state is guaranteed to beÿnite time.The proposed terminal sliding mode controller is then applied to the control of n-link rigid manipulators.Simulation results are presented to validate the analysis.?2002Elsevier Science Ltd.All rights reserved.Keywords:Terminal sliding mode control;Singularity;Robotic manipulator;Robust control;Lyapunov stability1.IntroductionVariable structure systems(VSS)are well known for their robustness to system parameter variations and external disturbances(Slotine&Li,1991;Utkin,1992; Yurl&James,1988).VSS have been widely used in many applications,such as robots,aircrafts,DC and AC motors, power systems,process control and so on.An aspect of VSS that is of particular interest is the sliding mode control,which is designed to drive and constrain the system states to lie within a neighborhood of the pre-scribed switching manifolds that exhibit desired dynam-ics.When in the sliding mode,the closed-loopresp onse becomes totally insensitive to both internal parameter un-certainties and external disturbances.A characteristic of conventional VSS is that the convergence of the system states to the equilibrium point is usually asymptotical due to the asymptotical convergence of the linear switching manifolds that are commonly chosen.Recently,a terminal sliding mode(TSM)controller was developed(Man&Yu,1997;Yu&Man,1996;Wu,Yu,& This paper was not presented at any IFAC meeting.This paper was recommended for publication in revised form by Associate Editor Jurek Z.Sasiadek under the direction of Editor Mituhiko Araki.∗Corresponding author.E-mail addresses:yfeng@(Y.Feng),x.yu@.au(X.Yu).Man,1998).TSM has been used in the control of rigid ma-nipulators(Man et al.,1994;Tang,1998).The TSM con-cept is related to theÿnite time control(Haimo,1986; Bhat&Bernstein,1997).Compared with linear hyperplane-based sliding modes,TSM o ers some superior properties such as fast,ÿnite time convergence.This controller is par-ticularly useful for high precision control as it speeds up the rate of convergence near an equilibrium point.However,the existing TSM controller design methods still have a singu-larity problem.An initial discussion to avoid the singularity in TSM control systems was presented(Wu et al.,1998). In this paper,a global non-singular terminal sliding mode (NTSM)controller is presented for a class of nonlinear dy-namical systems with parameter uncertainties and external disturbances.A new NTSM manifold is proposed to over-come the singularity problem.The time taken to reach the manifold from any initial state and the time taken to reach the equilibrium point in the sliding mode can be guaran-teed to beÿnite time.The proposed NTSM controller is then applied to the control of n-degree-of-freedom rigid ma-nipulators.Simulation results are presented to validate the analysis.2.Conventional terminal sliding mode controlThe basic principle of TSM control can be brie y sum-marized as follows:consider a second-order uncertain0005-1098/02/$-see front matter?2002Elsevier Science Ltd.All rights reserved. PII:S0005-1098(02)00147-42160Y.Feng et al./Automatica 38(2002)2159–2167nonlinear dynamical system ˙x 1=x 2;˙x 2=f (x )+g (x )+b (x )u;(1)where x =[x 1;x 2]T is the system state vector,f (x )and b (x )=0are smooth nonlinear functions of x ,and g (x )represents the uncertainties and disturbances satisfying g (x ) 6l g where l g ¿0,and u is the scalar control in-put.The conventional TSM is described by the following ÿrst-order terminal sliding variables =x 2+ÿx q=p1;(2)where ÿ0is a design constant,and p and q are positive odd integers,which satisfy the following condition:p ¿q:(3)The su cient condition for the existence of TSM is 12d d ts 2¡−Á|s |;(4)where Á¿0is a constant.For system (1),a commonly used control design isu =−b −1(x ) f (x )+ÿq px q=p −11x 2+(l g +Á)sgn(s );(5)which ensures that TSM occurs.It is clear that if s (0)=0,the system states will reach the sliding mode s =0within the ÿnite time t r ,which satisÿes t r 6|s (0)|Á:(6)When the sliding mode s =0is reached,the system dy-namics is determined by the following nonlinear di erential equation:x 2+ÿx q=p 1=˙x 1+ÿx q=p1=0;(7)where x 1=0is the terminal attractor of the system (7).The ÿnite time t s that is taken to travel from x 1(t r )=0to x 1(t s +t r )=0is given byt s =−ÿ−1x 1(t r )d x 1x q=p 1=p ÿ(p −q )|x 1(t r )|1−q=p :(8)This means that,in the TSM manifold (7),both the system states x 1and x 2converge to zero in ÿnite time.It can be seen in the TSM control (5)that the secondterm containing x q=p −11x 2may cause a singularity to occur if x 2=0when x 1=0.This situation does not occur inthe ideal sliding mode because when s =0;x 2=−ÿx q=p1hence as long as q ¡p ¡2q ,i.e.1¡p=q ¡2,the term x q=p −11x 2is equivalent to x (2q −p )=p 1which is non-singular.The singularity problem may occur in the reaching phase when there is insu cient control to ensure that x 2=0while x 1=0.The TSM controller (5)cannot guarantee a bounded controlsignal for the case of x 2=0when x 1=0before the system states reach the TSM s =0.Furthermore,the singularity may also occur even after the sliding mode s =0is reached since,due to computation errors and uncertain factors,the system states cannot be guaranteed to always remain in the sliding mode especially near the equilibrium point (x 1=0;x 2=0),and the case of x 2=0while x 1=0may occur from time to time.This underlines the importance of addressing the singularity problem in conventional TSM systems.3.Non-singular terminal sliding mode controlIn order to overcome the singularity problem in the con-ventional TSM systems,several methods have been pro-posed.For example,one approach is to switch the sliding mode between TSM and linear hyperplane based sliding mode (Man &Yu,1997).Another approach is to transfer the trajectory to a pre-speciÿed open region where TSM control is not singular (Wu et al.,1998).These methods are adopting indirect approaches to avoid the singularity.In this paper,a simple NTSM is proposed,which is able to avoid this problem completely.The proposed NTSM model is de-scribed as follows:s =x 1+1ÿx p=q 2;(9)where ÿ;p and q have been deÿned in (2).One can easilysee that when s =0,the NTSM (9)is equivalent to (2)so that the time taken to reach the equilibrium point x 1=0when in the sliding mode is the same as in (8).Note that in using (9)the derivative of s along the system dynamics does not result in terms with negative (fractional)powers.This can be seen in the following theorem about the NTSM control.Theorem 1.For system (1)with the NTSM (9),if the control is designed asu =−b −1(x ) f (x )+ÿq px 2−p=q2+(l g +Á)sgn(s );(10)where 1¡p=q ¡2;Á¿0,then the NTSM manifold (9)will be reached in ÿnite time.Furthermore ,the states x 1and x 2will converge to zero in ÿnite time .Proof.For the NTSM (9),its derivative along the system dynamics (1)is ˙s =˙x 1+1ÿp q x p=q −12˙x 2=x 2+1ÿp q x p=q −12˙x 2=x 2+1ÿp q x p=q −12(f (x )+g (x )+b (x )u )Y.Feng et al./Automatica38(2002)2159–21672161=x2+1ÿpqx p=q−12g(x)−ÿqpx2−p=q2−(l g+Á)sgn(s)=1ÿpqx p=q−12(g(x)−(l g+Á)sgn(s))thens˙s=1ÿpqx p=q−12(g(x)s−(l g+Á)sgn(s)s)6−1ÿpqÁx p=q−12|s|:Since p and q are positive odd integers and1¡p=q¡2,there is x p=q−12¿0for x2=0.Let (x2)=(1=ÿ)(p=q)Áx p=q−12.Then it hass˙s6− (x2)|s|(x2)¿0for x2=0:(11)Therefore,for the case x2=0,the condition for Lya-punov stability is satisÿed.The system states can reach the sliding mode s=0withinÿnite ing the following ar-guments can easily prove this:substituting the control(10) into system(1)yields˙x2=−ÿqpx2−p=q2+g(x)−(l g+Á)sgn(s):Then,for x2=0,it is obtained˙x2=g(x)−(l g+Á)sgn(s):For both s¿0and s¡0,it is obtained˙x26−Áand ˙x2¿Á,respectively,showing that x2=0is not an attractor.It also means that there exists a vicinity of x2=0such that for a small ¿0such that|x2|¡ ,there are˙x26−Áfor s¿0 and˙x2¿Áfor s¡0,respectively.Therefore,the crossing of the trajectory from the boundary of the vicinity x2= to x2=− for s¿0,and from x2=− to x2= for s¡0occurs inÿnite time.For other regions where|x2|¿ ,it can be easily concluded from(11)that the switching line s=0can be reached inÿnite time since we have˙x26−Áfor s¿0 and˙x2¿Áfor s¡0.The phase plane plot of the system is shown in Fig.1.Therefore,it is concluded that the sliding mode s=0can be reached from anywhere in the phase plane inÿnite time.Once the switching line is reached,one can easily see that NTSM(9)is equivalent to the TSM(2),so the time taken to reach the equilibrium point x1=0in the sliding mode is the same as in(8).Therefore,the NTSM manifold(9)can be reached inÿnite time.The states in the sliding mode will reach zero inÿnite time.This completes the proof.Remark1.It should be noted that the NTSM control(10) is always non-singular in the state space since1¡p=q¡2.Remark2.In order to eliminate chattering,a saturation function sat can be used to replace the sign function sgn.The1Fig.1.The phase plot of the system.relationshipbetween the steady-state errors of the NTSM system and the width of the layer surrounding the NTSM manifold s(t)=0is given by(Feng,Han,Stonier,&Man, 2000;Feng,Yu,&Man,2001)|s(t)|6’⇒|x(t)|6’and|x(t)|6(2ÿ’)q=p for t→∞:(12)4.Non-singular terminal sliding mode control for rigid manipulatorsIn this section,a non-singular terminal sliding mode con-trol is designed for the rigid n-link robot manipulatorM(q) q+C(q;˙q)+g(q)= (t)+d(t);(13) where q(t)is the n×1vector of joint angular position,M(q) the n×n symmetric positive deÿnite inertia matrix,C(q;˙q) the n×1vector containing Coriolis and centrifugal forces, g(q)the n×1gravitational torque,and (t)n×1vector of applied joint torques that are actually the control inputs,and d(t)n×1bounded input disturbances vector.It is assumed that rigid robotic manipulators have uncertainties,i.e.:M(q)=M0(q)+ M(q);C(q;˙q)=C0(q;˙q)+ C(q;˙q);g(q)=g0(q)+ g(q);where M0(q);C0(q;˙q)and g0(q)are the estimated terms; M(q); C(q;˙q)and g(q)are uncertain terms.Then, the dynamic equation of the manipulator can be written in the following form:M0(q) q+C0(q;˙q)+g0(q)= (t)+ (t)(14)2162Y.Feng et al./Automatica 38(2002)2159–2167with(t )=− M (q ) q − C (q ;˙q )q − g (q ):(15)The following assumptions are made about the robot dy-namics: M (q ) ¡ 0;(16) C (q ;˙q ) ¡ÿ0+ÿ1 q +ÿ2 ˙q 2;(17) g (q ) ¡ 0+ 1 q ;(18) (t ) ¡ 0+ 1 q + 2 ˙q 2;(19) (t ) ¡b 0+b 1 q +b 2 ˙q 2;(20)where 0;ÿ0;ÿ1;ÿ2; 0; 1; 0; 1; 2;b 0;b 1;b 2are positivenumbers.Suppose that q r is the desired input of the robot mani-pulator and ˙q r is the derivative of q r .Deÿne ”(t )=q −q r ;˙”(t )=˙q −˙q r ;e (t )=[”T (t )˙”T (t )]T .Then,the error equation of the rigid robotic manipulator can be obtained as follows:˙e (t )=Ae +B ;(21)whereA = 0I 00 ;B =0I;=M −10(q )(−C 0(q ;˙q )−g 0(q )−M 0(q ) q r + (t )+ (t )):It can be observed that the error dynamics (21)is of form (13).The NTSM control strategy developed in Section 3can be applied.The result is summarized in the following theorem.Before proceeding further,the notation of the frac-tional power of vectors is introduced.For a variable vector z ∈R n ,the fractional power of vectors is deÿned asz q=p =(z q=p 1;z q=p 2;:::;z q=p n )T;˙z q=p =(˙z q=p 1;˙z q=p 2;:::;˙zq=p n )T:Theorem 2.For the rigid n -link manipulator (14),if the NTSM manifold is chosen as s =”+C 1˙”p=q ;(22)where C 1=diag [c 11;:::;c 1n ]is a design matrix ,and the NTSM control is designed as follows ,then the system tracking error ”(t )will converge to zero in ÿnite time . = 0+u 0+u 1;(23)where0=C 0(q ;˙q )+g 0(q )+M 0(q ) q r ;(24)u 0=−q pM 0(q )C −11˙”2−p=q;(25)u 1=−q p [s T C 1diag (˙”p=q −1)M −10(q )]T s T C 1diag (˙”p=q −1)M −10(q )×[ s C 1diag (˙”p=q −1)M −10(q ) (b 0+b 1 q+b 2 ˙q 2)];(26)where b 0;b 1;b 2are supposed to be known parameters as in (20).Proof.Consider the following Lyapunov functionV =12s Ts :Di erentiating V with respect to time,and substituting (23)–(26)into it yields˙V =s T ˙s =s T ˙”+p qC 1diag (˙”p=q −1) ”=s T ˙”+p q C 1diag (˙”p=q −1)M −10(q )(u 1(t )+u 0(t ))+ (t ))=s T p q C 1diag (˙”p=q −1)M −10(q )(u 1(t )+ (t )) =−p qs C 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2)+p qs T C 1diag (˙”p=q −1)M −10(q ) (t )6−p qs C 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2)+p qs C 1diag (˙”p=q −1)M −10(q ) (t ) =−p qC 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2− (t ) ) s that is˙V 6−Á(t ) s ¡0for s =0;(27)where Á(t )=p qC 1diag (˙”p=q −1)M −10(q ) ×{(b 0+b 1 q +b 2 q 2)− (t ) }¿0:Therefore,according to the Lyapunov stability criterion,the NTSM manifold s (t )in (22)converges to zero in ÿ-nite time.On the other hand,if s =”+C 1˙”p=q =0are reached as shown in Theorem 1,then the output trackingY.Feng et al./Automatica38(2002)2159–21672163 error of the robot manipulator”(t)=q−q r will convergeto zero inÿnite time.This completes the proof.Remark3.The NTSM control proposed in Theorem2solves the control of the rigid n-link manipulator,that repre-sents a special class of problems.The method proposed canbe extended to a class of n-order(n¿2)nonlinear dynam-ical systems,that represents a broader class of problems:˙x1=f1(x1;x2);˙x2=f2(x1;x2)+g(x1;x2)+B(x1;x2)u;(28)where x1=(x11;x12;:::;x1n)T∈R n;x2=(x21;x22;:::;x2n)T∈R n;f1and f2are smooth vector functions and g rep-resents the uncertainties and disturbances satisfyingg(x1;x2) 6l g where l g¿0;B is a non-singular ma-trix and u=(u1;u2;:::;u n)T∈R n is the control vector.It is further assumed that(x1;x2)=(0;0)if and only if(x1;˙x1)=(0;0).Note that many practical dynamical sys-tems satisfy this condition,for example,the mechanicalsystems.Robotic systems are certainly a special case of(28).Actually,the robotic system(14)is not in the form of(28),but it can be transformed to such form by the coordi-nates change.So,the proposed algorithm in the paper can beapplied to any plant,which can be transformed to(28).TheNTSM for system(28)can be designed as follows.Chooses=x1+ ˙x p=q1;(29)where =diag( 1;:::; n);( i¿0)for i=1;:::;n,and˙x p=q1is represented as˙x p=q1=(x p1=q111;:::;x p n=q n1n)T:If the NTSM control is designed as in(30),then the high-order nonlinear dynamical systems(28)will converge to the NTSM and the equilibrium point inÿnite time,re-spectively,u=−@f1@x2B(x1;x2)−1l g@f1@x2+Áss+@f1@x1f1(x1;x2)+@f1@x2f2(x1;x2)+ −1 −1diag(x2−p1=q q11;:::;x2−p n=q n1n);(30)where =diag(p1=q1;:::;p n=q n);p i and q i are positive odd integers and q i¡p i¡2q i for i=1;:::;n.5.Simulation studiesThe section presents two studies:one is the comparison study of performance between NTSM and TSM,and the other an application to a robot control problem.-0.0500.050.10.150.20.250.3-0.4-0.20.20.40.60.81.0x1x2Fig.2.Phase plot of NTSM system.parison studyIn order to analyze the e ectiveness of the NTSM control proposed and to compare NTSM with TSM,consider the simple second-order dynamical system below:˙x1=x2;˙x2=0:1sin20t+u:(31) The NTSM and TSM are chosen as follows:s NTSM=x1+x5=32;s TSM=x2+x3=51:Three control approaches are adopted:NTSM control, TSM control,and indirect NTSM control.The NTSM con-trol is designed according to(10)and NTSM(9),and TSM control is designed according to(5)and TSM(2).The in-direct NTSM control is designed in the same way as TSM, with only one di erence,that is when|x1|¡ ,let p=q, and is selected as0.001(Man&Yu,1997).Three sys-tems achieve the same terminal sliding mode behavior.So, only the phase plane response of the NTSM control system is provided,as shown in Fig.2.The control signals for the three kinds of systems are shown in Figs.3–5.It can be ob-viously seen some valuable facts.No singularity occurs at all in the case of NTSM control.When the trajectory crosses the x1=0axis,singularity occurs in the case of TSM con-trol.For the indirect NTSM control,although singularity is avoided by switching from the TSM to linear sliding mode, the e ect of the singularity can be seen,especially when decreases to zero.However when is relatively large, the sliding mode of the system is switching between TSM and the linear plane based sliding mode,and the advantage of TSM system is lost.Therefore,from the results of the above simulations,the occurrence of singularity problem in the TSM system,the drawback of the indirect NTSM,and the e ectiveness of the NTSM in avoiding singularity,are observed,respectively.2164Y.Feng et al./Automatica 38(2002)2159–21670.51.0 1.52.02.5-8-7-6-5-4-3-2-1012time (sec.)uFig.3.Control signal of NTSM system.0.51.0 1.52.02.5-90-80-70-60-50-40-30-20-10010time(sec.)uFig.4.Control signal of TSM system.5.2.Control of a robotA simulation with a two-link rigid robot manipulator (seeFig.6)is performed for the purpose of evaluating the perfor-mance of the proposed NTSM control scheme.The dynamic equation of the manipulator model in Fig.6is given by a 11(q 2)a 12(q 2)a 12(q 2)a 22q 1 q 2 +−ÿ12(q 2)˙q 21−2ÿ12(q 2)˙q 1˙q 2ÿ12(q 2)˙q 22+ 1(q 1;q 2)g 2(q 1;q 2)g =1 2;(32)0.51.0 1.52.02.5-8-7-6-5-4-3-2-1012time(sec.)uFig.5.Control signal of indirect TSMsystem.Fig.6.Two-link robot manipulator model.wherea 11(q 2)=(m 1+m 2)r 21+m 2r 22+2m 2r 1r 2cos(q 2)+J 1;a 12(q 2)=m 2r 22+m 2r 1r 2cos(q 2);a 22=m 2r 22+J 2;ÿ12(q 2)=m 2r 1r 2sin(q 2);1(q 1;q 2)=((m 1+m 2)r 1cos(q 2)+m 2r 2cos(q 1+q 2)); 2(q 1;q 2)=m 2r 2cos(q 1+q 2):The parameter values are r 1=1m ;r 2=0:8m ;J 1=5kg m ;J 2=5kg m ;m 1=0:5kg ;m 2=1:5kg.The desired reference signals are given by q r 1=1:25−(7=5)e −t +(7=20)e −4t ;q r 2=1:25+e −t −(1=4)e −4t :The initial values of the system are selected as q 1(0)=1:0;q 2(0)=1:5;˙q 1(0)=0:0;˙q 2(0)=0:0:Y.Feng et al./Automatica 38(2002)2159–216721650123456789100.20.40.60.81.01.21.41.6time(sec)O u t p u t t r a c k i n g o f j o i n t 1( r a d )Fig.7.Output tracking of joint 1using a boundary layer.123456789101.21.31.41.51.61.71.81.92.0time(sec)O u t p u t t r a c k i n g o f j o i n t 2( r a d )Fig.8.Output tracking of joint 2using a boundary layer.The nominal values of m 1and m 2are assumed to be ˆm 1=0:4kg ;ˆm 2=1:2kg :The boundary parameters of system uncertainties in (20)are assumed to be b 0=9:5;b 1=2:2;b 2=2:8:Suppose the tracking error and the 1st tracking error are tobe |˜q i |60:001and |˙˜q i |60:024;i =1,2,where ˜q i =q i −q riand ˙˜q i =˙q i −˙q ri ;i =1,ing the above performance index,it can be determined the parameters of NTSM manifolds.According to (12),it is obtained that |˜q i |6’i ;i =1;2:Let ’i =0:001;i =1;2(33)012345678910-15-10-5051015202530time(sec)C o n t r o l i n p u t o f j o i n t 1( N m )Fig.9.Control of joint 1using a boundary layer.12345678910-14-12-10-8-6-4024time(sec)C o n t r o l i n p u t o f j o i n t 2 (N m )Fig.10.Control of joint 2using a boundary layer.the tracking error of the system |˜q i |can be guaranteed.Onthe other hand,according to (12),it is obtained that |˙˜q i |6(2ÿ’i )q=p ;i =1;2:Let(2ÿ’i )q=p 60:024;i =1;2;thenq p6log 0:024log(2ÿ’i );i =1;2:(34)For simplicity,let ÿi =1;i =1;2.Then from (34),it is obtained thatq p 6log 0:024log(2×1×0:001)=0:60015;i =1;2:(35)2166Y.Feng et al./Automatica 38(2002)2159–2167-0.100.10.20.30.40.50.60.70.80.9-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.1e1(t)(rad)d e 1/d t (r a d /s )Fig.11.Phase plot of tracking error of joint 1.-0.5-0.4-0.3-0.2-0.10.100.20.30.40.50.6e2(t)(rad)d e 2/d t (r a d /s )Fig.12.Phase plot of tracking error of joint 2.Let qp=0:6:Now,the parameters of the TSM can be obtained as:q =3;p =5(there are many other options as well).Finally,the NTSM models are obtained as follows:s 1=˜q 1+˙˜q 5=31=0;s 2=˜q 2+˙˜q 5=32=0:In order to eliminate the chattering,the boundary layermethod is adopted (Slotine &Li,1991)in the NTSM con-trol.The simulation results are shown in Figs.7–12.Figs.7and 8show the output tracking of joints 1and 2.Figs.9and 10depict the control signals of joints 1and 2,respec-tively.Figs.11and 12show the phase plot of tracking error of joints 1and 2,respectively.One can easily see that the system states track the desired reference signals.First,theoutput tracking errors of the system reach the terminal slid-ing mode manifold s =0in ÿnite time,then they converge to zero along s =0in ÿnite time.It can be clearly seen that neither singularity nor chattering occurs in the two control signals.6.ConclusionsIn this paper,a global non-singular TSM controller for a second-order nonlinear dynamic systems with parameter uncertainties and external disturbances has been proposed.The time taken to reach the manifold from any initial sys-tem states and the time taken to reach the equilibrium point in the sliding mode have been proved to be ÿnite.The new terminal sliding mode manifold proposed can enable the elimination of the singularity problem associated with con-ventional terminal sliding mode control.The global NSTM controller proposed has been used for the control design of an n -degree-of-freedom rigid manipulator.Simulation results are presented to validate the analysis.The proposed controller can be easily applied to practical control of robots as given the advances of microprocessors,the vari-ables with fractional power can be easily built into control algorithms.ReferencesBhat,S.P.,&Bernstein, D.S.(1997).Finite-time stability of homogeneous systems.Proceedings of American control conference (pp.2513–2514).Feng,Y.,Han,F.,Yu,X.,Stonier,D.,&Man,Z.(2000).Tracking precision analysis of terminal sliding mode control systems with saturation functions.In X.Yu,J.-X.Xu (Eds.),Advances in variable structure systems :Analysis,integration and applications (pp.325–334).Singapore:World Scientiÿc.Feng,Y.,Yu,X.,&Man,Z.(2001).Non singular terminal sliding mode control and its applications to robot manipulators.Proceedings of 2001IEEE international symposium on circuits and systems ,Vol.III (pp.545–548).Sydney,May 2001.Haimo,V.T.(1986).Finite time controllers.SIAM Journal of Control and Optimization ,24(4),760–770.Man,Z.,Paplinski,A.P.,&Wu,H.(1994).A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators.IEEE Transactions on Automatic Control ,39(12),2464–2469.Man,Z.,&Yu,X.(1997).Terminal sliding mode control of mimo linear systems.IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications ,44(11),1065–1070.Slotine,J.E.,&Li,W.(1991).Applied non-linear control .Englewood Cli s,NJ:Prentice-Hall.Tang,Y.(1998).Terminal sliding mode control for rigid robots.Automatica ,34(1),51–56.Utkin,V.I.(1992).Sliding modes in control optimization .Berlin,Heidelberg:Springer.Wu,Y.,Yu,X.,&Man,Z.(1998).Terminal sliding mode control design for uncertain dynamic systems.Systems and Control Letters ,34,281–288.Yu,X.,&Man,Z.(1996).Model reference adaptive control systems with terminal sliding modes.International Journal of Control ,64(6),1165–1176.Yurl,B.S.,&James,M.B.(1988).Continuous sliding mode control.Proceedings of American Control Conference (pp.562–563).Y.Feng et al./Automatica 38(2002)2159–21672167Yong Feng received the B.S.degree from the Department of Control Engineering in 1982,and M.S.degree from the Depart-ment of Electrical Engineering in 1985and Ph.D.degree from the Department of Con-trol Engineering in 1991,in Harbin Insti-tute of Technology,China,respectively.He has been with the Department of Electri-cal Engineering,Harbin Institute of Tech-nology since 1985,and is currently a Pro-fessor.He was a visiting scholar in the Faculty of Informatics and Communication,Australia,from May 2000to November 2001.He has authored and co-authored over 50journal and conference papers.He has published 3books.He has completed over 10research projects,including process control,arc welding robot,climbing wall robot,CNC system,a direct drive motor and its control system,the electronics and simulation of CCD digital camera,and so on.His current research interests are nonlinear control systems,sampled data systems,robot control,digital camera modelling andsimulation.Xinghuo Yu received B.Sc.(EEE)and M.Sc.(EEE)from the University of Sci-ence and Technology of China in 1982and 1984respectively,and Ph.D.degree from South-East University,China in 1987.From 1987to 1989,he was Research Fellow with Institute of Automation,Chi-nese Academy of Sciences,Beijing,China.From 1989to 1991,he was a Postdoctoral Fellow with the Applied Mathematics De-partment,University of Adelaide,Australia.From 1991to 2002,he was with CentralQueensland University,Rockhampton,Australia where he was Lecturer,Senior Lecturer,Associate Professor then Professor of Intelligent Sys-tems and the Associate Dean (Research)of the Faculty of Informatics and Communication.Since March 2002,he has been with the School of Electrical and Computer Engineering at Royal Melbourne Institute of Technology,Australia,where he is a Professor,Director of Software and Networks,and Deputy Head of School.He has also held Visiting Profes-sor positions in City University of Hong Kong and Bogazici University(Turkey).He has recently been conferred as Honorary Professor of Cen-tral Queensland University.He is Guest Professor of Harbin Institute of Technology (China),Huazhong University of Science and Technology (China),and Southeast University (China).Professor Yu’s research inter-ests include sliding mode and nonlinear control,chaos and chaos control,soft computing and applications.He has published over 200refereed pa-pers in technical journals,books and conference proceedings.He has also coedited four research books “Complex Systems:Mechanism of Adapta-tion”(IOS Press,1994),“Advances in Variable Structure Systems:Anal-ysis,Integration and Applications”(World Scientiÿc,2001),“Variable Structure Systems:Towards the 21st Century”(Springer-Verlag,2002),“Transforming Regional Economies and Communities with Information Technology”(Greenwood,2002).Prof.Yu serves as an Associate Editor of IEEE Trans Circuits and Systems Part I and is on the Editorial Board of International Journal of Applied Mathematics and Computer Science.He was General Chair of the 6th IEEE International Workshopon Variable Structure Systems held in December 2000on the Gold Coast,Australia.He was the sole recipient of the 1995Central Queensland University Vice Chancellor’s Award forResearch.Zhihong Man received the B.E.degree from Shanghai Jiaotong University,China,the M.S.degree from the Chinese Academy of Sciences,and the Ph.D.from the Uni-versity of Melbourne,Australia,all in electrical and electronic engineering,in 1982,1986and 1993,respectively.From 1994to 1996,he was a Lecturer in the Department of Computer and Commu-nication Engineering,Edith Cowan Uni-versity,Australia.From 1996to 2000,he was a Lecturer and then a SeniorLecturer in the Department of Electrical Engineering,the University of Tasmania,Australia.In 2001,he was a Visiting Senior Fellow in the School of Computer Engineering,Nanyang Technological University,Singapore.Since 2002,he has been an Associate Professor of Computer Engineering at Nanyang Technological University.His research interests are in robotics,fuzzy logic control,neural networks,sliding mode control and adaptive signal processing.He has published more than 120journal and conference papers in these areas.。