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第21頁 翻譯部分 英文部分 CONTROL OF A PNEUMATIC SYSTEM WITH ADAPTIVE NEURAL NETWORK COMPENSATION BY SASAN TAGHIZADEH A thesis submitted to the Department of Mechanical and Materials Engineering in conformity with the requirements for the degree of Master of Applied Science ABSTRACT Sasan Taghizadeh: Control of a Pneumatic System with Adaptive Neural Network Compensation. M.A.Sc. Thesis, Queen’s University, August, 2010. Considerable research has been conducted on the control of pneumatic systems due to their potential as a low-cost, clean, high power-to-weight ratio actuators. However, nonlinearities such as those due to compressibility of air continue to limit their accuracy. Among the nonlinearities in a pneumatic system, friction can have a significant effect on tracking performance, especially in applications that use rodless cylinders which have higher Coulomb friction than rodded cylinders. Compensation for nonlinearities in pneumatic systems has been a popular area of research in pneumatic system control. Most advanced nonlinear control strategies are based on a detailed mathematical model of the system. If a simplified mathematical model is used, then performance is sensitive to uncertainties and parameter variations in the robot. Although they show relatively good results, the requirement for model parameter identification has made these methods difficult to implement. This highlights the need for an adaptive controller that is not based on a mathematical model. The objective of this thesis was to design and evaluate a position and velocity controller for application to a pneumatic gantry robot. An Adaptive Neural Network (ANN) structure was mplemented as both a controller and as a compensator. The implemented ANN had online training as this was considered to be the algorithm that had the greatest potential to enhance the performance of the pneumatic system. One axis of the robot was used to obtain results for the cases of velocity and position control. Seven different velocity controllers were tested and their performance compared. For position control, only two controllers were examined: conventional PID and PID with an ANN Compensator (ANNC). The position controllers were tuned for step changes in the setpoint. Their performance was evaluated as applied to sinusoid tracking. It was shown that the addition of ANN as a compensator could improve the performance of both position and velocity control. For position control, the ANNC improved the tracking performance by over 20%. Although performance was better than with conventional PID control, it was concluded that the level of improvement with ANNC did not warrant the extra effort in tuning and implementation. Chapter 1 Introduction Control systems exist in a virtually infinite variety, both in type of application and level of sophistication. Control Engineering can be summed up as the design and implementation of automatic control systems to achieve specified objectives under given constraints. For a complex system, the overall objectives and constraints will need to be translated into performance specifications for the various subsystems – ultimately into control systems specifications for low-level subsystems. Control engineering practice includes the use of better and more efficient control design strategies for improving manufacturing processes, the efficiency of energy use, advanced automobiles, among others. The present challenge to control engineers is the modeling and control of modern, complex, interrelated systems such as navigation control systems, chemical processes and robotic systems. 1.1 Problem Overview PID is the acronym for the classical and most heavily used control algorithm. Proportional plus Integral plus Derivative (PID) control is sufficient for many control problems, particularly when there are benign process dynamics and modest performance requirements. However, there are numerous control situations in which PID control with constant gains fails to meet the requirements. For example, systems with large parameter variations are candidates for more sophisticated control structures. Considerable research has been conducted on the control of pneumatic systems due to their potential as a low-cost, clean, high power-to-weight ratio actuators. However, nonlinearities such as those due to compressibility of air continue to limit their accuracy. Among the nonlinearities in a pneumatic system, friction can have a significant effect on tracking performance, especially in applications that use rodless cylinders which have higher Coulomb friction than rodded cylinders. Compensation for nonlinearities in pneumatic systems has been a popular area of research in pneumatic system control. Most of the compensation strategies use model based algorithms. Although they show relatively good results, the requirement for model parameter identification has made these methods difficult to implement. A pneumatic gantry robot is an example of a pneumatic system that typically requires a controller more sophisticated than PID. Most advanced nonlinear control strategies are based on a detailed mathematical model of the system. If a simplified mathematical model is used, then performance is sensitive to uncertainties and parameter variations. This highlights the need for an adaptive controller that is not based on a mathematical model. 1.2 Objectives Abu-Mallouh and Surgenor (2008) conducted research on force/velocity control of a pneumatic gantry robot for contour tracking with NN compensation. They used two Proportional Pressure Control (PPC) valves. Both simulation and experimental results were presented. However, the NN compensator was only tested by simulation. They concluded that their work demonstrated the value of NN for online compensation of nonlinear elements in a pneumatic system, but experimental verification was required. The underlying purpose of the thesis is to provide that verification. The objective of this thesis is to design and evaluate a position and velocity controller for application to one axis of the pneumatic gantry robot. An adaptive NN will be tested as both a controller and as a compensator. The implemented NN will be adaptive with online training as this is an algorithm that appears to have the greatest potential to enhance the performance of a pneumatic system. Performance of the NN will be reported quantitatively. Comparison will be made with the performance of a conventional PID controller, in order to provide a benchmark. 1.3 Thesis Outline The organization of the thesis is as follows: Chapter 2 presents a literature review on six subjects: 1) pneumatic system control, 2) pneumatic control with compensation, 3) Neural Network (NN), 4) NN as a controller, 5) NN as a compensator and 6) online versus offline NN. Chapter 3 provides background on the apparatus including sensor calibration. Details on the Adaptive Neural Network (ANN) algorithm will also be given including the implementation and tuning. In Chapter 4 the apparatus is used to obtain results for the case of velocity control, in order to evaluate the performance of ANN as applied to one axis of the gantry robot. Seven different controllers are tested and their performance compared: 1) P-only, 2) PI, 3) PI+ΔP, 4) ANN, 5) ANN+ΔP, 6) P-only+ANNC (ANN compensator) and 7) PI+ANNC. For ANN and ANN+ΔP, ANN is applied as a stand-alone controller. For P-only+ANNC and PI+ANNC, ANN is applied as a compensator. In Chapter 5 the apparatus is used to obtain results for the case of position control, in order to evaluate the performance of ANN as a compensator. It provides the tuning methodology and comparative performance results. Two controllers are tested: 1) PID and 2) PID+ANNC. The controllers are tuned for step changes in the setpoint. Their performance is evaluated as applied to sinusoid tracking. Chapter 2 Literature Review This chapter presents a literature review on six subjects: 1) pneumatic system control, 2) pneumatic control with compensation, 3) Neural Network (NN), 4) NN as a controller, 5) NN as a compensator and 6) online versus offline NN. 2.1 Pneumatic System Control Pneumatic actuators are difficult to control because of low bandwidth and high nonlinearity due mainly to air compressibility and Coulomb friction effects. However, relative to electrically actuated systems, pneumatic systems are cheaper and easier to maintain. This observation has led to considerable interest and research on pneumatic system control. Two specific examples will be given in this section. They were chosen as they gave quantitative and comparative performance results for different control schemes. van Varseveld and Bone (1997) implemented a fast, accurate, and inexpensive position-controlled pneumatic actuator. Figure 2-1 illustrates the pneumatic system that they used. The system used a standard rodded pneumatic cylinder (stroke = 152 mm, diameter = 27 mm) with two on/off solenoid valves. The valves were pulsed using a novel Pulse Width Modulation (PWM) algorithm which produced a very linear open-loop velocity response. Four different schemes of PWM were examined. Figure 2-1 Schematic of pneumatic control system with solenoid valves (van Varseveld and Bone, 1997) Figure 2-2 Closed-loop position controller step responses for PWM schemes (van Varseveld and Bone, 1997) Figure 2-2 shows the closed-loop position controller step response for the different PWM schemes. The results led them to use PWM Scheme 4 due to its better transient response. Then, they added basic friction compensation to the PID controller with PWM Scheme 4. Figure 2-3 illustrates the results for the PID position controller with and without friction compensation. They reported that adding a friction compensator could reduce the average of steady-state error by 40%, from 0.19 mm without the compensator to 0.11 mm with the compensator. Figure 2-3 PID position controller result with and without friction compensation (van Varseveld and Bone, 1997) Figure 2-4 Fuzzy control with ΔP feedback for sine wave input (Chillari et al, 2001) Chillari et al (2001) conducted several experiments on pneumatic system control. They examined PID, Fuzzy, Sliding mode and Neuro-Fuzzy controllers. Experimental results for these controllers applied to different setpoint trajectories were presented. Main parts of the apparatus were: rodded pneumatic cylinder (stroke = 200 mm, diameter = 25 mm) and two pairs of on/off solenoid valves. The controllers were tested on sinusoidal, square, saw-tooth and staircase input signals. In their work, Chillari et al adopted a differential pressure (ΔP) feedback signal in order to compensate for external disturbances and also friction forces that would act against the motion. Figure 2-4 shows the Fuzzy control with ΔP feedback for a sine wave. Unfortunately, they did not present a figure which shows the controller without the ΔP feedback. In addition, they introduced a NN which was able to estimate the ΔP feedback and could be used instead of the differential pressure sensor. Figure 2-5 presents a quantitative performance comparison of the different controllers based on the standard deviation between the desired and the actual position signal in m. According to Figure 2-5, the error increases as the frequency of the signal increases. The Fuzzy controller showed slightly better performance than the PID controller. Adoption of the ΔP feedback improved the performance of the Fuzzy controller still further. The performance of the Fuzzy controller with the NN estimate of ΔP was comparable to that of the Fuzzy controller with real ΔP feedback. Figure 2-5 Performance comparison for different controllers and setpoints (Chillari et al, 2001) 2.1.1 Pneumatic Control with Compensation As discussed in the previous section, van Varseveld and Bone (1997) used a basic Coulomb friction compensation combined with bounded integral control which was found to substantially reduce the steady-state error due to stiction. At zero velocity, the friction force known as stiction is largely responsible for any steady-state error. Friction compensation was disabled once the steady-state error was within a specified tolerance. The results of applying the controllers on step input and S-curve were shown in Figure 2-2 and Figure 2-3, respectively. One of the common compensators in pneumatic controls is deadzone (dead time) compensation. The deadzone is an inherent nonlinearity in pneumatic servo valves, where for a range of input control values, the valve gives no output flow. From Ning and Bone (2002), Figure 2-6 illustrates the measured relationship between the maximum cylinder force versus the valve input (Part a) and the schematic of a servo pneumatic valve showing chambers A and B (Part b). There are three situations for the spool of valve based on the valve input. First, if the valve input is greaterthan , chamber A is filling. Second, if the valve input is less than , chamber B is filling. Third, if the valve input is in between and , the applied force is less than the static friction force. In the third case, the cylinder does not move and this is called the deadzone. （a）Measured relationship between the (b) Schematic of a servo pneumatic maximum cylinder force and the valve input valve showing chambers A and B Figure 2-6 Working principle of a servo pneumatic valve (Ning and Bone, 2002) Figure 2-7 gives the block diagram of the control system with friction compensation used in their paper. A Proportional plus Velocity plus Acceleration (PVA) position controller was adopted. A friction compensation block was added as a feedforward signal to the PVA output. Unfortunately, the authors did not provide any mechanical specifications for the apparatus. They did mention that a rodless cylinder was used. Figure 2-7 Block diagram of the PVA controller with friction compensation (Ning and Bone, 2002) In Ning and Bone (2002), when the cylinder was in the deadzone, a friction compensation term was added to the control signal to make the cylinder move until it reached the desired steady-state error value. The friction compensation parameters had to be tuned by the user. However, no tuning procedure was presented. They deployed both PV and PVA controllers. Since they used double differentiation for the acceleration feedback, significant noise was seen in the signal. The controller was set to PVA initially. It would be switched back to PV when the piston was 5 mm away from the setpoint. Figure 2-8 illustrates the experimental position and error responses of PVA/PV control where they could get a steady-state accuracy of ±0.01 mm. The proportional, velocity and acceleration gains are given. No comparison of performance was presented between the controller with and without the friction compensation. Figure 2-8 Experimental position and error signals of PVA/PV position control (Ning and Bone, 2002) Ning and Bone (2005) conducted an experimental comparison of two servo pneumatic position control algorithms: PVA + feedforward (FF) + deadzone compensation (DZC) and Sliding Mode Control (SMC). They used a rodless cylinder with a Proportional Flow Control (PFC) valve. The DZC was the same as the one used in Ning and Bone (2002). Figure 2-9 gives the block diagram of the PVA+FF+DZC controller. The tracking performances were evaluated by the RMSE . The PVA+FF+DZC controller had a RMSE of 0.910 mm for a sinusoid at 0.5 Hz. For the same sinusoid tracking, SMC could Figure 2-9 Block diagram of PVA+FF+DZC controller (Ning and Bone, 2005) reduce the RMSE to 0.375 mm. Andrighetto and Bavaresco (2009) reported success in using deadzone compensation for their pneumatic apparatus. They used a pneumatic rodless cylinder (stroke = 500 mm, diameter = 25 mm) and a 5 port 3 way PFC valve (the same valve used in this thesis). Figure 2-10 shows the experimental result for a sinusoidal input where deadzone compensation is added to a tuned P-only position controller. Specifically, the input was a sinusoidal wave at 1.6 Hz and amplitude of 200 mm. The maximum error was around 70 mm without deadzone compensation which was reduced to 20 mm with the compensation (70% reduction in the error). They claimed that the deadzone compensation was fairly easy to implement. Despite this statement, they mentioned that this method is only applicable when the deadzone is known and the valve dynamics are fast enough to be neglected. Figure 2-10 P-only position controller with and without deadzone compensation 中文部分 自適應(yīng)神經(jīng)網(wǎng)絡(luò)補(bǔ)償氣動(dòng)系統(tǒng)的控制 Sasan Taghizadeh 本論文符合機(jī)械與材料工程學(xué)院應(yīng)用科學(xué)碩士學(xué)位要求 皇后大學(xué) 加拿大 安大略省 金斯頓 2010年9月 摘 要 Sasan Taghizadeh：自適應(yīng)神經(jīng)補(bǔ)償氣動(dòng)系統(tǒng)的控制，應(yīng)用科學(xué)碩士論文，皇后大學(xué)，2010年8月。 氣動(dòng)系統(tǒng)的控制具有作為低成本、清潔、執(zhí)行元件功重比高等優(yōu)點(diǎn),因而在氣動(dòng)控制上有越來越多的研究。然而，其精準(zhǔn)度仍受到如壓縮空氣等一些非線性因素的限制。在氣動(dòng)系統(tǒng)的非線性因素中，摩擦?xí)Ω櫺阅墚a(chǎn)生顯著的影響，特別是在使用系列無桿氣缸的應(yīng)用上，因?yàn)闊o桿氣缸具有比有桿氣缸更高的庫侖摩擦。 在氣動(dòng)系統(tǒng)控制方面，氣動(dòng)系統(tǒng)的非線性補(bǔ)償已成為一種熱門的研究領(lǐng)域。最先進(jìn)的非線性調(diào)控方式是基于一個(gè)系統(tǒng)的復(fù)雜的數(shù)學(xué)模型。如果使用一個(gè)簡化的數(shù)學(xué)模型，那么它的性能對自動(dòng)裝置的不確定性和參數(shù)變化是非常敏感的。雖然他們可以表現(xiàn)出相對較好的結(jié)果，但是對模型參數(shù)辨識的要求使得這些方法難以實(shí)現(xiàn)。這說明自適應(yīng)控制器需要的并不是基于一個(gè)數(shù)學(xué)模型。 本論文的目標(biāo)是設(shè)計(jì)和評價(jià)位置和速度控制器并應(yīng)用于氣動(dòng)門式自動(dòng)裝置。自適應(yīng)神經(jīng)網(wǎng)絡(luò)(ANN)結(jié)構(gòu)同時(shí)作為控制器和補(bǔ)償器來實(shí)現(xiàn)。實(shí)施的ANN作為算法進(jìn)行網(wǎng)上測試從而最大化地提高氣動(dòng)系統(tǒng)的性能。 自動(dòng)裝置的一個(gè)軸用于獲得每組速度和位置控制的結(jié)果。七個(gè)不同的速度控制器進(jìn)行測試和性能比較。對于位置控制，只需要檢測兩個(gè)控制器：傳統(tǒng)的PID和具有ANN補(bǔ)償(ANNC)的PID。位置控制器在定位點(diǎn)調(diào)整步長變化。他們的性能評價(jià)是適用于正弦曲線跟蹤。 結(jié)果表明，ANN作為補(bǔ)償器的引入能夠改善位置和速度控制這兩者的性能。對于位置控制，ANNC提高跟蹤性能超過20%。盡管性能優(yōu)于傳統(tǒng)PID控制，但結(jié)論是，ANNC改進(jìn)的水平在優(yōu)化和實(shí)現(xiàn)中并沒有起到額外的作用。 第一章 介紹 控制系統(tǒng)的存在形式多種多樣，包括應(yīng)用類型和復(fù)雜程度?？刂乒こ炭梢愿爬樵诮o定的約束條件下，設(shè)計(jì)和實(shí)現(xiàn)自動(dòng)控制系統(tǒng)去實(shí)現(xiàn)指定的目標(biāo)。對于一個(gè)復(fù)雜的系統(tǒng)，所有的目標(biāo)和約束條件都需要轉(zhuǎn)化為各種子系統(tǒng)的性能指標(biāo)——最終為底層子系統(tǒng)的控制系統(tǒng)規(guī)范?？刂乒こ虒?shí)踐包括使用更好、更有效的控制對策來改善制造工藝、提高能源使用效率、獲得先進(jìn)的發(fā)動(dòng)機(jī)等。目前的挑戰(zhàn)來控制工程師是建模和控制的現(xiàn)代的、復(fù)雜的、相互關(guān)聯(lián)的系統(tǒng),如導(dǎo)航控制系統(tǒng)、化工過程和機(jī)器人系統(tǒng)?？刂圃O(shè)計(jì)策略為提高制造過程的效率的能源使用,先進(jìn)的汽車等。對工程師而言，目前的挑戰(zhàn)是是建模和控制新式的、復(fù)雜的、相互關(guān)聯(lián)的系統(tǒng)，如導(dǎo)航控制系統(tǒng)、化工過程和自動(dòng)機(jī)械系統(tǒng)。 1.1問題綜述 PID是傳統(tǒng)的和最頻繁使用的控制算法Proportional plus Integral plus Derivative的縮寫。比例積分微分(PID)控制是可以用于解決許多控制問題，特別是在良性過程動(dòng)力學(xué)和適度的性能要求的時(shí)候。然而，還有許多控制情況下，具有恒定增益的PID控制不能滿足要求。例如，對于更復(fù)雜的控制結(jié)構(gòu)，具有大參數(shù)變化的系統(tǒng)通常只是作為備用。 由于氣動(dòng)系統(tǒng)的控制具有作為低成本、清潔、高強(qiáng)度比的致動(dòng)器的潛在應(yīng)用而受到相當(dāng)多的研究。然而，其精準(zhǔn)度仍受到如壓縮空氣等一些非線性因素的限制。在氣動(dòng)系統(tǒng)的非線性因素中，摩擦?xí)Ω櫺阅墚a(chǎn)生顯著的影響，特別是在使用系列無桿氣缸的應(yīng)用上，因?yàn)闊o桿氣缸具有比有桿氣缸更高的庫侖摩擦。 在氣動(dòng)系統(tǒng)控制方面，氣動(dòng)系統(tǒng)的非線性補(bǔ)償已成為一種熱門的研究領(lǐng)域。大多數(shù)的補(bǔ)償策略是適用基于算法的模型。雖然他們可以表現(xiàn)出相對較好的結(jié)果，但是對模型參數(shù)辨識的要求使得這些方法難以實(shí)現(xiàn)。 氣動(dòng)門式自動(dòng)裝置是典型的需要比PID更復(fù)雜的控制器的氣動(dòng)系統(tǒng)的例子。最先進(jìn)的非線性調(diào)控方式是基于一個(gè)系統(tǒng)的復(fù)雜的數(shù)學(xué)模型。如果使用一個(gè)簡化的數(shù)學(xué)模型，那么它的性能對自動(dòng)裝置的不確定性和參數(shù)變化是非常敏感的。這說明自適應(yīng)控制器需要的并不是基于一個(gè)數(shù)學(xué)模型。 1.2 目標(biāo) Abu-Mallouh和Surgenor (2008)研究了通過NN補(bǔ)償對氣動(dòng)門式自動(dòng)裝置的力/速度控制進(jìn)行輪廓跟蹤。他們使用兩個(gè)比例壓力控制(PPC)閥門。同時(shí)給出了模擬和實(shí)驗(yàn)結(jié)果。然而，NN補(bǔ)償僅是通過模擬測試。結(jié)論是他們的工作證明了NN在氣動(dòng)系統(tǒng)非線性環(huán)節(jié)網(wǎng)絡(luò)補(bǔ)償中的值，但是需要實(shí)驗(yàn)驗(yàn)證。論文的根本目的就是提供這個(gè)驗(yàn)證。 本論文的目標(biāo)是設(shè)計(jì)和評價(jià)位置和速度控制器并應(yīng)用于氣動(dòng)門式自動(dòng)裝置的一軸。自適應(yīng)NN將作為控制器和補(bǔ)償器來進(jìn)行測試。實(shí)施的自適應(yīng)NN作為算法進(jìn)行進(jìn)行網(wǎng)上測試從而最大化地提高氣動(dòng)系統(tǒng)的性能。NN的性能將會(huì)定量給出。與傳統(tǒng)PID控制器性能作比較，以提供一個(gè)基準(zhǔn)。 1.3 論文提綱 論文組織如下： 第二章給出六個(gè)主題的文獻(xiàn)綜述：1) 氣動(dòng)系統(tǒng)控制，2) 補(bǔ)償?shù)臍鈩?dòng)控制，3) 神經(jīng)網(wǎng)絡(luò)(NN)，4) 神經(jīng)網(wǎng)絡(luò)作為控制器，5) NN作為補(bǔ)償器和 6) 在線與離線NN比較。 第三章給出裝置的背景，包括傳感器標(biāo)定。詳細(xì)的自適應(yīng)神經(jīng)網(wǎng)絡(luò)(ANN)算法也將給出，包括實(shí)現(xiàn)和優(yōu)化。 在第四章，設(shè)備用于獲取速度控制情況的結(jié)果，用來評價(jià)是ANN應(yīng)用于門式自動(dòng)裝置一軸的性能。七個(gè)不同的控制器進(jìn)行測試和性能比較：1) P-only，2) PI， 3) PI+ΔP， 4) ANN，5) ANN+ΔP，6) P-only+ANNC (ANN 補(bǔ)償器) 和 7) PI+ANNC。對于ANN和ANN+ΔP，ANN用來作為獨(dú)立控制器。對于P-only+ANNC和PI+ANNC，ANN用來作為補(bǔ)償器。 在第五章，設(shè)備是用于獲取位置控制情況的結(jié)果，為了評價(jià)ANN作為補(bǔ)償器的性能。給出優(yōu)化方法和性能對比結(jié)果。兩個(gè)控制器進(jìn)行測試:1) PID 和 2) PID +ANNC。位置控制器在定位點(diǎn)調(diào)整步長變化。他們的性能評價(jià)是適用于正弦曲線跟蹤。 第二章 文獻(xiàn)綜述 第二章給出六個(gè)主題的文獻(xiàn)綜述：1) 氣動(dòng)系統(tǒng)控制，2) 補(bǔ)償?shù)臍鈩?dòng)控制，3) 神經(jīng)網(wǎng)絡(luò)(NN)，4) 神經(jīng)網(wǎng)絡(luò)作為控制器，5) NN作為補(bǔ)償器和 6) 在線與離線NN比較。 2.1 氣動(dòng)系統(tǒng)控制 主要由空氣壓縮性和庫侖摩擦效應(yīng)導(dǎo)致的低帶寬、高非線性使得氣壓傳動(dòng)裝置難以控制。然而，相對于電動(dòng)系統(tǒng)，氣動(dòng)系統(tǒng)更便宜和更容易維護(hù)。這一點(diǎn)引起人們對氣動(dòng)系統(tǒng)控制相當(dāng)大的興趣，對其進(jìn)行研究。在這一節(jié)中將會(huì)給出兩個(gè)具體的例子。選這兩個(gè)例子是因?yàn)樗麄儗Σ煌刂品桨附o出了定量的性能對比的結(jié)果。 van Varseveld和Bone (1997)實(shí)現(xiàn)了一個(gè)快速、準(zhǔn)確而且便宜的位置控制氣壓傳動(dòng)裝置。圖2-1是他們用到的氣動(dòng)系統(tǒng)的原理圖。該系統(tǒng)使用一個(gè)帶有兩個(gè)開/關(guān)電磁閥的標(biāo)準(zhǔn)有桿氣缸（沖程=152mm，直徑=27mm）。閥門是脈沖使用新穎的脈沖寬度調(diào)制(PWM)算法，產(chǎn)生一個(gè)完全線性的開環(huán)速度響應(yīng)。測試四個(gè)不同的PWM方案。 圖2-1 帶有電磁閥的氣動(dòng)控制系統(tǒng)的原理圖。 圖2-2是不同的脈寬調(diào)制方案的閉環(huán)位置控制器的階躍響應(yīng)。得出的結(jié)果讓他們選擇PWM方案4，因其具有更好的瞬態(tài)響應(yīng)。然后，他們基本摩擦補(bǔ)償加入PWM方案4的PID控制器。圖2-3顯示的是有摩擦補(bǔ)償和無摩擦補(bǔ)償?shù)腜ID位置控制器的結(jié)果。他們得出，添加一個(gè)摩擦補(bǔ)償器可以減少40%的平均穩(wěn)態(tài)誤差，從無補(bǔ)償器時(shí)的0.19mm到有補(bǔ)償器時(shí)的0.11mm。 圖2-2 PWM方案下的閉環(huán)位置控制器的階躍響應(yīng)（Van Varseveld和Bone，1997） 圖2-3 有摩擦補(bǔ)償和無摩擦補(bǔ)償?shù)腜ID位置控制器的結(jié)果（van Varseveld和Bone，1997） Chillari 等人(2001)對氣動(dòng)系統(tǒng)控制進(jìn)行了幾次實(shí)驗(yàn)。他們測試了PID、模糊、滑動(dòng)模態(tài)和神經(jīng)模糊控制器。給出了這些應(yīng)用到不同定位點(diǎn)軌跡的控制器的實(shí)驗(yàn)結(jié)果。設(shè)備的主要部分是:有桿氣缸（沖程=200mm，直徑=25mm）和兩對開/關(guān)電磁