分数阶Pl~λD~μ控制器参数整定方法与设计研究
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摘要
进入21世纪以来，随着分数阶微积分理论研究不断取得突破，分数阶微积分控制理论研究开始成为控制领域中一个新的研究热点。研究发现基于分数阶微积分方程描述的实际系统或非线性系统物理意义更清晰，物理特性更精确；而基于分数阶Pl~λD~μ控制器参数整定方法因其多了两个整定参数（积分阶次λ和微分阶次μ），使系统控制更灵活，控制效果更好。然而，由于分数阶控制理论尚处于理论研究阶段，分数阶Pl~λD~μ控制器参数整定方法主要还是采用整数阶PID控制器参数整定方法，分数阶Pl~λD~μ控制器设计与实现方法比较复杂，对计算能力要求高，因此，分数阶Pl~λD~μ控制器的理论和应用研究有待进一步深入和完善。本文针对分数阶Pl~λD~μ控制器参数整定方法、分数阶微积分算子（s±α）的离散化方法、分数阶Pl~λD~μ控制器设计与实现以及分数阶控制系统动态响应特性仿真实验进行了深入系统的研究。主要研究工作及创新性成果如下：
1.针对被控对象模型已知的稳定最小相位被控对象，在相位裕度和幅值裕度等传统整数阶PID控制器参数整定方法的基础上，提出了控制系统开环传递函数相位Bode图在截止频率ω c附近相对平坦（flat-phase）的对增益变化具有鲁棒性的分数阶Pl~λD~μ控制器参数整定方法。而针对一类未知模型、稳定的最小相位被控对象，通过采用一种继电反馈测试实验的方法，提出了具有等阻尼（Iso-damping）特性的自整定分数阶Pl~λD~μ控制器参数整定方法。实验结果表明本文提出的分数阶Pl~λD~μ控制器参数整定方法不仅可以改善控制系统动态响应特性，而且可以获得比整数阶PID控制器参数整定方法更强的鲁棒性。
2.采用本文提出的分数阶PI λ Dμ控制器参数整定方法完成了面向不同被控对象的分数阶PI λ Dμ控制器设计。（1）提出并设计了面向一阶、二阶及三阶等被控对象的IOPID、FOPI、FO[PI]、FOPD、FO[PD]和FOPID等控制器。（2）提出并设计了面向滞后被控对象的IOPID、FOPI、FO[PI]、FOPD、FO[PD]和FOPID等控制器。（3）提出并设计了面向分数阶被控对象的IOPID、FOPI、FO[PI]和FOPID等控制器。（4）提出并设计了面向一组未知模型、稳定且具有最小相位的被控对象、系统具有等阻尼特性的自整定FOPI和FO[PI]控制器。
3.系统研究了分数阶微积分算子s±α(α∈R)的解析数值近似法、直接离散化近似法和间接离散化近似法。重点研究了近似效果较好的Al-Alaoui+CFE直接离散化方法以及Oustaloup和改进Oustaloup间接离散化方法。并利用这些方法实现了分数阶Pl~λD~μ控制器数值离散化处理。
4.基于Matlab/Simulink符号工具箱，在分数阶Pl~λD~μ控制器离散化数值实现的基础上，设计得到不同分数阶PI λ Dμ控制器封装模块和控制系统仿真原理图，运行仿真原理图得到不同控制系统的单位阶跃响应特性，并进行系统动态响应特性的对比分析。
5．首次在基于LabVIEW的半实物实验平台上实现了分数阶Pl~λD~μ控制器参数整定方法的实验研究。通过将数据采集板、控制软件、外设接口、放大器和物理设备的合理配置，搭建了实验仿真平台，并在该平台上进行了基于IOPID、FOPD、FO[PD]等控制器的系统动态特性仿真实验研究。进一步验证了采用分数阶Pl~λD~μ控制器参数整定方法不仅可以改善系统动态响应特性，而且可以获得好于传统PID控制器参数整定方法的系统鲁棒性及抗干扰性。推动了分数阶PI λ Dμ控制器在实时控制系统中的应用。
最后，对本文所做的工作进行总结，并对今后的研究工作进行展望。
Since the begging of the21st century, the fractional order calculus theory hasachieved lots of breakthough. The fractional order calculus control theory has becomda new hotspot in control field. It is found that the physical meaning of the practicalsystem or nonlinear system described by the fractional order calculus equation isclearer and the physical characteristics of those systems are more precise. Besides,due to the two additional tuning parameters of the fractional orderPl~λD~μcontrollerparameter tuning method, namely the integral order λ and the differential μ, thesystem control is more flexible and the control effect is better. However, since thefractional order control theory is still in the theoretical study stage, the fractionalorderPl~λD~μcontroller parameter tuning method mainly exploits the integer orderPID controller parameter tuning method. The design and the implementation of thefractional order controller is complex and high computing capabilities are required.Therefore, the theory and the application of the fractional orderPl~λD~μcontrollerneed to be further improved. This dissertation studied the fractional orderPI λ Dμcontroller parameter tuning method, the discretization method of the fractional ordercalculus operators (s±α), the design and the implementation of the fractional orderPl~λD~μcontroller and the dynamic response charactieristics simulation experimentsof the fractional order control system in details, the main research work andachievements are as follows:
1. For the stable minimum phase controlled plant of the known plant model, onthe basis of the traditional integral order PID controller parameter tuning method suchas the phase margin and the gain margin, we propose the fractional orderPI λ Dμcontroller parameter tuning method of which the control system open-loop transferfunction phase Bode diagram has flat-phase near the cutoff frequencyω cand isrobust to the gain variation. But for the stable minimum phase controlled plant of the unknown plant model, by using a relay feedback test experimental method, wepropose the autotuing fractional orderPl~λD~μcontroller parameter tuning methodwhich has the iso-damping characteristics. The simulation results indicate that theproposed method not only improve the dynamic response characteristics of thecontrolled system, but also obtain stronger robustness compared to the integer orderPID controller parameter tuning method.
2. By employing the proposed fractional orderPl~λD~μcontroller parametertuning method, we achieved the fractional orderPl~λD~μcontroller design orientedfor different plants.(1) We propose and design the IOPID, FOPI, FO[PI], FOPD,FO[PD] and FOPID controller for the first order, the second order and the third orderplants respectively.(2) We propose and design the IOPID, FOPI, FO [PI], FOPD,FO[PD] and FOPID controller for the time-delay plants.(3)We propose and designthe IOPID, FOPI, FO [PI], FOPD, FO[PD] and FOPID controller for the fractionalorder plants.(4) We propose and design the autotuning FOPI and FO[PI] controllerfor the stable and minimum-phase plants for which the model is unknown and thesystem has the iso-damping characteristics.
3. We systematically study the analytical numerical approximation method, thedirect and indirect discrete approximation method for the fractional order calculusoperator s±α(α∈R). Furthermore, we study the Al-Alaoui+CFE direct discretediscretization method and the Oustaloup and the modified Oustloup indirectdiscretization method all of which have good approximation effect and accomplishthe numerical discretization for the fractional orderPl~λD~μcontroller by employingthese methods.4. On the basis of the numerical discretization for the fractional orderPI λ Dμcontroller, we design the simulation systematic diagram for different fractional orderPl~λD~μcontroller by exploiting Matlab/Simulink toolbox and obtain the unit stepresponse characteristics of the different control system. We compare and analyze thedynamic response characteristics for different systems.
5. We achieve the experimental study for the parameter tuning method of thefractional orderPl~λD~μcontroller based on the LabVIEW platform for the first time.We establish the experimental simulation platform through the rational configuration of the data acquisition board, control software, peripheral interface, amplifier andphysical device and do the system simulation and experiments based on IOPID,FOPD, FO[PD] controllers on this platform. We further verify that by employing theparameter tuning method of the fractional orderPl~λD~μcontroller, we not onlyimprove the system dynamic response characteristics but also obtain better systemrobustness and interference immunity than traditional PID controller parameter tuningmethod, which promote the application of the fractional orderPl~λD~μcontroller inthe real-time control system.
Finally, we summarize the done work and look forward to the future researchwork.
1.针对被控对象模型已知的稳定最小相位被控对象，在相位裕度和幅值裕度等传统整数阶PID控制器参数整定方法的基础上，提出了控制系统开环传递函数相位Bode图在截止频率ω c附近相对平坦（flat-phase）的对增益变化具有鲁棒性的分数阶Pl~λD~μ控制器参数整定方法。而针对一类未知模型、稳定的最小相位被控对象，通过采用一种继电反馈测试实验的方法，提出了具有等阻尼（Iso-damping）特性的自整定分数阶Pl~λD~μ控制器参数整定方法。实验结果表明本文提出的分数阶Pl~λD~μ控制器参数整定方法不仅可以改善控制系统动态响应特性，而且可以获得比整数阶PID控制器参数整定方法更强的鲁棒性。
2.采用本文提出的分数阶PI λ Dμ控制器参数整定方法完成了面向不同被控对象的分数阶PI λ Dμ控制器设计。（1）提出并设计了面向一阶、二阶及三阶等被控对象的IOPID、FOPI、FO[PI]、FOPD、FO[PD]和FOPID等控制器。（2）提出并设计了面向滞后被控对象的IOPID、FOPI、FO[PI]、FOPD、FO[PD]和FOPID等控制器。（3）提出并设计了面向分数阶被控对象的IOPID、FOPI、FO[PI]和FOPID等控制器。（4）提出并设计了面向一组未知模型、稳定且具有最小相位的被控对象、系统具有等阻尼特性的自整定FOPI和FO[PI]控制器。
3.系统研究了分数阶微积分算子s±α(α∈R)的解析数值近似法、直接离散化近似法和间接离散化近似法。重点研究了近似效果较好的Al-Alaoui+CFE直接离散化方法以及Oustaloup和改进Oustaloup间接离散化方法。并利用这些方法实现了分数阶Pl~λD~μ控制器数值离散化处理。
4.基于Matlab/Simulink符号工具箱，在分数阶Pl~λD~μ控制器离散化数值实现的基础上，设计得到不同分数阶PI λ Dμ控制器封装模块和控制系统仿真原理图，运行仿真原理图得到不同控制系统的单位阶跃响应特性，并进行系统动态响应特性的对比分析。
5．首次在基于LabVIEW的半实物实验平台上实现了分数阶Pl~λD~μ控制器参数整定方法的实验研究。通过将数据采集板、控制软件、外设接口、放大器和物理设备的合理配置，搭建了实验仿真平台，并在该平台上进行了基于IOPID、FOPD、FO[PD]等控制器的系统动态特性仿真实验研究。进一步验证了采用分数阶Pl~λD~μ控制器参数整定方法不仅可以改善系统动态响应特性，而且可以获得好于传统PID控制器参数整定方法的系统鲁棒性及抗干扰性。推动了分数阶PI λ Dμ控制器在实时控制系统中的应用。
最后，对本文所做的工作进行总结，并对今后的研究工作进行展望。
Since the begging of the21st century, the fractional order calculus theory hasachieved lots of breakthough. The fractional order calculus control theory has becomda new hotspot in control field. It is found that the physical meaning of the practicalsystem or nonlinear system described by the fractional order calculus equation isclearer and the physical characteristics of those systems are more precise. Besides,due to the two additional tuning parameters of the fractional orderPl~λD~μcontrollerparameter tuning method, namely the integral order λ and the differential μ, thesystem control is more flexible and the control effect is better. However, since thefractional order control theory is still in the theoretical study stage, the fractionalorderPl~λD~μcontroller parameter tuning method mainly exploits the integer orderPID controller parameter tuning method. The design and the implementation of thefractional order controller is complex and high computing capabilities are required.Therefore, the theory and the application of the fractional orderPl~λD~μcontrollerneed to be further improved. This dissertation studied the fractional orderPI λ Dμcontroller parameter tuning method, the discretization method of the fractional ordercalculus operators (s±α), the design and the implementation of the fractional orderPl~λD~μcontroller and the dynamic response charactieristics simulation experimentsof the fractional order control system in details, the main research work andachievements are as follows:
1. For the stable minimum phase controlled plant of the known plant model, onthe basis of the traditional integral order PID controller parameter tuning method suchas the phase margin and the gain margin, we propose the fractional orderPI λ Dμcontroller parameter tuning method of which the control system open-loop transferfunction phase Bode diagram has flat-phase near the cutoff frequencyω cand isrobust to the gain variation. But for the stable minimum phase controlled plant of the unknown plant model, by using a relay feedback test experimental method, wepropose the autotuing fractional orderPl~λD~μcontroller parameter tuning methodwhich has the iso-damping characteristics. The simulation results indicate that theproposed method not only improve the dynamic response characteristics of thecontrolled system, but also obtain stronger robustness compared to the integer orderPID controller parameter tuning method.
2. By employing the proposed fractional orderPl~λD~μcontroller parametertuning method, we achieved the fractional orderPl~λD~μcontroller design orientedfor different plants.(1) We propose and design the IOPID, FOPI, FO[PI], FOPD,FO[PD] and FOPID controller for the first order, the second order and the third orderplants respectively.(2) We propose and design the IOPID, FOPI, FO [PI], FOPD,FO[PD] and FOPID controller for the time-delay plants.(3)We propose and designthe IOPID, FOPI, FO [PI], FOPD, FO[PD] and FOPID controller for the fractionalorder plants.(4) We propose and design the autotuning FOPI and FO[PI] controllerfor the stable and minimum-phase plants for which the model is unknown and thesystem has the iso-damping characteristics.
3. We systematically study the analytical numerical approximation method, thedirect and indirect discrete approximation method for the fractional order calculusoperator s±α(α∈R). Furthermore, we study the Al-Alaoui+CFE direct discretediscretization method and the Oustaloup and the modified Oustloup indirectdiscretization method all of which have good approximation effect and accomplishthe numerical discretization for the fractional orderPl~λD~μcontroller by employingthese methods.4. On the basis of the numerical discretization for the fractional orderPI λ Dμcontroller, we design the simulation systematic diagram for different fractional orderPl~λD~μcontroller by exploiting Matlab/Simulink toolbox and obtain the unit stepresponse characteristics of the different control system. We compare and analyze thedynamic response characteristics for different systems.
5. We achieve the experimental study for the parameter tuning method of thefractional orderPl~λD~μcontroller based on the LabVIEW platform for the first time.We establish the experimental simulation platform through the rational configuration of the data acquisition board, control software, peripheral interface, amplifier andphysical device and do the system simulation and experiments based on IOPID,FOPD, FO[PD] controllers on this platform. We further verify that by employing theparameter tuning method of the fractional orderPl~λD~μcontroller, we not onlyimprove the system dynamic response characteristics but also obtain better systemrobustness and interference immunity than traditional PID controller parameter tuningmethod, which promote the application of the fractional orderPl~λD~μcontroller inthe real-time control system.
Finally, we summarize the done work and look forward to the future researchwork.
引文
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