摘要
随着分数阶微积分理论逐渐渗透到越来越多的研究领域,它与控制理论的结合引起了人们对分数阶系统的研究热情。实际证明,一些复杂系统难以用传统的整数阶模型来描述,分数阶微积分作为传统整数阶微积分的推广,是一个很好的数学工具,可以准确地建立对象的分数阶模型。分数阶系统建模已成为一个开放的问题,吸引了众多学者致力于这方面的研究。同时,相关研究表明,采用分数阶控制器可以获得比传统整数阶控制器更好的性能,分数阶控制器设计成为当前的研究热点之一。
本文主要研究了分数阶系统辨识与分数阶控制器设计问题。在分数阶系统辨识方面,基于分数阶传递函数模型,提出了同元次分数阶系统的极大似然频域辨识算法；基于分数阶状态空间模型,分别研究了非时滞和时滞分数阶系统的频域子空间辨识方法；基于正交基理论,研究了分数阶正交基函数在分数阶系统辨识中的应用。在分数阶控制器设计方面,针对具有高度不确定性与干扰的系统,结合定量反馈理论(Quantitative Feedback Theory, QFT)提出了分数阶QFT鲁棒控制器设计方法。
极大似然方法因其良好的统计特性在系统辨识中颇受青睐。本文第3章研究了同元次分数阶系统频域辨识问题,基于单输入单输出(Single-Input Single-Output, SISO)同元次分数阶系统的传递函数模型,在频域中推导了分数阶系统极大似然辨识算法,并给出了详细的优化计算方法。在考虑噪声的情况下,通过数值仿真实验验证了所提方法的有效性,同时分析了噪声对辨识结果的影响。
多数实际系统具有多输入多输出(Multi-Input Multi-Output, MIMO)特性,因而有必要进一步研究MIMO系统的辨识问题。一般SISO系统的辨识方法不易直接用于MIMO系统,且基于优化计算的方法通常较难推广到MIMO系统。子空间方法在MIMO系统辨识中具有更大的优势。本文第4章基于分数阶状态空间模型,分别研究了非时滞和时滞分数阶系统的频域子空间辨识问题,给出了分数阶子空间辨识算法的基本框架。同时,指出了分数阶子空间辨识结果不唯一的问题并给出了解决方法,讨论了系统阶次与同元阶次的确定方法,以及噪声情况下相关加权矩阵的选取方法。针对时滞分数阶系统辨识中分数阶同元阶次和时滞参数的优化问题,提出了子空间与差分进化算法相结合的辨识方法。通过数值仿真,验证了算法的有效性。正交基理论在系统建模与控制中得到越来越多的应用,一般的有理正交基也逐渐推广到了分数阶情形。本文第5章研究了分数阶正交基在分数阶系统辨识中的应用,给出了基于分数阶生成函数的辨识方法,避免了正交化过程的复杂计算。通过将待辨识参数分组,给出了一种PSO与最小二乘相结合的算法。数值仿真结果验证了方法的有效性。
针对具有高度不确定性与干扰的系统,QFT是从工程应用角度提出的一种基于图形的鲁棒控制器设计理论。本文第6章基于QFT,研究了分数阶QFT控制器的设计方法,包括分数阶反馈控制器设计和分数阶前置滤波器设计。在分数阶反馈控制器设计中,给出了在Nichols图上进行自动回路成形设计的方法,同时考虑了系统超调对开环增益变化的鲁棒性,从而设计具有Iso-damping特性的控制器；基于辨识的思想,提出了两种分数阶前置滤波器的设计方法：FOB(Fractional Order Basis,分数阶基函数)法和FCT (Fractional Complex Term,分数阶复合项)法。通过数值仿真,验证了所提方法的有效性。
Fractional order calculus has gained much attention in lots of research areas during recent years. It has also been applied to control area, where fractional order systems attract many research interests. In fact, some actual complicated systems are hard to be modeled as traditional integer order models. While, FOC (Fractional Order Calculus) is a good mathematical tool, which can describe these systems precisely. Therefore, modeling of fractional order systems becomes an open question, and many works associated with this problem have been done. At the same time, researches have demonstrated that fractional order controllers have better performance and design of fractional order controllers becomes a hotspot.
This dissertation focuses on the fractional order system identification and fractional order controller design. In the part of fractional modeling, several system identification methods are studied. Based on commensurate fractional order transfer function, a fractional order maximum likelihood algorithm is presented in frequency domain. A frequency domain subspace method is studied based on fractional state space model, and it is convenient for the identification of MIMO fractional order systems. Then, this subspace method is extended to the fractional order systems with time delay. With the aid of orthogonal basis theory, fractional order orthogonal basis is applied to fractional order system identification. In the part of fractional controller design, systems with high uncertainties and disturbances are considered, and a fractional order controller design method combined with QFT (Quantitative Feedback Theory) is presented.
Maximum likelihood method is popular for its excellent statistic characteristics, and the maximum likelihood identification of fractional order systems is studied in Chapter 3. The identification algorithm is deduced in frequency domain based on SISO commensurate fractional order transfer function. Detailed optimization computing is also presented. Numerical simulations validate the proposed algorithm with existence of noise, impact of which is analyzed at the same time.
Many actual systems are MIMO systems, and it is necessary to research the identification of MIMO fractional order systems. Usually, identification methods for SISO systems can not be applied to MIMO systems directly, and it is hard to extend those algorithms based on optimization to MIMO cases. Subspace method is a good choice for MIMO system identification. Therefore, a frequency domain subspace method for fractional MIMO system identification is presented in Chapter 4. The problem, that fractional subspace identification result is not unique, is proposed, and an efficient solution is drawn. Determination of system order and commensurate order is discussed. When noise is considered, the choice of weighting matrix is also discussed in this chapter. Considering the common existence of input time delay, the presented fractional subspace method is extended to fractional order systems with time delay. Numerical simulations are performed for both non-delay system and delay system, and the results validate the algorithm.
Orthogonal basis theory has been applied to system modeling and control lately. Traditional reasonable orthogonal basis is extended to fractional cases. In Chapter 5, the application of fractional orthogonal basis in fractional system identification is discussed. A method of using fractional generating functions is presented, and complex orthogonization computation is avoided. After parameter decoupling, an optimization method combined PSO and LS is presented. Simulation results validate the presented algorithm.
For the systems with high uncertainties and disturbance, QFT is a robust theory proposed from the engineering point of view. In Chapter 6, a fractional QFT controller design is studied. Based on several typical fixed structure fractional controllers, an automatic loop shaping method on Nichols chart is presented. The fractional feedback controller is designed automatically during the process of loop shaping, during which, robustness to gain variation is considered, and the controller is designed with iso-damping characteristic. In the design of pre-filter, two design methods of fractional pre-filters are proposed:one is FOB (Fractional-order Orthogonal Basis) method and the other is FCT (Fractional. Complex Term) method. A classical example is performed, and the effectiveness of the new design method is proved.
引文
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