不确定分数阶系统的自适应控制研究

发布时间：2018-09-08 16:43

【摘要】：随着工程技术的发展,越来越多的场合对控制提出了更高的要求。分数阶微积分作为整数阶微积分延伸和推广,对于相当一部分复杂系统,能够描述得更为简洁有效,进而降低对控制器鲁棒性的要求；且分数阶微积分的引入能够增加控制器设计的自由度,改善控制品质。然而,随着实际系统的运行,环境变化、元器件老化等原因使得所建模型不再能精确地描述实际系统,所以研究不确定分数阶系统的分数阶控制问题有着重要的理论价值和实践意义。自适应方法在解决不确定整数阶系统控制问题中,已经取得了丰硕的成果,然而,在将其拓展到分数阶情形时,存在很多困难和挑战。分数阶微积分可以看作是整数阶微积分的某种“连续过渡”。但从整数阶到分数阶,系统发生了某些本质的改变,如系统特征函数由单值变为多值,状态空间由有限维变为无穷维,使其相应的理论体系发生了本质变化,这也正是分数阶自适应控制研究进展缓慢的原因之一。因此,本文将把握分数阶系统的本质特性,借助间接李雅普诺夫方法,开展分数阶自适应控制的研究,为不确定分数阶系统的分数阶控制问题提供有效解决方案。首先,本文对分数阶直接模型参考自适应控制方法进行了改进。对于阶次0α≤1的SISO情形,为降低参数估计对跟踪误差的依赖性和避免不希望的跳变,本文在参数更新律中加入预测误差项,得到了改进的控制策略,并将结果推广至1α2的情形。对于MIMO情形,则利用右增益矩阵去除了与高频增益矩阵相关的正定性的苛刻假设,提出了一个能适用于任意相对阶对象的控制方案。同时,基于连续频率分布模型和间接李雅普诺夫方法,本文证明了闭环控制系统的稳定性、输出跟踪误差以及参数估计误差的渐近收敛特性。其次,为获得更好的参数收敛特性和控制效果,本文首次提出了分数阶间接模型参考自适应控制方案,并进行了深入研究。首先讨论了分数阶系统的参数估计问题,针对系统参数有无约束的两种情况,分别给出了参数估计方案。基于这一结果,分别针对SISO单变量分数阶系统和SISO多变量分数阶系统给出了分数阶间接模型参考自适应控制器设计方法,解决了参考模型的选择、控制器结构的构建和控制器参数的整定等问题。然后,考虑到上述两种方法通常只能适用于线性参数化模型,本文提出了可用于非线性系统的分数阶自适应Backstepping控制方法。针对全状态可测的情形,本文首先通过合适的坐标变换,将被控对象变换为归一化的严格反馈系统；然后构造新的误差变量,设计分数阶自适应Backstepping状态反馈控制器。针对部分状态可测的情形,首先设计状态观测器,然后构造新的误差变量。通过引入新颖的李雅普诺夫函数,解决了在观测误差渐近收敛的情况下闭环控制系统稳定性证明的问题。并基于所提出的分数阶跟踪微分器,给出了分数阶自适应Backstepping输出反馈控制器的一般化设计流程和实现方法。另外,巧妙地基于辨识的思想,从两个角度研究了分数阶算子的逼近问题：最高精度逼近和最低阶次逼近。针对第一种情形,考虑到Oustaloup递推逼算法是从不精确的幅频特性出发得到的,且不能取复极点,所以并不是严格意义上的变极点方法,本文则基于矢量拟合方法,实现真正意义上的变极点有理逼近。针对第二种情形,提出了定极点逼近方法,将逼近问题转化为一个线性最小二乘问题,并考虑了纯积分环节的特性,给出了较优的初始极点选择方法。最后,在分数阶积分算子逼近的基础上,实现了对分数阶系统的逼近,指出了分数阶伪状态空间模型与其频率分布模型之间的关系,讨论并解决了非零初始值的系统响应问题。上述相关工作为本文所提出的分数阶自适应控制策略的验证提供了有效的方法。
[Abstract]:With the development of Engineering technology, more and more occasions put forward higher requirements for control. Fractional calculus, as an extension and extension of integer-order calculus, can be described more concisely and effectively for a considerable number of complex systems, thereby reducing the requirements for controller robustness; and the introduction of fractional calculus can increase control. However, with the operation of the actual system, environmental changes, component aging and other reasons, the model can no longer accurately describe the actual system, so the study of Fractional-order Control of uncertain fractional-order systems has important theoretical value and practical significance.
Adaptive methods have achieved fruitful results in solving the control problems of uncertain integer-order systems. However, there are many difficulties and challenges in extending them to fractional-order systems. Qualitative changes, such as the change of system characteristic function from single-valued to multi-valued, and the change of state space from finite-dimensional to infinite-dimensional, make the corresponding theoretical system change essentially, which is one of the reasons why the research progress of fractional-order adaptive control is slow. The research on fractional-order adaptive control is carried out to provide an effective solution to the problem of Fractional-order Control for uncertain fractional-order systems.
Firstly, the fractional order direct model reference adaptive control method is improved. For SISO with order 0 alpha < 1, in order to reduce the dependence of parameter estimation on tracking error and avoid undesirable jump, the predictive error term is added to the parameter update law, and the improved control strategy is obtained, and the result is extended to 1 alpha2. In the case of MIMO, the righthand gain matrix is used to remove the rigorous assumption of positive definiteness associated with the high frequency gain matrix, and a control scheme for any relative order plant is proposed. Based on the continuous frequency distribution model and the indirect Lyapunov method, the stability of the closed-loop control system is proved. Asymptotic convergence properties of tracking error and parameter estimation error.
Secondly, in order to obtain better parameter convergence characteristics and control effect, the fractional-order indirect model reference adaptive control scheme is proposed for the first time and studied deeply in this paper. Firstly, the problem of parameter estimation for fractional-order systems is discussed. Results The design methods of fractional-order indirect model reference adaptive controller for SISO single-variable fractional-order systems and SISO multivariable fractional-order systems are presented respectively. The problems such as the selection of reference model, the construction of controller structure and the tuning of controller parameters are solved.
Then, considering that the above two methods can only be applied to linear parameterized models, a fractional-order adaptive backstepping control method for nonlinear systems is proposed. A new error variable is constructed and a fractional-order adaptive backstepping state feedback controller is designed. In the case of partial state measurability, a state observer is first designed, and then a new error variable is constructed. Based on the proposed fractional order tracking differentiator, the general design flow and implementation method of the fractional order adaptive backstepping output feedback controller are given.
In addition, based on the idea of identification, the approximation problem of fractional operators is studied from two aspects: the highest precision approximation and the lowest order approximation. For the second case, a fixed pole approximation method is proposed, which transforms the approximation problem into a linear least squares problem. Considering the characteristics of the pure integral link, an optimal initial pole selection method is given. On the basis of integral operator approximation, the approximation of fractional-order systems is realized. The relationship between fractional pseudo-state space model and its frequency distribution model is pointed out. The system response problem with non-zero initial value is discussed and solved. Method.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TP13

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