时滞分数阶微积分若干理论及其在水轮机调节系统中应用

发布时间：2019-07-08 20:31

【摘要】：水轮机调节系统由压力引水管道、水轮发电机机组、液压调速系统和发电机端口侧电力负荷组成,是一个集水机电磁为一体的非线性控制系统。生产实践表明,液压随动系统具有时延性,水轮机传递系数随机组工况变化表现时变性,并且发电机端口侧电力负荷存在随机扰动。上述因素的存在不仅威胁着水轮机调节系统的稳定可靠运行,而且难以保证电网系统的供电质量。鉴于此,非常有必要对系统的不稳定性因素进行考虑来建立系统的非线性数学模型,进而探究系统保持稳定运行的充分条件。本毕业论文通过将时滞引入液压随动系统、将分数阶微积分非线性理论引入压力引水管道系统,建立了一个较符合工程实际的水轮机调节系统非线性动力学模型。此外,研究了一类时滞分数阶非线性系统解的存在性和唯一性定理、系统有限时间稳定和渐进稳定的充分条件。论文主要内容和结论如下:(1)研究系统稳定性的最基本条件是要保证系统解的存在性和唯一性。将水轮机调节系统数学模型归纳为一类时滞分数阶非线性系统,利用分数阶微积分性质和广义Gronwall不等式,给出该类系统解存在的充分必要条件以及系统存在唯一解的充分条件,并推导系统解的预估计值。(2)运用拉普拉斯变换、Mittag-Leffler函数及其性质,给出时滞分数阶非线性系统满足有限时间稳定的充分条件,即当系统满足该条件时,不论系统初始状态如何,该系统总能在有限时间内趋于稳定状态。通过数值仿真,验证所得稳定性理论的有效性。(3)研究一类离散时滞分数阶非线性系统满足渐进稳定的充分条件,并将其与已有的稳定性理论进行比较,得出本定理的优越性;进而,给定两个三维混沌非线性系统,应用本定理使其满足渐近稳定条件,验证所推理论。(4)在压力引水管道为复杂管系情况下,将分数阶微积分理论引入压力引水管道系统,建立分数阶复杂管系混流式水轮机调节系统的非线性数学模型。利用分数阶非线性系统稳定性定理,我们给出随分数阶阶次变化时系统分岔点的变化规律;同时,详尽分析系统稳定域随分数阶阶次的变化情况。通过分数阶分岔图、时域图、相轨迹图、庞加莱映射图、功率谱图以及频谱图,系统地分析不同阶次下系统的具体动力学行为,从而得出机组振动情况。(5)考虑到液压随动系统主配压阀死区造成接力器静止不动,以及接力器活塞速度响应滞后等因素,将时滞引入液压随动系统,建立时滞分数阶非线性动态模型。进而利用改进的ABM(Admas-Bashforth-Moulton)算法,基于MATLAB进行数值仿真,结合统计物理学原理,研究系统在时滞和分数阶共同作用下的稳定域变化趋势。本论文虽然对水轮机调节系统的稳定性特征进行了一定研究,也给出了一类离散时滞分数阶非线性系统的有限时间稳定性定理和渐近稳定性定理,但将该定理应用在水轮机调节系统中以控制系统的动态特征使其保持稳定尚需继续研究。
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[Abstract]:The water turbine regulating system is composed of a pressure water diversion pipe, a hydro-generator unit, a hydraulic speed regulating system and a generator port side power load, and is a non-linear control system integrated with the water machine. The production practice shows that the hydraulic follow-up system has the time ductility, the transfer coefficient of the water turbine is modified with the change of the working condition of the unit, and there is a random disturbance on the power load on the side of the generator port. The existence of the above factors not only threatens the stable and reliable operation of the turbine regulation system, but also makes it difficult to guarantee the power supply quality of the power grid system. In view of this, it is necessary to consider the unstable factors of the system to establish the nonlinear mathematical model of the system, and then to explore the sufficient conditions for the stable operation of the system. In this thesis, by introducing the time-delay into the hydraulic follow-up system, the nonlinear theory of the fractional-order calculus is introduced into the pressure-diversion pipe system, and a non-linear dynamic model of the turbine governing system is established. In addition, the existence and uniqueness theorems of solutions of a class of time-delay fractional-order nonlinear systems are studied, and the sufficient conditions of the system's finite-time stability and asymptotic stability are given. The main contents and conclusions are as follows: (1) The most basic condition of the system stability is to ensure the existence and uniqueness of the system solution. The mathematical model of the hydraulic turbine governing system is generalized to a class of time-delay fractional-order nonlinear systems, and the sufficient and necessary conditions for the existence of the system and the sufficient conditions for the existence of the system are given by using the fractional-order calculus and the generalized Gronwall inequality, and the pre-estimated value of the system solution is derived. (2) Using the Laplace transform, the Mittag-Leffler function and its properties, a sufficient condition for the time-delay order nonlinear system to satisfy the finite-time stability is given, that is, when the system satisfies this condition, the system can always be stable in a limited time, regardless of the initial state of the system. The validity of the stability theory is verified by numerical simulation. (3) The sufficient conditions for a class of discrete time-delay fractional-order nonlinear systems to satisfy the asymptotic stability are studied, and compared with the existing stability theory, the advantage of this theorem is obtained. This theorem is applied to satisfy the asymptotic stability conditions and to verify the theory of push. (4) Under the condition of complex piping, the fractional calculus theory is introduced into the pressure water diversion pipe system, and the nonlinear mathematical model of the mixed-flow turbine governing system of the fractional-order complex piping system is established. Using the stability theorem of fractional order nonlinear system, we give the rule of the bifurcation point of the system as the order of the fractional order is changed, and the variation of the system stability field with the order of the fractional order is analyzed in detail. The dynamic behavior of the system under different orders is systematically analyzed by the fractional-order bifurcation diagram, the time-domain diagram, the phase trace diagram, the Pincare map, the power spectrum diagram and the frequency spectrum graph, so as to obtain the vibration condition of the unit. (5) Considering that the dead zone of the main pressure valve of the hydraulic follow-up system causes the servomotor to be stationary, and the speed response of the servomotor piston is lagging and other factors, the time-delay is introduced into the hydraulic follow-up system, and the nonlinear dynamic model with the time-delay order is established. In this paper, the modified ABM (Admas-Bashar-Moulton) algorithm is used to simulate the numerical simulation based on MATLAB, and the variation trend of stability of the system under the common action of time-delay and fractional order is studied in combination with the principle of statistical physics. In this paper, the stability characteristics of the turbine governing system are studied, and the finite-time stability theorem and the asymptotic stability theorem of a class of discrete time-delay fractional-order nonlinear systems are also given. However, the application of this theorem in the water turbine regulation system to keep the stability of the control system remains to be studied.
【学位授予单位】：西北农林科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TV734

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