含有分数阶PID控制器的随机动力系统可靠性分析

RELIABILITY ANALYSIS OF STOCHASTIC DYNAMICAL SYSTEMS WITH FRACTIONAL ORDER PID CONTROLLER

随着粘弹性材料在工程结构中的广泛应用,刻画工程结构中粘弹性材料遗传特性和长记忆性的分数阶微积分成为研究的热点,特别是具有分数阶微积分特点的PID控制器更是从理论上和应用上受到关注.本文研究高斯白噪声激励下含有分数阶PID控制器的随机结构动力系统的可靠性问题.利用慢变过程的特征以及广义积分的性质,对分数阶PID控制器在数学上进行了近似处理,之后应用能量包络随机平均法确定了可靠性函数满足的后向Kolmogorov方程以及首次穿越时间统计矩满足的广义Pontryagin方程.结果表明:在分数阶控制器中,较小的分数阶α和较大的分数阶β均可以得到较为理想的可靠性结果,并且这些均与蒙特卡洛仿真结果一致,验证了方法的有效性和正确性.

With the widely application of viscoelastic material in engineering structures, fractional calculus that characterizes its feature of hereditary and long-run memory associating with stochastic engineering systems becomes the hot issue. Specially, fractional PID controller is more attractive in theory and application. In this paper, fractional order PID (FOPID) control on reliability of stochastic dynamical systems subjected to Gaussian white-noise excitation is investigated. FOPID controller is mathematically approximated based on the characteristics of slow-varying process and the properties of generalized integral. After that, stochastic averaging method of energy envelope is applied to determine Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time. The numerical results illustrate that both the smaller fractional order α and greater fractional valueβ in PID controller can obtain better reliable results, besides, these results are in very agreement with the simulations from Monte Carlo method, which verifies the correction and efficiency of our proposed methods.