Utility of probability density evolution method for experimental reliability-based active vibration control

Abstract

Summary The utility of probability density evolution method for reliability‐based active vibration control of a cantilevered flexible beam is experimentally investigated. In this respect, an optimal linear quadratic regulator (LQR) is utilized together with an observer to design an online full‐state feedback controller. In order to design a well‐performing controller and to simulate the controller performance, a system model is obtained via identification techniques. Reliability tests are consequently performed to verify the effectiveness of the presented reliability assessment method as a foundation for reliability‐based control. Subsequently, a hybrid metaheuristic optimization scheme of continuous ant colony system is used for simultaneous integrated tuning of the LQR and observer parameters through all processes. Optimization of the controller and the observer is performed with the objective of system structural reliability. An optimized version of classical LQR controller is also developed for making comparisons between the two control methods. Finally, a comparison is made between the mean power consumption of the 2 controllers. It is realized that the reliability‐based controller decreases operating costs without compromising system safety.

Time‐varying reliability analysis based on hybrid Kalman filtering and probability density evolution

“…Then, having estimated the desired states over time together with a proper definition of the failure criteria (from case studies, standards, or risk and safety analyses 44 ), we used the PDEM to estimate the time‐varying PDF of the structural response to be used for calculating the probability of crossing the limit state (i.e., structural failure probability). The joint PDF of the displacement ( X ), uncertainty vector ( θ ) of the physical parameters, and time ( t ) were estimated by solving the partial differential equation of PDEM (Equation ) 42 . The step‐by‐step derivation of the PDEM is as follows 45 : $$$$\begin{equation}\frac{{\partial {{\bm{P}}}_{{\bm{X\theta }}}\left( {{\bm{X}},{\bm{\theta }},t} \right)}}{{\partial t}} + \dot{\bm{X}} \ \frac{{\partial {{\bm{P}}}_{{\bm{X\theta }}}\left( {{\bm{X}},{\bm{\theta }},t} \right)}}{{\partial {\bm{x}}}} = 0,\end{equation}$$$$…”

“…They presented the PDEM formula according to the principle of probability preservation, which allowed for randomness propagations in dynamic structures. The PDEM has received increasing attention during the recent decade and has been verified via several theoretical and experimental investigations 41,42 . The applicability of the PDEM has also been validated through comparisons with other methods, such as the Monte Carlo simulation of the reliability assessment of complex structures 43 .…”

“…The PDEM has received increasing attention during the recent decade and has been verified via several theoretical and experimental investigations. 41,42 The applicability of the PDEM has also been validated through comparisons with other methods, such as the Monte Carlo simulation of the reliability assessment of complex structures. 43 Although the PDEM has facilitated the analysis of complex and nonlinear stochastic structures, it still requires an efficient method to solve the equations of motion of structural systems to predict the time-varying derivative of the structural response, which is the PDEM input.…”

Probability Density Evolution for Time-Varying Reliability Assessment of Wing Structures

“…The current study utilizes the probability of measured dynamic response to determine the WSC reliability. In this regard, a damage detection method based on statistical moments of the physical parameters (Xu, Zhang, Li, & Xia, 2009;Zhang, Xu, Xia, & Li, 2008), is employed together with PDE equation to find the probability of safe operation based on structural response (Li & Chen, 2004;Saraygord Afshari & Pourtakdoust, 2018). This method is implemented for a WSC via a single limit state function as well as the work recently published in (Raouf & Pourtakdoust, 2017).…”

A deep learning approach for the solution of probability density evolution of stochastic systems

“…In addition, by deriving the General Density Evolution Equation (GDEE) they proved that in stochastic structural systems, the equations governing the evolution of the probability density can be decoupled by introducing the physical solution of the system [3]. Over the past decade, PDEM has been used successfully in many numerical studies on the stochastic behavior of the structures [4][5][6] and has been validated in experimental studies [7,8]. Although PDEM decouples the probability density evolution PDE in the probability space [9], numerical treatment of the governing equations is still stringent and impractical for real-time applications.…”