统计代写|随机控制代写Stochastic Control代考|Preliminary Remarks

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

Modern control theory was originally from the Wiener filter theory and the Wiener control theory in mid-twentieth century (Wiener 1948,1949 ). It became a fairly complete formulation through a rapid development in the subsequent 20 years. Beginning with the early of 1970 s, the deterministic optimal control and stochastic optimal control of structures have been developed into almost independent two branches.

Structural control is devised to reduce or mitigate the responses of structures by utilizing the specified devices or facilities deployed in structures that are capable of shifting, dissipating, absorbing, and supplying energy. A diagram of typical logic of structural control is shown in Fig. 1.1. It is seen that a complete control system involves the sensor for monitoring the external excitation and the structural response, the controller for deducing the feedback law to regulate the structural state, and the actuator for implementing the control gain. According to whether a moderate power supply system is required for the operation of control system, structural control can be categorized into four modalities, i.e., the passive control, the active control, the semiactive control, and the hybrid control (Housner et al. 1997; Ou 2003; Teng 2009).
Passive control refers to a modality that allocates base isolation systems or energy dissipation devices to a structure so as to enhance the damping, stiffness, and strength of the structure. The control gain provided by passive systems just replies upon the structural response to the external excitation, and generally does not need power supplies. In early of 1970 s, Kelly and his colleagues proposed and experimentally proved the value of metal yield and energy dissipation component in the seismic mitigation of structures (Kelly et al. 1972; Skinner et al. 1975). The pioneering works played a significant role in the development of ductile design of seismic structures, which later formulated an important branch of structural control using energy dissipation schemes. Thereafter, an amount of energy dissipation components and devices were invented and applied into the vibration control of engineering structures under earthquakes and high winds, such as friction-type energy dissipation devices derived as the principle of automobile brakes (Pall and Marsh 1982; Li and Reinhorn 1995; Qu et al. 2001; Bhaskararao and Jangid 2006), viscoelastic dampers early used in the fatigue control of plane body (Zhang et al. 1989; Zhang and Soong 1992; Shen et al. 1995; Palmeri and Ricciardelli 2006; Xu 2007), viscous dampers or damping walls widely applied in the fields of military and aviation (Constantinou et al. 1993; Reinhorn et al. 1995; Museros and Martinez-Rodrigo 2007), tuned mass dampers (TMDs) and multiple-tuned mass dampers (MTMDs) (Villaverde 1994; Setareh 1994; Li 2000; Guo and Chen 2007; Wong and Johnson 2009), and tuned fluid dampers (Wakahara et al. 1992; Tamura et al. 1995; Tait et al. 2008). Until now, the research and development of energy dissipation devices with excellent performance are still one of the hotspot issues in the field of structural control (Chan and Albermani 2008; Jung et al. 2010; Zhang et al. 2013; Berardengo et al. 2015; Amjadian and Agrawal 2018).

统计代写|随机控制代写Stochastic Control代考|Stochastic Optimal Control of Structures

Since randomness inherent in the external excitations, the structural systems, and the control devices, the traditional deterministic control cannot guarantee the safety of structures. Stochastic optimal control of structures has thus become a challenging issue in the area of structural control (Housner et al. 1997). This topic can be traced back to the earlier researchers on the relevant fundamental theory. In 1940s, the stochastic differential theory had been established on the basis of the Itô calculus (Sobczyk 1991). Until to the late of 1950s and the early of 1960s, however, the development of the state-space method (Kalman 1960a), the maximum principle (Pontryagin 1962), the dynamic programming (Bellman 1957), and the Kalman-Bucy filtering theory (Kalman 1960b; Bucy and Kalman 1961) were eventually proposed and developed, which underlined the establishment of the modern optimal control theory. Stochastic optimal control then formed as a new branch of optimal control discipline.

统计代写|随机控制代写Stochastic Control代考|Classical Stochastic Optimal Control

The motivation of stochastic optimal control is to define the optimal control so that the transition probability density function (Spencer and Bergman 1993) or the statistical moments (Wojtkiewicz et al. 1996) are limited in a specified range of errors. In the theoretical framework of classical LQG control, there have formed the stochastic maximum principle (Yong and Zhou 1999) and the stochastic dynamic programming (Stengel 1994). Meanwhile, a series of design criteria for defining the stochastic optimal control law in accordance with the optimal control gain have been developed. The minimum variance criterion, for example, seeks for a control force with the objective of minimizing the variance of performance functions subjected to the mean constraint (Sain 1966). The optimal neighborhood feedback criterion involves an interaction scheme based on an initial costate estimation in view of the control equation and the performance objective (Stengel 1994). The statistical moment assessment criterion seeks for an optimal control law in the sense of trade-off, between the statistics of quantities of concern (Zhang and Xu 2001). The reliability-based design criterion is to attain a control law leading to a minimum failure probability of structural systems, through iteratively solving the limit state equation pertaining to the performance objective (Spencer et al. 1994a; May and Beck 1998). The probability density tracking criterion employs the Markov process theory to attain the analytical solution

of probability density of a family of specified systems, so as to define the control law approaching the desired probability density (Elbeyli and Sun 2002; Elbeyli et al. 2005; Sun 2006).

In aspect of stochastic optimal control of linear systems, the present moment reliability-based structural optimal control has a wide application. For example, Spencer et al. addressed the design and optimization method of a single-degreeof-freedom system attached with active tendon systems. In their work, a performance function directly relevant to the failure probability was defined, and the firstcrossing principle and a stationary white noise model of seismic ground motions were employed. Utilizing the first-order reliability method/the second-order reliability method (FORM/SORM), the reliability-based control of structures was then implemented so as to minimize the failure probability of structural displacement (Spencer et al. 1994a). Battaini et al. further employed the FORM/SORM to carry out the reliability assessment and experimental analysis of a controlled multi-degreeof-freedom system (Battaini et al. 2000). It was indicated in their work that the computational efficiency of moment-based reliability method was better than the Monte Carlo simulation, while the former merely attained an approximate solution in the case that the performance function of structures featured a high nonlinearity in the neighborhood of checking point, or that the distribution of basic random variables was far from the Gaussian distribution. The non-Gaussian properties of structural response might enlarge the error between the approximate solution and the accurate solution due to the high-order nonlinear mapping from the basic random variables to the structural responses. Therefore, the moment-based reliability method cannot be extended into the large-scale and complex structural systems. In 1998, May and Beck explored the first generation of Benchmark model pertaining to the probabilistic control problem, so as to maximize the reliability of the uncertain structures and control systems subjected to seismic ground motions. In this work, an acceleration feedback control scheme was employed aiming at the design of optimal active mass damper; the performance measure on the design objective includes the interstory drift, story acceleration of structures, and the deployment positions of sensors. An approximate expansion estimating the failure probability of structures was then proposed based on the level-crossing process theory (May and Beck 1998). However, their work involved the search iterations of the optimal value with respect to the approximate expansion and of the stochastic optimal control law, which will result in an increased computational cost.

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