Chaotic system optimal tracking using data-based synchronous method with unknown dynamics and disturbances

We develop an optimal tracking control method for chaotic system with unknown dynamics and disturbances. The method allows the optimal cost function and the corresponding tracking control to update synchronously. According to the tracking error and the reference dynamics, the augmented system is constructed. Then the optimal tracking control problem is defined. The policy iteration(PI) is introduced to solve the min-max optimization problem. The off-policy adaptive dynamic programming(ADP) algorithm is then proposed to find the solution of the tracking Hamilton–Jacobi–Isaacs(HJI) equation online only using measured data and without any knowledge about the system dynamics. Critic neural network(CNN), action neural network(ANN), and disturbance neural network(DNN) are used to approximate the cost function, control, and disturbance. The weights of these networks compose the augmented weight matrix, and the uniformly ultimately bounded(UUB) of which is proven. The convergence of the tracking error system is also proven. Two examples are given to show the effectiveness of the proposed synchronous solution method for the chaotic system tracking problem.     

Angular quadrature perturbations in radiative transfer theory

A new numerical method is presented for solving the general equation of radiative transfer. The approximation, which replaces the integral term over angle in the transfer equation by a quadrature sum, is studied; an estimate of the error involved is obtained and this error, which may be thought of as a further source or sink of photons (depending upon the sign), can then be used to evaluate a corection to the radiation field originally determined. This process may then be continued as a perturbation series. The method is found to give a final solution, when starting from the Eddington approximation, at least as accurate as that obtained using variable Eddington factors. Furthermore, the technique involves very little extra computing over that required using the Eddington approximation, and may be trivially generalized to any radiative transfer problem. It can also be used in conjunction with any of the existing methods for solving the equation of transfer. Examples are given in the context of spectral line formation in slab geometry.     

基于无记忆状态观测器的多时滞不确定非线性系统的自适应控制

针对一类多时滞不确定非线性系统,研究了基于无记忆状态观测器的自适应控制问题.时滞状态扰动的上界未知,在控制中通过自适应律估计未知参数,并利用估计值设计了不依赖于时滞的无记忆状态观测器和控制器,基于Lyapunov-Krasovskii函数证明了观测误差渐近收敛到零.最后仿真结果说明了该方法的有效性.     

PD-Based Optimal ADRC with Improved Linear Extended State Observer

Taking dead-zone nonlinearlity and external disturbances into account, an active disturbance rejection optimal controller based on a proportional-derivative (PD) control law is proposed by connecting the proportional-integral-derivative (PID) control, the active disturbance rejection control (ADRC) and particle swarm optimization (PSO), with the purpose of providing an efficient and practical technology, and improving the dynamic and steady-state control performances. Firstly, in order to eliminate the negative effects of the dead-zone, a class of 2-order typical single-input single-out system model is established after compensating the dead-zone. Following that, PD control law is introduced to replace the state error feedback control law in ADRC to simplify the control design. By analyzing the characteristics of the traditional linear extended state observer, an improved linear extended state observer is designed, with the purpose of improving the estimation performance of disturbances. Moreover, employing PSO with a designed objective function to optimize parameters of controller to improve control performance. Finally, ten comparative experiments are carried out to verify the effectiveness and superiority of the proposed controller.     

具有死区和饱和输入的自适应混沌控制

研究了一类具有死区饱和输入的不确定时滞混沌系统的跟踪问题.基于滑模控制原理，提出了一种自适应控制的新方案.该方法不仅克服了已有结果的设计缺陷，还取消了死区斜率已知和死区对称的假设，并从理论上分析证明了跟踪误差渐近收敛到零.最后，Duffing-Holme系统的仿真结果表明了该方案的有效性.     

Two-stage feedforward tracking control system with error-based disturbance observer for optical discs

Recently, the scaling up of the storage capacity and data transfer rate of digital storage media has been required. However, increasing in the storage capacity and transfer rate makes optical head control more difficult. Thus, a tracking control system for optical discs must exhibit a high degree of precision control. Consequently, a new two-stage feedforward control (TSFFC) system for high-precision control is proposed in this paper. The proposed system is constructed using two zero phase error tracking (ZPET) control systems based on error prediction and an error-based disturbance observer (EDOB) that uses a notch filter to suppress non periodic disturbances. The proposed control system is designed for DDU-1000 for digital versatile discs (DVDs). The experimental results demonstrate that the proposed system effectively suppresses tracking errors.     

随机扰动下时滞复杂动力网络的一致性

研究了随机扰动下一般时滞复杂动力网络的一致性问题,此复杂动力网络不仅具有随机扰动而且时变时滞同时出现在耦合项和节点系统中,所以这样的网络更具有一般性.基于随机Lyapunov稳定性理论、线性反馈控制理论和线性矩阵不等式,从理论上提出了此网络各个节点与孤立系统达到时滞无关和时滞相关一致性的充分条件.最后的数值模拟验证了理论结果的正确性和有效性.     

Robust chaos synchronization using input-to-state stable control

In this paper, we propose a new input-to-state stable (ISS) synchronization method for a general class of chaotic systems with disturbances. Based on Lyapunov theory and linear matrix inequality (LMI) approach, for the first time, the ISS synchronization controller is presented not only to guarantee the asymptotic synchronization but also to achieve the bounded synchronization error for any bounded disturbance. The proposed controller can be obtained by solving a convex optimization problem represented by the LMI. Simulation studies are presented to demonstrate the effectiveness of the proposed ISS synchronization scheme.     

A novel method to obtain a real-time control force strategy using genetic algorithms for dynamic systems subjected to external arbitrary excitations

The paper deals with a discrete differential dynamic programming type of problem. It is an optimal control problem where an external disturbance is controlled over the time horizon by a control force constituted with the well-known convolution approach. The paper presents a simple and novel idea to achieve an optimally controlled response when a linear system is subjected to an arbitrary external disturbance. The proposed approach uses the convolution concept and states that if a control method can be established to restore a unit external disturbance, then the convolution integral can be applied to generate an overall control strategy to control the system when it is subjected to an arbitrary external disturbance. In spite of its simplicity, such a strategy has not been encountered in the literature. The only requirement for this method to be useful is to obtain an optimal control strategy to suppress the vibration of the system when it is subjected to unit response disturbance. To accomplish this, a method from classical optimal control theory such as linear quadratic regulator (LQR) that involves solving the Riccati equation of the associated system can be used. However, genetic algorithm (GA) can be adopted as an alternative way to obtain an optimal control strategy against impulse input. As any arbitrary excitation can be divided into impulses, the convolution concept will constitute the overall optimal control strategy for any arbitrary excitation with simply shifting, scaling and summation (or integration) of the GA-optimized control strategy for each impulse of the arbitrary excitation. The proposed method can be used for real time control applications. Once the control strategy for the impulse disturbance is established, the results can then be used at each time step when online control is performed. Computer simulations were carried out to control the response of a quarter-vehicle active suspension system using the proposed method. The obtained results were compared to those of linear quadratic regulator (LQR) and passive suspension applications. The overall results demonstrated the effectiveness of the proposed method for active suspension systems, especially in suppressing the vehicle body displacement when compared to both the LQR based and passive systems. Furthermore, such a control system proves to be simpler requiring less information to process, which is crucial for real-time applications.     

Time-dependent aerodynamic non-LTE radiative transfer

A method is developed for solving simultaneously in one dimension the equation of transfer for non-LTE spectral line radiation and the time-dependent equations specifying conservation of mass, energy and linear momentum. In particular, we illustrate the method on a ‘simple’ time-dependent problem in which a pulsating disturbance at some point in a model homogeneous atmosphere propagates towards the surface and steepens into a shock. The resulting emergent intensities show rather dramatic changes over very small time intervals due to the effect of the velocity, density and temperature distributions on the radiative absorption properties of the gas, and thus emphasises the need to solve the above-mentioned four basic equations if one is to obtain physically realistic model atmospheres experiencing initial disturbances.     

Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can often be approximated by the inverse Hessian of the cost functional. Here we focus on highly nonlinear dynamics, in which case this approximation may not be valid. The equation relating the optimal solution error and the errors of the input data is used to construct an approximation of the optimal solution error covariance. Two new methods for computing this covariance are presented: the fully nonlinear ensemble method with sampling error compensation and the ‘effective inverse Hessian’ method. The second method relies on the efficient computation of the inverse Hessian by the quasi-Newton BFGS method with preconditioning. Numerical examples are presented for the model governed by Burgers equation with a nonlinear viscous term.     

NON-FUCHSIAN SINGULARITIES IN QUANTUM MECHANICS

A modification of the spiked harmonic oscillator is studied in the case for which the perturbation potential contains both an inverse power and a linear term. It is then possible to evaluate trial functions by solving an integral equation due to the occurrence of the linear term. The general form of such integral equation is obtained by using a Green-function method, and adding a modified Bessel function of second kind which solves an homogeneous problem with Dirichlet boundary condition at the origin.     

Approximation-error-ADP-based optimal tracking control for chaotic systems with convergence proof

In this paper, an optimal tracking control scheme is proposed for a class of discrete-time chaotic systems using the approximation-error-based adaptive dynamic programming （ADP） algorithm. Via the system transformation, the optimal tracking problem is transformed into an optimal regulation problem, and then the novel optimal tracking control method is proposed. It is shown that for the iterative ADP algorithm with finite approximation error, the iterative performance index functions can converge to a finite neighborhood of the greatest lower bound of all performance index functions under some convergence conditions. Two examples are given to demonstrate the validity of the proposed optimal tracking control scheme for chaotic systems.     

On the possibility of using the Galerkin projection method to model the spatial distribution of minority charge carriers generated by an electron probe in a semiconductor

In this paper, the results of using the Galerkin projection method to model the spatial distribution of minority charge carriers generated by an electron probe in a semiconductor material are presented. The problem is solved in the cylindrical coordinate system. The sought distribution is found in the form of the partial sum of the double Fourier series using the system of modified Laguerre functions. An order estimate of the residual error corresponding to approximate solution of the steady-state equation of their diffusion is given. The approximate calculated results are compared with the exact solution of the one-dimensional equation.     

柱状导体衍射级数

本文对于任意形状的光滑柱状理想导体的衍射提出一种级数解法。方法的原理与层变媒质的Bremmer级数相似:先以内接多面稜柱代替上述光滑柱体;将此稜柱产生的衍射场展为一个级数。级数之首项为几何光学场;级数之第二项为稜柱的所有各稜产生的元衍射场之和,其中每个元衍射场皆取Sommerfeld问题的解,即将该稜之两侧面视为半无限大的平面。上述每一元衍射场皆投射在其相邻稜上,并在相邻稜上发生衍射;这一衍射场随之又投射在下一个相邻稜上而发生衍射;依此类推。按此方式依次被各稜所衍射的场称为“主掠射元场”。级数之第三项即为这些主掠射元场之和。被某一稜A衍射而后又在相邻的稜B上衍射的某一元场,同样会回射到A上;然后以上述“主掠射”方式传递下去,这样的场称为“一次反射元场”。级数的第四项即为这些一次反射元场之和。依此类推。一般说来,级数之第m项(m>3)为m-3次反射元场之和。元场在任何一稜上的衍射皆取Sommerfeld解。当内接多面稜柱之面数趋向无穷,且每面之宽度趋向零时,多面稜柱即趋于光滑柱体,且级数每一项的求和变为一个积分。这时该级数总和之极限即为原问题之解。对级数之前三项单独进行了推导。对于一般的第m项(m>3),导出了一个递推公式。最后,对该级数之收敛条件进行了探讨。     

不确定动态混沌系统的最优控制

对于混沌系统的控制问题,考虑到控制系统能量限制的要求,首先确立一个二次目标函数,然后给出了求解最优控制律的一个简单方法,该方法通过求解线性二次最优控制问题,获得了混沌系统的最优控制律,避免了求解非线性Hamilton-Jacobi-Bellman 偏微分方程(HJB方程)的困难.利用Lyapunov方法证明了闭环系统的稳定性.对统一混沌系统和Liu混沌系统的仿真结果表明了控制策略的有效性.     

不确定动态混沌系统的最优控制

对于混沌系统的控制问题,考虑到控制系统能量限制的要求,首先确立一个二次目标函数,然后给出了求解最优控制律的一个简单方法,该方法通过求解线性二次最优控制问题,获得了混沌系统的最优控制律,避免了求解非线性Hamilton-Jacobi-Bellman偏微分方程(HJB方程)的困难.利用Lyapunov方法证明了闭环系统的稳定性.对统一混沌系统和Liu混沌系统的仿真结果表明了控制策略的有效性.     

A theorem on the representation of the field of forced vibrations of a composite elastic system

A theorem on the representation of a vibration field of an elastic system comprising two subsystems is proved in a general form. It is assumed that the system is linear and the subsystems are rigidly connected and interact along a continuous surface S. According to the theorem, forced vibrations of the system can be represented in the form of the sum of two components, which are the solutions of two simpler auxiliary boundary-value problems. The first component is the field of vibrations of the isolated (separated or blocked along S) subsystems under the effect of preset external forces. The second component represents the forced vibrations of the junction of the subsystems, where the external forces are taken equal to zero and only the reaction forces obtained in solving the first auxiliary problem act at the surface S. The theorem is applied to the problem on the reflection and transmission of elastic waves through a junction of two media. It is demonstrated that the theorem utilization reduces the amount of calculations. Other applications are discussed.     

Burgers方程初边值条件的控制

为将优化控制技巧用于复杂的地球物理环流模型及其它领域,以Burgers方程为模型,描述了初边值条件的优化控制.在一般化的意义上,给出了连续问题及其相应的离散形式.引入伴随变量,并由此导出伴随方程.比较初始控制中不同频率误差对优化控制的影响,分析了用有限观测数据作空间插值后的数据、方法的优化控制能力及效果,并进行了数值实验.     

Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method

A novel approach to solve optimal control problems dealing simultaneously with fractional differential equations and time delay is proposed in this work. More precisely, a set of global radial basis functions are firstly used to approximate the states and control variables in the problem. Then, a collocation method is applied to convert the time-delay fractional optimal control problem to a nonlinear programming one. By solving the resulting challenge, the unknown coefficients of the original one will be finally obtained. In this way, the proposed strategy introduces a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The algorithm’s performance has been analyzed for several non-trivial examples, and the obtained results have shown that this scheme is more accurate, robust, and efficient than most previous methods.