多自由度拟不可积哈密顿系统的随机分数阶最优控制

推广了适用于分数阶系统控制的随机分数阶最优控制策略,提出了高斯白噪声激励下多自由度拟不可积哈密顿系统以响应最小化为目标的随机分数阶最优控制策略.首先,应用拟不可积哈密顿系统随机平均法,将受控系统简化为关于能量的部分平均伊藤方程.然后,将控制性能指标中关于控制力的部分表示为分数阶形式,结合随机动态规划原理,建立并求解部分平均系统的无界遍历控制的随机动态规划方程,获得了随机分数阶最优控制律.最后,采用一个算例验证了随机分数阶控制策略的控制效果和控制效率.研究表明,随机分数阶最优控制策略对传统的整数阶随机动力学系统同样适用,能比传统的整数阶控制策略取得更好的控制效果.另外,随着激励强度增加,整数阶控制策略的控制效率显著降低;而分数阶控制策略的控制效率虽比整数阶控制策略的控制效率略低,但随着激励强度的增加,分数阶控制策略的控制效率缓慢上升并趋于平稳,可以有效地缓解控制效率与控制效果之间的矛盾.     

随机扰动下简单电力系统的可靠度反馈最大化

研究随机扰动下简单电力系统的可靠度反馈最大化.应用拟不可积哈密顿系统随机平均法和随机动态规划原理,导出以可靠度最大为目标的动态规划方程和以平均首次穿越时间最长为目标的动态规划方程.通过分别求解相应的动态规划方程,得到最优控制律,受控与未控系统的条件可靠性函数及平均首次穿越时间.最后应用Monte Carlo模拟验证结果的准确性.     

不确定随机离散分布时滞神经网络的鲁棒稳定性*

采用Its微分公式和不等式分析技巧,研究了一类不确定随机离散分布时滞神经网络的鲁棒稳定性问题。该模型同时考虑了神经网络模型的两种扰动因素,即随机扰动与不确定性扰动。通过构造适当的Lyapunov泛函,以线性矩阵不等式形式给出了系统在均方根意义下的全局鲁棒稳定性判据,能够利用LMI工具箱很容易地进行检验。此外,仿真结果进一步证明了结论的有效性。     

不确定噪声下确保控制性能的鲁棒LQG调节器

本文讨论了一类具有不确定噪声的离散时间随机线性系统的鲁棒ＬＱＧ问题，文章给出了确保控制性能的不确定噪声协方差矩阵的扰动上界，以及极小极大鲁棒ＬＱＧ调节器的设计方法，采用这种调节器不仅能极小化不确定下的最坏性能，而且也能确保控制性能指标达到给定的自由度内。     

具有不确定Wiener噪声随机非线性系统的自适应逆最优控制

针对一类具有不确定Wiener噪声扰动和未知定常参数的随机非线性系统,采用随 机微分方程描述系统,基于Backstepping算法,利用随机控制Lyapunov函数,研究了自适 应逆最优控制问题的可解定理,系统地给出了全局依概率渐近稳定和自适应逆最优控制策略 的设计方法.这种方法可同时获得控制律和自适应律,仿真结果表明该控制算法的有效性.     

一种基于随机平均的最优时滞控制方法

基于时滞系统的随机平均法与随机动态规划原理,提出一种非线性系统的随机最优时滞控制方法．先应用时滞随机平均法,将非线性系统的随机最优时滞控制问题变换成非时滞的最优控制问题;再根据随机动态规划原理,建立其动态规划方程;由此确定最优时滞控制律;最后,通过一个例子说明该时滞控制方法的控制效果．     

随机Rayleigh振子的首次穿越和最优控制

研究随机Rayleigh振子的首次穿越和最优控制问题.利用随机平均法给出了系统运动方程的随机平均微分方程,并对平均方程建立了条件可靠性函数的后向Kolmogorov方程,得出相应的首次穿越条件概率密度函数,并利用Lyapunov指数法对受控系统的平均方程进行了随机稳定化,得到了最优控制率.     

具分布参数的随机Hopfield神经网络的指数稳定

基于随机Fubini定理,将随机偏微分方程描述的Hopfield神经网络系统转化为用相应的随机常微分方程来描述.利用关于空间变量平均的Lyapunov函数与Ito^公式,通过对所构造的Lyapunov函数在Ito^微分规则下对相应系统求导的方法,获得了系统指数稳定的代数判据及其Lyapunov指数估计.实现了运用Lyapunov直接法对分布参数系统稳定性的研究.     

随机激励的非线性Markov跳变系统的稳态响应

大量实际工程问题需要用同时包含连续和离散变量的Markov跳变系统来描述.本文介绍了一类随机激励的单自由度(强)非线性Markov跳变系统的稳态响应的研究方法.首先,基于随机平均法导出具有Markov跳变参数的平均It随机微分方程,原系统方程的维数得到降低.接着,根据跳变过程原理,建立Fokker-Planck-Kolmogorov(FPK)方程组,方程组中的方程与系统的结构状态一一对应且互相耦合.求解该FPK方程组,得到Markov跳变系统的稳态随机响应及其统计量.最后,以一个高斯白噪声激励的Markov跳变Duffing振子为例,计算得到不同跳变规律下系统的稳态响应.研究结果表明,Markov跳变系统的稳态响应可以看作是各结构状态子系统稳态响应的加权和,加权值由跳变规律决定.     

基于状态反馈精确线性化的三相四桥臂逆变器的控制

根据三相四桥臂逆变器的工作原理,应用开关函数建立了控制系统数学模型,引入开关周期平均算子将离散的系统转化为连续系统.根据系统的主要控制目标选取状态变量、输入变量和输出变量,得到适合于微分几何方法的3输入3输出的仿射非线性系统模型.根据非线性微分几何理论,从理论上证明了该模型满足多输入、多输出系统精确线性化的条件,推导出非线性状态反馈控制律.对非线性坐标变换后得到的线性系统,利用二次型最优控制策略时,根据无源性控制方法的思想,提出一种闭环系统能量函数,并推导出权矩阵的参数形式.将最优化得到的控制律进行逆变换来实现原系统的优化控制设计.仿真结果验证了该方法的有效性和正确性.     

Stochastic minimax optimal control strategy for uncertain quasi-Hamiltonian systems using stochastic maximum principle

A stochastic minimax optimal control strategy for uncertain quasi-Hamiltonian systems is proposed based on the stochastic averaging method, stochastic maximum principle and stochastic differential game theory. First, the partially completed averaged Itô stochastic differential equations are derived from a given system by using the stochastic averaging method for quasi-Hamiltonian systems with uncertain parameters. Then, the stochastic Hamiltonian system for minimax optimal control with a given performance index is established based on the stochastic maximum principle. The worst disturbances are determined by minimizing the Hamiltonian function, and the worst-case optimal controls are obtained by maximizing the minimal Hamiltonian function. The differential equation for adjoint process as a function of system energy is derived from the adjoint equation by using the Itô differential rule. Finally, two examples of controlled uncertain quasi-Hamiltonian systems are worked out to illustrate the application and effectiveness of the proposed control strategy.     

Stochastic minimax control for stabilizing uncertain quasi- integrable Hamiltonian systems

A procedure for designing feedback control to asymptotically stabilize, with probability one, quasi-integrable Hamiltonian systems with bounded uncertain parametric disturbances is proposed. First, the partially averaged Itô stochastic differential equations are derived from given system by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Second, the Hamilton-Jacobi-Issacs (HJI) equation for the ergodic control problem of the averaged system and a performance index with undetermined cost function is established based on the principle of optimality. This equation is then solved to yield the worst disturbances and the associated optimal controls. Third, the asymptotic Lyapunov stability with probability one of the optimally controlled system with worst disturbances is analyzed by evaluating the maximal Lyapunov exponent of the fully averaged Itô equations. Finally, the cost function and feedback control are determined by the requirement of stabilizing the worst-disturbed system. A simple example is worked out to illustrate the application of the proposed procedure and the effects of optimal control on stabilizing the uncertain system.     

A stochastically averaged optimal control strategy for quasi-Hamiltonian systems with actuator saturation

A stochastic optimal control strategy for quasi-Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method and stochastic dynamical programming principle. First, the partially completed averaged Itô stochastic differential equations for the energy processes of individual degree of freedom are derived by using the stochastic averaging method for quasi-Hamiltonian systems. Then, the dynamical programming equation is established by applying the stochastic dynamical programming principle to the partially completed averaged Itô equations with a performance index. The saturated optimal control consisting of unbounded optimal control and bounded bang-bang control is determined by solving the dynamical programming equation. Numerical results show that the proposed control strategy significantly improves the control efficiency and chattering attenuation of the corresponding bang-bang control.     

Robustness of non-linear stochastic optimal control for quasi-Hamiltonian systems with parametric uncertainty

The robustness of non-linear stochastic optimal control for quasi-Hamiltonian systems with uncertain parameters is studied. Based on the independence of uncertain parameters and stochastic excitations, the non-linear stochastic optimal control for the nominal quasi-Hamiltonian system with average-value parameters is first obtained by using the stochastic averaging method and stochastic dynamical programming principle. Then, the means and standard deviations of root-mean-square responses, control effectiveness and control efficiency for the uncertain quasi-Hamiltonian system are calculated by using the stochastic averaging method and the probabilistic analysis. By introducing the sensitivity of the variation coefficients of controlled root-mean-square responses, control effectiveness and control efficiency to those of uncertain parameters, the robustness of the non-linear stochastic optimal control is evaluated. Two examples are given to illustrate the proposed control procedure and its robustness.     

Probability‐weighted nonlinear stochastic optimal control strategy of quasi‐integrable Hamiltonian systems with uncertain parameters

The nonlinear stochastic optimal control problem of quasi‐integrable Hamiltonian systems with uncertain parameters is investigated. The uncertain parameters are described by using a random vector with λ probability density function. First, the partially averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi‐integrable Hamiltonian systems. Then, the dynamical programming equation is established based on stochastic dynamical programming principle. By minimizing the dynamical programming equation with respect to control forces, the optimal control forces can be derived, which are functions of the uncertain parameters. The final optimal control forces are then determined by probability‐weighted average of the obtained control forces with the probability density of the uncertain parameters as weighting function. The mean control effectiveness and mean control efficiency are used to evaluate the proposed control strategy. The robustness of the proposed control is measured by using the ratios of the variation coefficients of mean control effectiveness and mean control efficiency to the variation coefficients of uncertain parameters. Finally, two examples are given to illustrate the proposed control strategy and its effectiveness and robustness. Copyright © 2014 John Wiley & Sons, Ltd.     

On the worst-case disturbance of minimax optimal control

We consider a minimax optimal control problem for uncertain stochastic systems. The uncertainty in the underlying stochastic system is formulated in terms of probability measure perturbations satisfying a relative entropy constraint. By characterizing the worst-case measure for a related stochastic minimax game, it is shown that the worst-case uncertain system can be represented in the form of a parametric perturbation of the nominal system. A numerical example is presented to illustrate theoretical results developed in this paper.     

Finite Horizon Minimax Optimal Control of Stochastic Partially Observed Time Varying Uncertain Systems

We consider a linear-quadratic problem of minimax optimal control for stochastic uncertain control systems with output measurement. The uncertainty in the system satisfies a stochastic integral quadratic constraint. To convert the constrained optimization problem into an unconstrained one, a special S-procedure is applied. The resulting unconstrained game-type optimization problem is then converted into a risk-sensitive stochastic control problem with an exponential-of-integral cost functional. This is achieved via a certain duality relation between stochastic dynamic games and risk-sensitive stochastic control. The solution of the risk-sensitive stochastic control problem in terms of a pair of differential matrix Riccati equations is then used to establish a minimax optimal control law for the original uncertain system with uncertainty subject to the stochastic integral quadratic constraint. Date received: May 13, 1997. Date revised: March 18, 1998.     

Stochastic Uncertain Systems Subject to Relative Entropy Constraints: Induced Norms and Monotonicity Properties of Minimax Games

Entropy and relative entropy are fundamental concepts on which information theory is founded on, and in general, telecommunication systems design. On the other hand, dissipation inequalities, minimax strategies, and induced norms are the basic concepts on which robustness of uncertain control and estimation of systems are founded on. In this paper, the precise relation between these notions is investigated. In particular, it will be shown that the higher the dissipation the higher the entropy of the system, which has implications in computing the induced norm associated with robustness. These connections are obtained by considering stochastic optimal uncertain control systems, in which uncertainty is described by a relative entropy constraint between the nominal and uncertain measures, while the pay-off is a linear functional of the uncertain measure. This is a minimax game, in which the controller measure seeks to minimize the pay-off, while the disturbance measure aims at maximizing the pay-off. Salient properties of the minimax solution are derived, including a characterization of the optimal sensitivity reduction, computation of the induced norm, monotonicity properties of minimax solution, and relations between dissipation and relative entropy of the system. The theory is developed in an abstract setting and then applied to nonlinear partially observable continuous-time uncertain controlled systems, in which the nominal and uncertain systems are described by conditional distributions. In addition, existence of the optimal control policy among the class of policies known as wide-sense control laws is shown, and an explicit formulae for the worst case conditional measure is derived. The results are applied to linear-quadratic-Gaussian problems