时变系数随机乘积噪声系统的算子谱及其在分析与控制中的应用

Stochastic systems with time-varying coefficients and multiplicative noise can simultaneously characterize the time-varying performance and the influence of exogenous random disturbance, and have become a class of widely used dynamical models in the domain of automatic control. Currently, stability criteria for such type of systems are mostly derived by use of generalized time-varying Lyapunov equation or inequality-based method, which can answer the question whether the examined system is stable, but can not yield quantized analysis about the stationary behavior. Besides, since the existing Grammian criteria of observability and detectability are hard to be combined with control strategy, the internal stability problem of the infinite horizon control remains unsolved up to now. That is why the obtained control results for time-varying stochastic systems mostly focused on the finite horizon case. This project aims to develop an operator-spectral technique for time-varying stochastic systems and establish the spectral criterion of stability. According to the spectral placement, the convergence rate of state trajectory can be intuitively analyzed and the theoretical basis will be supplied for investigating regional stability and spectral assignment. Further, we will present the spectral test of uniform observability(detectability) for the concerned systems and thereby prove that exponential mean square stability is equivalent to the existence and uniqueness of the positive (semi-)definite solution to a generalized time-varying Lyapunov equation. Thus, the difficulty of stabilization will be overcome in infinite horizon LQ and H2/H∞ control problems. Moreover, we will set up a mean square small-gain theorem of time-varying stochastic systems. By which, the stability radius can be determined, and then the robust state-feedback stabilizing controller can be devised.

时变系数随机乘积噪声系统在建模时能够考虑到系统随时间变化的动态特性以及随机因素的干扰，是自动控制领域应用非常广泛的一类动态系统。目前，这类系统的稳定性判据普遍基于广义时变Lyapunov方程方法或不等式方法得到，只能定性的判断系统稳定或不稳定，不能给出稳态性能的量化分析结果。并且，现有的能观（检）性的Gram型判据难以与控制设计相结合，无法解决无限时域控制器的内部稳定性问题，使得控制理论成果多集中在有限时域情形。本课题拟发展时变随机系统的算子谱方法，得到稳定性的谱判据，依据谱分布直观分析系统收敛的快慢，为探讨区域稳定性和谱配置问题提供基本理论依据；给出一致能观（检）性的谱判据，由此证明系统指数均方稳定等价于广义时变Lyapunov方程存在惟一的（半）正定解，进而克服无限时域LQ、H2/H∞控制的镇定难题；建立时变随机系统的均方小增益定理，确定稳定半径，在此基础上设计鲁棒状态反馈镇定控制器。

时变系数随机乘积噪声系统在建模时能够考虑到系统随时间变化的动态特性以及随机因素的干扰，在自动控制、生物基因网络设计、金融安全、智能电网等领域具有广泛应用。本项目分析了连续时间时变系数随机系统的鲁棒性能，建立了随机界实引理，将H∞控制器的存在性问题转化为广义耦合Riccati方程解的存在性问题，并给出了求解Riccati方程的倒向迭代算法；针对状态、扰动和控制同时依赖噪声的随机乘积噪声系统，在能检条件下发展了二人非零和博弈理论，证明了Nash平衡点的存在性，在此基础上解决了无限时域H2/H∞控制器设计问题；分析了含时滞的时变参数不确定随机系统的条件指数均方稳定性、渐近均方稳定性、随机稳定性之间的关系，给出了条件指数均方稳定判据；针对时变系数随机时滞脉冲系统，提出了鲁棒状态反馈镇定控制器的设计方法。本项目所采用的Lyapunov算子分析方法、Nash博弈方法克服了以往随机时变系统稳定性分析与控制设计的不足，并且所研究的系统模型更具一般性，获得的结果便于工程实践，因此结论的适用范围更加广泛。