【作者】 于淼；

【导师】 刘建昌；

【作者基本信息】 东北大学 ， 控制理论与控制工程， 2019， 博士

【摘要】 连续时间随机系统是指各个变量是时间的连续函数并且状态和输出向量受到噪声干扰的动态系统。由于生物学、经济学以及物理学等领域都存在连续随机的现象,连续时间随机系统的辨识受到了各领域专家学者的广泛关注。此外,许多工业过程也具有连续随机属性,特别是高炉冶金、石油化工等流程工业,它们的模型大多数是由微分方程描述的连续时间随机系统。目前,连续时间随机系统的辨识研究还存在着许多亟待解决的难题,例如怎样降低辨识方法的计算负荷和提高辨识模型的精确性。显然,针对连续时间随机系统的辨识问题提出相应的解决方案,无论在理论上,还是在系统的实际应用方面都具有非常重要的研究价值和意义。本文在连续时间随机系统的框架下,在子空间辨识的方法实现、方法性能以及方法应用等方面进行了深入的分析和研究,并取得了一系列具有创新性的研究成果。本文的主要工作和研究成果如下:(1)为了降低连续时间随机系统噪声的影响以及获得系统的最优模型阶数,提出了基于卡尔曼滤波器的核范数子空间辨识方法。生物学上的鼠疫病模型是一个典型的连续时间随机系统,我们以连续时间随机鼠疫病模型为研究背景。首先,对连续时间随机系统进行卡尔曼滤波器的设计,以估计误差的均方差最小为前提,得到其中待定矩阵,从而获得随机系统的等价新息形式。其次,对于广义状态空间模型进行矩阵投影运算,得到由输入输出数据构成的投影矩阵,对其构造核范数最优化问题,通过交替方向乘子法对增广拉格朗日问题进行求解,得到最优输出序列。然后,将得到的输出序列代入投影矩阵中进行奇异值分解,得到连续时间随机系统最优模型阶数,进而估计系统矩阵和噪声强度。最后,研究了连续时间随机鼠疫病模型的仿真实验。(2)针对连续时间系统辨识过程中易产生的零极点转换不一致、没有直接解决时间导数等问题,提出了基于分布理论的核范数子空间辨识方法。首先,利用分布理论中伊藤积分的概念,对系统输出方程进行微积分数值处理,从而建立分布意义下的输入输出矩阵方程。其次,通过卡尔曼滤波器获得最优状态估计,从而得到广义状态空间模型及其投影矩阵,并利用核范数最小化方法对投影矩阵进行优化。然后,通过交替方向乘子法对构造的增广拉格朗日问题进行求解,并对得到的投影矩阵进行奇异值分解,进而估计系统矩阵和噪声强度。最后,在高斯测试函数的基础上,研究了两类连续时间随机系统的数值仿真。(3)针对利用输入输出数据建立的离线模型不能有效跟踪系统的动态变化的问题,提出了基于分布理论的递推子空间辨识方法。田纳西伊斯曼化工过程具有连续性、随机性以及动态性等特性,我们以田纳西伊斯曼化工过程的数据作为研究基础。首先,通过分部积分法得到时变连续随机分布函数在分布意义下的时间导数,并在某时刻周围的小区间内建立输入输出矩阵方程。其次,为了提高辨识模型的精确性,引入一个正定的加权函数,得到加权输入输出矩阵方程。然后,对输入输出数据组成的等价方程进行QR分解,采用将矩阵“R”规模固定的方法达到数据压缩的目的。对得到的投影矩阵进行特征值分解,从而递推估计系统矩阵和噪声强度。最后,研究了田纳西伊斯曼化工过程的仿真实验。(4)针对实际生产过程中一些变量难以在线预测的问题,提出了基于子空间主角旋转的子空间预测方法。连续搅拌釜反应过程具有连续性,并且系统也频繁地受到确定性的震荡干扰,我们以连续搅拌釜反应过程的数据作为研究基础。首先,采用随机分布理论得到连续时间随机系统的广义能观测性矩阵,并利用广义能观测性矩阵列空间的过去与现在时刻列向量的旋转得到列空间之间的主角,从而得到其角速度。通过角速度预测现在和将来列向量之间的主角,得到将来的广义能观测性矩阵列空间。其次,对于系统矩阵变化快速的情况,通过过去与现在时刻列向量之间的角速度得到角加速度,预测系统将来的信号列空间,并得到基于子空间主角旋转的递推子空间预测方法。最后,研究了连续搅拌釜反应器系统的仿真实验。更多还原

【Abstract】 Continuous-time stochastic systems are dynamic systems in which each variable is a continuous function of time and the state and output vectors are disturbed by noise.Due to the existent phenomenon of continuous stochastic in biology,economics and physics,the identification of continuous-time stochastic systems has attracted much attention by experts and scholars in various fields.In addition,many industrial processes are of the continuous stochastic nature,especially in the blast furnace metallurgy and petrochemical industry.Most of their models axe the continuous-time stochastic systems described by differential equations.At present,there are still many problems to be solved in the identification of continuous-time stochastic systems,such as how to reduce the computational load of the identification method and improve the accuracy of the identification model.Obviously,how to solve the identification problem of the continuous-time stochastic systems is of theoretical and practical values.Based on the structure of the continuoustime stochastic systems,this dissertation deeply researches the implementation,the property analysis and the application of the subspace identification method,and provides lots of important theoretical results.The main work and research results of this dissertation are as follows:(1)In order to reduce the influence of system noise and optimize the system order,the nuclear norm subspace identification method based on the Kalman filter is studied.The biology of plague model is a typical continuous-time stochastic system.We take the continuous-time stochastic plague model as the research background.First of all,the Kalman filter of the continuous-time stochastic system is assumed with the undetermined matrices.The equivalent innovation form of the system is obtained on condition that mean square error is minimum.Secondly,the projection matrix is obtained by calculating the extended observability matrix.Constructing nuclear norm minimization with the projection matrix,the augmented Lagrangian of the problem is solved by alternating direction multiplier method,and the most optimal output sequences are obtained.Thirdly,substituting the output sequences into the projection matrix,the most optimal system order is given by singular value decomposition,and system matrices and noise intensity are obtained.Finally,the simulation experiment of the continuous-time stochastic model of plague is studied.(2)For the zeros of the identification process of the continuous-time stochastic system are not easily translatable to the poles,the nuclear norm subspace identification via distribution-based approach is proposed.Firstly,random distribution theory in the sense of Ito-Schwartz is introduced to differential and integral calculus of the output equation,and we have the input-output algebraic relationship.Secondly,the optimal state estimation is obtained by the Kalman filter,and an extended state space model and the projection matrix is given.Thirdly,the nuclear norm minimization is used to optimize the projection matrix.The augmented Lagrangian of the problem is solved by alternating direction multiplier method.System matrices and noise intensity are obtained by performing the singular value decomposition on the projection matrix.Finally,the numerical simulations of the scalar stochastic system and continuous-time stochastic systems are studied.(3)In view of the off-line model cannot effectively track the dynamic of the systems,the recursive subspace identification based on random distribution theory is researched.The Tennessee-Eastman process has the characteristic of continuity,randomness and dynamism.The data of the Tennessee-Eastman process is used as the research basis.Firstly,the time-derivative in sense of the distribution of the time-varying continuous stochastic process is calculated by using integration by parts.The input-output matrix equation is given in a small interval.By introducing a positive-definite the weight function,we have the weighted input-output algebraic equation.Secondly,performing the QR decomposition on the equivalent equation composed by input-output data and fixing the size of the "R" data matrices,the system matrices are obtained by performing the eigenvalue decomposition recursively.Finally,the simulation experiment of the Tennessee-Eastman chemical process is studied.(4)Aiming at some variables are hard to prediction on-line in production processes,the recursive subspace prediction based on the rotation of the principal angle is proposed.The continuous stirred tank heater process is of the continuous stochastic nature.In addition,the system is also subjected to the deterministic oscillatory disturbances frequently.The data of the continuous stirred tank heater process are used as the research basis.Firstly,the extended observability matrices of continuous-time stochastic systems are obtained by random distribution theory.Secondly,by using the angle between past and present subspaces spanned by the extended observability matrices,the future signal subspace is predicted by rotating the present subspace in the geometrical sense.In order to predict the future signal subspace,the angular velocity and acceleration of the signal subspace have been derived.For the changes of the system matrices are so fast,the recursive subspace prediction based on the rotation of the principal angle is presented.Finally,the simulation experiment of the continuous stirred tank heating system is studied.更多还原