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Comparison of uncertainty in state equation based on probabilistic approach and interval analysis method
QIU Jingbo , REN Zhang , LI Qingdong , DONG Xiwang
School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2016-01-06; Accepted: 2016-04-29; Published online: 2016-05-23 09:00
Foundation item: Innovation Found of Aviation Industry Corporation of China (cxy2012BH01)
Corresponding author. E-mail:liqingdong@buaa.edu.cn
Abstract: Based on the solution algorithm of state equation in modern control theory, analysis and comparison between interval analysis method and stochastic process are proposed to solve control system with uncertain but bounded parameters. After the definition and influence of uncertainty in engineering practice are known, the uncertain parameters were expressed in the forms of interval and stochastic process. To obtain the response of the system, uncertain variables are divided into the one related to initial condition and the other concerned in system input:zero input response and zero state response. According to extension principle of interval function in interval analysis and Chebyshev's inequality in probability and statistics theory, based on mathematical proof and numerical calculation, the problem of compatibility of using non-probabilistic interval analysis method and probabilistic approach is resolved. The results illustrate that with the uncertain input interval vector which is acquired by probabilistic approach, the system's response interval acquired by non-probabilistic interval analysis method contains the one obtained by probabilistic approach.
Key words: statespace analysis     uncertainty     stochastic process     Chebyshev's inequality     interval analysis     extension principle of interval function

﻿自动控制理论源自于力学，经过长期的发展已经形成了一门独立的学科。20世纪60年代，随着计算机的发展，控制理论从经典的控制方法向以状态空间为标志的现代控制理论发展，状态空间分析成为研究最优控制、滤波问题和系统辨识的基础。在实际应用中，闭环控制系统的性能，也就是系统动力学响应输出，是工程界关注的重点。不管是飞行力学、结构动力学、振动主动控制还是飞行控制等研究领域都需要研究系统的动力响应输出问题[1-2]。Taghipour等[3]利用状态空间模型方法研究了时域与频域混合情况下的系统动力学响应问题。Johansson[4]基于状态空间模型研究了含不确定时变参数的模型确认问题。

1 问题的描述

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2 区间分析和随机过程求解方程 2.1 区间系统中求解状态方程

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wc(t)为与输入相关的区间中值;ΔwI(t)为其扰动区间; Δw(t)为区间半径。

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xc(t0)为与初始值相关的区间中值;ΔxI(t0)为其扰动区间;Δx(t0)为区间半径。

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2.2 随机系统中求解状态方程

1) 假设随机扰动δw(τ)为连续时间高斯白噪声，则其功率密度为常数。为简化分析，也符合大多数实际情况，一般假设其均值为零，即

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2) 假设状态初值δx(t0)为高斯随机向量，即

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3) 假设δw(τ)δx(t0)互相独立，即满足：

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2.3 随机系统和区间系统状态方程解的相容性

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2.3.1 与初始条件相关的随机系统和区间系统状态方程解的比较研究

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2.3.2 与输入相关的随机系统和区间系统状态方程解的比较研究

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3 数值算例

 图 1 无扰动量的第一自由度位移响应曲线 Fig. 1 First degree of freedom of displacement responsecurves without uncertainty
 图 2 随机过程的系统响应区间 Fig. 2 System response interval of state equation byprobabilistic approach
 图 3 区间分析的系统响应区间 Fig. 3 System response interval of state equation byinterval analysis method
 图 4 随机过程和区间分析的系统响应区间比较 Fig. 4 Comparison of system response intervals of state equation obtained by probabilistic approachand interval analysis method

4 结 论

1) 非概率集合理论可求出系统的最大可能响应和最小可能响应，计算量小。在很高的可靠度上，其响应界限要比概率模型所得的更实际，可作为概率模型研究不确定性问题的补充手段和方法。

2) 随机过程理论方法的应用具有一定程度的普适性。目前的工程实践中也习惯把不确定性归结为随机性。应用这种方法需要获得充分的统计数据，且估计的边界一般较为保守。

3) 无论是零输入响应，还是零状态响应的状态方程解的响应区间都遵循同样的规律，即非概率区间分析方法得到的响应区间包含由随机过程得到的响应区间。区间分析所得的响应区间比随机过程的更宽、更保守。

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#### 文章信息

QIU Jingbo, REN Zhang, LI Qingdong, DONG Xiwang

Comparison of uncertainty in state equation based on probabilistic approach and interval analysis method

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(1): 151-158
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0021