Comparison of uncertainty in state equation based on probabilistic approach and interval analysis method
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摘要:
基于现代控制理论中状态方程的求解算法，对具有参数不确定性的控制系统采用非概率区间分析方法与随机控制理论进行研究。首先明确实际工程应用中不确定性的概念和影响，分别建立了区间值和随机过程2种描述方法，求解系统的响应区间，并分为与初始条件和输入相关的零输入和零状态两部分不确定量。根据区间数学中的区间函数扩张原理和概率统计理论中的切比雪夫不等式，从数学证明和数值计算2个方面，分别用非概率区间分析和概率统计方法求解不确定系统的响应，并对二者进行比较，分析其相容性。结果表明，在由概率统计信息得到不确定性变量的区间向量为系统输入的情况下，非概率区间分析方法得到的响应区间包含由随机控制理论得到的响应区间。
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.
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图 1 无扰动量的第一自由度位移响应曲线
Figure 1. First degree of freedom of displacement responsecurves without uncertainty
图 2 随机过程的系统响应区间
Figure 2. System response interval of state equation byprobabilistic approach
图 3 区间分析的系统响应区间
Figure 3. System response interval of state equation byinterval analysis method
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