The uniform distribution: A rigorous justification for its use in robustness analysis
The uniform distribution: A rigorous justification for its use in robustness analysis
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DOI：
10.1007/bf01211503
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发表时间：
1996-12
期刊：
影响因子：
--
通讯作者：
B. Barmish;C. Lagoa
中科院分区：
文献类型：
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作者：
B. Barmish;C. Lagoa
考虑一个控制系统，其不确定参数的容许值超过经典鲁棒性理论所规定的界限。在这种情况下，重要的是量化性能下降的风险和增加的不确定性容忍度之间的权衡。如果可以确定不确定性界限的大幅增加，则可以接受的小风险通常是合理的。由于鲁棒性问题的公式不包括不确定性的统计描述，问题是是否有可能提供这样的保证在一个“分布免费”的方式。换句话说，如果f * 表示不确定性q的一类可能的概率分布，我们寻找具有以下性质的最坏情况f *ε *：f * 下的性能满意度概率小于任何其他f ε * 下的概率。换句话说，f* 提供了最好的保证。这个新的框架说明了鲁棒稳定性问题与Kharitonov定理和边缘定理。主要结果是直接描述的：令p（s，q）表示所考虑的不确定多项式，并取P（ω）为不确定值p（jω，q）的频率依赖凸目标集（在复平面上）。与值集分析一致，P（ω）被假设为关于名义p（jω，0）对称。不确定性参数qi取为零均值独立随机变量，且支持区间已知。对于每一个不确定性，假设类的密度函数是对称的，在零的每一侧非递增。然后，对于固定频率ω，第一定理表明p（jω，q）在P（ω）中的概率由q的均匀分布最小化。第二个定理是第一个定理的推广，它表明同样的结果对于频率一致成立。然后在第三定理中给出了鲁棒稳定性的概率保证。事实证明，在许多情况下，经典的鲁棒性裕度可以远远超过，同时保持不稳定的风险出奇地小。最后，对于一类更一般的不确定性结构，本文还建立了f * 可以由截断均匀分布估计的事实。
Consider a control system which is operated with admissible values of uncertain parameters which exceed the bounds specified by classical robustness theory. In this case it is important to quantify the tradeoffs between risk of performance degradation and increased tolerance of uncertainty. If a large increase in the uncertainty bound can be established, an acceptably small risk may often be justified. Since robustness problem formulations do not include statistical descriptions of the uncertainty, the question arises whether it is possible to provide such assurances in a “distribution-free” manner. In other words, if ℱ denotes a class of possible probability distributions for the uncertaintyq, we seek some worst-casef*ε ℱ having the following property: The probability of performance satisfaction underf*is smaller than the probability under any otherfε ℱ. Said another way,f*provides the best possible guarantee. This new framework is illustrated on robust stability problems associated with Kharitonov's theorem and the Edge Theorem. The main results are straightforward to describe: Letp(s, q)denote the uncertain polynomial under consideration and takeP(ω) to be a frequency-dependent convex target set (in the complex plane) for the uncertain valuesp(jω, q). Consistent with value set analysis,P(ω) is assumed to be symmetric with respect to the nominalp(jω, 0). The uncertain parametersqiare taken to be zero-mean independent random variables with known support interval. For each uncertainty, the class ℱ is assumed to consist of density functions which are symmetric and nonincreasing on each side of zero. Then, for fixed frequencyω, the first theorem indicates that the probability thatp(jω, q)is inP(ω) is minimized by the uniform distribution forq. The second theorem, a generalization of the first, indicates that the same result holds uniformly with respect to frequency. Then probabilistic guarantees for robust stability are given in the third theorem. It turns out that in many cases, classical robustness margins can be far exceeded while keeping the risk of instability surprisingly small. Finally, for a much more general class of uncertainty structures, this paper also establishes the fact thatf*can be estimated by a truncated uniform distribution.