线性滤波和预测理论的新成果

New Results in Linear Filtering and Prediction Theory'R. 
线性滤波与预测理论的新结果[j]。
E. KALMAN Study, Baltimore, MarylandResearch Institute for AdvancedR. 
E.卡尔曼研究，巴尔的摩，马里兰州高级研究所。
S. BUCYA nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. 
针对最优滤波误差的协方差矩阵，导出了Riccati型的S. BUCYA非线性微分方程。
The solution of this "variance equation" completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics.
该“方差方程”的解完全指定了有限或无限平滑区间以及平稳或非平稳统计量的最优滤波器。
The Johns Hopkins Applied Physics Laboratory, Silver Spring, MarylandThe variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. 
方差方程与变分学中的哈密顿(正则)微分方程密切相关。
Analytic solutions are available in some cases. 
在某些情况下，可以使用解析解。
The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
通过重复、简化或扩展该领域早期结果的例子来说明方差方程的重要性。
The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. 
关于随机估计和确定性控制问题的对偶原理在理论结果的证明中起着重要的作用。
In several examples, the estimation problem and its dual are discussed side-by-side.
在几个例子中，对估计问题及其对偶问题进行了并行讨论。
Properties of the variance equation are of great interest in the theory of adaptive syslems. 
在自适应系统理论中，方差方程的性质引起了人们极大的兴趣。
Some aspects of this are considered briefly.
本文简要地考虑了其中的一些方面。
 

第二段

1 IntroductionAT PRESENT, a nonspecialist might well regard the Wiener-Kolmogorov theory of filtering and prediction [1, 2]3 as "classical'—in short, a field where the techniques are well established and only minor improvements and generalizations can be expected.
目前，非专业人士很可能将Wiener-Kolmogorov滤波和预测理论[1,2]3视为“经典”——简而言之，这是一个技术已经建立良好的领域，只有微小的改进和推广可以期待。
a new approach to the standard filtering and prediction problem. 
标准滤波和预测问题的一种新方法。
The novelty consisted in combining two well-known ideas:(i) the "state-transition" method of describing dynamical systems [12-14], and(ii) linear filtering regarded as orthogonal projection in Hilbert space [15, pp. 150-155].
新颖之处在于结合了两个众所周知的思想:(i)描述动力系统的“状态转移”方法[12-14]，以及(ii)希尔伯特空间中被视为正交投影的线性滤波[15，第150-155页]。
That this is not really so can be seen convincingly from recentresults of Shinbrot [3], Steeg [4], Pugachev [5, 6], and Parzen [7].
从Shinbrot[3]、Steeg[4]、Pugachev[5,6]和Parzen[7]最近的研究结果可以令人信服地看出，事实并非如此。
Using a variety of time-domain methods, these investigators have solved some long-standing problems in nonstationary filtering and prediction theory. 
利用多种时域方法，这些研究者解决了非平稳滤波和预测理论中一些长期存在的问题。
We present here a unified account of our own independent researches during the past two years (which overlap with much of the work [3-7] just mentioned), as well as numerous new results. 
在这里，我们对过去两年中我们自己的独立研究(与刚才提到的许多工作[3-7]重叠)以及许多新结果进行了统一的描述。
We, too, use time-domain methods, and obtain major improvements and generalizations of the conventional Wiener theory. 
我们也使用时域方法，对传统的维纳理论进行了重大改进和推广。
In particular, our methods apply without modification to multivariate problems.
特别是，我们的方法无需修改即可适用于多变量问题。
As an important by-product, this approach yielded the Duality Principle [11, 16] which provides a link between (stochastic) filtering theory and (deterministic) control theory. 
作为一个重要的副产品，这种方法产生了对偶原理[11,16]，它提供了(随机)滤波理论和(确定性)控制理论之间的联系。
Because of the duality, results on the optimal design of linear control systems [13, 16, 17] are directly applicable to the Wiener problem. 
由于对偶性，线性控制系统的优化设计结果[13,16,17]直接适用于Wiener问题。
Duality plays an important role in this paper also.
对偶在本文中也起着重要的作用。
When the authors became aware of each other's work, it was soon realized that the principal conclusion of both investigations was identical, in spite of the difference in methods:The following is the historical background of this paper.
当作者意识到彼此的工作，它很快就意识到，这两个调查的主要结论是相同的，尽管在方法上有所不同。
In an extension of the standard Wiener filtering problem, Follin[8] obtained relationships between time-varying gains and error variances for a given circuit configuration. 
在标准维纳滤波问题的扩展中，Follin[8]得到了给定电路配置的时变增益与误差方差之间的关系。
Later, Hanson [9] proved that Follin's circuit configuration was actually optimal for the assumed statistics; 
后来，Hanson[9]证明对于假设的统计量，Follin的电路配置实际上是最优的;
moreover, he showed that the differential equations for the error variance (first obtained by Follin) follow rigorously from the Wiener-Hopf equation. 
此外，他还证明了误差方差的微分方程(首先由Follin得到)严格遵循Wiener-Hopf方程。
These results were then generalized by Bucy [10], who found explicit relationships between the optimal weighting functions and the error variances; 
Bucy对这些结果进行了推广[10]，他发现了最优加权函数与误差方差之间的明确关系;
he also gave a rigorous derivation of the variance equations and those of the optimal filter for a wide class of nonstationary signal and noise statistics.
他还给出了方差方程的严格推导，以及对一类广泛的非平稳信号和噪声统计的最优滤波器的推导。
Rather than to attack the Wiener-Hopf integral equation directly, it is better to convert it into a nonlinear differential equation, whose solution vields the covariance matriz of the minimum filtering error, which in turn contains all necessary information for the design of the optimal filter.2 
与其直接攻击Wiener-Hopf积分方程，不如将其转化为一个非线性微分方程，其解为最小滤波误差的协方差矩阵，而协方差矩阵又包含了设计最优滤波器所需的所有信息
Summary of Results: DescriptionThe problem considered in this paper is stated precisely in Section 4. 
结果总结:描述本文所考虑的问题在第4节中进行了精确的说明。
There are two main assumptions:(A) A sufficiently accurate model of the message process is given by a linear (possibly time-varying) dynamical system excited by white noise.(A) Every observed signal contains an additive white noise component.
有两个主要的假设:(A)信息过程的一个足够精确的模型是由一个受白噪声激励的线性(可能时变的)动力系统给出的。(A)每个观测到的信号都包含一个加性白噪声成分。
Independently of the work just mentioned, Kalman [11] gaveiThis research was partially supported by the United States Air Force under Contracts AF 49(638)-382 and AF 33(616)-6952 and by the Bureau of Naval Weapons under Contract NOrd-73861.7212 Bellona AvenueAssumption (As) is unnecessary when the random processes in question are sampled (discrete-time parameter); 
独立于刚才提到的工作，Kalman[11]给出了该研究部分由美国空军根据合同AF 49(638)-382和AF 33(616)-6952以及海军武器局根据合同NOrd-73861.7212 Bellona avene支持。当所讨论的随机过程被采样(离散时间参数)时，假设(As)是不必要的;
see [11). 
见(11)。
Even in the continuous-time case, (Aa) is no real restriction since it can be removed in various ways as will be shown in a future paper. 
即使在连续时间的情况下，(Aa)也不是真正的限制，因为它可以通过各种方式去除，这将在以后的论文中展示。
Assumption (Ai), however, is quite basic; 
然而，假设(Ai)是非常基本的;
it is analogous to but somewhat less restrictive than the assumption of rational spectra in the conventional theory.
它类似于传统理论中的有理谱假设，但限制较少。
Numbers in brackets designate References at the end of paper.
括号内的数字表示论文末尾的参考文献。
Contributed by the Instruments and Regulators Division of THe AMERICAN SoCIETY OP MECHANICAL ENGINEERS and presented at the Joint Automatic Controls Conference, Cambridge, Mass. 
由美国机械工程师协会仪器和调节器部门贡献，并在马萨诸塞州剑桥举行的联合自动控制会议上发表。
September 7-9, 1960. 
1960年9月7日至9日。
Manuscript received at ASME Headquarters, May 31, 1960. 
手稿于1960年5月31日在ASME总部收到。
Paper No. 60—JAC-12.
论文号:60-JAC-12。
Within these assumptions, we seek the best linear estimate of the message based on past data lying in either a finite or infinite time-interval.
在这些假设中，我们寻求基于有限或无限时间间隔内的过去数据的消息的最佳线性估计。
The fundamental relations of our new approach consist of five equations:Journal of Basic EngineeringMARCH 196195
我们的新方法的基本关系由五个方程组成:基础工程杂志196195三月