Total least squares for affinely structured matrices and the noisy realization problem
Total least squares for affinely structured matrices and the noisy realization problem
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DOI：
10.1109/78.330370
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发表时间：
1994-11
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影响因子：
--
通讯作者：
B. Moor
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文献类型：
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作者：
B. Moor
结构化秩亏矩阵出现在信号处理、系统辨识和控制理论的许多应用中。作者讨论了结构化总体最小二乘（STLS）问题，该问题是通过类似结构的秩亏矩阵来逼近仿射结构矩阵（即参数中的仿射矩阵），同时最小化 L/sub 2/-误差准则的问题。结果表明，最优性条件导致非线性广义奇异值分解，可以通过受逆迭代启发的算法来求解。接下来作者集中讨论所谓的 L/sub 2/-最优噪声实现问题，该问题相当于通过给定阶数的有限维、时不变线性系统的脉冲响应来逼近给定的数据序列。这可以作为结构化总体最小二乘问题来解决。一些简单的反例表明，“经典”算法，例如 Steiglitz-McBride (1965)、迭代二次最大似然法和 Cadzow's (1988) 迭代，并没有收敛到最优 L/sub 2/ 解，尽管文献中存在误导性的说法。 >
Structured rank-deficient matrices arise in many applications in signal processing, system identification, and control theory. The author discusses the structured total least squares (STLS) problem, which is the problem of approximating affinely structured matrices (i.e., matrices affine in the parameters) by similarly structured rank-deficient ones, while minimizing an L/sub 2/-error criterion. It is shown that the optimality conditions lead to a nonlinear generalized singular value decomposition, which can be solved via an algorithm that is inspired by inverse iteration. Next the author concentrates on the so-called L/sub 2/-optimal noisy realization problem, which is equivalent with approximating a given data sequence by the impulse response of a finite dimensional, time invariant linear system of a given order. This can be solved as a structured total least squares problem. It is shown with some simple counter examples that "classical" algorithms such as the Steiglitz-McBride (1965), iterative quadratic maximum likelihood and Cadzow's (1988) iteration do not converge to the optimal L/sub 2/ solution, despite misleading claims in the literature. >