This paper presents two efficient algorithms to determine whether a bivariate polynomial, possibly with complex coefficients, does not vanish in the cross product of two closed right-half planes (is “2-C stable”). A 2-C stable polynomial in the denominator of a two-dimensional analog filter has been proved (not long ago) to imply bounded-input bounded-output (BIBO) stability. The two algorithms are entirely different but both rely on a recently proposed fraction-free (FF) Routh test for complex polynomials in this transaction. The first algorithm tests the 2-C stability of a bivariate polynomial of degree $(n_{1},n_{2})$ in order $n^{6}$ of elementary operations (when $n_{1}=n_{2}=n$ ). It is a “tabular type” two-dimensional stability test that can be regarded as a “Routh table” whose scalar entries were replaced by univariate polynomials. The second 2-C stability test is obtained from the first by its telepolation. It carries out the 2-C stability test by a finite collection of FF Routh tests and requires only order $n^{4}$ elementary operations. Both algorithms possess an integer-preserving property that enhances them with additional merits including numerical error-free decision on 2-C stability.

中文翻译：

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二维连续时间系统的有界输入有界输出稳定性测试

本文提出了两种有效的算法来确定可能具有复数系数的双变量多项式在两个闭合的右半平面的叉积（“ 2-C稳定”）中是否不消失。二维模拟滤波器分母中的2-C稳定多项式已被证明（不久前），它暗示了有界输入有界输出（BIBO）的稳定性。两种算法完全不同，但是都依赖于最近提出的针对此事务中的复杂多项式的无分数（FF）Routh测试。第一种算法测试度数的二元多项式的2-C稳定性 $（n_ {1}，n_ {2}）$ 为了 $ n ^ {6} $ 基本操作数（当 $ n_ {1} = n_ {2} = n $ ）。这是“表格类型”二维稳定性测试，可以看作是“嘴表格”，其标量条目已被单变量多项式代替。第二个2-C稳定性测试是从第一个通过遥测获得的。它通过有限的FF Routh测试来执行2-C稳定性测试，仅需订购 $ n ^ {4} $ 基本操作。两种算法均具有保留整数的特性，并具有其他优点，包括2C稳定性的无数字错误判定，从而增强了它们的性能。