This paper solves the problem of exact computation of the phase and gain margins of multivariable control systems. These stability margins are studied using the concept of the Davis–Wielandt shell of complex matrices. Calculation of the phase and gain margins requires solving a quadratically constrained quadratic program (QCQP) and a parametrized quadratically constrained problem, respectively, which are known to be difficult in general. The semidefinite relaxation (SDR) technique is often used as a computationally efficient approximation technique to solve QCQPs. It turns out that the QCQPs formulated from the phase and gain margin problems in this paper fall under the class of quadratic optimization problem, for which the SDR is exact. This is so in the sense that optimal values of the QCQP and its SDR are equal, and an optimal solution for the original QCQP problem can be obtained from an optimal solution of its SDR. Thus, we are able to propose computationally efficient algorithms to compute the phase and gain margins with an arbitrarily high precision. Numerical examples are given to illustrate the effectiveness of the proposed algorithms.

本文解决了多变量控制系统相位和增益裕度的精确计算问题。这些稳定性裕度是使用复杂矩阵的 Davis-Wielandt 壳的概念研究的。相位和增益裕度的计算需要分别解决二次约束二次规划 (QCQP) 和参数化二次约束问题，这通常是众所周知的困难。半定松弛 (SDR) 技术通常用作求解 QCQP 的计算高效的近似技术。事实证明，本文中由相位和增益裕度问题制定的 QCQP 属于二次优化问题类，其 SDR 是精确的。从某种意义上说，QCQP 及其 SDR 的最优值是相等的，并且可以从其SDR的最优解得到原始QCQP问题的最优解。因此，我们能够提出计算效率高的算法，以任意高精度计算相位和增益裕度。给出了数值例子来说明所提出算法的有效性。