计算离散时间LTI系统的缩放相对图：一种频域方法

中文摘要 AI 辅助

缩放相对图（SRG）在复平面中表示输入-输出算子的联合增益和相位特性。本文刻画了因果、稳定、方阵离散时间线性时不变（LTI）系统在单边平方可和序列上的SRG闭包。该闭包的Beltrami-Klein像等于变换后的频率响应矩阵数值范围的凸包。对于实系数单输入单输出系统，这产生了离散时间奈奎斯特轨迹的双曲凸包。证明处理了单边输入所施加的Hardy空间限制，该限制排除了持续正弦波，通过用显式构造的有限持续时间正弦波替代它们来解决。这种构造展示了频域SRG中的点如何作为由可容许输入生成的算子SRG点的极限出现。该刻画使得从状态空间实现出发，利用频率响应评估、数值范围和平面凸包进行闭包的基于模型的计算成为可能，而无需求解线性矩阵不等式。

英文摘要

The scaled relative graph (SRG) represents the joint gain and phase properties of an input--output operator in the complex plane. This paper characterizes the SRG closure of causal, stable, square discrete-time linear time-invariant (LTI) systems on one-sided square-summable sequences. The Beltrami--Klein image of this closure equals the convex hull of the numerical ranges of the transformed frequency-response matrices. For real-coefficient single-input single-output systems, this yields the hyperbolic convex hull of the discrete-time Nyquist locus. The proof addresses the Hardy-space restriction imposed by one-sided inputs, which excludes persistent sinusoids, by replacing them with explicitly constructed finite-duration sinusoids. This construction shows how points in frequency-wise SRGs arise as limits of operator SRG points generated by admissible inputs. The characterization enables model-based computation of the closure from a state-space realization using frequency-response evaluations, numerical ranges, and planar convex hulls, without solving linear matrix inequalities.

发表机构

• Politecnico di Milano（米兰理工大学）

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