发表机构
Shenzhen University（深圳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究静态输出反馈镇定的计算复杂性，证明其与实数存在性理论等价，并刻画有效增益维数，在固定维数下多项式时间可解，同时给出NP-hard实例及鲁棒性结果。
AI 中文摘要
我们研究连续时间线性系统的静态输出反馈（SOF）镇定的计算复杂性。对于有理数型被控对象数据，我们证明该问题对于实数存在性理论是完备的：即在多项式时间变换下，它等价于判定一个具有整数系数的多项式方程组和不等式组是否存在实数解。这一结果甚至对具有两个输入的可控且可观测的整数被控对象也成立。我们将有效增益维数刻画为确定特征多项式（对每个增益而言）所需的增益项线性组合的最小数目。当该维数固定时，可行性检验以及（若存在）构造有理数镇定增益均可在多项式时间内完成。SOF镇定对于一族极小的单输入整数被控对象仍然是NP困难的。该族中的每个可行被控对象都存在一个增益项属于$\{-1,1\}$的镇定增益。对于这些增益，只要每个扰动的幅度不超过一个与被控对象维数无关的均匀正常数，稳定性在所有增益项同时扰动下得以保持。
英文摘要
We study the computational complexity of static output feedback (SOF) stabilization for continuous-time linear systems. For rational plant data, we prove that this problem is complete for the existential theory of the reals: it is equivalent, under polynomial-time transformations, to deciding whether a system of polynomial equations and inequalities with integer coefficients has a real solution. This result holds even for controllable and observable integer plants with two inputs. We characterize the effective gain dimension as the minimum number of linear combinations of gain entries that determine the characteristic polynomial for every gain. When this dimension is fixed, both feasibility testing and the construction of a rational stabilizing gain, whenever one exists, can be performed in polynomial time. SOF stabilization also remains strongly NP-hard for a family of minimal single-input integer plants. Every feasible plant in this family admits a stabilizing gain with entries in $\{-1,1\}$. For these gains, stability is preserved under simultaneous perturbations of all entries, provided that each perturbation has magnitude at most a uniform positive constant independent of the plant dimensions.
Comments21 pages
