We model and study the problem of localizing a set of sparse forcing inputs for linear dynamical systems from noisy measurements when the initial state is unknown. This problem is of particular relevance to detecting forced oscillations in electric power networks. We express measurements as an additive model comprising the initial state and inputs grouped over time, both expanded in terms of the basis functions (i.e., impulse response coefficients). Using this model, with probabilistic guarantees, we recover the locations and simultaneously estimate the initial state and forcing inputs using a variant of the group LASSO (linear absolute shrinkage and selection operator) method. Specifically, we provide a tight upper bound on: (i) the probability that the group LASSO estimator wrongly identifies the source locations, and (ii) the $\ell_2$-norm of the estimation error. Our bounds explicitly depend upon the length of the measurement horizon, the noise statistics, the number of inputs and sensors, and the singular values of impulse response matrices. Our theoretical analysis is one of the first to provide a complete treatment for the group LASSO estimator for linear dynamical systems under input-to-output delay assumptions. Finally, we validate our results on synthetic models and the IEEE 68-bus, 16-machine system.

翻译：在初始状态不明时,我们模拟和研究将一组线性动态系统微弱的强制输入数据定位于初始状态不明时,对一组线性动态测量系统进行本地化的问题。这个问题与探测电力网络中强制振动特别相关。我们将测量数据作为一个添加模型,包括初始状态和按时间分组的输入数据,两者均以基本功能(即脉冲反应系数)为基础加以扩大。我们利用这一模型,以概率保障,恢复位置,同时使用LASSO(线性绝对缩水和选择操作员)组的变异方法估算初始状态和强制输入数据。我们首先进行理论分析,目的是为LASSO组的测深器提供一种完整的处理方法。具体地是:(一) LASSOS估计器错误地辨别源位置的可能性,以及(二) 估算错误的美元-2美元-日元-日值。我们的界限明确取决于测量范围长度、噪音统计、输入和传感器的数量以及脉冲反应矩阵的单值。我们首先对16个合成动力-EEE系统验证模型的直线性动态-I-Simal-imal-I-I-I-I-I-I-I-BI-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-