We consider the problem of identification of linear dynamical systems from a single trajectory. Recent results have predominantly focused on the setup where no structural assumption is made on the system matrix $A^* \in \mathbb{R}^{n \times n}$, and have consequently analyzed the ordinary least squares (OLS) estimator in detail. We assume prior structural information on $A^*$ is available, which can be captured in the form of a convex set $\mathcal{K}$ containing $A^*$. For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm which depend on the local size of the tangent cone of $\mathcal{K}$ at $A^*$. To illustrate the usefulness of this result, we instantiate it for the settings where, (i) $\mathcal{K}$ is a $d$ dimensional subspace of $\mathbb{R}^{n \times n}$, or (ii) $A^*$ is $k$-sparse and $\mathcal{K}$ is a suitably scaled $\ell_1$ ball. In the regimes where $d, k \ll n^2$, our bounds improve upon those obtained from the OLS estimator.

翻译：我们考虑从单条轨迹识别线性动力系统的问题。近期研究主要聚焦于系统矩阵 $A^* \in \mathbb{R}^{n \times n}$ 无结构假设的设定，并由此详细分析了普通最小二乘估计器。本文假设 $A^*$ 具有可被凸集 $\mathcal{K}$（包含 $A^*$）捕获的先验结构信息。针对由此产生的约束最小二乘估计器的解，我们推导了依赖于 $\mathcal{K}$ 在 $A^*$ 处切锥局部大小的 Frobenius 范数非渐近误差界。为展示该结果的有效性，我们将其具体应用于以下两种情形：(i) $\mathcal{K}$ 是 $\mathbb{R}^{n \times n}$ 的 $d$ 维子空间，或 (ii) $A^*$ 为 $k$-稀疏且 $\mathcal{K}$ 是适当缩放的 $\ell_1$ 球。在 $d, k \ll n^2$ 的范围内，我们的误差界优于普通最小二乘估计器所得结果。