We consider the problem of finite-time identification of linear dynamical systems from $T$ samples of 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 that depend on the local size of $\mathcal{K}$ at $A^*$. To illustrate the usefulness of these results, we instantiate them for four examples, namely when (i) $A^*$ is sparse and $\mathcal{K}$ is a suitably scaled $\ell_1$ ball; (ii) $\mathcal{K}$ is a subspace; (iii) $\mathcal{K}$ consists of matrices each of which is formed by sampling a bivariate convex function on a uniform $n \times n$ grid (convex regression); (iv) $\mathcal{K}$ consists of matrices each row of which is formed by uniform sampling (with step size $1/T$) of a univariate Lipschitz function. In all these situations, we show that $A^*$ can be reliably estimated for values of $T$ much smaller than what is needed for the unconstrained setting.

翻译：我们考虑从单条轨迹的$T$个样本中有限时间识别线性动力系统的问题。近期研究主要聚焦于对系统矩阵$A^* \in \mathbb{R}^{n \times n}$不作任何结构假设的情形，并因此详细分析了普通最小二乘（OLS）估计量。本文假设$A^*$的可用先验结构信息可被凸集$\mathcal{K}$（包含$A^*$）所刻画。针对由此产生的约束最小二乘估计量的解，我们推导了依赖于$\mathcal{K}$在$A^*$处局部大小的Frobenius范数非渐近误差界。为说明这些结果的有效性，我们将其应用于四个实例：（i）$A^*$稀疏且$\mathcal{K}$为适当缩放的$\ell_1$球；（ii）$\mathcal{K}$为子空间；（iii）$\mathcal{K}$由在均匀$n \times n$网格上采样二元凸函数生成的矩阵构成（凸回归）；（iv）$\mathcal{K}$由每行通过均匀采样（步长为$1/T$）一元Lipschitz函数生成的矩阵构成。在所有情形中，我们证明当$T$远小于无约束设置所需值时，$A^*$仍可被可靠估计。