AI 中文总结
本文为非线性数据驱动预测控制中的轨迹流形建立几何基础，证明有限时域行为是嵌入子流形，维数为n+Nm，并给出显式全局坐标及精确编码器-解码器表示，无需可控性等假设。
AI 中文摘要
本文为行为框架下确定性非线性系统的轨迹流形表示建立了几何基础，该框架由数据驱动预测控制所激发。对于具有测量状态和$C^r$转移映射（$r\geq 1$）的离散时间系统$x_{k+1}=f(x_k,u_k)$，我们考虑由预测时域$N$内所有允许的状态-输入轨迹组成的终端状态增广有限时域行为。我们证明该行为是环境轨迹空间的一个$C^r$嵌入子流形，其内蕴维数为$n+Nm$，其中$n$和$m$分别为状态和输入维数。此外，从允许的初始状态和输入坐标$(x_0,\mathbf u)$出发的展开映射是到行为流形上的$C^r$微分同胚，提供了显式的全局光滑坐标。这产生了规范的精确编码器-解码器表示，并意味着完整行为的任何精确可微潜表示必须具有至少$n+Nm$的潜维数。该几何结果不要求可控性、可镇定性或动力学可逆性。对于零阶保持采样连续时间系统和固定步长数值转移映射，给出了相应结果。这些结果为后续的数据驱动逼近和预测控制发展提供了确定性几何基础。
英文摘要
This note establishes a geometric foundation for trajectory-manifold representations of deterministic nonlinear systems in a behavioral setting motivated by data-enabled predictive control. For a discrete-time system $x_{k+1}=f(x_k,u_k)$ with measured state and a $C^r$ transition map, $r\geq 1$, we consider the terminal-state-augmented finite-horizon behavior consisting of all admissible state-input trajectories over a prediction horizon $N$. We prove that this behavior is a $C^r$ embedded submanifold of the ambient trajectory space with intrinsic dimension $n+Nm$, where $n$ and $m$ are the state and input dimensions. Moreover, the rollout map from the admissible initial-state and input coordinates $(x_0,\mathbf u)$ is a $C^r$ diffeomorphism onto the behavior manifold, providing explicit global smooth coordinates. This yields a canonical exact encoder--decoder representation and implies that any exact differentiable latent representation of the full behavior must have latent dimension at least $n+Nm$. The geometric result does not require controllability, stabilizability, or invertibility of the dynamics. Corresponding results are given for zero-order-hold sampled continuous-time systems and fixed-step numerical transition maps. These results provide the deterministic geometric foundation for subsequent data-driven approximation and predictive-control development.
Comments15 pages
