Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended DMD and DMD with control, two popular methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. Both hard asymptotic stability constraints and system norm regularizers are considered as methods to improve the numerical conditioning of the approximate Koopman operator. In particular, the $\mathcal{H}_\infty$ norm is used as a regularizer to penalize the input-output gain of the linear system defined by the Koopman operator. Weighting functions are then applied to penalize the system gain at particular frequencies.

翻译：在考虑许多提升功能时,从数据中接近库普曼操作员在数字上具有挑战性。即使是低维系统也会在高维提升空间中产生不稳定或条件差的结果。在本论文中,扩展 DMD和DMD 具有控制权的扩展 DMD 和 DMD 两种接近库普曼操作员的流行方法被重新拟订为线性矩阵不平等制约的螺旋优化问题。硬性非同步稳定性限制和系统规范规范被视作改进大约库普曼操作员的数字调节的方法。特别是,用$\mathcal{H ⁇ infty$标准作为常规,惩罚库普曼操作员定义的线性系统输入-输出收益。然后运用加权功能来惩罚特定频率的系统收益。