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 Dynamic Mode Decomposition (DMD) and DMD with control, two popular methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. 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 H-infinity 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 specific frequencies. Experimental results using data from an aircraft fatigue structural test rig and a soft robot arm highlight the advantages of the proposed regression methods.
翻译:在考虑许多提升功能时,从数据中接近Koopman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,扩展动态模式分解(DMD)和DMD(DMD)具有控制性,两种接近Koopman操作员的流行方法被重新拟订为线性矩阵不平等制约的共通优化问题。弱性稳定制约和系统规范规范规范被视作改进近似Koopman操作员的数字调节的方法。特别是,H-无限规范被用作一种常规手段,用以惩罚Koopman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。使用飞机疲劳结构测试机和软机器人臂的数据进行的实验结果突出了拟议的回归方法的优点。