We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is founded on the density function based almost everywhere stability certificate that is dual to the Lyapunov function for dynamic systems. Unlike Lyapunov based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently by invoking the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory of the dynamical system can be understood using linear Perron-Frobenius and Koopman operators. This connection allows us to use data-driven methods developed to approximate these operators combined with the SOS techniques for the convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

中文翻译：

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非线性控制综合的凸数据驱动方法

我们考虑一类非线性控制综合问题，其中潜在的数学模型不明确。我们提出了一种数据驱动的方法，当只有动态的样本轨迹可访问时，可以稳定系统。我们的方法建立在基于几乎处处稳定性证明的密度函数上，该证明与动态系统的李雅普诺夫函数是对偶的。与基于李雅普诺夫的方法不同，密度函数导致用于联合搜索控制策略和稳定性证书的凸公式。这种类型的凸问题可以通过调用平方和 (SOS) 机制来有效解决。对于数据驱动部分，我们利用可以使用线性 Perron-Frobenius 和 Koopman 算子来理解动态系统稳定性理论中的二元性结果这一事实。这种联系使我们能够使用数据驱动的方法来近似这些算子，并结合 SOS 技术用于控制综合的凸公式。通过几个例子证明了所提出方法的有效性。