In this paper, we consider the problem of reference tracking in uncertain nonlinear systems. A neural State-Space Model (NSSM) is used to approximate the nonlinear system, where a deep encoder network learns the nonlinearity from data, and a state-space component captures the temporal relationship. This transforms the nonlinear system into a linear system in a latent space, enabling the application of model predictive control (MPC) to determine effective control actions. Our objective is to design the optimal controller using limited data from the \textit{target system} (the system of interest). To this end, we employ an implicit model-agnostic meta-learning (iMAML) framework that leverages information from \textit{source systems} (systems that share similarities with the target system) to expedite training in the target system and enhance its control performance. The framework consists of two phases: the (offine) meta-training phase learns a aggregated NSSM using data from source systems, and the (online) meta-inference phase quickly adapts this aggregated model to the target system using only a few data points and few online training iterations, based on local loss function gradients. The iMAML algorithm exploits the implicit function theorem to exactly compute the gradient during training, without relying on the entire optimization path. By focusing solely on the optimal solution, rather than the path, we can meta-train with less storage complexity and fewer approximations than other contemporary meta-learning algorithms. We demonstrate through numerical examples that our proposed method can yield accurate predictive models by adaptation, resulting in a downstream MPC that outperforms several baselines.

翻译：摘要：本文研究不确定性非线性系统的参考轨迹跟踪问题。采用神经状态空间模型（NSSM）逼近非线性系统，其中深度编码器网络从数据中学习非线性特征，状态空间组件捕获时序关系。该方法将非线性系统转化为潜在空间中的线性系统，使模型预测控制（MPC）能够确定有效控制作用。我们的目标是通过有限的目标系统（目标系统）数据设计最优控制器。为此，我们采用隐式模型无关元学习（iMAML）框架，利用源系统（与目标系统具有相似性的系统）的信息加速目标系统的训练过程并提升控制性能。该框架包含两个阶段：（离线）元训练阶段利用源系统数据学习聚合NSSM，（在线）元推理阶段仅需少量数据点和几次在线训练迭代，基于局部损失函数梯度将聚合模型快速适应至目标系统。iMAML算法通过隐函数定理在训练过程中精确计算梯度，无需依赖完整优化路径。通过仅关注最优解而非优化路径，相较于其他当代元学习算法，我们的方法在元训练时具有更低存储复杂度和更少近似误差。数值实验表明，所提方法通过自适应可生成精确预测模型，使下游MPC性能优于多种基线方法。