Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) is a nonlinear control technique that assigns a port-Hamiltonian (pH) structure to a controlled system using a state-feedback law. While IDA-PBC has been extensively studied and applied to many systems, its practical implementation often remains confined to academic examples and, almost exclusively, to stabilization tasks. The main limitation of IDA-PBC stems from the complexity of analytically solving a set of partial differential equations (PDEs), referred to as the matching conditions, which enforce the pH structure of the closed-loop system. However, this is extremely challenging, especially for complex physical systems and tasks. In this work, we propose a novel numerical approach for designing IDA-PBC controllers without solving the matching PDEs exactly. We cast the IDA-PBC problem as the learning of a neural ordinary differential equation. In particular, we rely on sparse dictionary learning to parametrize the desired closed-loop system as a sparse linear combination of nonlinear state-dependent functions. Optimization of the controller parameters is achieved by solving a multi-objective optimization problem whose cost function is composed of a generic task-dependent cost and a matching condition-dependent cost. Our numerical results show that the proposed method enables (i) IDA-PBC to be applicable to complex tasks beyond stabilization, such as the discovery of periodic oscillatory behaviors, (ii) the derivation of closed-form expressions of the controlled system, including residual terms in case of approximate matching, and (iii) stability analysis of the learned controller.

翻译：互联与阻尼分配无源控制（IDA-PBC）是一种非线性控制技术，通过状态反馈律为受控系统赋予端口哈密顿（pH）结构。尽管IDA-PBC已被广泛研究并应用于众多系统，其实践实现往往局限于学术案例，且几乎完全集中于镇定任务。IDA-PBC的主要限制源于解析求解一组偏微分方程（即匹配条件）的复杂性，该组方程用于强制闭环系统保持pH结构。然而，这对于复杂物理系统及任务而言极具挑战性。本文提出一种新颖的数值方法，用于设计IDA-PBC控制器而无需精确求解匹配偏微分方程。我们将IDA-PBC问题转化为神经常微分方程的学习问题。具体而言，我们借助稀疏字典学习，将期望的闭环系统参数化为非线性状态依赖函数的稀疏线性组合。控制器参数的优化通过求解一个多目标优化问题实现，该问题的代价函数由任务相关的通用代价与匹配条件相关代价共同构成。数值结果表明，所提方法能够：（i）使IDA-PBC适用于镇定之外的复杂任务，例如周期振荡行为的发现；（ii）推导出受控系统的闭式表达式（在近似匹配情况下包含残差项）；（iii）对学习得到的控制器进行稳定性分析。