This work targets the identification of a class of models for hybrid dynamical systems characterized by nonlinear autoregressive exogenous (NARX) components, with finite-dimensional polynomial expansions, and by a Markovian switching mechanism. The estimation of the model parameters is performed under a probabilistic framework via Expectation Maximization, including submodel coefficients, hidden state values and transition probabilities. Discrete mode classification and NARX regression tasks are disentangled within the iterations. Soft-labels are assigned to latent states on the trajectories by averaging over the state posteriors and updated using the parametrization obtained from the previous maximization phase. Then, NARXs parameters are repeatedly fitted by solving weighted regression subproblems through a cyclical coordinate descent approach with coordinate-wise minimization. Moreover, we investigate a two stage selection scheme, based on a l1-norm bridge estimation followed by hard-thresholding, to achieve parsimonious models through selection of the polynomial expansion. The proposed approach is demonstrated on a SMNARX problem composed by three nonlinear sub-models with specific regressors.

中文翻译：

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切换马尔可夫多项式 NARX 模型的估计

这项工作的目标是识别一类混合动力系统模型，其特征是非线性自回归外生 (NARX) 分量、有限维多项式展开和马尔可夫切换机制。模型参数的估计是在概率框架下通过期望最大化进行的，包括子模型系数、隐藏状态值和转移概率。离散模式分类和 NARX 回归任务在迭代中解开。软标签通过对状态后验进行平均分配给轨迹上的潜在状态，并使用从前一个最大化阶段获得的参数化进行更新。然后，通过具有坐标最小化的循环坐标下降方法解决加权回归子问题，反复拟合 NARX 参数。此外，我们研究了基于 l1 范数桥估计和硬阈值的两阶段选择方案，以通过选择多项式展开来实现简约模型。在由具有特定回归量的三个非线性子模型组成的 SMNARX 问题上演示了所提出的方法。