loop stability
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Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1912
Zhe Wu ◽  
David Rincon ◽  
Quanquan Gu ◽  
Panagiotis D. Christofides

Recurrent neural networks (RNNs) have been widely used to model nonlinear dynamic systems using time-series data. While the training error of neural networks can be rendered sufficiently small in many cases, there is a lack of a general framework to guide construction and determine the generalization accuracy of RNN models to be used in model predictive control systems. In this work, we employ statistical machine learning theory to develop a methodological framework of generalization error bounds for RNNs. The RNN models are then utilized to predict state evolution in model predictive controllers (MPC), under which closed-loop stability is established in a probabilistic manner. A nonlinear chemical process example is used to investigate the impact of training sample size, RNN depth, width, and input time length on the generalization error, along with the analyses of probabilistic closed-loop stability through the closed-loop simulations under Lyapunov-based MPC.

2021 ◽  
Vol 4 (5) ◽  
pp. 1-18
Laurent Baratchart ◽  
Sylvain Chevillard ◽  
Adam Cooman ◽  
Martine Olivi ◽  

<abstract><p>We study the properties of electronic circuits after linearization around a fixed operating point in the context of closed-loop stability analysis. When distributed elements, like transmission lines, are present in the circuit it is known that unstable circuits can be created without poles in the complex right half-plane. This undermines existing closed-loop stability analysis techniques that determine stability by looking for right half-plane poles. We observed that the problematic circuits rely on unrealistic elements with an infinite bandwidth. In this paper, we therefore define a class of realistic linearized components and show that a circuit composed of realistic elements is only unstable with poles in the complex right half-plane. Furthermore, we show that the amount of right half-plane poles in a realistic circuit is finite, even when distributed elements are present. In the second part of the paper, we provide examples of component models that are realistic and show that the class includes many existing models, including ones for passive devices, active devices and transmission lines.</p></abstract>

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