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There has been an increasing interest in using neural networks in closed-loop control systems to improve performance and reduce computational costs for on-line implementation. However, providing safety and stability guarantees for these systems is challenging due to the nonlinear and compositional structure of neural networks. In this paper, we propose a novel forward reachability analysis method for the safety verification of linear time-varying systems with neural networks in feedback interconnection. Our technical approach relies on abstracting the nonlinear activation functions by quadratic constraints, which leads to an outer-approximation of forward reachable sets of the closed-loop system. We show that we can compute these approximate reachable sets using semidefinite programming. We illustrate our method in a quadrotor example, in which we first approximate a nonlinear model predictive controller via a deep neural network and then apply our analysis tool to certify finite-time reachability and constraint satisfaction of the closed-loop system.
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Neural networks in feedback for flow analysis and control
This work presents a methodology for analysis and control of nonlinear fluid systems using neural networks. The approach is demonstrated in four different study cases: the Lorenz system, a modified version of the Kuramoto-Sivashinsky equation, a streamwise-periodic two-dimensional channel flow, and a confined cylinder flow. Neural networks are trained as models to capture the complex system dynamics and estimate equilibrium points through a Newton method, enabled by back-propagation. These neural network surrogate models (NNSMs) are leveraged to train a second neural network, which is designed to act as a stabilizing closed-loop controller. The training process employs a recurrent approach, whereby the NNSM and the neural network controller are chained in closed loop along a finite time horizon. By cycling through phases of combined random open-loop actuation and closed-loop control, an iterative training process is introduced to overcome the lack of data near equilibrium points. This approach improves the accuracy of the models in the most critical region for achieving stabilization. Through the use of L1 regularization within loss functions, the NNSMs can also guide optimal sensor placement, reducing the number of sensors from an initial candidate set. The data sets produced during the iterative training process are also leveraged for conducting a linear stability analysis through a modified dynamic mode decomposition approach. The results demonstrate the effectiveness of computationally inexpensive neural networks in modeling, controlling, and enabling stability analysis of nonlinear systems, providing insights into the system behavior and offering potential for stabilization of complex fluid systems.
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- Award ID(s):
- 2238770
- PAR ID:
- 10557735
- Publisher / Repository:
- Physical Review Fluids
- Date Published:
- Journal Name:
- Physical Review Fluids
- Volume:
- 9
- Issue:
- 6
- ISSN:
- 2469-990X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1103/PhysRevFluids.9.063904
