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1. Lyapunov Neural Network with Region of Attraction Search

   https://doi.org/10.23919/ACC60939.2024.10644877  

   Wang, Zili; Andersson, Sean B; Tron, Roberto (July 2024, Proceedings of the American Control Conference)

   Deep learning methods have been widely used in robotic applications, making learning-enabled control design for complex nonlinear systems a promising direction. Although deep reinforcement learning methods have demonstrated impressive empirical performance, they lack the stability guarantees that are important in safety-critical situations. One way to provide these guarantees is to learn Lyapunov certificates alongside control policies. There are three related problems: 1) verify that a given Lyapunov function candidate satisfies the conditions for a given controller on a region, 2) find a valid Lyapunov function and controller on a given region, and 3) find a valid Lyapunov function and a controller such that the region of attraction is as large as possible. Previous work has shown that if the dynamics are piecewise linear, it is possible to solve problem 1) and 2) by solving a Mixed-Integer Linear Program (MILP). In this work, we build upon this method by proposing a Lyapunov neural network that considers monotonicity over half spaces in different directions. We 1) propose a specific choice of Lyapunov function architecture that ensures non-negativity and a unique global minimum by construction, and 2) show that this can be leveraged to find the controller and Lyapunov certificates faster and with a larger valid region by maximizing the size of a square inscribed in a given level set. We apply our method to a 2D inverted pendulum, unicycle path following, a 3-D feedback system, and a 4-D cart pole system, and demonstrate it can shorten the training time by half compared to the baseline, as well as find a larger ROA. 

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2. Estimating the Region of Attraction Using Polynomial Optimization: A Converse Lyapunov Result

   https://doi.org/978-1-5090-2872-6/17  

   Jones, M.; Mohammadi, H.; Peet, M. (December 2017, Proceedings of the IEEE Conference on Decision & Control)

   In this paper, we propose an iterative method for using SOS programming to estimate the region of attraction of a polynomial vector field, the conjectured convergence of which necessitates the existence of polynomial Lyapunov functions whose sublevel sets approximate the true region of attraction arbitrarily well. The main technical result of the paper is the proof of existence of such a Lyapunov function. Specifically, we usetheHausdorffdistancemetrictoanalyzeconvergenceandin the main theorem demonstrate that the existence of an n-times continuously differentiable maximal Lyapunov function implies that for any ε >0, there exists a polynomial Lyapunov function and associated sub-level set which together prove stability of a set which is within ε Hausdorff distance of the true region of attraction. The proposed iterative method and probably convergence is illustrated with a numerical example. 

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3. Approximation of compositional functions with ReLU neural networks

   https://doi.org/10.1016/j.sysconle.2023.105508  

   Gong, Q.; Kang, W.; Fahroo, F. (May 2023, Systems control letters)

   The power of DNN has been successfully demonstrated on a wide variety of high-dimensional problems that cannot be solved by conventional control design methods. These successes also uncover some fundamental and pressing challenges in understanding the representability of deep neural networks for complex and high dimensional input–output relations. Towards the goal of understanding these fundamental questions, we applied an algebraic framework developed in our previous work to analyze ReLU neural network approximation of compositional functions. We prove that for Lipschitz continuous functions, ReLU neural networks have an approximation error upper bound that is a polynomial of the network’s complexity and the compositional features. If the compositional features do not increase exponentially with dimension, which is the case in many applications, the complexity of DNN has a polynomial growth. In addition to function approximations, we also establish ReLU network approximation results for the trajectories of control systems, and for a Lyapunov function that characterizes the domain of attraction. 

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4. Robustness of networked systems to unintended interactions with application to engineered genetic circuits

   https://doi.org/10.1109/tcns.2021.3078144  

   Qian, Yili; Del Vecchio, Domitilla (May 2021, IEEE Transactions on Control of Network Systems)

   null (Ed.)

   A networked dynamical system is composed of subsystems interconnected via prescribed interactions. In many engineering applications, however, one subsystem can also affect others via unintended interactions that can significantly change the intended network's behavior. Although unintended interactions can be modeled as disturbance inputs to the subsystems, these disturbances depend on the network's states. Thus, a disturbance attenuation property of each subsystem is insufficient to ensure that the network is robust to unintended interactions. Here, we provide conditions on subsystem dynamics and interaction maps, such that a network is robust to unintended interactions. These conditions require that each subsystem asymptotically attenuates constant external disturbances, is monotone or near-monotone, the unintended interactions are monotone, and the prescribed interactions do not contain feedback loops. We apply this result to guide the design of resource-limited genetic circuits composed of feedback-regulated subsystems. 

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5. Structured Neural-PI Control with End-to-End Stability and Output Tracking Guarantees

   Wenqi Cui, Yan Jiang (December 2023, Advances in Neural Information Processing Systems 36 (NeurIPS 2023))

   A. Oh and T. Neumann and A. Globerson and K. Saenko and M. Hardt and S. Levine (Ed.)

   We study the optimal control of multiple-input and multiple-output dynamical systems via the design of neural network-based controllers with stability and output tracking guarantees. While neural network-based nonlinear controllers have shown superior performance in various applications, their lack of provable guarantees has restricted their adoption in high-stake real-world applications. This paper bridges the gap between neural network-based controllers and the need for stabilization guarantees. Using equilibrium-independent passivity, a property present in a wide range of physical systems, we propose neural Proportional-Integral (PI) controllers that have provable guarantees of stability and zero steady-state output tracking error. The key structure is the strict monotonicity on proportional and integral terms, which is parameterized as gradients of strictly convex neural networks (SCNN). We construct SCNN with tunable softplus-β activations, which yields universal approximation capability and is also useful in incorporating communication constraints. In addition, the SCNNs serve as Lyapunov functions, giving us end-to-end performance guarantees. Experiments on traffic and power networks demonstrate that the proposed approach improves both transient and steady-state performances, while unstructured neural networks lead to unstable behaviors. 

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