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1. Forward Invariance in Neural Network Controlled Systems

   https://doi.org/10.1109/LCSYS.2023.3341980  

   Harapanahalli, Akash; Jafarpour, Saber; Coogan, Samuel (January 2023, IEEE Control Systems Letters)

   We present a framework based on interval analysis and monotone systems theory to certify and search for forward invariant sets in nonlinear systems with neural network controllers. The framework (i) constructs localized first-order inclusion functions for the closed-loop system using Jacobian bounds and existing neural network verification tools; (ii) builds a dynamical embedding system where its evaluation along a single trajectory directly corre- sponds with a nested family of hyper-rectangles provably converging to an attractive set of the original system; (iii) utilizes linear transformations to build families of nested paralleletopes with the same properties. The framework is automated in Python using our interval analysis tool- box npinterval, in conjunction with the symbolic arith- metic toolbox sympy, demonstrated on an 8-dimensional leader-follower system. 

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2. Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming

   https://doi.org/10.1109/CDC42340.2020.9304296  

   Hu, Haimin; Fazlyab, Mahyar; Morari, Manfred; Pappas, George J. (December 2020, IEEE Conference on Decision and Control)

   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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3. Signal Estimation and System Identification With Nonlinear Dynamic Sensors

   https://doi.org/10.1109/CCTA.2019.8920592  

   Berberich, Julian; Sznaier, Mario; Allgower, Frank (August 2019, 2019 IEEE Conference on Control Technology and Applications (CCTA))

   We consider the problem of estimating the output of an unknown discrete-time linear time-invariant system and identifying a model of the system, where only measurements via a nonlinear dynamic sensor with known dynamics are available. The main result of this paper is a rank-constrained semidefinite program, which provides an equivalent characterization of this identification and estimation problem. This extends existing results from Wiener system identification to the more general case that the nonlinear block exhibits dynamic behavior, which is a commonly found scenario in practical applications. Notably, the result can be applied in the presence of nonlinear sensors with general non-invertible system dynamics. Two examples are used to illustrate the applicability of our approach. 

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4. Adaptive Robust Model Predictive Control with Matched and Unmatched Uncertainty

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   Sinha, Rohan; Harrison, James; Richards, Spencer M.; Pavone, Marco (June 2022, American Control Conference)

   We propose a learning-based robust predictive control algorithm that compensates for significant uncertainty in the dynamics for a class of discrete-time systems that are nominally linear with an additive nonlinear component. Such systems commonly model the nonlinear effects of an unknown environment on a nominal system. We optimize over a class of nonlinear feedback policies inspired by certainty equivalent "estimate-and-cancel" control laws pioneered in classical adaptive control to achieve significant performance improvements in the presence of uncertainties of large magnitude, a setting in which existing learning-based predictive control algorithms often struggle to guarantee safety. In contrast to previous work in robust adaptive MPC, our approach allows us to take advantage of structure (i.e., the numerical predictions) in the a priori unknown dynamics learned online through function approximation. Our approach also extends typical nonlinear adaptive control methods to systems with state and input constraints even when we cannot directly cancel the additive uncertain function from the dynamics. Moreover, we apply contemporary statistical estimation techniques to certify the system’s safety through persistent constraint satisfaction with high probability. Finally, we show in simulation that our method can accommodate more significant unknown dynamics terms than existing methods. 

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5. A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

   https://doi.org/10.3934/jcd.2022019  

   Jones, Morgan; Peet, Matthew M. (January 2022, Journal of Computational Dynamics)

   Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose \begin{document}$ 1 $$\end{document}$-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator. 

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