2000 character limit reached
Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation (2401.02511v1)
Published 4 Jan 2024 in eess.SY, cs.AI, cs.LG, cs.SY, math.DS, and math.OC
Abstract: To stabilize PDE models, control laws require space-dependent functional gains mapped by nonlinear operators from the PDE functional coefficients. When a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent, a gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. The GS version of PDE backstepping employs gains obtained by solving a PDE at each value of the state. Performing such PDE computations in real time may be prohibitive. The recently introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value, without requiring a PDE solution. In this paper we introduce NOs for GS-PDE backstepping. GS controllers act on the premise that the state change is slow and, as a result, guarantee only local stability, even for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation using both a "full-kernel" approach and the "gain-only" approach to gain operator approximation. Numerical simulations illustrate stabilization and demonstrate speedup by three orders of magnitude over traditional PDE gain-scheduling. Code (Github) for the numerical implementation is published to enable exploration.
- A. Banach and W. Baumann, “Gain-scheduled control of nonlinear partial differential equations,” in 29th IEEE Conference on Decision and Control, 1990, pp. 387–392 vol.2.
- N. Bekiaris-Liberis and R. Vazquez, “Nonlinear bilateral output-feedback control for a class of viscous Hamilton–Jacobi PDEs,” Automatica, vol. 101, pp. 223–231, 2019.
- L. Bhan, Y. Shi, and M. Krstic, “Neural operators for bypassing gain and control computations in PDE backstepping,” IEEE Transactions on Automatic Control, pp. 1–16, 2023.
- T. Chen and H. Chen, “Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems,” IEEE Transactions on Neural Networks, vol. 6, no. 4, pp. 911–917, 1995.
- J. Coron, R. Vazquez, M. Krstic, and G. Bastin, “Local exponential H2superscript𝐻2H^{2}italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT stabilization of a 2×2222\times 22 × 2 quasilinear hyperbolic system using backstepping,” SIAM Journal on Control and Optimization, vol. 51, no. 3, pp. 2005–2035, 2013.
- B. Deng, Y. Shin, L. Lu, Z. Zhang, and G. E. Karniadakis, “Approximation rates of deeponets for learning operators arising from advection–diffusion equations,” Neural Networks, vol. 153, pp. 411–426, 2022.
- E. Fernández-Cara and E. Zuazua, “Null and approximate controllability for weakly blowing up semilinear heat equations,” Annales de l’I.H.P. Analyse non linéaire, vol. 17, no. 5, pp. 583–616, 2000.
- I. Karafyllis, N. Espitia, and M. Krstic, “Event-triggered gain scheduling of reaction-diffusion PDEs,” SIAM Journal on Control and Optimization, vol. 59, no. 3, pp. 2047–2067, 2021.
- I. Karafyllis and M. Krstic, “Global stabilization of a class of nonlinear reaction-diffusion partial differential equations by boundary feedback,” SIAM Journal on Control and Optimization, vol. 57, no. 6, pp. 3723–3748, 2019.
- ——, “Spill-free transfer and stabilization of viscous liquid,” IEEE Transactions on Automatic Control, vol. 67, no. 9, pp. 4585–4597, 2022.
- M. Krstic and A. Smyshlyaev, “Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays,” Systems &\&& Control Letters, vol. 57, no. 9, pp. 750–758, 2008.
- M. Krstic, L. Bhan, and Y. Shi, “Neural operators of backstepping controller and observer gain functions for reaction-diffusion PDEs,” 2023, arXiv:2303.10506.
- M. Krstic, L. Magnis, and R. Vazquez, “Nonlinear stabilization of shock-like unstable equilibria in the viscous Burgers PDE,” IEEE Transactions on Automatic Control, vol. 53, no. 7, pp. 1678–1683, 2008.
- S. Lanthaler, Z. Li, and A. M. Stuart, “The nonlocal neural operator: Universal approximation,” 2023, arXiv:2304.13221.
- Z. Li, H. Zheng, N. Kovachki, D. Jin, H. Chen, B. Liu, K. Azizzadenesheli, and A. Anandkumar, “Physics-informed neural operator for learning partial differential equations,” 2023, arXiv:2111.03794.
- L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis, “Learning nonlinear operators via deeponet based on the universal approximation theorem of operators,” Nature Machine Intelligence, vol. 3, no. 3, pp. 218–229, 2021.
- L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, “DeepXDE: A deep learning library for solving differential equations,” SIAM Review, vol. 63, no. 1, pp. 208–228, 2021.
- J. Qi, J. Zhang, and M. Krstic, “Neural operators for delay-compensating control of hyperbolic PIDEs,” 2023, arXiv:2307.11436.
- M. Raissi, P. Perdikaris, and G. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics, vol. 378, pp. 686–707, 2019.
- W. J. Rugh and J. S. Shamma, “Research on gain scheduling,” Automatica, vol. 36, no. 10, pp. 1401–1425, 2000.
- W. Rugh, “Analytical framework for gain scheduling,” IEEE Control Systems Magazine, vol. 11, no. 1, pp. 79–84, 1991.
- J. Shamma and M. Athans, “Analysis of gain scheduled control for nonlinear plants,” IEEE Transactions on Automatic Control, vol. 35, no. 8, pp. 898–907, 1990.
- A. A. Siranosian, M. Krstic, A. Smyshlyaev, and M. Bement, “Gain scheduling-inspired boundary control for nonlinear partial differential equations,” Journal of dynamic systems, measurement, and control, vol. 133, no. 5, p. 12, 2011.
- R. Vazquez and M. Krstic, “Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design,” Automatica, vol. 44, no. 11, pp. 2778–2790, 2008.
- ——, “Control of 1-D parabolic PDEs with Volterra nonlinearities, Part II: Analysis,” Automatica, vol. 44, no. 11, pp. 2791–2803, 2008.
- S. Wang, M. Diagne, and M. Krstić, “Deep learning of delay-compensated backstepping for reaction-diffusion PDEs,” 2023, arXiv:2308.10501.
- S. Wang, H. Wang, and P. Perdikaris, “Learning the solution operator of parametric partial differential equations with physics-informed deeponets,” Science Advances, vol. 7, no. 40, p. eabi8605, 2021.