Robust Boundary Stabilization of Stochastic Hyperbolic PDEs (2312.16636v1)
Abstract: This paper proposes a backstepping boundary control design for robust stabilization of linear first-order coupled hyperbolic partial differential equations (PDEs) with Markov-jumping parameters. The PDE system consists of 4 X 4 coupled hyperbolic PDEs whose first three characteristic speeds are positive and the last one is negative. We first design a full-state feedback boundary control law for a nominal, deterministic system using the backstepping method. Then, by applying Lyapunov analysis methods, we prove that the nominal backstepping control law can stabilize the PDE system with Markov jumping parameters if the nominal parameters are sufficiently close to the stochastic ones on average. The mean-square exponential stability conditions are theoretically derived and then validated via numerical simulations.
- Exponential stability of switched linear hyperbolic initial-boundary value problems. IEEE Transactions on Automatic Control, 57(2):291–301, 2011.
- H. Anfinsen and O. M. Aamo. Adaptive control of linear2×\times× 2 hyperbolic systems. Automatica, 87:69–82, 2018.
- Delay-robust control design for two heterodirectional linear coupled hyperbolic pdes. IEEE Transactions on Automatic Control, 63(10):3551–3557, 2018.
- J. Auriol and F. Di Meglio. Minimum time control of heterodirectional linear coupled hyperbolic pdes. Automatica, 71:300–307, 2016.
- Mean-square exponential stabilization of coupled hyperbolic systems with random parameters. IFAC-PapersOnLine, pages 8823–8828, 2023.
- G. Bastin and J.-M. Coron. Stability and boundary stabilization of 1-d hyperbolic systems, volume 88. Springer, 2016.
- On almost sure stability of continuous-time markov jump linear systems. Automatica, 42(6):983–988, 2006.
- Stop-and-go suppression in two-class congested traffic. Automatica, 125:109381, 2021.
- Local exponential h2superscriptℎ2h^{2}italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT stabilization of a 2×\times×2 quasilinear hyperbolic system using backstepping. SIAM Journal on Control and Optimization, 51(3):2005–2035, 2013.
- Boundary feedback control in networks of open channels. Automatica, 39(8):1365–1376, 2003.
- Stabilization of a system of n+1𝑛1n+1italic_n + 1 coupled first-order hyperbolic linear pdes with a single boundary input. IEEE Transactions on Automatic Control, 58(12):3097–3111, 2013.
- Continuous-time Markov jump linear systems. Springer Science & Business Media, 2012.
- E. B. Dynkin. Theory of Markov processes. Courier Corporation, 2012.
- A. Hoyland and M. Rausand. System reliability theory: models and statistical methods. John Wiley & Sons, 2009.
- Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes. IEEE Transactions on Automatic Control, 61(11):3301–3314, 2015.
- I. Kolmanovsky and T. L. Maizenberg. Mean-square stability of nonlinear systems with time-varying, random delay. Stochastic analysis and Applications, 19(2):279–293, 2001.
- Robust output regulation of 2×2222\times 22 × 2 hyperbolic systems: Control law and input-to-state stability. In 2018 Annual American Control Conference (ACC), pages 1732–1739. IEEE, 2018.
- Stability of switched linear hyperbolic systems by lyapunov techniques. IEEE Transactions on Automatic control, 59(8):2196–2202, 2014.
- S. M. Ross. Introduction to probability models. Academic press, 2014.
- Stochastically exponential stability and stabilization of uncertain linear hyperbolic pde systems with markov jumping parameters. Automatica, 48(3):569–576, 2012.
- H. Yu and M. Krstic. Traffic Congestion Control by PDE Backstepping. Springer, 2022.
- L. Zhang and C. Prieur. Stochastic stability of markov jump hyperbolic systems with application to traffic flow control. Automatica, 86:29–37, 2017.