Control of 2×2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking

Abstract

We consider the problems of trajectory generation and tracking for general 2 × 2 systems of firstorder linear hyperbolic PDEs with anti-collocated boundary input and output. We solve the trajectory generation problem via backstepping. The reference input, which generates the desired output, incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by Vazquez, Krstic, and Coron for the backstepping stabilization of 2 × 2 linear hyperbolic systems. We apply our approach to a wave PDE with indefinite in-domain and boundary damping. For tracking the desired trajectory we employ a PI control law on the tracking error of the output. We prove exponential stability of the closed-loop system, under the proposed PI control law, when the parameters of the plant and the controller satisfy certain conditions, by constructing a novel "non-diagonal" Lyapunov functional. We demonstrate that the proposed PI control law compensates in the output the effect of in-domain and boundary disturbances. We illustrate our results with numerical examples.

Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

“…It should be noted that the assumed form of (1a) can always be obtained from the general case, in which Λ(z) and A(z) are arbitrary matrices with elements in C 1 [0, 1]. This is possible by making use of the transformations given in (Debnath, 2012;Lamare & Bekiaris-Liberis, 2015;Vazquez & Krstic, 2014, Ch. 6.9).…”

Prediction of traffic convective instability with spectral analysis of the Aw–Rascle–Zhang model

“…Analytical solutions to these non-linear equations are difficult to derive but linearization facilitates design of efficient control schemes with multiple inputs and outputs. Such schemes have for example been set up in this framework in [55] assuming Riemann invariants are proportional on the left boundary. We do not make such an assumption but pay particular attention to the formulation of boundary conditions to guarantee the well-posedness of the problem.…”

PDE-Based Feedback Control of Freeway Traffic Flow via Time-Gap Manipulation of ACC-Equipped Vehicles

“…Since second-order, PDE traffic flow models (i.e., systems that incorporate two PDE states, one for traffic density and one for traffic speed) constitute realistic descriptions of the traffic dynamics, capturing important phenomena, such as, for example, stop-and-go traffic, capacity drop, etc. [15], [28], [33], boundary control designs are recently developed for such systems [6], [26], [28], [49], [50], [53], [54] some of which are based on techniques originally developed for control of systems of hyperbolic PDEs, such as, for example, [12], [18], [25], [29], [31], [36], [46]. Even though simpler, first-order traffic flow models, in conservation law or Hamilton-Jacobi PDE formulation, are also important for modeling purposes.…”