Stabilization of coupled linear heterodirectional hyperbolic PDE–ODE systems

Abstract

We solve the problem of stabilizing a linear ODE having a system of a linearly coupled hyperbolic PDEs in the actuating and sensing paths. The system is exponentially stabilized by mapping it to a target system with a cascade structure using a Volterra transformation.Key words: Stabilization, Distributed Parameter Systems IntroductionThe interest for coupled Ordinary Differential EquationsPartial Differential Equations (ODE-PDE) systems has first emerged when considering delays in the actuating and sensing paths of ODE. Delays can be seen as first-order hyperbolic PDEs. There are many approaches to deal with input or measurement delays, usually divided into two categories: memoryless controllers, which extend standard control techniques without explicitly accounting for the delay in the control design [16,25,9]; and prediction-based controllers aiming at explicitly compensating the delay [20,4,2].The use of Lyapunov and backstepping methods enabled dealing with more involved PDEs in the actuating and sensing paths. In [13], an output feedback control law is derived for an ODE having a heat equation in the actuating and sensing paths. The coupled PDE-ODE system is stabilized using an observer-controller structure relying on a backstepping approach. The same approach has been used to deal with ODEs coupled (rather than cascaded) with parabolic PDEs The first application of the backstepping approach to deal with hyperbolic PDE-ODE couplings is [14] where actuator and sensor delays are explicitly compensated. While this problem had already been tackled by, e.g., the Smith predictor [20], the reformulation of the delay as a linear ODE enabled numerous related problem to be tackled, most notably non-constant and uncertain delays [2,4]. In [22], the problem of stabilizing a multi-input ODE with distinct delays is tackled using a backstepping approach. In [6], an observer is designed for an ODE having a homodirectional 1 hyperbolic PDE in the sensing path, relying on a Lyapunov analysis requiring to solve Linear Matrix Inequalities (LMI). As will appear, the systems of [22] are particular cases of the system considered here, although the control design approaches are different.Here, we solve the problem of stabilizing an ODE with a system of first-order linear hyperbolic PDEs in the sensing 1 i.e. where all the states transport in the same direction

Compensation of actuator dynamics governed by quasilinear hyperbolic PDEs

“…A topic of future research may be the problem of boundary stabilization of general, quasilinear systems of first-order hyperbolic PDEs coupled with nonlinear ODE systems, as it is done in [18] for the case in which both the PDE and ODE parts of the system are linear.…”

“…From (19) it is evident that the positivity assumption of v guarantees that the delay is always positive, i.e., it guarantees the causality of system (18), and thus, also of system (1)-(3). Moreover, relation (4) guarantees the boundness of the delay, i.e., it guarantees that the control signal eventually reaches the plant (18), and thus, also (1).…”

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.…”

A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

“…Other techniques, such as backstepping method [12], [13], [14], [15] and sliding mode control method [16], [17], rely on the use of predetermined Lyapunov functional which is chosen in an ad hoc manner. Other methods, such as [5], [6], [7], [8] use LMIs to search over a given set of Lyapunov functionals.…”