Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations

Abstract

The paper deals with the control and regulation by integral controllers for the nonlinear systems governed by scalar quasi-linear hyperbolic partial differential equations. Both the control input and the measured output are located on the boundary. The closed-loop stabilization of the linearized model with the designed integral controller is proved first by using the method of spectral analysis and then by the Lyapunov direct method. Based on the elaborated Lyapunov function we prove local exponential stability of the nonlinear closed-loop system with the same controller. The output regulation to the set-point with zero static error by the integral controller is shown upon the nonlinear system. Numerical simulations by the Preissmann scheme are carried out to validatethe robustness performance of the designed controller to face tounknown constant disturbances.

Adding Integral Action for Open-Loop Exponentially Stable Semigroups and Application to Boundary Control of PDE Systems

“…This bound is better then the one obtained in [20] for the linear transport equation (its maximal value is obtained for µ = 1 and is 1 √ e . Note however that similar to the bound of [20], the result obtained with our novel Lyapunov functional is far from the value we get following a frequency approach ( π 2 in this case). Recently in [7], the Lyapunov functional obtained in [20] has been modified to reach this optimal value of the integral gain.…”

“…Also, an interesting aspect of this Lyapunov approach is that explicit values of the supremum value of the gain k * i may be given. For instance, as in [20] consider the very particular case of a transport equation. In this case the system is simply…”

Regulation of Inhomogeneous Drilling Model With a P-I Controller

“…Moreover, it has been shown in [9] or [26] that it was possible to design a P-I for boundary control for different classes of hyperbolic systems. Following the approach of recent contributions in [23], [24], [19], [2], we prove that regulation and stabilization can be achieved using a Lyapunov approach.…”

PI boundary control of linear hyperbolic balance laws with stabilization of ARZ traffic flow models

“…Backstepping method is exploited in [13] to elaborate the PI-based trajectory tracking. Moreover, Lyapunov approach is considered in [14,15,16] for linear, nonlinear and network hyperbolic systems of conservation laws following the idea introduced in [17]. However, only few results appear in this direction due to the difficulties of constructing an appropriate Lyapunov candidate with PI boundary control.…”

“…PI boundary feedback controller is designed to stabilize the system by using integrated on-ramp metering and variable speed control. With respect to [15], we do not consider only a scalar conservation law, but rather a hyperbolic system of balance laws.…”