Delphine Bresch-Pietri

Output-feedback adaptive control of a wave PDE with boundary anti-damping ✩

Abstract We develop an adaptive output-feedback controller for a wave PDE in one dimension with actuation on one boundary and with an unknown anti-damping term on the opposite boundary. This model is representative of a torsional stick–slip instability in drillstrings in deep oil drilling, as well as of various acoustic instabilities. The key feature of the proposed controller is that it requires only the measurements of boundary values and not of the entire distributed state of the system. Our approach is based on employing Riemann variables to convert the wave PDE into a cascade of two delay elements, with the first of the two delay elements being fed by control and the same element in turn feeding into a scalar ODE. This enables us to employ a prediction-based design for systems with input delays, suitably converted to the adaptive output-feedback setting. The result’s relevance is illustrated with simulation example.

Robust compensation of a chattering time-varying input delay

We investigate the design of a prediction-based controller for a linear system subject to a time-varying input delay, not necessarily causal. This means that the information feeding the system can be older than ones previously received. We propose to use the current delay value in the prediction employed in the control law. Modeling the input delay as a transport Partial Differential Equation, we prove asymptotic tracking of the system state, providing that the average ℒ2-norm of the delay time-derivative is sufficiently small. This result is obtained by generalizing Halanay inequality to time-varying differential inequalities.

Adaptive output-feedback for wave PDE with anti-damping - application to surface-based control of oil drilling stick-slip instability

We develop an adaptive output-feedback controller for a wave PDE in one dimension with actuation and measurement on one boundary and with an unknown anti-damping dynamics on the opposite boundary. This model is representative of drill string torsional instabilities arising in deep oil drilling, for which the model of bottom interaction with the rock is poorly known. The key achievement of the proposed controller is that it requires only the measurements of top-boundary values and not of the entire distributed state of the system. Our approach is based on employing Riemann variables to convert the wave PDE into a cascade of two delay elements and to reconstruct a delayed version of the unmeasured boundary. This enables us to employ a prediction-based design for systems with output and input delays, suitably converted to the adaptive output-feedback setting. The results relevance and ability to suppress undesirable torsional vibrations of the drill string in oil well drilling systems is illustrated with simulation example.

Implicit Integral Equations for Modeling Systems with a Transport Delay

In this chapter, we present a particular class of transport delay systems (e.g. systems involving transportation of material), in which the delay is defined through an implicit integral equation. To illustrate the practical interest of this class, experimental use of such models is presented for two different examples of physical systems, both from the field of automotive gasoline engines (specifically, exhaust gas recirculation and exhaust catalyst thermal dynamics). We also discuss related control challenges, together with some solutions.

Practical delay modeling of externally recirculated burned gas fraction for Spark-Ignited Engines

In this chapter, the authors provide an overview and study of the low-pressure burned gas recirculation in spark-ignited engines for automotive powertrain. It is shown, at the light of supportive experimental results, that a linear delay system permits to capture the dominant effects of the system dynamics. The modeled transport delay is defined by implicit equations stemming from first principles and can be calculated online. This model is shown to be sufficiently accurate to replace a sensor that would be difficult and costly to implement on commercial engines.

Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping

This paper presents a backstepping control design for a one-dimensional wave PDE with in-domain viscous damping, subject to a dynamical anti-damped boundary condition. Its main contribution is the design of an observer-based control law which stabilizes the wave PDE velocity, using only boundary mesurements. Numerical simulations on an oil-inspired example show the relevance of our result and illustrate the merits of this control design.

Robustness of an adaptive output feedback for an anti-damped boundary wave PDE in presence of in-domain viscous damping

This paper presents a robustness result for the use of a previously developed adaptive output-feedback controller designed for a pure wave PDE. This one-dimensional wave PDE has a boundary actuation opposite to an unknown anti-damping dynamics boundary. We prove that stabilization is still valid in the case of in-domain viscous damping using a controller designed neglecting the damping, provided that the damping coefficient is small enough. Our robustness proof grounds on the use of an alternative pure wave PDE, neglecting the indomain damping. We compare these two systems and apply a tailored backstepping transformation to carry out Lyapunov analysis. Numerical simulations show the relevance of our result for drilling applications and illustrate the merits of this control design.

Prediction-based control for nonlinear state- and input-delay systems with the aim of delay-robustness analysis

This paper investigates prediction-based control for nonlinear systems subject to both pointwise input- and (potentially) distributed state-delays. We address infinity-norm stability analysis of the corresponding closed-loop system reformulating both delays as transport Partial Differential Equations (PDEs) and transforming the resulting distributed state. We show how the performed analysis can be extended to establish robustness to delay uncertainties. We illustrate the merit of this design with numerical simulation of a prey-predator population dynamics.

Adaptive compensation of diffusion-advection actuator dynamics using boundary measurements

For (potentially unstable) Ordinary Differential Equation (ODE) systems with actuator delay, delay compensation can be obtained with a prediction-based control law. In this paper, we consider another class of PDE-ODE cascade, in which the Partial Differential Equation (PDE) accounts for diffusive effects. We investigate compensation of both convection and diffusion and extend a previously proposed control design to handle both uncertainty in the ODE parameters and boundary measurements. Robustness to small perturbations in the diffusion and convection coefficients is also proved.

Estimation for decentralized safety control under communication delay and measurement uncertainty

This paper addresses the design of a decentralized safety controller for two agents, subject to communication delay and imperfect measurements. The control objective is to ensure safety, meaning that the state of the two-agent system does not enter an undesired set in the state space. Assuming that we know a feedback map designed for the delay free-case, we propose a state estimation strategy which guarantees control agreement between the two agents. We present an estimation technique for bounded communication delays, assuming that the agents share the same internal clock, and extend it for infinitely-distributed communication delays by determining a lower bound for the probability of safety. We also explain how the proposed approach can be extended to a general system of N agents and discuss efficient computation of our estimation strategy. Performance of the controller and relevance of the proposed approach are discussed in light of simulations performed for a collision avoidance problem between two semi-autonomous vehicles at an intersection.