Mamadou Diagne

An Adaptive Observer Design for

In this paper, we use swapping design filters to bring systems of n+1 partial differential equations of the hyperbolic type to static form. Standard parameter identification laws can then be applied to estimate unknown parameters in the boundary conditions. Proof of boundedness of the adaptive laws are offered, and the results are demonstrated in simulations.

n+1

Abstract In this paper, we extend recent results on state and boundary parameter estimation in coupled systems of linear partial differential equations (PDEs) of the hyperbolic type consisting of n rightward and one leftward convecting equations, to the general case which involves an arbitrary number of PDEs convecting in both directions. Two adaptive observers are derived based on swapping design, where one introduces a set of filters that can be used to express the system states as linear, static combinations of the filter states and the unknown parameters. Standard parameter identification laws can then be applied to estimate the unknown parameters. One observer which requires sensing at both boundaries, generates estimates of the boundary parameters and system states, while the second observer estimates the parameters from sensing limited to the boundary anti-collocated with the uncertain parameters. Proof of boundedness of the adaptive laws is offered, and sufficient conditions ensuring exponential convergence are derived. The theory is verified in simulations.

Coupled Linear Hyperbolic PDEs Based on Swapping

Abstract This paper deals with the stability study of the nonlinear Saint-Venant Partial Differential Equation (PDE). The proposed approach is based on the multi-model concept which takes into account some Linear Time Invariant (LTI) models defined around a set of operating points. This method allows describing the dynamics of this nonlinear system in an infinite dimensional space over a wide operating range. A stability analysis of the nonlinear Saint-Venant PDE is proposed both by using Linear Matrix Inequalities (LMIs) and an Internal Model Boundary Control (IMBC) structure. The method is applied both in simulations and real experiments through a microchannel, illustrating thus the theoretical results developed in the paper.

Estimation of boundary parameters in general heterodirectional linear hyperbolic systems

Using backstepping design, exponential stabilization of the linearized Saint-Venant–Exner (SVE) model of water dynamics in a sediment-filled canal with arbitrary values of canal bottom slope, friction, porosity, and water–sediment interaction, is achieved. The linearized SVE model consists of two rightward convecting transport Partial Differential Equations (PDEs) and one leftward convecting transport PDE. A single boundary input control strategy with actuation located only at the downstream gate is employed. A full state feedback controller is designed which guarantees exponential stability of the desired setpoint of the resulting closed-loop system. Using the reconstruction of the distributed state through a backstepping observer, an output feedback controller is established, resulting in the exponential stability of the closed-loop system at the desired setpoint. The proposed state and output feedback controllers can deal with both subcritical and supercritical flow regimes without any restrictive conditions.

A multi-model approach to Saint-Venant equations

In this paper a delay-compensated Bang-Bang control law for the stabilization of the nozzle output flow rate of an isothermal screw-extruder-based 3D printing process is designed. The presented application has a great potential to move beyond the common 3D printing processes such as Fused Deposition Modeling (FDM) and Syringe Based Extrusion (SBE), improving the build speed and the print precision. A geometrical decomposition of the screw extruder in a partially and a fully filled regions (PFZ and FFZ) allows to describe the material convection in the extruder chamber by a 1D hyperbolic Partial Differential Equation (PDE) coupled with an Ordinary Differential Equation (ODE). After solving the hypercolic PDE by the Method of Characteristics (MC), the coupled PDE-ODE system is transformed into a nonlinear state-dependent input delay system. The Global Exponential Stability (GES) of the nonlinear free-delay plant is established with a piecewise exponential feedback control law. Combining the “Bang-Bang”-like controller with a nonlinear predictor feedback control law, the Global Asymptotic Stability (GAS) of the plant with respect to any setpoint in the physical domain is ensured.

Backstepping stabilization of the linearized Saint-Venant–Exner model

In this article, we study the stabilization problem for an extrusion process in the isothermal case. The model expresses the mass conservation in the extruder chamber and consists of a hyperbolic Partial Differential Equation (PDE) and a nonlinear Ordinary Differential Equation (ODE) whose dynamics describes the evolution of a moving interface. By using a Lyapunov approach, we obtain the exponential stabilization for the closed-loop system under natural feedback controls through indirect measurements. Numerical simulations are also provided with a comparison between the proposed approach and linear PI feedback controller.

State-dependent input delay-compensated Bang-Bang control: Application to 3D printing based on screw-extruder

Abstract This paper is an analysis of 1D hyperbolic partial differential equation with moving interface as a delay system. The model derives from the mass balance of an extrusion process that describes the strong coupling between the mass transport equation and an ordinary differential equation which represents the interface motion. Solving the transport equation by the method of characteristics, we obtain an state-dependent-input-delay control problem. The stabilization of the whole system around an equilibrium is done by using a state predictor.

Feedback Stabilization for the Mass Balance Equations of an Extrusion Process

In this paper, a backstepping control of the one-phase Stefan Problem, which is a 1-D diffusion Partial Differential Equation (PDE) defined on a time varying spatial domain described by an ordinary differential equation (ODE), is studied. A new nonlinear backstepping transformation for moving boundary problem is utilized to transform the original coupled PDE-ODE system into a target system whose exponential stability is proved. The full-state boundary feedback controller ensures the exponential stability of the moving interface to a reference setpoint and the ℋ1-norm of the distributed temperature by a choice of the setpint satisfying given explicit inequality between initial states that guarantees the physical constraints imposed by the melting process.

Mass Transport Equation with Moving Interface and Its Control As an Input Delay System.

In this paper, we study the well-posedness and exact controllability of a physical model for an extrusion process in the isothermal case. The model expresses the mass balance in the extruder chamber and consists of a hyperbolic partial differential equation (PDE) and a nonlinear ordinary differential equation (ODE) whose dynamics describes the evolution of a moving interface. By suitable change of coordinates and fixed point arguments, we prove the existence, uniqueness, and regularity of the solution and finally, the exact controllability of the coupled system. Copyright

Backstepping control of the one-phase stefan problem

The topic of this paper is to present and analyse a physical model of the extrusion process which is expressed two systems of conservation laws (with source terms) coupled by a moving interface whose relation is derived from the conservation of momentum. After a change of variables on the spatial variables is performed in order to transform the time-varying spatial domain in fixed one, the linearisation of the model around an equilibrium profile is given, the well-posedness in the sense of the existence of a C0-semigroup of those the coupled systems is proven.