Henrik Anfinsen

Disturbance Rejection in the Interior Domain of Linear 2

In this technical note, we develop a full state feedback law for disturbance rejection in systems described by linear 2 × 2 partial differential equations of the hyperbolic type, with the disturbance modelled as an autonomous, finite dimensional linear system affecting the PDEs left boundary, and actuation limited to the right boundary. The effect of the disturbance is rejected at an arbitrary point in the domain within a finite time. The performance is demonstrated through simulation.

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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.

2 Hyperbolic Systems

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.

An Adaptive Observer Design for

In this paper, we design a state feedback control law to adaptively stabilize a 2 × 2 linear hyperbolic system of partial differential equations (PDEs) with uncertain in-domain coupling coefficients. We do this using swapping design, where filters are designed so that the system states can be expressed as static, linear combinations of the filters and the uncertain parameters. Standard parameter identification methods can then be used to estimate the parameters. The adaptive observer is combined with a backstepping-based controller and boundedness in the ℒ2-sense of all signals in the closed loop system, as well as integrability of the system states are shown. The theory is demonstrated in a simulation.

n+1

We design a state feedback control law to adaptively stabilize a 2 × 2 linear hyperbolic system of partial differential equations (PDEs) with uncertain in-domain coupling coefficients. We do this by combining an identifier with a backstepping-based controller and show boundedness and integrability in the ℒ2-sense of all signals in the closed loop system. The theory is demonstrated in a simulation.

Coupled Linear Hyperbolic PDEs Based on Swapping

Abstract We solve a disturbance rejection problem for a general class of heterodirectional 1-D linear hyperbolic partial differential equations (PDEs). The disturbance enters at one boundary, while sensing and actuation is limited to the opposite boundary. We do this by combining a separately designed controller and observer into an output feedback controller, and proving stability of the overall closed loop system and exponential regulation of the control objective to zero. The theory is demonstrated in a simulation.

Estimation of boundary parameters in general heterodirectional linear hyperbolic systems

Abstract We design an adaptive control law stabilizing a class of systems of the form of n + 1 linear partial differential equations (PDEs) of the hyperbolic type from boundary sensing only. We do this by combining a recently derived observer based on swapping design, with a backstepping-based adaptive control law with time-varying gains. Boundedness of all signals in the closed loop system and asymptotic convergence to zero pointwise in space for the system states are proved. The theory is demonstrated in a simulation.

Stabilization of linear 2 × 2 hyperbolic systems with uncertain coupling coefficients - part II: Swapping design

Abstract We construct a control law that manages to adaptively stabilize a class of linear 2 × 2 hyperbolic systems of partial differential equations (PDEs) from a single boundary sensing anti-collocated with the boundary where actuation takes place. We do this by introducing a series of invertible transformations that bring the system into an observer canonical form, for which adaptive control design becomes feasible. We establish pointwise boundedness of all signals in the closed loop system, and pointwise convergence of the system states to zero. The theory is demonstrated in a simulation.

Stabilization of linear 2 × 2 hyperbolic systems with uncertain coupling coefficients - part I: Identifier-based design

Using a series of invertible transformations, we bring a system of 2 × 2 linear hyperbolic partial differential equations (PDEs) to an observer canonical form. With transport delays as the only information assumed known about the system parameters, a filter based adaptive output feedback control law is designed for the transformed system. Pointwise boundedness of all signals in the closed loop system and pointwise convergence of the system states to zero are established. The theory is demonstrated in a simulation.

Disturbance rejection in general heterodirectional 1-D linear hyperbolic systems using collocated sensing and control ☆

We use a backstepping technique with time-varying kernels to derive an observer that estimates unknown boundary parameters and the states in 2×2 linear hyperbolic PDEs from sensing anti-collocated with the uncertain parameters. The theory is demonstrated in a simulation.