Backstepping stabilization of 2×2 linear hyperbolic PDEs coupled with potentially unstable actuator and load dynamics (original) (raw)
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arXiv: Optimization and Control, 2020
In this paper, a linear hyperbolic system of balance laws with boundary disturbances in one dimension is considered. An explicit candidate Input-to-State Stability (ISS)-Lyapunov function in $ L^2- $norm is considered and discretised to investigate conditions for ISS of the discrete system as well. Finally, experimental results on test examples including the Saint-Venant equations with boundary disturbances are presented. The numerical results demonstrate the expected theoretical decay of the Lyapunov function.
Numerical boundary feedback stabilisation of non-uniform hyperbolic systems of balance laws
International Journal of Control, 2018
In this paper, numerical boundary stabilisation of a non-uniform hyperbolic system of balance laws is studied. For the numerical discretisation of the balance laws, a first order explicit upwind scheme is used for the spatial discretisation; and for the temporal discretisation a splitting technique is employed. A discrete L 2 −Lyapunov function is employed to investigate conditions for the stability of the system. After constructing discrete numerical Lyapunov functionals, we prove an asymptotic exponential stability result for a class of non-uniform linear hyperbolic systems of balance laws. Convergence of the solution to its equilibrium is proved. Further application of the approach to practical problems through concrete examples is presented together with suggestions for numerical implementation. The numerical computations also demonstrate the stability of the numerical scheme with parameters chosen to satisfy the stability requirements.
Applied Mathematics & Optimization, 2020
In this paper, a linear hyperbolic system of balance laws with boundary disturbances in one dimension is considered. An explicit candidate Input-to-State Stability (ISS)-Lyapunov function in L 2 −norm is considered and discretised to investigate conditions for ISS of the discrete system as well. Finally, experimental results on test examples including the Saint-Venant equations with boundary disturbances are presented. The numerical results demonstrate the expected theoretical decay of the Lyapunov function.
arXiv: Optimization and Control, 2020
A boundary feedback stabilisation problem of non-uniform linear hyperbolic systems of balance laws with additive disturbance is discussed. A continuous and a corresponding discrete Lyapunov function is defined. Using an input-to-state-stability (ISS) $ L^2- Lyapunovfunction,thedecayofsolutionsoflinearsystemsofbalancelawsisproved.Inthediscreteframework,afirst−orderfinitevolumeschemeisemployed.Insuchcases,thedecayratescanbeexplicitlyderived.ThemainobjectiveistoprovetheLyapunovstabilityfortheLyapunov function, the decay of solutions of linear systems of balance laws is proved. In the discrete framework, a first-order finite volume scheme is employed. In such cases, the decay rates can be explicitly derived. The main objective is to prove the Lyapunov stability for the Lyapunovfunction,thedecayofsolutionsoflinearsystemsofbalancelawsisproved.Inthediscreteframework,afirst−orderfinitevolumeschemeisemployed.Insuchcases,thedecayratescanbeexplicitlyderived.ThemainobjectiveistoprovetheLyapunovstabilityfortheL^2$-norm for linear hyperbolic systems of balance laws with additive disturbance both analytically and numerically. Theoretical results are demonstrated by using numerical computations.
Dynamic boundary stabilization of linear and quasi-linear hyperbolic systems
2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012
Systems governed by hyperbolic partial differential equations with dynamics associated with their boundary conditions are considered in this paper. These infinite dimensional systems can be described by linear or quasi-linear hyperbolic equations. By means of Lyapunov based techniques, some sufficient conditions are derived for the exponential stability of such systems. A polytopic approach is developed for quasilinear hyperbolic systems in order to guarantee stability in a region of attraction around an equilibrium point, given specific bounds on the parameters. The main results are illustrated on the model of an isentropic inviscid flow.
Lyapunov exponential stability of linear hyperbolic systems of balance laws
a b s t r a c t Explicit boundary dissipative conditions are given for the exponential stability in L 2 -norm of onedimensional linear hyperbolic systems of balance laws ∂ t ξ + Λ∂ x ξ − Mξ = 0 over a finite interval, when the matrix M is marginally diagonally stable. The result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised Saint-Venant-Exner equations.
Backstepping stabilization of an underactuated 3× 3 linear hyperbolic system of fluid flow equations
2011
We investigate the boundary stabilization of a particular subset of 3×3 linear hyperbolic systems with varying coefficients on a bounded domain. The system is underactuated since only one of the three hyperbolic PDEs is actuated at the boundary. The setup considered in the paper occurs in control of multiphase flows on oil production systems. We use a backstepping approach to design a full-state feedback law yielding exponential stability of the origin.
Feedback stabilization of bilinear coupled hyperbolic systems
Discrete & Continuous Dynamical Systems - S, 2021
This paper studies the problem of stabilization of some coupled hyperbolic systems using nonlinear feedback. We give a sufficient condition for exponential stabilization by bilinear feedback control. The specificity of the control used is that it acts on only one equation. The results obtained are illustrated by some examples where a theorem of Mehrenberger has been used for the observability of compactly perturbed systems [18].