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Boundary exponential stabilization of 1-D inhomogeneous quasilinear hyperbolic systems
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Abstract
This paper deals with the problem of boundary stabilization of first-order n\times n inhomogeneous quasilinear hyperbolic systems. A backstepping method is developed. The main result supplements the previous works on how to design multi-boundary feedback controllers to realize exponential stability of the original nonlinear system in the spatial H^2 sense.
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We analyse dissipative boundary conditions for nonlinear hyperbolic systems
in one space dimension. We show that a previous known sufficient condition for
exponential stability with respect to the C^1-norm is optimal. In particular a
known weaker sufficient condition for exponential stability with respect to the
H^2-norm is not sufficient for the exponential stability with respect to the
C^1-norm. Hence, due to the nonlinearity, even in the case of classical
solutions, the exponential stability depends strongly on the norm considered.
We also give a new sufficient condition for the exponential stability with
respect to the W^{2,p}-norm. The methods used are inspired from the theory of
the linear time-delay systems and incorporate the characteristic method.
In this work, we consider the problem of boundary stabilization for a
quasilinear 2X2 system of first-order hyperbolic PDEs. We design a new
full-state feedback control law, with actuation on only one end of the domain,
which achieves H^2 exponential stability of the closedloop system. Our proof
uses a backstepping transformation to find new variables for which a strict
Lyapunov function can be constructed. The kernels of the transformation are
found to verify a Goursat-type 4X4 system of first-order hyperbolic PDEs, whose
well-posedness is shown using the method of characteristics and successive
approximations. Once the kernels are computed, the stabilizing feedback law can
be explicitly constructed from them.
We give a new sufficient condition on the boundary conditions for the exponential stability of one-dimensional nonlinear hyperbolic systems on a bounded interval. Our proof relies on the construction of an explicit strict Lyapunov function. We compare our sufficient condition with other known sufficient conditions for nonlinear and linear one-dimensional hyperbolic systems.
We consider the problem of generating and tracking a trajectory between two arbitrary parabolic profiles of a periodic 2D channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poiseuille flow, this problem is frequently cited as a paradigm for transition to turbulence. Our procedure consists in generating an exact trajectory of the nonlinear system that approaches exponentially the objective profile. Using a backstepping method, we then design boundary control laws guaranteeing that the error between the state and the trajectory decays exponentially in , , and norms. The result is first proved for the linearized Stokes equations, then shown to hold locally for the nonlinear Navier-Stokes system.
We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel
This paper is concerned with boundary dissipative conditions that guarantee the exponential stability of classical solutions of one-dimensional quasi-linear hyperbolic systems. We present a comprehensive review of the results that are available in the literature. The main result of the paper is then to supplement these previous results by showing how a new Lyapunov stability approach can be used for the analysis of boundary conditions that are known to be dissipative for the C1-norm.
We solve the problem of stabilization of a class of linear first-order hyperbolic systems featuring n rightward convecting transport PDEs and one leftward convecting transport PDE. We design a controller, which requires a single control input applied on the leftward convecting PDE's right boundary, and an observer, which employs a single sensor on the same PDE's left boundary. We prove exponential stability of the origin of the resulting plant-observer-controller system in the spatial L-2-sense.
Research on stabilization of coupled hyperbolic PDEs has been dominated by
the focus on pairs of counter-convecting ("heterodirectional") transport PDEs
with distributed local coupling and with controls at one or both boundaries. A
recent extension allows stabilization using only one control for a system
containing an arbitrary number of coupled transport PDEs that convect at
different speeds against the direction of the PDE whose boundary is actuated.
In this paper we present a solution to the fully general case, in which the
number of PDEs in either direction is arbitrary, and where actuation is applied
on only one boundary (to all the PDEs that convect downstream from that
boundary). To solve this general problem, we solve, as a special case, the
problem of control of coupled "homodirectional" hyperbolic linear PDEs, where
multiple transport PDEs convect in the same direction with arbitrary local
coupling. Our approach is based on PDE backstepping and yields solutions to
stabilization, by both full-state and observer-based output feedback,
trajectory planning, and trajectory tracking problems.
We study the asymptotic stabilization of the origin for the two-dimensional (2-D) Euler equation of incompressible inviscid ∞uid in a bounded,domain. We assume that the controls act on an arbitrarily small nonempty,open subset of the boundary. We prove the null global asymptotic stabilizability by means of explicit feedback laws if the domain,is connected and simply connected. Key words. inviscid ∞uid, stabilization, nonlinear control AMS subject classications. 76C99, 93B52, 93C20, 93D15 PII. S036301299834140X
Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains Tn of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid”, 0≤ξn≤ξn−1⋯≤ξ1≤x≤1. We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L2 and H1 local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.
We consider a problem of boundary feedback stabilization of first-order hyperbolic partial differential equations (PDEs). These equations serve as a model for physical phenomena such as traffic flows, chemical reactors, and heat exchangers. We design controllers using a backstepping method, which has been recently developed for parabolic PDEs. With the integral transformation and boundary feedback the unstable PDE is converted into a “delay line” system which converges to zero in finite time. We then apply this procedure to finite-dimensional systems with actuator and sensor delays to recover a well-known infinite-dimensional controller (analog of the Smith predictor for unstable plants). We also show that the proposed method can be used for the boundary control of a Korteweg–de Vries-like third-order PDE. The designs are illustrated with simulations.
Conditions for boundary feedback stabilizability of non-uniform linear 2×22×2 hyperbolic systems over a bounded interval are investigated. The main result is to show that the existence of a basic quadratic control Lyapunov function requires that the solution of an associated ODE is defined on the considered interval. This result is used to give explicit conditions for the existence of stabilizing linear boundary feedback control laws. The analysis is illustrated with an application to the boundary feedback stabilization of open channels represented by linearized Saint–Venant equations with non-uniform steady-states.
Abstract We develop output-feedback adaptive controllers for two benchmark,parabolic PDEs motivated by a model,of thermal instability in solid propellant rockets. Both benchmark plants are unstable, have infinite relative degree, and are controlled from the boundary. One plant has an unknown,parameter,in the PDE and the other in the boundary,condition. In both cases the unknown,parameter multiplies the measured output of the system, which is obtained with a boundary sensor located on the “opposite side” of the domain from the actuator. In comparison with the Lyapunov output-feedback adaptive controllers in Krstic and Smyshlyaev,[(2005). Adaptive boundary,control for unstable parabolic PDEs—Part I: Lyapunov design. IEEE Transactions on Automatic Control, submitted for publication], the controllers presented here employ much,simpler update laws and do not require a priori knowledge,about the unknown,parameters. We show how,our two benchmarks,examples can be combined,and illustrate the adaptive stabilization design by simulation. 2007 Elsevier Ltd. All rights reserved. Keywords: Adaptive control; Boundary control; Distributed parameter systems
For a class of stabilizing boundary controllers for nonlinear 1D parabolic PDEs introduced in a companion paper, we derive bounds for the gain kernels of our nonlinear Volterra controllers, prove the convergence of the series in the feedback laws, and establish the stability properties of the closed-loop system. We show that the state transformation is at least locally invertible and include an explicit construction for computing the inverse of the transformation. Using the inverse, we show L2 and H1 exponential stability and explicitly construct the exponentially decaying closed-loop solutions. We then illustrate the theoretical results on an analytically tractable example.
Finite-time backstepping stabilization of 3 × 3 hyperbolic systems
- Long Hu
- Florent Di Meglio
Long Hu and Florent Di Meglio. Finite-time backstepping stabilization of 3 × 3 hyperbolic
systems. IEEE European Control Conference, pages 67-72, 2015.
Boundary control of PDEs
- Miroslav Krstic
- Andrey Smyshlyaev
Miroslav Krstic and Andrey Smyshlyaev. Boundary control of PDEs, volume 16 of Advances
in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia,
PA, 2008. A course on backstepping designs.
Exact boundary controllability for quasi-linear hyperbolic systems
- Tatsien Li
- Bopeng Rao
Tatsien Li and Bopeng Rao. Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J. Control Optim., 41(6):1748-1755, 2003.