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Finite-time backstepping boundary stabilization of 3 × 3 hyperbolic systems
Abstract
We consider the problem of boundary stabilization of 3 × 3 linear first-order hyperbolic systems with one positive and two negative transport speeds by using backstepping. The main result of the paper is to supplement the previous works on how to choose multi-boundary feedback inputs applied on the states corresponding to the negative velocities to obtain finite-time stabilization of the original system in the spatial L 2 sense. Our method is still valid for boundary stabilization of general n × n hyperbolic system with arbitrary numbers of states traveling in either directions. 2010 Mathematics Subject Classification. 93D15, 35L04.
... c) Contribution: The first step towards this paper's general solution for m > 1 was presented (but not published as a paper) in [24] for m = 2 and n = 0. In conference paper [20], an extension to m = 2 and n = 1 is achieved. ...
... i.e. there are no (internal) diagonal coupling terms for v-system. Such coupling terms can be removed using a change of coordinates as presented in, e.g., [7] and [20]. This yields spatiallyvarying coupling terms, which is not an issue in the light of Remark 1. ...
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.
... The main focus of this paper is on the backstepping stabilization technique. While [10], [11], [14], [19], [34], [35] provide results in case of 2 × 2 or 3 × 3 coupled hyperbolic systems, there also exists literature if an arbitrary amount of linear coupled PDEs is considered [3], [12], [15], [20], [30]. In fact, the presented output feedback control of this work corresponds to the special case of the theoretical result in [20] for m = 3 and n = 1. ...
... where (19) and (20) assume that the same total traffic flow enters and leaves the track section which is given by the sum of the class 1 and class 2 equilibrium flows q * 1 and q * 2 . The traffic flow of class i is defined as ...
This paper develops boundary feedback control laws in order to damp out traffic oscillations in the congested regime of the linearized two-class Aw-Rascle (AR) traffic model. The macroscopic second-order two-class AR traffic model consists of four hyperbolic partial differential equations (PDEs) describing the dynamics of densities and velocities on freeway. The concept of area occupancy is used to express the traffic pressure and equilibrium speed relationship yielding a coupling between the two classes of vehicles. Each vehicle class is characterized by its own vehicle size and driver's behavior. The considered equilibrium profiles of the model represent evenly distributed traffic with constant densities and velocities of both classes along the investigated track section. After linearizing the model equations around those equilibrium profiles, it is observed that in the congested traffic one of the four characteristic speeds is negative, whereas the remaining three are positive. Backstepping control design is employed to stabilize the heterodirectional hyperbolic PDEs. The control input actuates the traffic flow at outlet of the investigated track section and is realized by a ramp metering. A full-state feedback is designed to achieve finite time convergence of the density and velocity perturbations to the equilibrium at zero. This result is then combined with an anti-collocated observer design in order to construct an output feedback control law that damps out stop-and-go waves in finite time by measuring the velocities and densities of both vehicle classes at the inlet of the investigated track section. The performance of the developed controllers is verified by simulation.
... c) Contribution: The first step towards this paper's general solution for m > 1 was presented (but not published as a paper) in [24] for m = 2 and n = 0. In conference paper [20], an extension to m = 2 and n = 1 is achieved. ...
... i.e. there are no (internal) diagonal coupling terms for v-system. Such coupling terms can be removed using a change of coordinates as presented in, e.g., [7] and [20]. This yields spatiallyvarying coupling terms, which is not an issue in the light of Remark 1. ...
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.
... The set F corresponds to the set of boundary conditions that are free to choose for the kernel equations. The freedom for the boundary condition (35) was already used in the works [HDM15,HDMVK16,HVDMK19] in order to give to (Γ A (M )) −− a structure of strictly lower triangular matrix. However, in the present paper this will not be used and it is the other boundary condition (36) that will turn out to be essential (see Section 6 below). ...
In this paper, we are interested in the minimal null control time of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls. Our main result is an explicit characterization of the smallest and largest values that this minimal null control time can take with respect to the internal coupling matrix. In particular, we obtain a complete description of the situations where the minimal null control time is invariant with respect to all the possible choices of internal coupling matrices. The proof relies on the notion of equivalent systems, in particular the backstepping method, a canonical LU-decomposition for boundary coupling matrices and a compactness-uniqueness method adapted to the null controllability property.
... In [11] the authors adapted this technique to obtain the first finite-time stabilization result for 2 × 2 linear hyperbolic system. This method was then developed, notably with a more careful choice of the target system, to treat 3 × 3 systems in [17] and then to treat general n × n systems in [18,20]. However, the control time obtained in these works was larger than the one in [29] and it was only shown in [1,5] that we can stabilize with the same time as the one of [29]. ...
The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order 2×2 linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.
... In [CVKB13] the authors adapted this technique to obtain the first finite-time stabilization result for 2 × 2 linear hyperbolic system. This method was then developed, notably with a more careful choice of the target system, to treat 3 × 3 systems in [HDM15] and then to treat general n × n systems in [HDMVK16,HVDMK19]. However, the control time obtained in these works was larger than the one in [Rus78b] and it was only shown in [ADM16,CHO17] that we can stabilize with the same time as the one of [Rus78b]. ...
The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.
In this paper we introduce a method to find the minimal control time for the null controllability of 1D first-order linear hyperbolic systems by one-sided boundary controls when the coefficients are regular enough.
Currently, scholars typically investigate the dynamics of two coupled partial differential systems. However, in practical engineering applications, there are often
N
coupled partial differential systems involved. In this paper, a novel
N
coupled hyperbolic-parabolic partial differential systems (HPPDS) is introduced. Firstly, the stabilization problem of the isolated HPPDS is addressed using Lyapunov functions and the backstepping method. Secondly, the synchronization problem of
N
coupled HPPDS is considered in the
sense. The synchronization problem of the target system is transformed into the stability problem of the decoupled system. The definition of synchronization error is provided, and the synchronization error system is derived. Subsequently, a boundary controller is designed using the backstepping method to achieve stability of the error system, i.e., synchronization of the
N
coupled HPPDS. Finally, two simulation examples are presented to demonstrate the validity of the obtained results.
In this paper, we are interested in the minimal null control time of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls. Our main result is an explicit characterization of the smallest and largest values that this minimal null control time can take with respect to the internal coupling matrix. In particular, we obtain a complete description of the situations where the minimal null control time is invariant with respect to all the possible choices of internal coupling matrices. The proof relies on the notion of equivalent systems, in particular the backstepping method, a canonical LU-decomposition for boundary coupling matrices and a compactness-uniqueness method adapted to the null controllability property.
This work focuses on the observer design for a first order ODE-transport PDE coupled at the boundary points. A novel anti-collocated observer and an output feedback boundary control law are designed for an under-actuated coupled system using the backstepping method. The homo-directional hyperbolic type PDE is considered with in-domain coupling between the states. The stabilization of the coupled systems are discussed by Lyapunov theory and linear matrix inequality (LMI) approach is implemented to design the gains. The obtained results show that the observer value coincides with the actual ones and it has been demonstrated through numerical examples. The effectiveness of the output feedback controller is also illustrated.
In this article we are interested in the boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space. We extend the so called “backstepping method” by introducing appropriate time-dependent integral transformations in order to map our initial system to a new one which has desired stability properties. The kernels of the integral transformations involved are solutions to non standard multi-dimensional hyperbolic PDEs, where the time dependence introduces several new difficulties in the treatment of their well-posedness. This work generalizes previous results of the literature, where only time-independent systems were considered.
This paper is concerned with the problem of boundary observer-based finite-time output feedback control for a heat system with Neumann boundary condition. In this system, it is endowed a single sensor at the left boundary and actuator at the right boundary. First, a backstepping finite-time observer is designed. Based on linear switching observer gains with state-dependent switching, the proposed observer estimates the system state variables in a finite-time only using one displacement boundary measurement. Then, an observer-based linear switching output feedback control with state-dependent switching is presented, which steers any solution of the output feedback closed-loop system to zero in a finite-time. Finally, the simulation results are provided to support the theoretical results.
This chapter deals with homogeneous stabilization of evolution systems. We design finite-time and fixed-time stabilizing homogeneous control laws for linear and nonlinear evolution equations in Hilbert and Euclidean spaces.
Based on the notion of generalized homogeneity, a new algorithm of feedback control design is developed for a plant modeled by a linear evolution equation in a Hilbert space with a possibly unbounded operator. The designed control law steers any solution of the closed-loop system to zero in a finite time. Method of homogeneous extension is presented in order to make the developed control design principles to be applicable for evolution systems with non-homogeneous operators. The design scheme is demonstrated for heat equation with the control input distributed on the segment [0,1].
The paper deals with boundary finite-time control for heat system. A linear switching control with state dependent switchings is designed based on backstepping procedure. It steers any solution of the heat system to zero in a finite time. The theoretical results are supported by numerical simulations.
Using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time-varying feedback laws. To this end, on the one hand, we provide a new proof of the well-posedness and the “optimal” bound with respect to damping constants for the solutions of the kernel equations; this allows one to deal with variable coefficients, even with a weak regularity of these coefficients. On the other hand, we establish the well-posedness and estimates for the heat equations with a nonlocal boundary condition at one side.
In this chapter, we address the problem of boundary stabilization of hyperbolic systems of balance laws by full state feedback
and by dynamic output feedback in observer-controller form
. We consider only the case of systems of two balance laws as in Section 5.3 The control design problem is solved by using a ‘backstepping’
method where the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The backstepping method for hyperbolic PDEs was initially introduced by Krstic and Smyshlyaev (2008a), Krstic and Smyshlyaev (2008b), and Smyshlyaev et al. (2010). This chapter is essentially based on Vazquez et al. (2011) and Coron et al. (2013).
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.
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 analyze the subcritical gas flow through fan-shaped networks of pipes, that is, through tree-shaped networks with exactly one node where more than two pipes meet. The gas flow in pipe networks is modeled by the isothermal Euler equations, a hyperbolic PDE system of balance laws. For this system we analyze stationary states and classical nonstationary solutions locally around a stationary state on a finite time interval. Furthermore, we present a Lyapunov function and boundary feedback laws to stabilize a fan-shaped network around a given stationary state.
Read More: http://epubs.siam.org/doi/abs/10.1137/100799824
Explicit boundary dissipative conditions are given for the exponential stability in L2-norm of one-dimensional linear hyperbolic systems of balance laws ∂tξ+Λ∂xξ−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.
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
In this work, we consider the problem of boundary stabilization for a quasilinear 2×2 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 H2 exponential stability of the closed-loop 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 4×4 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.
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.
This paper is an assessment of the current state of controllability and observability theories for linear partial differential equations, summarizing existing results and indicating open problems in the area. The emphasis is placed on hyperbolic and parabolic systems. Related subjects such as spectral determination, control of nonlinear equations, linear quadratic cost criteria and time optimal control are also discussed.
We consider the flow of gas through pipelines controlled by a compressor
station. Under a subsonic flow assumption we prove the existence
of classical solutions for a given finite time interval.
The existence result is used to construct Riemannian feedback laws and
to prove a stabilization result for a coupled system of gas pipes with a compressor
station. We introduce a Lyapunov function and prove exponential decay
with respect to the L2-norm.
Exponential decay of solutions to hyperbolic equations in bounded domains
- Jeffrey Rauch
- Michael Taylor