No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
On boundary stability of inhomogeneous 1-D hyperbolic systems for the norm.
Abstract
We study the exponential stability for the C¹ norm of general 2 × 2 1-D quasilinear hyperbolic systems with source terms and boundary controls. When the propagation speeds of the system have the same sign, any nonuniform steady-state can be stabilized using boundary feedbacks that only depend on measurements at the boundaries and we give explicit conditions on the gain of the feedback. In other cases, we exhibit a simple numerical criterion for the existence of basic C¹ Lyapunov function, a natural candidate for a Lyapunov function to ensure exponential stability for the C¹ norm. We show that, under a simple condition on the source term, the existence of a basic C¹ (or Cp , for any p ≥ 1) Lyapunov function is equivalent to the existence of a basic H² (or Hq , for any q ≥ 2) Lyapunov function, its analogue for the H² norm. Finally, we apply these results to the nonlinear Saint-Venant equations. We show in particular that in the subcritical regime, when the slope is larger than the friction, the system can always be stabilized in the C¹ norm using static boundary feedbacks depending only on measurements at the boundaries, which has a large practical interest in hydraulic and engineering applications.
... The authors also considered Lyapunov approach and LaSalle invariance principle to study the stabilization. It is also worth mentioning that, the Lyapunov approach has been successfully used in the recent works [31], [32] for linear and nonlinear hyperbolic model. One may also see the book [3] for other hyperbolic model. ...
... This proves the trace regularity (32). Thus to conclude the Proposition 3.3, we need to prove (33). ...
... This inequality forces us to take T > 1. Therefore we now have the desired inequality (33). By using the density of D(A) in L 2 (0, 1), we can prove all the three inequalities (31), (32) and (33) ...
In this paper, we study the controllability of a coupled first-order hyperbolic-elliptic system in the interval (0,1) by a Dirichlet boundary control acting at the left endpoint of the hyperbolic component only. Using the multiplier approach and compactness-uniqueness argument, we establish the exact controllability for the hyperbolic component of the model, at any time . We explore the method of moments to conclude the exact controllability at the critical time T=1. For the case of small time, that is for , we show that the system is not null controllable. Further, using a Gramian-based approach introduced by Urquiza, we prove the exponential stabilization of the corresponding closed-loop system with arbitrary prescribed decay rate by means of boundary feedback control law.
... In this article we present cutting-edge tools for stabilizing nonlinear 1D hyperbolic systems with boundary controls and for ensuring their exponential stability, or their ISS in some cases. We will state the results and only give ideas of the proofs, which can be found in detail in [91] and in [94,89,88,18,17,47,90,95,92,16], from which the results presented in this article are taken. While some methods presented have existed for decades, others are only very recent. ...
... We denote again A(0, x) = diag(Λ 1 , Λ 2 ) and, in the rest of this subsection, we assume that n = 2 unless specified otherwise. When Λ 1 and Λ 2 have the same sign, there always exists an energy-like Lyapunov function both for the C 1 norm and the H 2 norm, as shown in [88], Theorem 3.7. Assume that A, B and G are of class C 2 and Λ 1 Λ 2 > 0 and let M (x) = ∂ u B(0, x). ...
... In addition, we can see a strong similarity between the condition (3.27) and the condition (3.17) of Theorem 3.3 dealing with the H 2 norm. This enables the following comparison result [88] Corollary. Assume that A, B, G are of class C 3 and Λ 1 Λ 2 < 0 and let M = ∂ u B(0, ·). ...
Hyperbolic systems model the phenomena of propagations at finite speeds. They are present in many fields of science and, consequently, in many human applications. For these applications, the question of stability or stabilization of their stationary state is a major issue. In this paper we present state-of-the-art tools to stabilize 1-D nonlinear hyperbolic systems using boundary controls. We review the power and limits of energy-like Lyapunov functions; the particular case of density–velocity systems; a method to stabilize shock steady-states; an extraction method allowing to use the spectral information of the linearized system in order to stabilize the nonlinear system; and some results on proportional-integral boundary control. We also review open questions and perspectives for this field, which is still largely open.
... Lyapunov techniques allows to establish exponential stabilization in Sobolev or C p spaces when term M is supposed to be small. Applications to linearized Saint Venant systems are given in [4,10,14,19,20]. ...
... In order to conclude on the whole stability, we have to prove that the infinite-dimensional part in (20) is not destabilized by the control. To prove the stability of the infinite-dimensional part, it suffices to see the second equation of (20) as an inhomogeneous Cauchy problem on M : ...
In this work, the problem of stabilization of general systems of linear transport equations with in-domain and boundary couplings is investigated. It is proved that the unstable part of the spectrum is of finite cardinal. Then, using the pole placement theorem, a linear full state feedback controller is synthesized to stabilize the unstable finite-dimensional part of the system. Finally, by a careful study of semigroups, we prove the exponential stability of the closed-loop system. As a by product, the linear control constructed before is saturated and a fine estimate of the basin of attraction is given.
... Hence, we can see that the feedforward controller P c (s ) seems to achieve the desired purpose. There is however an important limitation: the result is obtained under the assumption that the initial condition (18) is uniform and that the initial output density H(0 , L ) is already at the set point H * L . If this assumption is not verified, then initial transients may appear. ...
... Furthermore, in this special case of Saint-Venant equations, it is worth noting that conditions (a1)-(a3) and (b) ensure exponential stability not only in L 2 but also in C 0 for the linearized system (and thus locally in C 1 for the nonlinear system). This property follows directly from Theorem 3.2 and Corollary 1 in [18] (see also [17] ). ...
Because they represent physical systems with propagation delays, hyperbolic systems are well suited for feedforward control. This is especially true when the delay between a disturbance and the output is larger than the control delay. In this paper, we address the design of feedforward controllers for a general class of 2×2 hyperbolic systems with a single disturbance input located at one boundary and a single control actuation at the other boundary. The goal is to design a feedforward control that makes the system output insensitive to the measured disturbance input. We show that, for this class of systems, there exists an efficient ideal feedforward controller which is causal and stable. The problem is first stated and studied in the frequency domain for a simple linear system. Then, our main contribution is to show how the theory can be extended, in the time domain, to general nonlinear hyperbolic systems. The method is illustrated with an application to the control of an open channel represented by Saint-Venant equations where the objective is to make the output water level insensitive to the variations of the input flow rate. Finally, we address a more complex application to a cascade of pools where a blind application of perfect feedforward control can lead to detrimental oscillations. A pragmatic way of modifying the control law to solve this problem is proposed and validated with a simulation experiment.
... Hence, we can see that the feedforward controller P c (s) seems to achieve the desired purpose. There is however an important limitation: the result is obtained under the assumption that the initial condition (18) is uniform and that the initial output density H(0, L) is already at the set point H * L . If this assumption is not verified, then initial transients may appear. ...
... Furthermore, in this special case of Saint-Venant equations, it is worth noting that conditions (a1), (a2), (a3) and (b) ensure exponential stability not only in L 2 but also in C 0 for the linearized system (and thus locally in C 1 for the nonlinear system). This property follows directly from Theorem 3.2 and Corollary 1 in [18] (see also [17]). This means that, if the disturbance d(t) is bounded, then all the internal signals of the control system are also guaranteed to be bounded. ...
Because they represent physical systems with propagation delays, hyperbolic systems are well suited for feedforward control. This is especially true when the delay between a disturbance and the output is larger than the control delay. In this paper, we address the design of feedforward controllers for a general class of 2 times 2 hyperbolic systems with a single disturbance input located at one boundary and a single control actuation at the other boundary. The goal is to design a feedforward control that makes the system output insensitive to the measured disturbance input. We show that, for this class of systems, there exists an efficient ideal feedforward controller which is causal and stable. The problem is first stated and studied in the frequency domain for a simple linear system. Then, our main contribution is to show how the theory can be extended, in the time domain, to general nonlinear hyperbolic systems. The method is illustrated with an application to the control of an open channel represented by Saint- Venant equations where the objective is to make the output water level insensitive to the variations of the input flow rate. Finally, we address a more complex application to a cascade of pools where a blind application of perfect feedforward control can lead to detrimental oscillations. A pragmatic way of modifying the control law to solve this problem is proposed and validated with a simulation experiment.
... The second approach relies on Lyapunov functions. Many results on the boundary stabilization of first-order hyperbolic systems, linear and nonlinear, have been obtained using this approach: see for example the book [24], and the recent results in [91,92]. However, this approach can be limited, as it is sometimes impossible to obtain an arbitrary decay rate using Lyapunov functions (see [53,Remark 12.9, page 318] for a finite dimensional example). ...
... To investigate the stabilization of infinite-dimensional systems, there are three main types of approaches: the Gramian method (see for example [153,151,108]), Lyapunov functions (see for example [55], the book [24], and the recent results in [91,92], which study the boundary stabilization of hyperbolic systems), and the backstepping method. The latter is derived from a method in finite dimension, also called backstepping, used to stabilize stabilizable systems with an added chain of integrators (see [110,53,143] for an overview of the finite-dimensional case, and [56] or [118] for applications to partial differential equations). ...
In this thesis we study controllability and stabilization questions for some hyperbolic systems in one space dimension, with an internal control. The first question we study is the indirect internal controllability of a system of two coupled semilinear wave equations, the control being a function of time and space. Using the so-called fictitious control method, we give sufficient conditions for such a system to be locally controllable around 0, and a natural condition linking the minimal control time to the support of the control. Then, we study a particular case where the aforementioned sufficient conditions are not satisfied, applying the return method.The second question in this thesis is the design of explicit scalar feedbacks to stabilize controllable systems. The method we use draws from the backstepping method for PDEs elaborated by Miroslav Krstic, and its most recent developments: thus, the controllability of the system under consideration plays a crucial role. The method yields explicit stationary feedbacks which stabilize the linear periodic transport equation exponentially, and even in finite time.Finally, we implement this method on a more complex system, the so-called water tank system. We prove that the linearized systems around constant acceleration equilibria are controllable if the acceleration is not too strong. Our method then yields explicit feedbacks which, although they remain explicit, are no longer stationary and require the addition of an integrator in the feedback loop.
... Using Lyapunov techniques, it is possible to establish exponential stabilization in Sobolev or C p spaces when the terms M •• are supposed to be small. Applications to Saint Venant systems are given in [16,17,8,5,12]. ...
In this paper, we introduce the numerical backstepping method by applying it to a problem of finite-time stabilization for a system of \begin{document}\end{document} balance laws discretized thanks to the upwind scheme. On the one hand, we illustrate on an example that the scheme used to compute the feedback control cannot be chosen arbitrarily. On the other hand, an algorithm is given to construct this control properly and an approached finite-time stabilization result is proven.
... There exists various way to design feedback laws for infinite dimensional systems: Riccati equations (see for instance [36,45] and the references therein), Gramian methods [31,46], Lyapunov functionals (see for instance [5,23,28,29,48]) or pole-shifting techniques [40,41], and among others. The backstepping method is also among these methods, and its use for infinite dimensional systems can be traced back to Russell [42] and to Balogh and Krstic [3] (we refer to [15,32,43] for an introduction to the finite dimensional backstepping method). ...
We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with Fredholm transformations on Laplace operators in a sharp functional setting, which is the main objective of this work. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.
... X ≤ Ce −γ t ||R 0 || X (2) where C, γ > 0 are constants independent on t and || · || X is a norm on a Banach space X . In [7,Chapter 1], typical examples of systems modeled by hyperbolic PDEs with such feedback boundary conditions are cited; the telegrapher equations for electrical lines, the shallow water (Saint-Venant) equations for open channels [20], the isothermal Euler equations for gas flow in pipelines or even the Aw-Rascle equations [4] for road traffic. However, it should be noted that, in previous examples, there is often an indomain coupling which is not present in our model. ...
In this paper, we establish the exponential BV stability of general systems of discretized scalar conservation laws with positive speed. The focus is on numerical approximation of such systems using a wide class of slope limiter schemes built from the upwind monotone flux. The proof is based on a Lyapunov analysis taken from the continuous theory (Coron et al. in J Differ Equ 262(1):1–30, 2017) and a careful use of Harten formalism.
... The second approach relies on Lyapunov functions. Many results on the boundary stabilization of rst-order hyperbolic systems, linear and nonlinear, have been obtained using this approach: see for example the book [2], and the recent results in [21,22]. However, this approach can be limited, as it is sometimes impossible to obtain an arbitrary decay rate using Lyapunov functions (see [15]*Remark 12.9, page 318 for a nite dimensional example). ...
We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval \begin{document} (0,L) \end{document}, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than \begin{document} 1 \end{document}, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate. The variant of the backstepping method used here relies mainly on the spectral properties of the linear transport equation, and leads to some original technical developments that differ substantially from previous applications.
... The second approach relies on Lyapunov functions. Many results on the boundary stabilization of firstorder hyperbolic systems, linear and nonlinear, have been obtained using this approach: see for example the book [2], and the recent results in [17,18]. However, this approach can be limited, as it is sometimes impossible to obtain an arbitrary decay rate using Lyapunov functions (see [14,Remark 12.9, page 318] for a finite dimensional example). ...
... In particular, the Lyapunov function they constructed is an extension of the entropy and can be made strictly negative definite by properly choosing the boundary conditions. This method has been later on widely used for hyperbolic conservation laws in the framework of C 1 solutions 11,20,21 or H 2 solutions 2,4,5,10,12,17,22 (see Ref. 3 for an overview of this method). ...
In this paper, we study the exponential stabilization of a shock steady state for the inviscid Burgers equation on a bounded interval. Our analysis relies on the construction of an explicit strict control Lyapunov function. We prove that by appropriately choosing the feedback boundary conditions, we can stabilize the state as well as the shock location to the desired steady state in H2-norm, with an arbitrary decay rate.
... The second approach relies on Lyapunov functions. Many results on the boundary stabilization of firstorder hyperbolic systems, linear and nonlinear, have been obtained using this approach: see for example the book [2], and the recent results in [16,17]. However, this approach can be limited, as it is sometimes impossible to obtain an arbitrary decay rate using Lyapunov functions (see [13,Remark 12.9, page 318] for a finite dimensional example). ...
We use the backstepping method to study the stabilization of a 1-D linear transport equation on the interval (0, L), by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate.
This paper is devoted to the discussion of the exponential stability of a networked hyperbolic system with a circle. Our analysis extends an example by Bastin and Coron about the limits of boundary stabilizability of hyperbolic systems to the case of a networked system that is defined on a graph which contains a cycle. By spectral analysis, we prove that the system is stabilizable while the length of the arcs is sufficiently small. However, if the length of the arcs is too large, the system is not stabilizable. Our results are robust with respect to small perturbations of the arc lengths. Complementing our analysis, we provide numerical simulations that illustrate our findings.
The Input-to-State Stability of the non-horizontal cascade channels with different arbitrary cross section, slope and friction modeled by Saint-Venant equations is addressed in this paper. The control input and measured output are both on the collocated boundary. The PI control is proposed to study both the exponential stability and the output regulation of the closed-loop systems with the aid of Lyapunov approach. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the non-uniform steady-states of different channels is constructed. We show that by a suitable choice of the boundary feedback controls, the local exponential stability and the Input-to-State Stability of the nonlinear Saint-Venant equations for the
norm are guaranteed, then validated with numerical simulations. Meanwhile, the output regulation and the rejection of constant disturbances are realized as well.
The saturated boundary stabilization problem for quasi-linear hyperbolic systems of balance laws is considered under H2-norm in this paper, where the boundary conditions of the system are subject to actuator saturations. The resulting closed-loop system is proven to be locally exponentially stable with respect to the steady states in the presence of saturations. To this end, the sector nonlinearity model is introduced to deal with the saturation term, and then sufficient conditions for ensuring the locally exponential stability are established in terms of a set of matrix inequalities by employing the Lyapunov function method along with a sector condition. Furthermore, these results are applied to the stabilization of the two-lane traffic flow dynamic represented by Lighthill-Whitham-Richards (LWR) model. By utilizing variable speed limit (VSL) devices, a saturated boundary feedback controller is designed to stabilize the two-lane traffic flow, and the exponential convergence of the quasi-linear traffic flow system in H2 sense is validated by numerical simulations.
We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. This classical framework allows us to present the backstepping method with Fredholm transformations for the Laplace operator in a sharp functional setting, which is the main objective of this work. We first prove that, under some assumptions on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. Then, we prove that the Fredholm transformation constructed for the Laplacian also leads to the local rapid stability of the viscous Burgers equation.
In this paper, we address the problem of the exponential stability of density-velocity systems with boundary conditions. Density-velocity systems are typical hyperbolic systems that are omnipresent in physics as they encompass all systems that consist in a flux conservation and a momentum equation. In this paper we show that any such system can be stabilized exponentially quickly in the H2 norm using simple local boundary feedbacks, provided a condition on the source term is valid. This condition holds for most physical systems, even when the source term is not dissipative. Besides, the feedback laws obtained only depend on the target values at the boundaries, which implies that they do not depend on the expression of the source term or the force applied on the system. This makes them both very easy to implement in practice and robust to model errors. For instance, for a river modeled by Saint-Venant equations this means that the feedback law does not require any information on the friction model, the slope or the shape of the channel considered. This feat is obtained by showing the existence of a basic H2 Lyapunov function. We apply it to several systems: the general Saint-Venant equations, the isentropic Euler equations, the motion of water in rigid-pipe, the osmosis phenomenon, the traffic flow, etc.
We give sufficient conditions for local Input-to-State Stability in C1 norm of general quasilinear hyperbolic systems with boundary input disturbances. In particular the derivation of explicit Input-to-State Stability conditions is discussed for the special case of 2 × 2 systems.
In this paper we study the global exponential stability in the L2 norm of semilinear 1-d hyperbolic systems on a bounded domain, when the source term and the nonlinear boundary conditions are Lipschitz. We exhibit two sufficient stability conditions: an internal condition and a boundary condition. This result holds also when the source term is nonlocal. Finally, we show its robustness by extending it to global Input-to State Stability in the L2 norm with respect to both interior and boundary disturbances.
We consider a 1-D linear transport equation on the interval (0,L), with an internal scalar control. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized in finite time, and we give an explicit feedback law.
We consider the problem of boundary stabilization and state estimation for a 2×2 system of first-order hyperbolic linear PDEs with spatially varying coefficients. First, we design a full-state feedback law with actuation on only one end of the domain and prove exponential stability of the closed-loop system. Then, we construct a collocated boundary observer which only needs measurements on the controlled end and prove convergence of observer estimates. Both results are combined to obtain a collocated output feedback law. The backstepping method is used to obtain both control and observer kernels. The kernels are the solution of a 4 × 4 system of first-order hyperbolic linear PDEs with spatially varying coefficients of Goursat type, whose well-posedness is shown.
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.
The stability problem of a system of conservation laws perturbed by non-homogeneous terms is investigated. These non-homogeneous terms are assumed to have a small C
1-norm. By a Riemann coordinates approach a sufficient stability criterion is established in terms of the boundary conditions. This criterion can be interpreted as a robust stabilization condition by means of a boundary control, for systems of conservation laws subject to external disturbances. This stability result is then applied to the problem of the regulation of the water level and the flow rate in an open channel. The flow in the channel is described by the Saint-Venant equations perturbed by small non-homogeneous terms that account for the friction effects as well as external water supplies or withdrawals.
This article deals with the regulation of water flow in open-channels modelled by Saint-Venant equations. By means of a Riemann invariants approach, we deduce stabilizing control laws for a single horizontal reach without friction. The stability condition is extended to a general class of hyperbolic systems which can describe canal networks with more general topologies. A control law design based on this condition is illustrated with a simple case study: two reaches in cascade. The proof of the main stability theorem is based on a previous result from Li Ta-tsien concerning the existence and decay of classical solutions of hyperbolic systems.
The exponential stability problem of the nonlinear Saint-Venant equations is addressed in this paper. We consider the general case where an arbitrary friction and space-varying slope are both included in the system, which lead to non-uniform steady-states. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the steady-states is constructed. Then we show that by a suitable choice of boundary feedback controls, that we give explicitly, the local exponential stability of the nonlinear Saint-Venant equations for the H2-norm is guaranteed.
We address the question of the exponential stability for the norm of general 1-D quasilinear systems with source terms under boundary conditions. To reach this aim, we introduce the notion of basic Lyapunov functions, a generic kind of exponentially decreasing function whose existence ensures the exponential stability of the system for the norm. We show that the existence of a basic Lyapunov function is subject to two conditions: an interior condition, intrinsic to the system, and a condition on the boundary controls. We give explicit sufficient interior and boundary conditions such that the system is exponentially stable for the norm and we show that the interior condition is also necessary to the existence of a basic Lyapunov function. Finally, we show that the results conducted in this article are also true under the same conditions for the exponential stability in the norm, for any .
A new explicit Lyapunov function allows to study the exponential stability for a class of physical ‘2 by 2’ hyperbolic systems with nonuniform steady states. In fluid dynamics, this class of systems involves isentropic Euler equations and Saint-Venant equations. The proposed quadratic Lyapunov function allows to analyze the local exponential stability of the system equilibria for suitable dissipative Dirichlet boundary conditions without additional conditions on the system parameters.
This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices.
The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control.
Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.
In this chapter we provide an introduction to the modeling of balance laws
by hyperbolic partial differential equations (PDEs). A balance law is the mathematical expression of the physical principle that the variation of the amount of some extensive quantity over a bounded domain is balanced by its flux through the boundaries of the domain and its production/consumption inside the domain. Balance laws are therefore used to represent the fundamental dynamics of many physical open conservative systems.
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 consider a class of density-flow systems, described by linear hyperbolic conservation laws, which can be monitored and controlled at the boundaries. These control systems are open-loop unstable and subject to unmeasured flow disturbances. We address the issue of feedback stabilization and disturbance rejection under PI boundary control. Explicit necessary and sufficient stability conditions in the frequency domain are provided.
This paper considers the modelling and control of a particular type of optical amplifier that finds application in long-haul wavelength division multiplexed optical communications systems. The objective of this consideration is to design and demonstrate a simple (and potentially cheap) linear control strategy for regulating amplifier output signal power across a range of signal wavelengths in the presence of upstream signal power uncertainty, where limited control authority is available. Although such amplifiers are highly nonlinear and distributed devices, the straight-forward application of linearization, model reduction and H∞ loopshaping design is demonstrated to be a promising approach for achieving the stated amplifier control objectives.
We consider the problem of output feedback stabilization for a quasilinear 2×2 system of first-order hyperbolic PDEs with boundary actuation and measurement. We design an output feedback control law, with actuation and measurement on only one end of the domain, and prove local H2 exponential stability of the closed-loop system. The proof of stability is based on the construction of a strict Lyapunov function which includes the observer states. The feedback law and output injection gains are found using the backstepping method for 2 × 2 system of first-order hyperbolic linear PDEs, developed by the authors in a previous work, which is briefly reviewed.
We introduce a new "second order" model of traffic flow. As noted in [C. Daganzo, Requiem for second-order fluid with approximation to traffic flow, Transportation Res. Part B, 29 (1995), pp. 277-286], the previous "second order" models, i.e., models with two equations (mass and "momentum"), lead to nonphysical effects, probably because they try to mimic the gas dynamics equations, with an unrealistic dependence on the acceleration with respect to the space derivative of the "pressure." We simply replace this space derivative with a convective derivative, and we show that this very simple repair completely resolves the inconsistencies of these models. Moreover, our model nicely predicts instabilities near the vacuum, i.e., for very light traffic.
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.
Penser globalement, agir localement
- J.-M Coron
- B Novel
- G Bastin
J.-M. Coron, B. d'Andrea Novel and G. Bastin, Penser globalement, agir localement. Recherche-Paris 417 (2008) 82.
- P M Dower
- P M Farrell
P.M. Dower and P.M. Farrell, On linear control of backward pumped Raman amplifiers. Vol. 39 of IFAC Proceedings (2006)
547-552.
- M Krstic
- A Smyshlyaev
M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Vol. 16. SIAM (2008).