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Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation
Abstract
This paper is devoted to a simple and new proof on the optimal finite control time for general linear coupled hyperbolic system by using boundary feedback on one side. The feedback control law is designed by first using a Volterra transformation of the second kind and then using an invertible Fredholm transformation. Both existence and invertibility of the transformations are easily obtained.
... Stabilization of large-scale systems of general, n + m heterodirectional linear hyperbolic PDEs can be achieved via backstepping, see, for example, [1][2][3][4][5][6][7][8]. Such large-scale systems of hyperbolic PDEs may be utilized to describe the dynamics of various systems with practical importance. ...
... The first result on backstepping stabilization of a class of continua of hyperbolic PDE systems was developed in [27], while a formal connection between the class of systems considered in [27] and the class of n + 1 linear hyperbolic systems [28] (for large n), as well as the application of the control design originally developed for the continuum system to the large-scale counterpart, were made in [26,29]. Therefore, besides [27] and [26,29], the present paper is related to the results on backstepping stabilization of n + m linear hyperbolic systems, see, for example, [1,[3][4][5][6][7], as well as to results in which PDE ensembles may arise as result of employment of Fourier transform, see, for example, [30] (that deals with parabolic PDEs). In addition, as the actual motivation for our developments is to address computational complexity of backstepping designs for large-scale hyperbolic systems, papers related to computation of backstepping kernels are also relevant, in particular, [31] that introduces a neural operators-based computation method, [32] that presents a late-lumpingbased approach, and [33] that relies on power series representations of the kernels (even though these results do not explicitly aim at addressing computational complexity with respect to increasing number of systems components). ...
... where L ∈ L ∞ (T ; R m×m ) and K ∈ L ∞ (T ; L 2 ([0, 1]; R m )) are the backstepping kernels. 4 Derivation of the continuum kernels equations is provided in Appendix A. We obtain that L and K need to satisfy the following kernel equations ...
We develop a backstepping control design for a class of continuum systems of linear hyperbolic PDEs, described by a coupled system of an ensemble of rightward transporting PDEs and a (finite) system of m leftward transporting PDEs. The key analysis challenge of the design is to establish well-posedness of the resulting ensemble of kernel equations, since they evolve on a prismatic (3-D) domain and inherit the potential discontinuities of the kernels for the case of n+m hyperbolic systems. We resolve this challenge generalizing the well-posedness analysis of Hu, Di Meglio, Vazquez, and Krstic to continua of general, heterodirectional hyperbolic PDE systems, while also constructing a proper Lyapunov functional. Since the motivation for addressing such PDE systems continua comes from the objective to develop computationally tractable control designs for large-scale PDE systems, we then introduce a methodology for stabilization of general n+m hyperbolic systems, constructing stabilizing backstepping control kernels based on the continuum kernels derived from the continuum system counterpart. This control design procedure is enabled by establishing that, as n grows, the continuum backstepping control kernels can approximate (in certain sense) the exact kernels, and thus, they remain stabilizing (as formally proven). This approach guarantees that complexity of computation of stabilizing kernels does not grow with the number n of PDE systems components. We further establish that the solutions to the n+m PDE system converge, as , to the solutions of the corresponding continuum PDE system. We also provide a numerical example in which the continuum kernels can be obtained in closed form (in contrast to the large-scale kernels), thus resulting in minimum complexity of control kernels computation, which illustrates the potential computational benefits of our approach.
... Although the methodology we propose uses the same ingredients presented in [6] and [50], extending such results to a chain of nonscalar subsystems is far from trivial. Indeed, when dealing with non-scalar systems, backstepping transformations cannot usually remove all the in-domain coupling terms [23]. Therefore, due to these remaining in-domain coupling terms, the approach presented in [50] does not work as the predictors cannot be adequately defined (causality problem). ...
... It implies that the dimension of the (virtual) input entering the subsystem i (which corresponds to the effect of the upstream subsystem) is equal to the number of the rightward propagating states n i of this subsystem. Such a condition is usually required to design stabilizing controllers for hyperbolic systems that do not have a specific structure (see, e.g., [23], [11], [36]). Indeed, to the best of our knowledge, only marginal results currently exist in the literature for stabilizing under-actuated systems (i.e., systems for which the dimension of the control input is smaller than the dimension of the boundary state) with no specific cascade structure (see [8]). ...
... Property 2: Consider the i th subsystem (i ∈ {1, . . . , N −1}) and consider that Property 1 holds, whereÛ i (t) is defined by equation (23). Then, there exist two constants κ i > 0 and η i > 0 such that for all t > t 0 + 1 ...
In this article, we detail the design of an output feedback stabilizing control law for an underactuated network of N subsystems of n + m heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. The network has a chain structure, as only one of the subsystems is actuated. The available measurements are located at the opposite extremity of the chain. The proposed approach introduces a new type of integral transformation to tackle in-domain couplings in the different subsystems while guaranteeing a ''clear actuation path'' between the control input and the different subsystems. Then, it is possible to state several essential properties of each subsystem: output trajectory tracking, input-to-state stability, and predictability (the possibility of designing a state prediction). We recursively design a stabilizing state-feedback controller by combining these properties. We then design a state-observer that reconstructs delayed values of the states. This observer is combined with the state-feedback control law to obtain an output-feedback controller. Simulations complete the presentation.
... For this class of systems, a stabilizing control law has been proposed in [18]. The proposed controller guarantees finite-time stabilization. ...
... Inspired by [18], we first combine two integral transformations to move the local coupling terms Σ ·· to the boundary (in the form of integral terms). Due to these transformations, non-local coupling terms and ODE terms may appear in the system. ...
... Due to these transformations, non-local coupling terms and ODE terms may appear in the system. Consider the following Volterra transformation, similar to the one introduced in [18,28,10] ...
In this paper, we present a filtering technique that robustifies stabilizing controllers for systems composed of heterodirectional linear first-order hyperbolic Partial Differential Equations (PDEs) interconnected with Ordinary Differential Equations (ODEs) through their boundaries. The actuation is either available through one of the ODE or at the boundary of the PDE. The proposed framework covers a broad general class of interconnected systems. Assuming that a stabilizing controller is available, we derive simple sufficient conditions under which appropriate low-pass filters can be combined with the control law to robustify the closed-loop system. Our approach is based on a rewriting of the distributed dynamics as a delay-differential algebraic equation and an analysis in the Laplace domain. The proposed technique will simplify the design of stabilizing controllers for the class of systems under consideration (which can now be done only on a case-by-case basis due to the complexity and generality of the underlying interconnections) since it dissociates the stabilization problem from the robustness aspects. Indeed, it becomes possible to use convenient (but non-robust) techniques for the stabilization of such systems (as the cancellation of the boundary coupling terms or the inversion of the ODE dynamics), knowing that the resulting control law can be made robust (to delays and uncertainties) using the proposed filtering methodology.
... For hyperbolic PDE systems, finite-time stabilization and estimation have been studied in, e.g., [3,8,34], and [10,11]. The latter two contributions deal with the problem of finitetime output regulation for hyperbolic systems by using the backstepping approach and by invoking the finite-time convergent observer design introduced in [13] (which is the first continuous, prescribed-time observer design for LTI systems, and exploits the infinite-dimensionality of an auxiliary, delayed state estimate to generate a determined algebraic system from which the exact state is reconstructed). ...
... where the function c(t) is defined according to (8), and where the coefficients r i , i = 1, . . . , n, are distinct positive real numbers (i.e., r i > 0, ...
... Theorem 1. Let Q be as in (3), a i−1 be as in (5), c(t − h) be defined according to (8), µ i−1 (t − h) be given by (14), (22)-(23) (for i = 1, . . . , n), and let T > 0 be fixed. ...
We present an observer for linear time-invariant (LTI) systems with measurement delay. Our design ensures that the observer error converges to zero within a prescribed terminal time. To achieve this, we employ time-varying output gains that approach infinity at the terminal time, which can be arbitrarily short but no shorter than the sensor delay time. We model the sensor delay as a transport partial differential equation (PDE) and build upon the cascade ODE–PDE setting while accounting for the infinite dimensionality of the sensor. To construct our time-varying gains, the observer design needs to be conducted in a particular system representation. For this reason, we employ a sequence of state transformations (and their inverses) mapping the original observer error model into (1) the observer form, (2) a sensor delay-compensated observer error form via backstepping, and (3) a particular diagonal form that is amenable to the selection of time-varying gains for prescribed-time stabilization. Our construction of the time-varying observer gains uses (a) generalized Laguerre polynomials, (b) elementary symmetric polynomials, and (c) polynomial-based Vandermonde matrices. A simulation illustrates the results.
... On one hand, for hyperbolic PDE systems, some contributions on stabilization in finite-time can be highlighted: see e.g. [33], [11], [2], [11], [13]. On the other hand, for linear parabolic PDEs, some works have addressed some relevant issues on null controllability and finite-time stabilization (e.g. ...
... On one hand, for hyperbolic PDE systems, some contributions on stabilization in finite-time can be highlighted: see e.g. [33], [11], [2], [11], [13]. On the other hand, for linear parabolic PDEs, some works have addressed some relevant issues on null controllability and finite-time stabilization (e.g. ...
... Proof: See Appendix B. The above formula (11) suggests to introduce the following powered "square" function for some r ∈ R, υ ∈ Z: ...
This paper deals first with the problem of prescribed-time stability of linear systems without delay. The analysis and design involve the Bell polynomials, the generalized Laguerre polynomials, the Lah numbers and a suitable polynomial-based Vandermonde matrix. The results can be used to design a new controller -with time-varying gains- ensuring prescribed-time stabilization of controllable LTI systems. Based on the preliminary results for the delay-free case, we study controllable LTI systems with single input and subject to a constant input delay. We design a predictor feedback with time-varying gains. To achieve this, we model the input delay as a transport PDE and build on the cascade PDE-ODE setting (inspired by Krstic. 2009) so as the design of the prescribed-time predictor feedback is carried out based on the backstepping approach which makes use of time-varying kernels. We guarantee the bounded invertibility of the backstepping transformation and we prove that the closed-loop solution converges to the equilibrium in a prescribed-time. A simulation example illustrates the results.
... More precisely, we show that imposing finite-time convergence by completely canceling the proximal reflection (i.e the reflection at the actuated boundary) yields, in some cases, zero robustness margins to arbitrarily small delays in the actuation path. In particular, the control laws in recent contributions (see for instance [9], [17], [18], [26], [31]) can have very poor to no robustness to delays due to the cancellation of the proximal reflection. To overcome this problem, we propose some changes in the design of target system to preserver a small amount of this reflection and ensure delay-robustness. ...
... The backstepping approach [18], [31] has enabled the design of stabilizing full-state feedback laws for these systems. The generalization of these stabilization results for a large number of systems has been a focus point in the recent literature (details in [9], [17], [18], [31]). The main objective of these controllers is to ensure convergence in the minimum achievable time (as defined in [36]), thereby neglecting the robustness aspects that are essential for practical applications. ...
We detail in this article the necessity of a change of paradigm for the delay-robust control of systems composed of two linear first order hyperbolic equations. One must go back to the classical trade-off between convergence rate and delay-robustness. More precisely, we prove that, for systems with strong reflections, canceling the reflection at the actuated boundary will yield zero delay-robustness. Indeed, for such systems, using a backstepping-controller, the corresponding target system should preserve a small amount of this reflection to ensure robustness to a small delay in the loop. This implies, in some cases, giving up finite time convergence.
... Later, a new target system, distinct from the one in [26], is presented in [28], that achieves the minimum-time convergence to its zero equilibrium. In [29], the authors introduce a straightforward and novel proof regarding the optimal finite control time for linear coupled hyperbolic systems with spatially-varying coefficients. ...
... To demonstrate the continuity of the solution, it is essential to observe the continuity of each individual term, i.e., ∀d = 0, 1, 2, . . . , ∆M d (z, w) ∈ C(E 1 ), which can be established on account of the uniform convergence of equation (29). First, for d = 0, it is easy to see that ∆M 0 = ϕ(z, w) ∈ C(E 1 ). ...
This paper presents bilateral control laws for one-dimensional(1-D) linear 2x2 hyperbolic first-order systems (with spatially varying coefficients). Bilateral control means there are two actuators at each end of the domain. This situation becomes more complex as the transport velocities are no longer constant, and this extension is nontrivial. By selecting the appropriate backstepping transformation and target system, the infinite-dimensional backstepping method is extended and a full-state feedback control law is given that ensures the closed-loop system converges to its zero equilibrium in finite time. The design of bilateral controllers enables a potential for fault-tolerant designs.
... Later this Lyapunov functions 3 approach was extended and can now be applied for general pairs .m; k/ in the linear case ( [2,8,10,12,16,27]). In [13], the authors obtained feedbacks leading to finite stabilization in time 1 C 2 with m D k D 1. ...
... In [27], the authors considered the case where † is constant and obtained feedback laws for null-controllability at time k C P m lD1 kCl . Later ( [2,8]), feedbacks leading to finite stabilization in time k C kC1 were derived. ...
Hyperbolic systems in one-dimensional space are frequently used in the modeling of many physical systems. In our recent works we introduced time-independent feedbacks leading to finite stabilization in optimal time of homogeneous linear and quasilinear hyperbolic systems. In this work we present Lyapunov’s functions for these feedbacks and use estimates for Lyapunov’s functions to rediscover the finite stabilization results.
... In [15], using the second kind Volterra transformation and reversible Fredholm transformation, optimal management problems for general linear hyperbolic systems are investigated. ...
... Note that in the works [8,10,[12][13][14][15] we study the issues related to the theoretical aspects of the solvability and stability of mixed problems for hyperbolic systems and the issues of constructing numerical solutions and the stability of difference schemes are not considered. During the numerical calculation of mixed problems for hyperbolic systems, the reason for the above is that the dimension of the linear algebraic equations system increases with an increasing dimension of the considered area. ...
The problem of numerical determination of Lyapunov-stable (exponential stability) solutions of the Saint-Venant equations system has remained open until now. The authors of this paper previously proposed an implicit upwind difference splitting scheme, but its practical applicability was not indicated there. In this paper, the problem is solved successfully, namely, an algorithm for calculating Lyapunov-stable solutions of the Saint-Venant equations system is developed and implemented using an upwind implicit difference splitting scheme on the example of the Big Almaty Canal (hereinafter BAC). As a result of the proposed algorithm application, it was established that:
1) we were able to perform a computational calculation of the numerical determination problem of the water level and velocity on a part of the BAC (10,000 meters) located in the Almaty region;
2) the numerical values of the water level height and horizontal velocity are consistent with the actual measurements of the parameters of the water flow in the BAC;
3) the proposed computational algorithm is stable;
4) the numerical stationary solution of the system of Saint-Venant equations on the example of the BAC is Lyapunov-stable (exponentially stable);
5) the obtained results (according to the BAC) show the efficiency of the developed algorithm based on an implicit upwind difference scheme according to the calculated time.
Since we managed to increase the values of the difference grid time step up to 0.8 for calculating the numerical solution according to the proposed implicit scheme.
... More recently, control design methods have been developed for other types of n + m systems in [16][17][18]. Moreover, statefeedback stabilization of various types of n+m systems has been considered, e.g., in [19][20][21][22][23]. Such approaches may result in high derivation complexity and computational burden of the respective control laws, as these increase with n and m (see, e.g., [17]), due to the requirements of obtaining the solutions to the control/observer kernels, as well as of implementing the observer dynamics. ...
We develop a non-collocated, observer-based output-feedback law for a class of continua of linear hyperbolic PDE systems, which are viewed as the continuum version of n+m, general heterodirectional hyperbolic systems as . The design relies on the introduction of a novel, continuum PDE backstepping transformation, which enables the construction of a Lyapunov functional for the estimation error system. Stability under the observer-based output-feedback law is established by using the Lyapunov functional construction for the estimation error system and proving well-posedness of the complete closed-loop system, which allows utilization of the separation principle. Motivated by the fact that the continuum-based designs may provide computationally tractable control laws for large-scale, n+m systems, we then utilize the control/observer kernels and the observer constructed for the continuum system to introduce an output-feedback control design for the original n+m system. We establish exponential stability of the resulting closed-loop system, which consists of a mixed n+m-continuum PDE system (comprising the plant-observer dynamics), introducing a virtual continuum system with resets, which enables utilization of the continuum approximation property of the solutions of the n+m system by its continuum counterpart (for large n). We illustrate the potential computational complexity/flexibility benefits of our approach via a numerical example of stabilization of a large-scale n+m system, for which we employ the continuum observer-based controller, while the continuum-based stabilizing control/observer kernels can be computed in closed form.
... Finally, the bounded invertibility of (5), similar to (41), verifies that x(z, t) → 0 for t → ∞ and z ∈ [0, 1]. ✷ Remark 7 It is well known that choosing B 0 , B 1 , B(z) andB(z) equal to zero ensures finite-time stability of (34) in minimum time T min = φ − n− (1) + φ + n+ (1) (see [1,3]), i.e.,χ(z, t) = 0 for t > T min , which implies x(z, t) = 0 for t > T min . Note that T min corresponds to the sum of the delays induced by the slowest transport velocities λ − n− (z) and λ + n+ (z) of system (2) in the negative and positive zdirection, respectively. ...
This paper systematically introduces dynamic extensions for the boundary control of general heterodirectional hyperbolic PDE systems. These extensions, which are well known in the finite-dimensional setting, constitute the dynamics of state feedback controllers. They make it possible to achieve design goals beyond what can be accomplished by a static state feedback. The design of dynamic state feedback controllers is divided into first introducing an appropriate dynamic extension and then determining a static feedback of the extended state, which includes the system and controller state, to meet some design objective. In the paper, the dynamic extensions are chosen such that all transport velocities are homogenized on the unit spatial interval. Based on the dynamically extended system, a backstepping transformation allows to easily find a static state feedback that assigns a general dynamics to the closed-loop system, with arbitrary in-domain couplings. This new design flexibility is also used to determine a feedback that achieves complete input-output decoupling in the closed loop with ensured internal stability. It is shown that the modularity of this dynamic feedback design allows for a straightforward transfer of all results to hyperbolic PDE-ODE systems. An example demonstrates the new input-output decoupling approach by dynamic extension.
... The extension of [33] to n + m systems in [34] was followed by major progress on adaptive control design [35]. The recent results in [36,37] allow for finite-time stabilization of linear and coupled hyperbolic systems with space and time dependent parameters. In light of the current context, it's worth noting the emergence of nonlinear controllers for nonlinear infinite-dimension systems of conservation laws [38,39]. ...
Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more gain kernels --- a PDE structure unaddressed thus far with DeepONet. In this note, we open the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants by considering a simple counter-convecting coupled system in whose control a kernel PDE systems in Goursat form arises.
Applications include oil drilling, Saint-Venant model of shallow water waves, and Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. In this paper we establish the continuity of the mapping from (a total of five) plant PDE functional coefficients to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and establish that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our \emph{-Globally-exponentially} stabilizing (GES) approximate gain kernel-based output feedback design implies the deep learning of both the controller's and the observer's gains. Moreover, the encoding of the output-feedback law into DeepONet ensures \emph{semi-global practical exponential stability (SG-PES).} The DeepONet operator speeds up the computation of the controller gains by multiple orders of magnitude. Its theoretically proven stabilizing capability is demonstrated through simulations.
... It is worth mentioning that numerous investigations [4][5][6] focus on the stability analysis of differential methods for hyperbolic systems. However, in all these studies, stability was evaluated using dissipative energy integrals. ...
In this paper, we introduce a numerical integration method for hyperbolic systems problems known as the splitting method, which serves as an effective tool for solving complex multidimensional problems in mathematical physics. The exponential stability of the upwind explicit–implicit difference scheme split into directions is established for the mixed problem of a linear two-dimensional symmetric t-hyperbolic system with variable coefficients and lower-order terms. It is noteworthy that there are control functions in the dissipative boundary conditions. A discrete quadratic Lyapunov function was devised to address this issue. A condition for the problem’s boundary data, ensuring the exponential stability of the difference scheme with directional splitting for the mixed problem in the l2 norm, has been identified.
... In recent years, many fundamental works [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] have been devoted to the study of the theory of the numerical solution of the Saint-Venant system of equations. ...
Citation: Berdyshev, A.; Aloev, R.; Bliyeva, D.; Dadabayev, S.; Baishemirov, Z. Stability Analysis of an Upwind Difference Splitting Scheme for Two-Dimensional Saint-Venant Equations. Symmetry 2022, 14, 1986. https://doi.
... In recent years, many fundamental works [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] have been devoted to the study of the theory of the numerical solution of the Saint-Venant system of equations. ...
The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint–Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modeling tides. The issues of formulation of the mixed problems for these equations are studied. The system of equations is reduced to a symmetrical form by transforming dependent variables. Then, the matrices of coefficients are represented as the sums of two symmetric semidefinite matrices. This transformation allows constructing an upwind difference scheme in spatial directions to determine the numerical solution of the initial boundary value problem. The stability of the proposed difference scheme in energy norms is rigorously proved. The results of numerical experiments conducted for a model problem are provided to confirm the stability of the proposed method.
... Unlike traditional approaches [21], we need to use a Fredholm integral transform to get rid of the integral coupling terms.Contrary to Volterra transforms, its existence and invertibility are not guaranteed. Several results in the literature deal with the invertibility of Fredholm transforms when kernels have a specific structure, for instance, lower diagonal matrices [22], or specific boundary conditions [23]. As these conditions are not fulfilled here, we use an operator framework inspired by [20]. ...
For most networked systems found in the literature, the actuated boundary is usually located at one end. In this article, we first consider the stabilization of a chain of two interconnected subsystems, actuated at the
in-between
boundary. Each subsystem corresponds to coupled hyperbolic partial differential equations. Such in-domain actuation leads to higher complexity, and represents a significant difference with existing results. Then, starting from a classical controllability condition, we design a state feedback control law for the considered class of systems. The proposed approach is based on the backstepping methodology. However, to deal with the complex structure of the system, we use Fredholm integral transforms instead of classical Volterra transforms. We prove the invertibility of such transforms using an original operator framework. The well posedness of the backstepping kernel equations defining the transformations is also shown with the same arguments. By using a similar procedure, we are then able to design a Luenberger-type observer. Finally, we use the state estimation in the stabilizing controller to obtain an output-feedback law. Some test cases complete the paper.
... Several works have broadened the scope of the method by considering general kernel operators, namely Fredholm transformations. This requires more work as a Fredholm transform is not always invertible, but has been successful in many cases: see [26] for the Korteweg-de Vries equation and [25] for a Kuramoto-Sivashinsky equation, [22] for a Schrödinger equation, [23] for integro-differential hyperbolic systems, and in [24] for general hyperbolic balance laws. ...
In this article we study the so-called water tank system. In this system, the behavior of water contained in a one dimensional tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we will focus on the linearized systems around non-uniform steady states, corresponding to a constant acceleration of the tank. We prove that these systems are controllable in Sobolev spaces, using the moments method and perturbative spectral estimates. Then, for steady states corresponding to small enough accelerations, we design an explicit Proportional Integral feedback law (obtained thanks to a well-chosen dynamic extension of the system) that stabilizes these systems exponentially with arbitrarily large decay rate. Our design relies on feedback equivalence/backstepping.
... For first order hyperbolic systems, the exact (null) controllability has been considered in [2,7,9,30,31] for solutions in L ∞ , L 2 or C 1 . Moreover, there are many studies on the optimal time for the exact (null) controllability for hyperbolic systems in recent years, see for example [1,[3][4][5][6] and references therein. For the results that will be frequently used in this paper, readers can refer to [26,27]. ...
In recent years there have been many in-depth researches on the boundary controllability and boundary synchronization for coupled systems of wave equations with various types of boundary conditions. In order to extend the study of synchronization from wave equations to a much larger range of hyperbolic systems, in this paper we will define and establish the exact boundary synchronization for the first order linear hyperbolic system based on previous work on its exact boundary controllability. The determination and estimate of exactly synchronizable states and some related problems are also discussed. This work can be applied to a great deal of diverse systems, and a new perspective to study the synchronization problem for the coupled system of wave equations can be also provided.
... In the recent years several studies started to look at other more general linear transforms such as Fredholm transforms [52,49,46,154,155,48,75,74]. These transforms are more general, and therefore potentially more powerful, but they are not always invertible and proving the invertibility of the candidate transform becomes one of the main difficulties. ...
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.
... One can cite [9,21] where authors use the backstepping method to locally stabilize quasilinear hyperbolic systems in H 2 . Additionally, backstepping can be used to stabilize system (1) in finite time [8]. For an introduction to this method, the book [23] gives a wide overview of the topic. ...
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.
... The backstepping approach [13,22] has enabled the design of stabilizing full-state feedback laws for these systems. The generalization of these stabilization results for a large number of systems has been a focus point in the recent literature (details in [6,10,13,22]). The main objective of these controllers is to ensure convergence in the minimum achievable time (as defined in [11,27]), thereby omitting the robustness aspects that are known to be the major limitation for practical applications. ...
We detail in this article the development of a delay-robust stabilizing state feedback control law for an underactuated network of two subsystems of heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the two subsystems is actuated. The proposed approach is based on the backstepping methodology. A backstepping transform allows us to construct a first feedback to tackle in-domain couplings present in the actuated PDE subsystem. Then, we introduce a predictive tracking controller to stabilize the second PDE subsystem. The stabilization of this subsystem implies the stabilization of the whole network. Finally, the proposed control law is combined with a low-pass filter to become robust with respect to small delays in the control signal and uncertainties on the system parameters.
... 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]. These works have recently been generalized to time-dependent systems in [6]. ...
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.
... Also, several results (e.g. [16]) were obtained using a backstepping approach, a very powerful method based on a Volterra transformation, developed mainly for PDE in [24], and generalized recently with a Fredholm transformation for hyperbolic systems [14,34,35]. One may look at [22] for a more detailed survey about this method and its use for the Saint-Venant equations. ...
We study the exponential stability in the norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single Proportional-Integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the Input-to-State stability of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.
In this paper, we design a stabilizing state-feedback control law for a system represented by a general class of integral delay equations subject to a pointwise and distributed input delay. The proposed controller is defined in terms of integrals of the state and input history over a fixed-length time window. We show that the closed-loop stability is guaranteed, provided the controller integral kernels are solutions to a set of Fredholm equations. The existence of solutions is guaranteed under an appropriate spectral controllability assumption, resulting in an implementable stabilizing control law. The proposed methodology appears simpler and more general compared to existing results in the literature. In particular, under additional regularity assumptions, the proposed approach can be expanded to address the degenerate case where only a distributed control term is present.
Finite-time stability means that the trajectory of dynamical system converges to a Lyapunov stable equilibrium state in finite time. The finite time stabilization of traffic flow model can improve the transportation efficiency and reduce the occurrence of traffic congestion. In this letter, the finite-time boundary stabilization problem for the Lighthill-Whitham-Richards (LWR) traffic flow model is considered, in which a variable speed limit (VSL) device is applied at the downstream boundary and a finite-time boundary controller is designed to drive the traffic density to the steady state in finite time. By utilizing the method of characteristics to illustrate the property of finite-time convergence and the Lyapunov function approach to prove stability, the local finite-time stability of LWR traffic flow system is ensured in
-norm. Moreover, the settling-time can be given depended on the initial value and the parameters of finite-time boundary controller. Lastly, a numerical simulation example is presented to verify the effectiveness of the theoretical results.
The problem of boundary output feedback for fixed-time stabilization of parabolic distributed parameter systems with space and time dependent reactivity is considered by utilizing the backstepping method. An observer is constructed by applying the time-varying observer gain and boundary measurements, where the gain is unbounded as time approaches the terminal time. However, the fixed-time stability of error system is guaranteed by comparing the time growth rate of the observer gain with the decay rate of target error system state. Then, an observer-based output feedback boundary controller is established to achieve the fixed-time stabilization of the closed-loop system by combining the fixed-time stabilizing state feedback boundary controller and the fixed-time observer based on separation principle. Finally, a numerical simulation is shown to illustrate the effectiveness of the theoretical results.
Based on the idea of Lyapunov function, we investigate the output finite-time stability of some classes of abstract bilinear systems using feedback control laws. We first design a stabilizing feedback control. Then, we study the well-posedness of the system in closed-loop by using the theory of maximal monotone operators. Then we proceed to the question of output finite-time stability in a settling time of the system at hand. The obtained results are also applied to derive output finite time stabilization results for linear systems. The algorithm of feedback control design is further applied to finite dimension systems and also to PDE of parabolic and hyperbolic types.
A prediction-based controller is shown to achieve prescribed-time stabilization of a nonlinear infinite-dimensional system, which consists of a general boundary controlled first-order semilinear hyperbolic PDE that is bidirectionally interconnected with nonlinear ODEs at its unactuated boundary. The approach uses a coordinate transformation to map the nonlinear system into a form suitable for control. In particular, this transformation is based on predictions of system trajectories, which can be obtained by solving a general nonlinear Volterra integro-differential equation. Then, a prediction-based controller is designed to stabilize the system in prescribed-time. Numerical simulations illustrate the performance of both the prescribed-time controller and an asymptotically stabilizing one, which follows as a special case.
This paper focuses on comprehensive analysis of fixed-time stability and energy consumed by controller in nonlinear neural networks with time-varying delays. A sufficient condition is provided to assure fixed-time stability by developing a global composite switched controller and employing inequality techniques. Then the specific expression of the upper of energy required for achieving control is deduced. Moreover, the comprehensive analysis of the energy cost and fixed-time stability is investigated utilizing a dual-objective optimization function. It illustrates that adjusting the control parameters can make the system converge to the equilibrium point under better control state. Finally, one numerical example is presented to verify the effectiveness of the provided control scheme.
This paper focuses on the problem of prescribedtime control for a class of uncertain nonlinear systems. First, a prescribed-time stability theorem is proposed by following the adaptive technology for the first time. Based on this theorem, a new state feedback control strategy is put forward by using the backstepping method for high-order nonlinear systems with unknown parameters to ensure the prescribed-time convergence. Moreover, the prescribed-time controller is obtained in the form of continuous time-varying feedback, which can render all system states converge to zero within the prescribed-time. It should be noted that the prescribed-time is independent of system initial conditions, which means that the prescribed-time can be set arbitrarily within the physical limitations. Finally, two simulation examples are provided to illustrate the effectiveness of our proposed algorithm.
The paper is devoted to the study of boundary finite-time control for a reaction-diffusion (RD) system with switching time-delayed input. The RD system with switching time-delay input is converted to a switching system of RD equation cascaded with a transport equation with non-delay boundary input. Next, a novel switching controller is designed for the cascaded RD-transport system based on the backstepping technique, and this causes the closed-loop system to be convergence in a finite-time. Simulation results are provided to exhibit the effectiveness of the proposed method.
In this paper we derive minimum time convergent observers for n+m 1 systems of linear coupled first-order 1-D hyperbolic PDEs, that use either unilateral (single boundary measured), bilateral (both boundaries measured) or pointwise in-domain sensing. First, a Volterra integral transformation is combined with a Fredholm integral transformation to derive a minimum time unilateral observer for n+m systems. Then, it is shown that an n+m system with bilateral sensing can be transformed to an (n+m)+(n+m) system with unilateral sensing via an invertible coordinate transformation. The n+m bilateral observer is subsequently obtained from the (n+m)+(n+m) minimum time unilateral observer, and it is shown that it converges in a theoretical minimum time for bilateral sensing. In a similar fashion, the observer using pointwise in-domain measurement is derived using the same techniques. The performances of the n+m bilateral and unilateral observers are demonstrated in simulations.
In contrast to equations in exterior domains where local decay takes place because signals radiate to infinity, in the bounded case there must be some direct dissipative mechanism. The paper discusses two types. In one kind the energy of a wave will decrease when it passes through some fixed subregion of the domain. The second sort of decay occurs at the boundary; energy is lost when a wave is reflected from a fixed subset of the boundary are investigated. 7 refs.
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.
In this paper a family of stabilizing boundary feedback control laws for a class of linear parabolic PDEs motivated by engineering
applications is presented. The design procedure presented here can handle systems with an arbitrary finite number of open-loop
unstable eigenvalues and is not restricted to a particular type of boundary actuation. Stabilization is achieved through the
design of coordinate transformations that have the form of recursive relationships. The fundamental difficulty of such transformations
is that the recursion has an infinite number of iterations. The problem of feedback gains growing unbounded as the grid becomes
infinitely fine is resolved by a proper choice of the target system to which the original system is transformed. We show how
to design coordinate transformations such that they are sufficiently regular (not continuous but L
∞). We then establish closed-loop stability, regularity of control, and regularity of solutions of the PDE. The result is accompanied
by a simulation study for a linearization of a tubular chemical reactor around an unstable steady state.
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 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 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.
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.
In the present article we study the stabilization of first-order linear
integro-differential hyperbolic equations. For such equations we prove that the
stabilization in finite time is equivalent to the exact controllability
property. The proof relies on a Fredholm transformation that maps the original
system into a finite-time stable target system. The controllability assumption
is used to prove the invertibility of such a transformation. Finally, using the
method of moments, we show in a particular case that the controllability is
reduced to the criterion of Fattorini.
We solve the problem of stabilization of a class of linear first-order
hyperbolic systems featuring n rightward convecting transport PDEs and m
leftward convecting transport PDEs. Using the backstepping approach yields
solutions to stabilization in minimal time and observer based output feedback.
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 devoted to the study of the rapid exponential stabilization
problem for a controlled Korteweg-de Vries equation on a bounded interval with
homogeneous Dirichlet boundary conditions and Neumann boundary control at the
right endpoint of the interval. For every noncritical length, we build a
feedback control law to force the solution of the closed-loop system to decay
exponentially to zero with arbitrarily prescribed decay rates, provided that
the initial datum is small enough. Our approach relies on the construction of a
suitable integral transform.
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.
In this paper, the authors define the strong (weak) exact boundary controllability and the strong (weak) exact boundary observability
for first order quasilinear hyperbolic systems, and study their properties and the relationship between them.
KeywordsStrong (weak) exact boundary controllability-Strong (weak) exact boundary observability-First order quasilinear hyperbolic system
2000 MR Subject Classification35B37-93B05-93B07
In this paper we study the problem of boundary feedback stabilization for the unstable heat equation u t(x, t) = u xx(x, t) + a(x)u(x, t). This equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused (mathematically due to the diffusive term u xx] but also the destabilizing heat is generating (mathematically due to the term au with a > 0). We show that for any given continuously differentiable function a and any given positive constant λ we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of λ. This is a continuation of the recent work of Boskovic, Krstic, and Liu [IEEE Trans. Automat. Control, 46 (2001), pp. 2022-2028] and Balogh and Krstic [European J. Control, 8 (2002), pp. 165-176].
In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.
A Chinese summary appears in Chinese Ann
- Chinese Ann
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Classical solutions for quasilinear hyperbolic systems (In Chinese). Thesis
- Yanchun Zhao
Yanchun Zhao. Classical solutions for quasilinear hyperbolic
systems (In Chinese). Thesis. Fudan University, 1986.
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.
Stabilization of control systems and nonlinearities
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Jean-Michel Coron. Stabilization of control systems and
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Congress on Industrial and Applied Mathematics, pages 17-40. Higher Ed. Press, Beijing, 2015.