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Output-feedback stabilization of an unstable wave equation
Abstract
We consider the problem of stabilization of a one-dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated “boundary damper” feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open-loop instability of the plant. The controller and observer gains are designed using the method of “backstepping,” which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We also derive the explicit compensators in frequency domain. The results are illustrated with simulations.
... In this paper, the system to be controlled is described by a one-dimensional wave equation, and it is related to those considered in [25], [16], [41] and others: ...
... The operator C is bounded on H thanks to the Sobolev embedding theorem. For a more detailed physical background of system (1.1), we refer to [25]. The case q = 0 of (1.1) has been investigated by [13] and [17]. ...
... The next step is to design a state feedback control law for the nominal system (2.2), described equivalently by (2.4), so that it becomes exponentially stable and moreover, its state (ε, ε t ) converges to zero even as the disturbance p acts on the system. Motivated by [25], the state feedback control law for system (2.4) is the following: choose c 3 , c 4 > 0 and take ...
We study the output regulation of an unstable wave equation subject to disturbances generated by an exosystem. The main challenges are that there is a destabilizing boundary condition and the scalar tracking error is the only measurement signal available to the controller. Moreover, all the coefficients coupling the exosystem to the wave system are unknown. We construct a nominal system by specially selecting nominal values for the unknown coefficients through which the disturbance enters the equations of the wave system. For this nominal system, an exponentially stabilizing state feedback control is designed by the backstepping method. Then, an observer is proposed to estimate the state of the exosystem and the state of the wave system, based on the tracking error and the control input only. The observer contains a copy of the exosystem, in accordance with the internal model principle. By replacing the states with their estimates in the state feedback law, the desired error feedback controller is obtained for the nominal system. Using the backstepping approach and -semigroup theory, we prove that this observer-based error feedback controller solves the output regulation problem also for the original wave system (with the unknown coefficients). Moreover, when the frequencies of the exosystem are also unknown, we propose to use magnitude phase-locked loops to identify these frequencies. Numerical simulations are presented to validate the main results.
... In contrast, when the actuator acts through one boundary, whereas the sensor is placed at the other boundary, the output feedback stabilization problem can become much more complicated and remains largely unexplored in the literature. As Krstic et al. pointed out in [18], the difficulty arises from the fact that "the input-output operator is no longer passive (…) which precludes the application of simple controllers". Anti-located sensing and actuation requires to use dynamic compensators that include delayed values of the output (directly or through state observers). ...
... Anti-located sensing and actuation requires to use dynamic compensators that include delayed values of the output (directly or through state observers). A meaningful example is the output feedback stabilization of a simple unstable wave equation addressed in [18] using a separation principle that combines a state feedback control with a state observer. This approach is extended to the adaptive stabilization of more general linear hyperbolic systems with unknown parameters in [2,7]. ...
... Our analysis will also reveal, however, that the effect on stability of adding an arbitrarily small diffusion is far from obvious, contrary to what one might expect. Indeed, while diffusion strengthens the robustness of the exponential stability for the 2 × 2 problem (18), it can also destroy the stability of similar simpler systems as we will see in Sect. 6. ...
We consider the problem of boundary feedback control of single-input-single-output one-dimensional linear hyperbolic systems when sensing and actuation are anti-located. The main issue of the output feedback stabilization is that it requires dynamic control laws that include delayed values of the output (directly or through state observers) which may not be robust to infinitesimal uncertainties on the characteristic velocities. The purpose of this paper is to highlight some features of this problem by addressing the feedback stabilization of an unstable open-loop system which is made up of two interconnected transport equations and provided with anti-located boundary sensing and actuation. The main contribution is to show that the robustness of the control against delay uncertainties is recovered as soon as an arbitrary small diffusion is present in the system. Our analysis also reveals that the effect of diffusion on stability is far from being an obvious issue by exhibiting an alternative simple example where the presence of diffusion has a destabilizing effect instead.
... In order to study both the well-posedness and the stability properties of the abstract Cauchy problem (10), it will be useful to consider a second inner product on H . Such an approach is generally employed to enforce a dissipative property of the studied operator in an adequate Hilbert space and belongs to the framework of energy multipliers (see, e.g., [19,23]). Let ε 1 , ε 2 ∈ R * + be the constants involved in the control law (7a-7b) and Ψ : H × H → R be defined for any ...
... Then, structural stiffness, along with the control parameters ε 1 , ε 2 can be adjusted to satisfy Assumption 2.Theorem 3 Assume that Assumption 2 holds. Then the augmented energy E defined by(19) exponentially decays to zero, i.e., there exists Λ ∈ R * + such that ∀t ≥ 0, E (t) ≤ E (0) exp(−Λt). ...
This paper addresses the boundary stabilization of a flexible wing model, both in bending and twisting displacements, under unsteady aerodynamic loads, and in presence of a store. The wing dynamics is captured by a distributed parameter system as a coupled Euler-Bernoulli and Timoshenko beam model. The problem is tackled in the framework of semigroup theory, and a Lyapunov-based stability analysis is carried out to assess that the system energy, as well as the bending and twisting displacements, decay exponentially to zero. The effectiveness of the proposed boundary control scheme is evaluated based on simulations.
... The stability analysis of equilibrium solutions is also a primary and challenging problems in studying the dynamics of systems that are modeled by infinite-dimensional systems involving PDEs. During the last decades, the analysis of the stability of the fixed points of various PDE systems and the response stabilization and stabilizability of distributed parameter systems modeled by PDEs have also been studied in the context of Lyapunov's direct method; see, for instance, [9,10,24,35,37,38,60]. In the PDE context, the Lyapunov function is replaced by a Lyapunov functional, typically the integration of the Lyapunov function in space. ...
... The theoretical results of this method can be extended, on a case-by-case basis, to other types of equations, such as PDEs, stochastic equations, and functional differential equations. In particular, Lyapunov functional-based techniques and Lapunov's direct method are often applied in the study of families of PDEs ranging from hyperbolic to parabolic systems [9,10,24,35,37,38,60]. The reaction-diffusion PDE models solved numerically in this work are an example of the latter class. ...
Reaction-diffusion equations model various biological, physical, sociological, and environmental phenomena. Often, numerical simulations are used to understand and discover the dynamics of such systems. Following the extension of the nonlinear Lyapunov theory applied to some class of reaction-diffusion partial differential equations (PDEs), we develop the first fully discrete Lyapunov discretizations that are consistent with the stability properties of the continuous parabolic reaction-diffusion models. The proposed framework provides a systematic procedure to develop fully discrete schemes of arbitrary order in space and time for solving a broad class of equations equipped with a Lyapunov functional. The new schemes are applied to solve systems of PDEs, which arise in epidemiology and oncolytic M1 virotherapy. The new computational framework provides physically consistent and accurate results without exhibiting scheme-dependent instabilities and converging to unphysical solutions. The proposed approach represents a capstone for developing efficient, robust, and predictive technologies for simulating complex phenomena.
... In contrast, when the actuator acts through one boundary, whereas the sensor is placed at the other boundary, the output feedback stabilization problem can become much more complicated and remains largely unexplored in the literature. As Krstic et al. pointed out in [18], the difficulty arises from the fact that "the input-output operator is no longer passive (...) which precludes the application of simple controllers". Anti-located sensing and actuation requires to use dynamic compensators that include delayed values of the output (directly or through state observers). ...
... Anti-located sensing and actuation requires to use dynamic compensators that include delayed values of the output (directly or through state observers). A meaningful example is the output feedback stabilization of a simple unstable wave equation addressed in [18] using a separation principle that combines a state feedback control with a state observer. This approach is extended to the adaptive stabilization of more general linear hyperbolic systems with unknown parameters in [2] and [7]. ...
We consider the problem of boundary feedback control of single-input-single-output (SISO) one-dimensional linear hyperbolic systems when sensing and actuation are anti-located. The main issue of the output feedback stabilization is that it requires dynamic control laws that include delayed values of the output (directly or through state observers) which may not be robust to infinitesimal uncertainties on the characteristic velocities. The purpose of this paper is to highlight some features of this problem by addressing the feedback stabilization of an unstable open-loop system which is made up of two interconnected transport equations and provided with anti-located boundary sensing and actuation. The main contribution is to show that the robustness of the control against delay uncertainties is recovered as soon as an arbitrary small diffusion is present in the system. Our analysis also reveals that the effect of diffusion on stability is far from being an obvious issue by exhibiting an alternative simple example where the presence of diffusion has a destabilizing effect instead.
... From the engineering point of view this will be not only useful but also so practical, first, because in practical it is "impossible" to localized for instance a point ξ as an irrational number on a string! Therefore, because our control design is based on the method of backstepping [7,24,25], the gain functions formula is explicit and can be calculated numerically via a scheme of successive approximation. This makes its implementation possible in real problems. ...
We consider the problem of pointwise stabilization of a one-dimensional wave equation with an internal spatially varying anti-damping term. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate.
... Introduction. Elastic string has been an important benchmark for the development of distributed parameter system theory for several decades [6,7]. The controllability and stabilization of elastic string systems have been widely studied in [10,11,12] and references therein and have been extended to the variable coefficients case [10,14] in the last several decades. ...
... Laypunov's direct method has been successfully used in the PDE context to, for example, study the stability of fixed points and the stabilization of distributed parameter PDEs [1][2][3][4][5][6][7]. In the PDEs context, the conventional Lyapunov function is replaced by a Lyapunov functional which is typically constructed by integrating an appropriate Lyapunov function over the spatial domain. ...
Convection-diffusion-reaction equations are a class of second-order partial differential equations widely used to model phenomena involving the change of concentration/population of one or more substances/species distributed in space. Understanding and preserving their stability properties in numerical simulation is crucial for accurate predictions, system analysis, and decision-making. This work presents a comprehensive framework for constructing fully discrete Lyapunov-consistent discretizations of any order for convection-diffusion-reaction models. We introduce a systematic methodology for constructing discretizations that mimic the stability analysis of the continuous model using Lyapunov's direct method. The spatial algorithms are based on collocated discontinuous Galerkin methods with the summation-by-parts property and the simultaneous approximation terms approach for imposing interface coupling and boundary conditions. Relaxation Runge-Kutta schemes are used to integrate in time and achieve fully discrete Lyapunov consistency. To verify the properties of the new schemes, we numerically solve a system of convection-diffusion-reaction partial differential equations governing the dynamic evolution of monomer and dimer concentrations during the dimerization process. Numerical results demonstrated the accuracy and consistency of the proposed discretizations. The new framework can enable further advancements in the analysis, control, and understanding of general convection-diffusion-reaction systems.
... To evaluate the effectiveness of the Order-Reduced Finite Difference (ORFD) approximation (15), which utilizes mid-point and average operators, compared to the standard Finite Difference (FD) approximation, we first conducted an experiment using identical meshing with parameters l 1 = l 2 = 1 meter and M = N = 30 (corresponding to h 1 = h 2 = 1/31). With the material and control parameters specified in Table 1, we ensured that the assumptions (6)-(7) of the hybrid controller design and Theorem 4.3 were satisfied. ...
This paper presents a model reduction technique for a system of partial differential equations (PDEs) modeling heat transfer, mechanical vibrations, and charge distributions in a magnetizable piezoelectric beam within a transmission line framework. Leveraging recent results in designing static and dynamic feedback controllers, we reduce the system to a lower-dimensional model while preserving uniform exponential stability. The proposed method incorporates thermal effects and mitigates high-frequency eigenmode issues using Finite Differences with average operators. Importantly, the decay rate of the discrete model is identical to that of the original PDE model and is independent of the discretization parameters and , ensuring uniform exponential stability. Stability is achieved through a discrete Lyapunov function, which provides an explicit decay rate for selecting feedback gains and ensures robust performance across different scales. This framework effectively applies Finite Difference techniques to complex systems with coupled thermal and electromagnetic dynamics, maintaining stability without requiring direct spectral methods.
... Concerning the boundary controls for the KdV equation, in addition to the work [4,12] mentioned previously, we refer [5,17,49] and the references therein. It is worth noting that the backstepping related technique used in [12] has been developed to study the stabilization for other settings such as hyperbolic systems [1,18], wave equations [24,43], heat equations [14,29], Kuramoto-Sivashinsky equations [13], water waves systems [10], Gribov operator [22]. An introduction of backstepping technique can be found in [25]. ...
We construct a static feedback control in a trajectory sense and a dynamic feedback control to obtain the local rapid boundary stabilization of a KdV system using Gramian operators. We also construct a time-varying feedback control in the trajectory sense and a time varying dynamic feedback control to reach the local finite-time boundary stabilization for the same system.
... one can easily show that the closed-loop system (7) is still dissipative. like in [4,6,21,22], designing a stabilizing output feedback control requires designing observers first: ...
This paper addresses the mathematical modeling of a magnetizable piezoelectric beam with free ends, described by partial differential equations (PDEs) capturing the interplay between longitudinal vibrations and charge accumulation at the electrodes. Departing from conventional collocated control design, a non-collocated controller and observer design are proposed, enabling state recovery for the implementation of boundary output feedback controllers at one end through estimates obtained from observers at the opposite end. Recent investigations have unveiled exponentially stable solutions within this model, motivating a rigorous model reduction approach that encapsulates this unique property. To this end, we introduce an order-reduction-based Finite Differences technique tailored for this specific model. Leveraging midpoints in uniform discretization in conjunction with average operators, our approach establishes the groundwork for a discrete Lyapunov function. Through meticulous analysis, we demonstrate that both the observer and observer error dynamics exhibit uniformly exponentially stable solutions as the discretization parameter approaches zero. Importantly, the decay rate maintains its independence from the discretization parameter, aligning seamlessly with the original PDE system.
... The detailed procedure is as follows: At first we rewrite the Timoshenko system with delayed control into an equivalence associated system of a transport equation and a Timoshenko equation, and then we construct an exponentially stable target system; Finally, we give the form of the parameterization feedback controller and obtain the stability of closed-loop system by the stability of target system. Such an idea has been used in [17][18][19][20][21][22] to design the feedback controller, in which it is called the backstepping method. ...
In this paper, we are concerned with rapid stabilization of Timoshenko beam system with the internal delay control. The main idea of solving the stabilization problem is transformation. The original time delay system is firstly transformed into the undelayed system, and then the feedback control law which can stabilize the undelayed system is found. Finally, we prove that the feedback control law can also exponentially stabilize the time delay system.
... whereẽ(ζ, t) := H(ζ)x(ζ, t). Replaceũ = u −û from (11) in (14) and since P 0 + P 0 ≤ 0 it is obtained thaṫ ...
This letter investigates the design of a class of infinite-dimensional observers for one dimensional (1D) boundary controlled port-Hamiltonian systems (BC-PHS) defined by differential operators of order . The convergence of the proposed observer depends on the number and location of available boundary measurements. \textcolor{hector}{Asymptotic convergence is assured for , and provided that enough boundary measurements are available, exponential convergence can be assured for the cases N=1 and N=2.} Furthermore, in the case of partitioned BC-PHS with N=2, such as the Euler-Bernoulli beam, \textcolor{hector}{it is shown} that exponential convergence can be assured considering less available measurements. The Euler-Bernoulli beam model is used to illustrate the design of the proposed observers and to perform numerical simulations.
... Uni-directional beam generation is most often associated with boundary sources, since then only the outside of the boundary region exists. In the context of wave control, boundary sources are used in active closed loop setups for wave suppression in the entire structure [45,46,30,47,48]. However, launching such beams from a domain interior, i.e. when no physical boundaries present, is more intricate. ...
We consider the problem of hiding non-stationary objects from acoustic detection in a two-dimensional environment, where both the object's impedance and the properties of the detection signal may vary during operation. The detection signal is assumed to be an acoustic beam created by an array of emitters, which scans the area at different angles and different frequencies. We propose an active control-based solution that creates an effective moving dead zone around the object, and results in an artificial quiet channel for the object to pass through undetected. The control principle is based on mid-domain generation of near uni-directional beams using only monopole actuators. Based on real-time response prediction, these beams open and close the dead zone with a minimal perturbation backwards, which is crucial due to detector observers being located on both sides of the object's route. The back action wave determines the cloak efficiency, and is traded-off with the control effort; the higher is the effort the quieter is the cloaking channel. We validate our control algorithm via numerical experiments in a two-dimensional acoustic waveguide, testing variation in frequency and incidence angle of the detection source. Our cloak successfully intercepts the source by steering the control beams and adjusting their wavelength accordingly.
... The backstepping method is another control strategy, which is applied to design controllers to stabilize systems in many aspects, such as to deal with PDEs with space-dependent diffusivity or time-dependent reactivity [24], a wave equation with an internal spatially varying antidamping term [23], coupled ODE-hyperbolic equations [25]. In [13], it is considered that the stabilization problem of a 1-d wave equation where the instability is at its free end and control is on the opposite end. By backstepping transformation, the controller and the observer are then designed. ...
This article concerns the internal stabilization problem of 1-D interconnected heatwave equations, where information exchange and the two actuators occur at the adjacent side of the two equations. By designing an inverse back-stepping transformation, the original system is converted into a dissipative target system. Moreover, we investigate the eigenvalues distribution and the corresponding eigenfunctions of the closed-loop system by an asymptotic analysis method. This shows that the spectrum of the system can be divided into two families: one distributed along the a line parallel to the left side of the imaginary axis and symmetric to the real axis, and the other on the left half real axis. Then we work on the properties of the resolvent operator and we verify that the root subspace is complete. Finally, we prove that the closed-loop system is exponentially stable.
... Literature Overview on PDE Backstepping Observers. 1 Following the introduction of the PDE backstepping observer design approach in [1] for parabolic 1-D PDEs, numerous extensions of this approach to hyperbolic and other PDE structures have followed. Extensions to wave PDEs were introduced in [2], and to beam models arising in atomic force microscopy in [3]. An adaptive observer for a single first-order hyperbolic PDE is reported in [4]. ...
State estimation is important for a variety of tasks, from forecasting to substituting for unmeasured states in feedback controllers. Performing real-time state estimation for PDEs using provably and rapidly converging observers, such as those based on PDE backstepping, is computationally expensive and in many cases prohibitive. We propose a framework for accelerating PDE observer computations using learning-based approaches that are much faster while maintaining accuracy. In particular, we employ the recently-developed Fourier Neural Operator (FNO) to learn the functional mapping from the initial observer state and boundary measurements to the state estimate. By employing backstepping observer gains for previously-designed observers with particular convergence rate guarantees, we provide numerical experiments that evaluate the increased computational efficiency gained with FNO. We consider the state estimation for three benchmark PDE examples motivated by applications: first, for a reaction-diffusion (parabolic) PDE whose state is estimated with an exponential rate of convergence; second, for a parabolic PDE with exact prescribed-time estimation; and, third, for a pair of coupled first-order hyperbolic PDEs that modeling traffic flow density and velocity. The ML-accelerated observers trained on simulation data sets for these PDEs achieves up to three orders of magnitude improvement in computational speed compared to classical methods. This demonstrates the attractiveness of the ML-accelerated observers for real-time state estimation and control.
... With the proposal of the backstepping approach, this method has been extensively used in stabilization problem for parabolic equa-tions [9] [10] [11], first-order hyperbolic equations [12] [13] [14], wave equations [15] [16] [17] and other partial differential equations [18] [19] [20]. In [21], in order to stabilize an unstable wave equation, using the backstepping method, not only the collocated Dirichlet boundary control but also the non-collocated Neumann boundary control is considered. The strong stabilization of unstable wave equation by using non-collocated boundary displacement can be found in [22]. ...
In this paper, we focus on the periodic output regulation for a one-dimensional wave equation where the performance output is non-collocated with the control input. The exosystems considered here are multi-periodic time-varying, capable of generating periodic reference signals and disturbances within the domain and at the boundaries, each with distinct periods. Initially, a backstepping-based state feedback regulator is designed by tackling an ill-posed periodic regulator equation. Subsequently, leveraging an external matrix, we design a novel time-varying observer tailored for the transformed wave equation and the exosystems. An output feedback regulator is then obtained by replacing the states with their estimations. The tracking error is shown to decay exponentially and the closed-loop system is proved to be bounded. Some numerical simulations are presented to validate the results.
We are considering the problem of designing observers for heat partial differential equations (PDEs) that are subject to sensor delay and parameter uncertainty. In order to get finite‐dimensional observers, described by ordinary differential equations (ODE), we develop a design method based on the modal decomposition approach. The approach is extended so that both parameter uncertainty and sensor delay effects are compensated for. To cope more effectively with sensor delay, an output predictor is designed and the online provided output predictions are substituted to the future output values in the observer. To compensate for parameter uncertainty, we design a parameter estimator providing online parameter estimates, which are substituted to the unknown parameters in the observer. The parameter estimator design is made decoupled from the observer gain design by using an appropriate decoupling transformation. Using an analysis of the small‐gain‐theorem type, the whole (state and parameter) estimation error system is shown to be exponentially stable, under well‐defined conditions on the observer dimension, the sensor delay, and signal persistent excitation (PE).
This study presents a novel approach for the state and output feedback exponential control of cascaded unstable ODE-wave equations with nonlinear boundary condition. The new approach can be also used to investigate the state and output feedback stabilization of other cascaded ODE-partial differential equation (PDE) systems with nonlinear boundary condition. The interconnection between the ODE and wave PDE with the nonlinear boundary is bi-directional at two points. First, an exponential controller is planned and the well-posed and exponential stability of the closed-loop system is derived. Next, an exponential observer is designed, whereby the exponential stability of the overall error system is shown. Then, an observer-based output feedback control that causes the overall real system to be exponentially stable is built. The existence and exponential stability of the solution to the overall closed-loop system are further deduced. Finally, simulation results are given to verify the effectiveness, feasibility and validity of the proposed technique.
A magnetizable piezoelectric beam is subjected to a dynamic load on the controlled boundary. With several non-collocated observers and controllers, an exponential stability result is achieved. The results show that the decay rate strongly depends on the mechanical and electrical wave propagation speeds. Numerical experiments are provided to show the strength of the observer and controller design.
In this paper, we consider parameter estimation and velocity signal extraction from a disturbed boundary velocity signal for an unstable wave equation. Firstly, an adaptive observer is designed based on the boundary displacement and the corrupted boundary velocity. Then the design of the feedback law adopts the backstepping method of infinite dimensional system. Finally, as time approaches infinity, the estimated parameters converge to the unknown parameters, the initial value disturbance can be obtained, and the velocity signal can be asymptotically recovered. Meanwhile, the asymptotic stability of the closed-loop system can be proved by -semigroup theory and Lyapunov method. Numerical simulation shows that the proposed scheme is reasonable.
This letter investigates the design of a class of infinite-dimensional observers for one dimensional (1D) boundary controlled port-Hamiltonian systems (BC-PHS) defined by differential operators of order
. The convergence of the proposed observer depends on the number and location of available boundary measurements. Asymptotic convergence is assured for
, and provided that enough boundary measurements are available, exponential convergence can be assured for the cases
N=1
and
N=2
. Furthermore, in the case of partitioned BC-PHS with
N=2
, such as the Euler-Bernoulli beam, it is shown that exponential convergence can be assured considering less available measurements. The Euler-Bernoulli beam model is used to illustrate the design of the proposed observers and to perform numerical simulations.
In this paper, we are concerned with the output feedback problem for coupled ODE-wave systems subject to spatially-varying coefficients. An original transformation is designed to turn the wave equation into an equation of the same type with constant coefficient and low order term. Taking advantage of the backstepping method, a state observer is constructed by non-collocated boundary displacement and velocity measurement. Output feedback on the based of the observer output feedback is then constructed, which makes use of the backstepping method to make the system exponentially stable. Finally, we demonstrate the well-posed and the exponential stability of the closed-loop system.
Mathematical control theory for a single partial differential equation (PDE) has dominated the research literature for quite a while, new, complex, and challenging issues have recently arisen in the context of coupled, or interconnected, PDE systems. This has led to a rapidly growing interest, and many unanswered questions, within the PDE community. By concentrating on systems of hyperbolic and parabolic coupled PDEs that are nonlinear, Mathematical Control Theory of Coupled PDEs seeks to provide a mathematical theory for the solution of three main problems: well-posedness and regularity of the controlled dynamics; stabilization and stability; and optimal control for both finite and infinite horizon problems along with existence/uniqueness issues of the associated Riccati equations.
These lectures are an outgrowth of recent research in the area of control theory for systems governed by coupled partial differential equations (PDEs). While mathematical control theory for a single PDE has long occupied a central place within a broader discipline often referred to as control of infinite-dimensional systems, more complex and challenging issues have arisen recently in the context of coupled or interconnected PDE systems. This is in response to modern technological demands where the interaction (coupling) between various components of the model is a central mechanism in controlling the system. On the other hand, mathematical tools developed in the context of single equations more often than not are inadequate for dealing with and accounting for the more complex interactive nature of the structures. This has propelled a rapidly growing interest within the PDE community, which has posed an array of new questions and problems, including, How can we take advantage of the coupling within the model in order to improve system's performance? This question is clearly tightly connected to the issue of (controlled) propagation of certain properties from one component of a system into another, more generally, creating of new phenomena via coupling.
The purpose of these lectures is to describe several classes of coupled PDE models that display the above properties and to present mathematical tools and a cogent framework by which to successfully analyze the associated control problems.
We concentrate on systems of coupled PDEs of hyperbolic and parabolic type, which, moreover, are nonlinear and may include thermal effects. (The latter further contribute to the mixing of various dynamical components of the overall dynamics.) The controls are rough and as such naturally arise in the context of smart material technology. Particular emphasis is paid to boundary and point controls whose mathematical treatment is more involved and leads to the theory of semigroups with unbounded inputs.
We consider a quasilinear partial integro-differential equation modelling the vibrations of a nonlinear string. In the presence of an appropriate boundary damping, we prove a global existence and uniqueness result, for “small” initial data. By the use of suitably chosen Lyapunov functionals, we also show the exponential decay of the energy for any strong solution and we give precise estimates of the rate of decay.
In this paper, we investigate the problem of finding the optimal location of sensors and actuators to achieve reduction of the noise field in an acoustic cavity. We offer two control strategies: the first is based on linear quadratic tracking where the offending noise is tracked, and the second considers the formulation of the harmonic control strategy as a periodic static output feedback control problem. The first method, which is based on full state information, is suitable for optimal location of actuators while the second strategy can extend the results to finding optimal location of sensors as well as actuators. For both methods we consider the optimization of an appropriate quadratic performance criterion with respect to the location of the actuators and/or the sensors. Numerical examples are presented to compare the effectiveness of each control strategy and also the effect of optimal placement of actuators and sensors.
The rapid development of the theories of Volterra integral and functional equations has been strongly promoted by their applications in physics, engineering and biology. This text shows that the theory of Volterra equations exhibits a rich variety of features not present in the theory of ordinary differential equations. The book is divided into three parts. The first considers linear theory and the second deals with quasilinear equations and existence problems for nonlinear equations, giving some general asymptotic results. Part III is devoted to frequency domain methods in the study of nonlinear equations. The entire text analyses n-dimensional rather than scalar equations, giving greater generality and wider applicability and facilitating generalizations to infinite-dimensional spaces. The book is generally self-contained and assumes only a basic knowledge of analysis. The many exercises illustrate the development of the theory and its applications, making this book accessible to researchers in all areas of integral and differential equations.
This note addresses the stabilization of a one-dimensional wave equation with control at one end and noncollocated observation at another end. A simple exponentially convergent observer is constructed. The dynamical stabilizing boundary output feedback is designed via the observed state. While the closed-loop system is nondissipative, we show its exponential stability using Riesz basis approach
We consider a general class of infinite-dimensional linear systems, called regular linear systems, for which convenient representations are known to exist both in time and in frequency domain. For this class of systems, we investigate the concepts of stabilizability and detectability, in particular, their invariance under feedback and their relationship to exponential stability. We introduce two concepts of dynamic stabilization, the first formulated as usual, with the plant and the controller connected in feedback, and the second with two feedback loops. Even for finite-dimensional systems, the second concept, stabilization with an internal loop in the controller, is more general. We argue that the more general concept is the natural one, and we derive sufficient conditions under which an observer-based stabilizing controller with an internal loop can be constructed.
. Issues regarding the experimental implementation of PDE-based controllers are discussed in this work. While the motivating application involves the reduction of vibration levels for a circular plate through excitation of surface-mounted piezoceramic patches, the general techniques described here will extend to a variety of applications. The initial step is the development of a PDE model which accurately captures the physics of the underlying process. This model is then discretized to yield a vector-valued initial value problem. Optimal control theory is used to determine continuous-time voltages to the patches, and the approximations needed to facilitate discrete-time implementation are addressed. Finally, experimental results demonstrating the control of both transient and steady-state vibrations through these techniques are presented. Key words. feedback control, piezoceramic actuators, PDE model AMS subject classifications. Primary, 93C20; Secondary, 93B40 PII. S036301299528590...
The aim of this book is to teach you the essentials of spectral collocation methods with the aid of 40 short MATLAB® programs, or “M-files.”* The programs are available online at http://www.comlab.ox.ac.uk/oucl/work/nick.trefethen, and you will run them and modify them to solve all kinds of ordinary and partial differential equations (ODEs and PDEs) connected with problems in fluid mechanics, quantum mechanics, vibrations, linear and nonlinear waves, complex analysis, and other fields. Concerning prerequisites, it is assumed that the words just written have meaning for you, that you have some knowledge of numerical methods, and that you already know MATLAB.
If you like computing and numerical mathematics, you will enjoy working through this book, whether alone or in the classroom—and if you learn a few new tricks of MATLAB along the way, that's OK too!
For the observation or control of solutions of second-order hyperbolic equation in Rt × Ω, Ralston's construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. 807-823] showed that it is necessary that the region of control meet every ray of geometric optics that has, at worst, transverse reflection at the boundary. For problems in one space dimension, the method of characteristics shows that this condition is essentially sufficient. For problems on manifolds without boundary, the sufficiency was proved in [J. Rauch and M. Taylor, Indiana Univ. Math. J., 24 (1974)]. The theorems regarding propagation of singularities [M. Taylor, Comm. Pure Appl. Math., 28 (1975), pp. 457-478], [R. Melrose, Acta Math., 147 (1981), pp. 149-236], [J. Sjostrand, Communications in Partial Differential Equations, 1980, pp. 41-94] allows the extension of the latter argument to the problem of interior control [C. Bardos, G. Lebeau, and J. Rauch, Rendiconti del Seminario Mathematico, Universita e Politecnico di Torino, 1988, pp. 11-32]. In this paper, the sufficiency is proved for problems of control and observation from the boundary. For multidimensional problems, the region of control must meet each ray in a nondiffractive point, and a new microlocal lower bound on the trade of solutions at the boundary at gliding points is required. This paper treats linear problems with variable coefficients and solutions of all Sobolev regularities. The regularity of the controls is precisely linked to the regularity of the solutions.
An energy decay rate is obtained for solutions of wave type equations in a bounded region in Rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface.
Presents one of the main directions of research in the area of design and analysis of feedback stabilizers for distributed parameter systems in structural dynamics. Important progress has been made in this area, driven, to a large extent, by problems in modern structural engineering that require active feedback control mechanisms to stabilize structures which may possess only very weak natural damping. Much of the progress is due to the development of new methods to analyze the stabilizing effects of specific feedback mechanisms.
Boundary Stabilization of Thin Plates provides a comprehensive and unified treatment of asymptotic stability of a thin plate when appropriate stabilizing feedback mechanisms acting through forces and moments are introduced along a part of the edge of the plate. In particular, primary emphasis is placed on the derivation of explicit estimates of the asymptotic decay rate of the energy of the plate that are uniform with respect to the initial energy of the plate, that is, on uniform stabilization results.
The method that is systematically employed throughout this book is the use of multipliers as the basis for the derivation of a priori asymptotic estimates on plate energy. It is only in recent years that the power of the multiplier method in the context of boundary stabilization of hyperbolic partial differential equations came to be realized. One of the more surprising applications of the method appears in Chapter 5, where it is used to derive asymptotic decay rates for the energy of the nonlinear von Karman plate, even though the technique is ostensibly a linear one.
In recent years important progress has been made in the design and analysis of feedback stabilizers for distributed parameter systems of importance in structural dynamics. Research in this area has been driven, to a large extent, by problems in modern structural engineering which require active feedback control mechanisms to stabilize structures that may be inherently unstable in the absence of control or that may possess only very weak natural damping. Much of the progress recently seen is due to the development of new methods to analyze the stabilizing effects of specific feedback mechanisms. The purpose of this book is to present one of the main directions of current research in this area in the context of an important structural entity, the thin plate. Specifically, this book is intended to provide a comprehensive and unified treatment of asymptotic stability of the motion of a thin plate when appropriate stabilizing feedback mechanisms acting through forces and moments are introduced along a part of the edge of the plate. In particular, primary emphasis is placed in the derivation of explicit estimates of the asymptotic decay rate of the energy of the plate that are uniform with respect to the initial energy of the plate, that is, on uniform stabilization results.
Exact controllability is studied for distributed systems, of hyperbolic type or for Petrowsky systems (like plate equations). The control is a boundary control or a local distributed control. Exact controllability consists in trying to drive the system to rest in a given finite time. The solution of the problems depends on the function spaces where the initial data are taken, and also depends on the function space where the control can be chosen. A systematic method (named HUM, for Hilbert Uniqueness Method) is introduced. It is based on uniqueness results (classical or new) and on Hilbert spaces constructed (in infinitely many ways) by using Uniqueness. A number of applications are indicated. Nonlinear Riccati type PDEs are obtained. Finally, we consider how all this behaves for perturbed systems.
We consider the wave equation y″−Δy=0 in Ω(0,∞) with boundary conditions y=∂/∂v(Gy″) on ∂Ω(0,∞) where G=(−Δ)−1:H−1(Ω))H01(Ω) and initial data in L2(Ω)H−1(Ω). We prove, by microlocal analysis techniques, that every solution {y(t), y″(t) decays exponentially to zero in L2(Ω)H−1(Ω) as t⟶+∞. In fact, we prove a stabilization result for a class of boundary conditions containing the one above and the classical y″+∂y/∂v=0. We also treat, by the same methods, the wave equation with absorbing boundary conditions.
Magnetic fields can be used to apply damping to a vibrating structure. Dampers of this type function through the eddy currents that are generated in a conductive material experiencing a time-changing magnetic field. The density of these currents is directly related to the velocity of the change in magnetic field. However following the generation of these currents, the internal resistance of the conductor causes them to dissipate into heat. Because a portion of the moving conductor's kinetic energy is used to generate the eddy currents, which are then dissipated, a damping effect occurs. This damping force can be described as a viscous force due to the dependence on the velocity of the conductor In a previous study, a permanent magnet was fixed in a location such that the poling axis was perpendicular to the beam's motion and the radial magnetic flux was used to passively suppress the beam's vibration. Using this passive damping concept and the idea that the damping force is directly related to the velocity of the conductor a new passive-active damping mechanism will be created. This new damper will function by allowing the position of the magnet to change relative to the beam and thus allow the net velocity between the two to be maximized and thus the damping force significantly increased. Using this concept, a model of both the passive and active portion of the system will be developed, allowing the beams response to be simulated. To verify the accuracy of this model, experiments will be performed that demonstrate both the accuracy of the model and the effectiveness of this passive-active control system for use in suppressing the transverse vibration of a structure.
In this paper we shall consider the question of uniform stabilization of thin, elastic plates through the action of forces and moments on the edge of the plate (or on a part of the edge of the plate). Two particular plate models will be considered: The classical fourth order Kirchoff model, but incorporating rotational inertia, and the sixth order Mindlin-Timoshenko model. The difference in the two models, from a physical point of view, is that the M-T model incorporates transverse shear effects while the Kirchhoff model does not. Actually, the M-T model is a hyperbolic system three coupled second order partial differential equations in two dependent variables. The unknowns, denoted by w, psi, phi are the vertical component w of displacement and angles which are measures of the amount of transverse shear. The three equations are coupled through terms which are multiples of a factor K called the coefficient of elasticity in shear.
In this paper we study open-loop stabilizability, a general notion of stabilizability for linear differential equations=Ax+Bu in an infinite-dimensional state space. This notion is sufficiently general to be implied by exact controllability, by optimizability, and by various general definitions of closedloop stabilizability. Here,A is the generator of a strongly continuous semigroup, and we make very few a priori restrictions on the class of controlsu. Our results hinge upon the control operatorB being smoothly left-invertible, which is a very mild restriction when the input space is finite-dimensional. Since open-loop stabilizability is a weak concept, lack of open-loop stability is quites strong. A focus of this paper is to give necessary conditions for open-loop stabilizability, thus identifying classes of systems which are not open-loops stabilizable.
First we give useful frequency domain conditions that are equivalent to our definitions of open-loop stabilizability, and lead to a version of the Hautus test for open-loop stabilizability. When the input space is finite-dimensional, we give necessary conditions for open-loop stabilizability which involve spectral properties ofA. We show that these results are not true if the conditions onB are weakened. We obtain analogous results for discrete-time systems. We show that, for a class of systems without spectrum determined growth, optimizability is impossible. Finally, we show that a system is open-loop stabilizable with a class of controlu if and only if the system with the sameA but a more boundedB is open-loop stabilizable with a larger class of controls.
In this paper we present the first extension of the backstepping methods that we have developed so far for control of parabolic PDEs (thermal, fluid, and chemical reaction dynamics) to second-order PDE systems (often referred loosely as hyperbolic) which model flexible structures and acoustics. We introduce controller and observer designs capable of adding damping to a model of beam dynamics using actuation only at the beam base and using sensing only at the beam tip. We present our designs for the Timoshenko beam model (the most advanced in the catalog of beam models, which also includes the simplest Euler-Bernoulli, as well as the Rayleigh and "shear" beam models) under the assumption that the beam is "slender." We allow the presence of a small amount of Kelvin-Voigt (KV) damping, which models internal material friction (rather than viscous interaction with the environment) and is present in every realistic material, though our method also applies in the completely undamped case. The closed-loop system with our backstepping boundary feedback is equivalent to a model of a string immersed in viscous fluid, with increased stiffness, supported on one end by a spring of high stiffness and on the other end by a damper. Such a closed loop system is very well damped and achieves the same excellent damping performance as the previous damping feedbacks which apply actuation at the free end of the beam. To ease the reader into the ideas, we first present the same method for a wave equation (string) with one free end and with a small amount of KV damping and then pursue the development for the Timoshenko beam model
The aim of this manuscript is to present an alternative method for designing state observers for second-order distributed parameter systems without resorting to a first-order formulation. This method has the advantage of utilizing the algebraic structure that second-order systems enjoy with the obvious computational savings in observer gain calculations. The proposed scheme ensures that the derivative of the estimated position is indeed the estimate of the velocity component and to achieve such a result, a parameter-dependent Lyapunov function was utilized to ensure the asymptotic convergence of the state estimation error.
This note addresses observer design for second-order distributed parameter systems in R2. Particularly, second-order distributed parameter systems without distributed damping are studied. Based on finite number of measurements, exponentially stable observer is designed. The existence, uniqueness and stability of solutions of the observers are based on semigroup theory.
In this paper we design exponentially convergent observers for a class of parabolic partial integro-differential equations (P(I)DEs) with only boundary sensing available. The problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system. Observer gain (output injection function) is shown to satisfy a well-posed hyperbolic PDE that is closely related to the hyperbolic PDE governing backstepping control gain for the state-feedback problem. For several physically relevant problems the observer gains are obtained in closed form. The observer gains are then used for an output-feedback design in both collocated and anti-collocated setting of sensor and actuator. The order of the resulting compensator can be substantially lowered without affecting stability. Explicit solutions of a closed loop system are found in particular cases.
In this paper, we present a technical overview of the design and analysis of active boundary controllers for distributed parameter (vibration and noise) systems. This presentation is done in the context of Lyapunov control design and stability analysis tools, which are commonly applied to non-linear finite dimensional systems. The main purpose of this presentation is to shed more light on this powerful control design philosophy for distributed parameter systems. The Lyapunov-based boundary control framework will be illustrated through three example systems—the transverse vibrating string, the acoustic noise duct, and the flexible rotor—each with an increasing level of complexity. To complement the theoretical content of the paper, the experimental implementation of the flexible rotor controller is also presented.
Incluye bibliografía e índice
In this paper, we design a control strategy for a cantilevered
Timoshenko beam with free-end mass/inertial dynamics. The control
strategy, which is composed of boundary force and torque inputs applied
to the beam's free-end requires measurements of the free-end
displacement, slope, rotation due to bending, slope of the rotation due
to bending and the time derivatives of these quantities. We initially
develop a model-based control law which exponentially stabilizes the
beam displacement and rotation. We then illustrate how the model-based
control law can be redesigned as an adaptive controller which
asymptotically stabilizes the beam displacement and rotation while
compensating for parametric uncertainty. Experimental results are
included to illustrate the control performance
We consider the problem of designing a boundary controller for a
flexible link robot arm with a payload mass at the link's free-end.
Specifically, we utilize a nonlinear, hybrid dynamic system model (the
model is hybrid in the sense that it comprises a distributed parameter,
dynamic field equation coupled to discrete, dynamic boundary equations)
to design a model-based control law which asymptotically stabilizes the
link displacement while driving the actuator hub's position to a desired
setpoint. We then illustrate how the control law can be redesigned as an
adaptive controller which achieves the same control objective while
compensating for parametric uncertainty including unknown payload mass.
The control strategy is composed of a boundary control torque applied to
the actuator hub and a boundary control force at the link's free-end.
Experimental results are presented to illustrate the performance of the
proposed control laws
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.
Stabilization of the wave equation by Dirichlet type boundary feedback
- C Bardos
- L Halpern
- G Lebeau
- J Rauch
- E Zuazua
Bardos, C., Halpern, L., Lebeau, G., Rauch, J., & Zuazua, E. (1991). Stabilization of the wave equation by Dirichlet type boundary feedback. Asymptotic Analysis, 4(4), 285–291
Boundary control of the Timoshenko beam with free-end mass/inertia
- Zhang
- Zhang
On damping structures with piezoelectric transducers
- A Preumont
- B De Marneffe
- A Deraemaeker
- F Bossens
Preumont, A., De Marneffe, B., Deraemaeker, A., & Bossens, F. (2005). On damping structures with piezoelectric transducers. In Mota Soares, C. A. et al. (Eds.), II ECCOMAS thematic conference on smart structures and materials, Lisbon.
(CCSD) at UCSD. He received his PhD in Electrical Engineering from University of California Krstic is a coauthor of the books Nonlinear and Adaptive Control Design He received the NSF Career, ONR YI, and PECASE Awards, as well as the Axelby and the Schuck paper prizes
Miroslav Krstic is the Sorenson Professor of
Mechanical and Aerospace Engineering and the
Director of the newly formed Center for Control, Systems, and Dynamics (CCSD) at UCSD.
He received his PhD in Electrical Engineering from University of California, Santa Barbara, in 1994. Krstic is a coauthor of the books
Nonlinear and Adaptive Control Design (1995),
Stabilization of Nonlinear Uncertain Systems
(1998), Flow Control by Feedback (2002), and
Real Time Optimization by Extremum Seeking
Control (2003). He received the NSF Career,
ONR YI, and PECASE Awards, as well as the Axelby and the Schuck paper
prizes. In 2005 he was the first engineering professor to receive the UCSD
Award for Research. Krstic is a Fellow of IEEE, a Distinguished Lecturer of
the Control Systems Society, and a former CSS VP for Technical Activities.
Stabilization of the wave equation by Dirichlet type boundary feedback
- Bardos