Now showing 1 - 6 of 6
  • Publication
    Boundary feedback stabilization of a reaction–diffusion equation with Robin boundary conditions and state-delay
    (Elsevier, 2020-06) ;
    This paper discusses the boundary feedback stabilization of a reaction–diffusion equation with Robin boundary conditions and in the presence of a time-varying state-delay. The proposed control design strategy is based on a finite-dimensional truncated model obtained via a spectral decomposition. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the design performed on the finite-dimensional truncated model achieves the exponential stabilization of the original infinite-dimensional system. In the presence of distributed disturbances, we show that the closed-loop system is exponentially input-to-state stable with fading memory.
    Scopus© Citations 22  200
  • Publication
    Feedback Stabilization of a Class of Diagonal Infinite-Dimensional Systems With Delay Boundary Control
    (IEEE Transactions on Automatic Control, 2021-01) ;
    This article studies the boundary feedback stabilization of a class of diagonal infinite-dimensional boundary control systems. In the studied setting, the boundary control input is subject to a constant delay while the open-loop system might exhibit a finite number of unstable modes. The proposed control design strategy consists of two main steps. First, a finite-dimensional subsystem is obtained by truncation of the original infinitedimensional system (IDS) via modal decomposition. It includes the unstable components of the IDS and allows the design of a finite-dimensional delay controller by means of the Artstein transformation and the pole-shifting theorem. Second, it is shown via the selection of an adequate Lyapunov function that: 1) the finite-dimensional delay controller successfully stabilizes the original IDS and 2) the closed-loop system is exponentially input-to-state stable (ISS) with respect to distributed disturbances. Finally, the obtained ISS property is used to derive a small gain condition ensuring the stability of an IDS-ODE interconnection.
    Scopus© Citations 28  301
  • Publication
    Input-to-State Stability of a Clamped-Free Damped String in the Presence of Distributed and Boundary Disturbances
    This note establishes the exponential input-to-state stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades, establishing ISS properties with respect to boundary disturbances remains challenging. One of the well-known methods for well-posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one. However, the resulting distributed disturbance involves time derivatives of the boundary perturbation. Thus, the subsequent ISS estimate depends on its amplitude, and may not be expressed in the strict form of ISS properties. To solve this problem, we show for a clamped-free damped string equation that the projection of the original system trajectories in an adequate Riesz basis can be used to establish the desired EISS property.
      289
  • Publication
    Control Law Realification for the Feedback Stabilization of a Class of Diagonal Infinite-Dimensional Systems With Delay Boundary Control
    Recently, a predictor feedback control strategy has been reported for the feedback stabilization of a class of infinite-dimensional Riesz-spectral boundary control systems exhibiting a finite number of unstable modes by means of a delay boundary control. Nevertheless, for real abstract boundary control systems exhibiting eigenstructures defined over the complex field, the direct application of such a control strategy requires the embedding of the control problem into a complexified state-space which yields a complex-valued control law. This letter discusses the realification of the control law, i.e., the modification of the design procedure for obtaining a real-valued control law for the original real abstract boundary control system. The obtained results are applied to the feedback stabilization of an unstable Euler-Bernoulli beam by means of a delay boundary control.
      156Scopus© Citations 9
  • Publication
    ISS Property with respect to boundary disturbances for a class of Riesz-spectral boundary control systems
    (Elsevier, 2019-11) ;
    This paper deals with the establishment of Input-to-State Stability (ISS) estimates for infinite dimen-sional systems with respect to both boundary and distributed disturbances. First, a new approach isdeveloped for the establishment of ISS estimates for a class of Riesz-spectral boundary control systemssatisfying certain eigenvalue constraints. Second, a concept of weak solutions is introduced in order torelax the disturbances regularity assumptions required to ensure the existence of classical solutions.The proposed concept of weak solutions, that applies to a large class of boundary control systemswhich is not limited to the Riesz-spectral ones, provides a natural extension of the concept of bothclassical and mild solutions. Assuming that an ISS estimate holds true for classical solutions, we showthe existence, the uniqueness, and the ISS property of the weak solutions.
      178
  • Publication
    Robustness of constant-delay predictor feedback for in-domain stabilization of reaction–diffusion PDEs with time- and spatially-varying input delays
    This paper discusses the in-domain feedback stabilization of reaction–diffusion PDEs with Robin boundary conditions in the presence of an uncertain time- and spatially-varying delay in the distributed actuation. The proposed control design strategy consists of a constant-delay predictor feedback designed based on the known nominal value of the control input delay and is synthesized on a finite-dimensional truncated model capturing the unstable modes of the original infinite-dimensional system. By using a small-gain argument, we show that the resulting closed-loop system is exponentially stable provided that the variations of the delay around its nominal value are small enough. The proposed proof actually applies to any distributed-parameter system associated with an unbounded operator that 1) generates a -semigroup on a weighted space of square integrable functions over a compact interval; and 2) is self-adjoint with compact resolvent.
    Scopus© Citations 15  147