scholarly journals Sampled-data finite-dimensional boundary control of 1D parabolic PDEs under point measurement via a novel ISS Halanay’s inequality

Automatica ◽  
2022 ◽  
Vol 135 ◽  
pp. 109966
Author(s):  
Rami Katz ◽  
Emilia Fridman
2017 ◽  
Vol 36 (2) ◽  
pp. 485-513
Author(s):  
Krishna Chaitanya Kosaraju ◽  
Ramkrishna Pasumarthy ◽  
Dimitri Jeltsema

Abstract It is well documented that shaping the energy of finite-dimensional port-Hamiltonian systems by interconnection is severely restricted due to the presence of dissipation. This phenomenon is usually referred to as the dissipation obstacle. In this paper, we show the existence of dissipation obstacle in infinite dimensional systems. Motivated by this, we present the Brayton–Moser formulation, together with its equivalent Dirac structure. Analogous to finite dimensional systems, identifying the underlying gradient structure is crucial in presenting the stability analysis. We elucidate this through an example of Maxwell’s equations with zero energy flows through the boundary. In the case of mixed-finite and infinite-dimensional systems, we find admissible pairs for all the subsystems while preserving the overall structure. We illustrate this using a transmission line system interconnected to finite dimensional systems through its boundary. This ultimately leads to a new passive map, using this we solve a boundary control problem, circumventing the dissipation obstacle.


2009 ◽  
Vol 54 (6) ◽  
pp. 1305-1310 ◽  
Author(s):  
Meng-Bi Cheng ◽  
V. Radisavljevic ◽  
Chung-Cheng Chang ◽  
Chia-Fu Lin ◽  
Wu-Chung Su

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1854
Author(s):  
Eduardo Cruz-Quintero ◽  
Francisco Jurado

There are physical phenomena, involving diffusion and structural vibrations, modeled by partial differential equations (PDEs) whose solution reflects their spatial distribution. Systems whose dynamics evolve on an infinite-dimensional Hilbert space, i.e., infinite-dimensional systems, are modeled by PDEs. The aim when designing a controller for infinite-dimensional systems is similar to that for finite-dimensional systems, i.e., the control system must be stable. Another common goal is to design the controller in such a way that the response of the system does not be affected by external disturbances. The controller design for finite-dimensional systems is not an easy task, so, the controller design for infinite-dimensional systems is even more challenging. The backstepping control approach is a dominant methodology for boundary feedback design. In this work, we try with the backstepping design for the boundary control of a reaction-advection-diffusion (R-A-D) equation, namely, a type parabolic PDE, but with constant coefficients and Neumann boundary conditions, with actuation in one of these latter. The heat equation with Neumann boundary conditions is considered as the target system. Dynamics of the open- and closed-loop solution of the PDE system are validated via numerical simulation. The MATLAB®-based numerical algorithm related with the implementation of the control scheme is here included.


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