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

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.

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A Sampled-Data Singularly Perturbed Boundary Control for a Heat Conduction System With Noncollocated Observation

IEEE Transactions on Automatic Control ◽

10.1109/tac.2009.2015522 ◽

2009 ◽

Vol 54 (6) ◽

pp. 1305-1310 ◽

Cited By ~ 32

Author(s):

Meng-Bi Cheng ◽

V. Radisavljevic ◽

Chung-Cheng Chang ◽

Chia-Fu Lin ◽

Wu-Chung Su

Keyword(s):

Heat Conduction ◽

Boundary Control ◽

Conduction System ◽

Singularly Perturbed ◽

Sampled Data

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Optimal state feedback boundary control of parabolic PDEs using SOS polynomials

2016 American Control Conference (ACC) ◽

10.1109/acc.2016.7525606 ◽

2016 ◽

Cited By ~ 1

Author(s):

Aditya Gahlawat ◽

Matthew M. Peet

Keyword(s):

Boundary Control ◽

State Feedback ◽

Parabolic Pdes ◽

Optimal State ◽

Optimal State Feedback

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Finite-dimensional guaranteed cost sampled-data fuzzy control of Markov jump distributed parameter systems via T–S fuzzy model

IET Control Theory and Applications ◽

10.1049/iet-cta.2017.1413 ◽

2018 ◽

Vol 12 (15) ◽

pp. 2098-2108 ◽

Cited By ~ 4

Author(s):

Huihui Ji ◽

He Zhang ◽

Baotong Cui

Keyword(s):

Fuzzy Control ◽

Fuzzy Model ◽

Distributed Parameter Systems ◽

Distributed Parameter ◽

Sampled Data ◽

Markov Jump ◽

Finite Dimensional ◽

Guaranteed Cost

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Boundary Control for a Certain Class of Reaction-Advection-Diffusion System

Mathematics ◽

10.3390/math8111854 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1854

Author(s):

Eduardo Cruz-Quintero ◽

Francisco Jurado

Keyword(s):

Boundary Conditions ◽

Boundary Control ◽

Distribution Systems ◽

Controller Design ◽

Neumann Boundary ◽

Neumann Boundary Conditions ◽

Infinite Dimensional ◽

Infinite Dimensional Systems ◽

Finite Dimensional ◽

Advection Diffusion

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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