Modeling and boundary control of infinite dimensional systems in the Brayton–Moser framework

2017 ◽  
Vol 36 (2) ◽  
pp. 485-513
Author(s):  
Krishna Chaitanya Kosaraju ◽  
Ramkrishna Pasumarthy ◽  
Dimitri Jeltsema
Keyword(s):  
Boundary Control ◽  
Zero Energy ◽  
Line System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Boundary Control Problem ◽  
Energy Flows ◽  
Admissible Pairs ◽  
The Stability

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.

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

Mathematics ◽  
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.

An adaptive PID-like controller for vibration suppression of piezo-actuated flexible beams

2017 ◽  
Vol 24 (12) ◽  
pp. 2656-2670 ◽  
Author(s):  
Teerawat Sangpet ◽  
Suwat Kuntanapreeda ◽  
Rüdiger Schmidt
Keyword(s):  
Vibration Suppression ◽  
Flexible Structures ◽  
Flexible Beam ◽  
Flexible Beams ◽  
Present Scheme ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Control Scheme ◽  
The Stability

Flexible structures have been increasingly utilized in many applications because of their light-weight and low production cost. However, being flexible leads to vibration problems. Vibration suppression of flexible structures is a challenging control problem because the structures are actually infinite-dimensional systems. In this paper, an adaptive control scheme is proposed for the vibration suppression of a piezo-actuated flexible beam. The controller makes use of the configuration of the prominent proportional-integral-derivative controller and is derived using an infinite-dimensional Lyapunov method. In contrast to existing schemes, the present scheme does not require any approximated finite-dimensional model of the beam. Thus, the stability of the closed loop system is guaranteed for all vibration modes. Experimental results have illustrated the feasibility of the proposed control scheme.

scholarly journals Adaptive Vibration Control of Piezoactuated Euler-Bernoulli Beams Using Infinite-Dimensional Lyapunov Method and High-Order Sliding-Mode Differentiation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Teerawat Sangpet ◽  
Suwat Kuntanapreeda ◽  
Rüdiger Schmidt
Keyword(s):  
Adaptive Control ◽  
Lyapunov Method ◽  
Sliding Mode ◽  
High Order ◽  
Control Law ◽  
Flexible Beams ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Sensor Configuration ◽  
The Stability

This paper presents an adaptive control scheme to suppress vibration of flexible beams using a collocated piezoelectric actuator-sensor configuration. A governing equation of the beams is modelled by a partial differential equation based on Euler-Bernoulli theory. Thus, the beams are infinite-dimensional systems. Whereas conventional control design techniques for infinite-dimensional systems make use of approximated finite-dimensional models, the present adaptive control law is derived based on the infinite-dimensional Lyapunov method, without using any approximated finite-dimension model. Thus, the stability of the control system is guaranteed for all vibration modes. The implementation of the control law requires a derivative of the sensor output for feedback. A high-order sliding mode differentiation technique is used to estimate the derivative. The technique features robust exact differentiation with finite-time convergence. Numerical simulation and experimental results illustrate the effectiveness of the controller.

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