On the stability uniformity of infinite-dimensional systems

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

IMA Journal of Mathematical Control and Information ◽

10.1093/imamci/dnx057 ◽

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.

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

Journal of Vibration and Control ◽

10.1177/1077546317692160 ◽

2017 ◽

Vol 24 (12) ◽

pp. 2656-2670 ◽

Cited By ~ 6

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.

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On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations

Journal of Dynamical and Control Systems ◽

10.1007/s10883-021-09587-6 ◽

2021 ◽

Author(s):

Abdulla Azamov ◽

Gafurjan Ibragimov ◽

Khudoyor Mamayusupov ◽

Marks Ruziboev

Keyword(s):

Infinite System ◽

Null Controllability ◽

Infinite Matrix ◽

Asymptotically Stable ◽

Controllability Problem ◽

Infinite Dimensional ◽

Infinite Dimensional Systems ◽

The Matrix ◽

Linear Differential ◽

The Stability

AbstractIn this work, the null controllability problem for a linear system in ℓ2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with $\lambda \in \mathbb {R}$ λ ∈ ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤− 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤− 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered $\ell ^{\infty }$ ℓ ∞ is not asymptotically stable if λ = − 1.

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Stabilization of Mechanical Structures With Distributed Sensors and Actuators: A Hamiltonian Approach

Volume 7B: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8008 ◽

1999 ◽

Author(s):

Kurt Schlacher ◽

Andreas Kugi

Keyword(s):

Control Systems ◽

Controller Design ◽

Active Vibration Control ◽

Sensors And Actuators ◽

Distributed Sensors ◽

Infinite Dimensional ◽

Infinite Dimensional Systems ◽

Spatially Distributed ◽

Active Vibration ◽

The Stability

Abstract Intelligent mechanical structures based on piezoelectricity form an important new group of actuators and sensors for active vibration control. Since this technology allows to construct spatially distributed devices, new design possibilities for the control systems open up. Now the design of the spatially distributed sensors and actuators becomes part of the controller design itself. Several well established approaches, like the PD-, H2- and H∞-design are adapted to solve this problem. They are based on infinite dimensional Hamiltonian systems in conjunction with collocated actuator and sensor pairing. Since this approach is based on the so called Poisson bracket only, one can unify the controller design for finite and for infinite dimensional systems. Of course, the stability investigations are much more complicated in the latter case. Finally, applications to beams and plates demonstrate the power and effectiveness of the proposed methods.

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