Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

This is an open access article under the CC BY-NC-ND license.In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate

A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

In this paper, we consider input-output properties of linear systems consisting of PDEs on a finite domain coupled with ODEs through the boundary conditions of the PDE. This framework can be used to represent e.g. a lumped mass fixed to a beam or a system with delay. This work generalizes the sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using the PQRS positive matrix parameterization of operators resulting in a finite-dimensional LMI, feasibility of which implies prima facie provable passivity or L2-gain of the system. No discretization or approximation is involved at any step and we use numerical examples to demonstrate that the bounds obtained are not conservative in any significant sense and that computational complexity is lower than existing methods involving finite-dimensional projection of PDEs

Modeling and Control of a Rotating Flexible Spacecraft: A Port-Hamiltonian Approach

In this brief, we develop a mathematical model of a flexible spacecraft system composed of a hub and two symmetrical beams using the port-Hamiltonian framework. This class of system has favorable properties, such as passivity for controller synthesis and stability analysis, where the global Hamiltonian plays the role of a Lyapunov function candidate. The spacecraft model is viewed as a power-conserving interconnection between an infinite (beam) and finite (hub) dimensional system. We show that the interconnection result has a port-Hamiltonian structure and is passive. The introduction of a nonlinear feedback term, which takes into account the beam’s flexibility, is developed using the control by an interconnection approach. The closed-loop stability is proven; then, through explicitly solving the partial differential equations of the system, asymptotic stability is obtained. Finally, the experimental results are carried out to assess the validity of the proposed design methodology

Modelling and adaptive parameter estimation for a piezoelectric cantilever beam

© 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other worksThis paper proposes a new adaptive estimation approach to online estimate the model parameters of a piezoelectric cantilever beam. The beam behavior is firstly modeled using partial differential equations (PDE) considering the Kelvin-Voigt damping. To facilitate the estimation of unknown model parameters, the Galerkin’s method is introduced to extract desired vibration modes by separating the time and space variables of the PDE. Then, considering two major vibration modes, the corresponding system model can be represented by a fourth-order ordinary differential equation (ODE). Finally, by using measured input and output information, a novel adaptive parameter estimation strategy is introduced to estimate the unknown parameters of the derived ODE model in real time. For the purpose of driving the parameter updating law, the estimation error is extracted by using an auxiliary variable and a time-varying gain. Consequently, the convergence of the parameter estimation error is rigorously proved based on the Lyapunov theory. Simulations and experimental results show the validity and practicability of the proposed estimation method.Peer ReviewedPostprint (author's final draft

A mechanics-based model for simulation and control of flexible needle insertion in soft tissue

AbstractIn needle-based medical procedures, beveled-tip exible needles are steered inside soft tissue with the aim of reaching pre-dened target locations. The efciency of needle-based interventions depends on accurate control of the needle tip. This paper presents a comprehensive mechanics-based model for simulation of planar needle insertion in soft tissue. The proposed model for needle deection is based on beam theory, works in real-time, and accepts the insertion velocity as an input that can later be used as a control command for needle steering. The model takes into account the effects of tissue deformation, needle-tissue friction, tissue cutting force, and needle bevel angle on needle deection. Using a robot that inserts a exible needle into a phantom tissue, various experiments are conducted to separately identify different subsets of the model parameters. The validity of the proposed model is veried by comparing the simulation results to the empirical data. The results demonstrate the accuracy of the proposed model in predicting the needle tip deection for different insertion velocities. I

Boundary control of a rotating shear beam with observer feedback

We consider a flexible structure modeled as a shear beam which is free to rotate on the horizontal plane. We first model the system by using partial differential equations and we propose boundary feedback laws to achieve set-point regulation of the rotation angle as well as to suppress elastic vibrations. The main advantage of the proposed design, which consists of a decoupling controller together with an observer, is that it is easy to implement. We utilize a coordinate transformation based on an invertible integral transformation by using Volterra form and backstepping techniques. We show that with the proposed controller, the control objectives are satisfied. © The Author(s) 2011 Reprints and permissions: sagepub.co.uk/ journalsPermissions.nav

Free Vibration Analysis of Rotating Beams Based on the Modified Couple Stress Theory and Coupled Displacement Field

In this paper, transverse vibration analysis of rotating micro-beam is investigated based on the modified couple stress theory. The simply-supported micro-beam is modeled utilizing Euler-Bernoulli and Timoshenko beam theories. The system is rotating around a fixed axis perpendicular to the axial direction of the beam. For the first time, displacement filed is introduced as a coupled field to the translational field. In other words, the mentioned rotational displacement field is expressed as a proportional function of translational displacement field using first (axial), second (lateral), and third (angular or rotational) velocity factors. Utilizing Hamilton’s approach as a variational method, dynamic-vibration equations of motion of the proposed model are derived. Galerkin’s method is adopted to solve the equation corresponding to the Euler–Bernoulli and Timoshenko beams. For the case considering shear deformation effects, Navier method is chosen. For evaluation of current results and models, they are compared with those available at the benchmark. In this paper; effects of slenderness ratio, axial, lateral, and angular velocity factors, and rotations of the beam on the frequency are reported. Based on the results presented, mentioned factors should be counted in the analysis and design of such rotating micro-systems