Ömer Morgül

On the Stabilization of a Flexible Beam with a Tip Mass

We study the stability of a flexible beam that is clamped at one end and free at the other; a mass is also attached to the free end of the beam. To stabilize this system we apply a boundary control force at the free end of the beam. We prove that the closed-loop system is well-posed and is exponentially stable. We then analyze the spectrum of the system for a special case and prove that the spectrum determines the exponential decay rate for the considered case.

Orientation and stabilization of a flexible beam attached to a rigid body: planar motion

The author considers a flexible structure modeled as a rigid body which rotates in inertial space; a light flexible beam is clamped to the rigid body at one end and free one is clamped at the other. It is assumed that the flexible beam performs only planar motion. The equations of motion are obtained by using free body diagrams. Two control problems are posed, namely the orientation and stabilization of the system. It is shown that suitable boundary controls applied to the free end of the beam and suitable control torques applied to the rigid body solve the problems posed above. The proofs are obtained by using the energy of the system as a Lyapunov functional. >

Dynamic boundary control of a Euler-Bernoulli beam

A flexible beam, clamped to a rigid base at one end and free at the other end is considered. To stabilize the beam vibrations, a dynamic boundary force control and a dynamic boundary torque control applied at the free end of the beam are proposed. It is proved that with the proposed controls, the beam vibrations decay exponentially. The proof uses a Lyapunov functional, based on the energy functional of the system. >

On the Synchronization of Chaos Systems by Using State Observers

We show that the synchronization of chaotic systems can be achieved by using the observer design techniques which are widely used in the control of dynamical systems. We prove that local synchronization is possible under relatively mild conditions and global synchronization is possible if the chaotic system has some special structures, or can be transformed into some special forms. We show that some existing synchronization schemes for chaotic systems are related to the proposed observer-based synchronization scheme. We prove that the proposed scheme is robust with respect to noise and parameter mismatch under some mild conditions. We also give some examples including the Lorenz and Rossler systems and Chuas oscillator which are known to exhibit chaotic behavior, and show that in these systems synchronization by using observers is possible.

A chaotic masking scheme by using synchronized chaotic systems

Abstract We present a new chaotic masking scheme by using synchronized chaotic systems. In this method, synchronization and message transmission phases are separated, and while synchronization is achieved in the synchronization phases, the message is only sent in message transmission phases. We show that if synchronization is achieved exponentially fast, then under certain conditions any message of any length could be transmitted and successfully recovered provided that the synchronization length is sufficiently long. We also show that the proposed scheme is robust with respect to noise and parameter mismatch under some mild conditions.

Stabilization and disturbance rejection for the beam equation

We consider a system described by the one-dimensional linear wave equation in a bounded domain with appropriate boundary conditions. To stabilize the system, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the proposed controller is a proper rational function of the complex variable and may contain a single pole at the origin and a pair of complex conjugate poles on the imaginary axis, provided that the residues corresponding to these poles are nonnegative; the rest of the transfer function is required to be a strictly positive real function. We then show that depending on the location of the pole on the imaginary axis, the closed-loop system is asymptotically stable. We also consider the case where the output of the controller is corrupted by a disturbance and show that it may be possible to attenuate the effect of the disturbance at the output if we choose the controller transfer function appropriately. We also present some numerical simulation results which support this argument.

On the stabilization of a cable with a tip mass

In this note, we consider a vertical cable which is pinched at the upper end. A mass is attached at the lower end where a control force is also applied. We show that this hybrid system is uniformly stabilized by choosing a suitable control law for the control force depending on the velocity and angular velocity at the free end. Moreover for specific values of the feedback coefficients, we obtain the rate of decay of the energy of the system. >

Observer-based control of a class of chaotic systems

Abstract We consider the control of a class of chaotic systems, which covers the forced chaotic oscillators. We focus on two control problems. The first one is to change the dynamics of the system to a new one which exhibits a desired behavior, and the second one is the tracking problem, i.e., to force the solutions of the chaotic system to track a given trajectory. To solve these problems we use observers which could be used to estimate the unknown states of the system to be controlled. We apply the proposed method to the control of Duffing equation and the Van der Pol oscillator and present some simulation results.

A dynamic control law for the wave equation

Abstract We consider a system described by the one-dimensional linear wave equation in a bounded domain with appropriate boundary conditions. To stabilize the system, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the proposed controller is restricted to be a positive real function which could be strictly proper. We then show that, if the transfer function of the controller is strictly proper, then the resulting closed-loop system is asymptotically stable, and if proper but not strictly proper, then the resulting closed-loop system is exponentially stable.

Boundary control of a Timoshenko beam attached to a rigid body: planar motion

A flexible spacecraft modelled as a rigid body which rotates in an inertial space is considered; a light flexible beam is clamped to the rigid body at one end and free at the other end. The equations of motion are obtained by using the geometrically exact beam model for the flexible beam, and it is then shown that under planar motion assumption, linearization of this model yields the Timoshenko beam model. It is shown that suitable boundary controls applied to the free end of the beam and a control torque applied to the rigid body stabilize the system. The proof is obtained by using a Lyapunov functional based on the energy of the system.