Browsing by Subject "Partial differential equations"

Now showing 1 - 7 of 7

Open Access
Control and stabilization of a rotating flexible structure
(Elsevier, 1994) Morgül, Ö.
We consider a flexible beam clamped to a rigid base at one end and free at the other end. We assume that the rigid base rotates with a constant angular velocity and that the motion of the flexible beam takes place on a plane. To suppress the beam vibrations, we propose dynamic control laws for boundary control force and torque, both applied to the free end of the beam. We show that, under some conditions, one of which is the strict positive realness of the actuator transfer functions which generate the boundary control force and torque, the beam vibrations asymptotically decay to zero if the rigid base angular frequency is sufficiently small. Moreover, if the transfer functions are proper but not strictly proper, we show that the decay is exponential. We also give a bound on the constant angular velocity above which the system becomes unstable.
Open Access
Dynamic boundary control of the timoshenko beam
(Pergamon Press, 1992) Morgül, Ö.
We consider a clamped-free Timoshenko beam. To stabilize the beam vibrations, we propose a dynamic boundary control law applied at the free end of the beam. We prove that with the proposed control law, the beam vibrations uniformly and exponentially decay to zero. The proof uses a Lyapunov functional based on the energy of the system. © 1992.
Open Access
A dynamic control law for the wave equation
(Elsevier, 1994) Morgül, Ö.
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.
Open Access
Infinite dimensional and reduced order observers for Burgers equation
(Taylor & Francis, 2005) Efe, M. Ö.; Özbay, Hitay; Samimy, M.
Obtaining a representative model in feedback control system design problems is a key step and is generally a challenge. For spatially continuous systems, it becomes more difficult as the dynamics is infinite dimensional and the well known techniques of systems and control engineering are difficult to apply directly. In this paper, observer design is reported for one-dimensional Burgers equation, which is a non-linear partial differential equation. An infinite dimensional form of the observer is demonstrated to converge asymptotically to the target dynamics, and proper orthogonal decomposition is used to obtain the reduced order observer. When this is done, the corresponding observer is shown to be successful under certain circumstances. The paper unfolds the connections between target dynamics, observer and their finite dimensional counterparts. A set of simulation results has been presented to justify the theoretical claims of the paper.
Open Access
On the control of two-link flexible robot arm with nonuniform cross section
(SAGE, 2010) Dogan, M.; Morgül, Ö.
We consider the motion of a two-link flexible arm with nonuniform cross section. We obtain the equations of motion by using the extended Hamiltons principle. These equations consist of coupled partial differential equations and (nonlinear) ordinary differential equations with appropriate boundary conditions. Our control problem is to achieve the given desired link angles and suppress the link vibrations. To solve this problem, we propose a novel control scheme which consists of a dominant control law together with a parallel controller. We show that with the proposed controller, the control objectives are satisfied. Our stability analysis is based on the Lyapunov approach and LaSalles invariance principle extended to infinite-dimensional systems. We also present some simulation results, which indicate that large parameter uncertainties such as tip and hub mass changes are also handled effectively by the proposed controller.
Open Access
PDE control of a rotating shear beam with boundary feedback
(IEEE, 2009-08) Doğan, M.; Morgül, Ömer
We consider a flexible structure modeled as a shear beam which is clamped to a rigid body at one end and is free at the other end. The whole structure is free to rotate on the horizontal plane. We first model the system by using Partial Differential Equations (PDE) and we propose boundary feedback laws to achieve set point regulation of the rotation angle as well as to suppress the elastic vibrations. The proposed control laws are based on PDE model, hence we do not resort to discretization of the system equations by available methods. We utilize a coordinate transformation based on an invertible integral transformation by using Volterra form and backstepping techniques. We also present some simulation results. © 2009 EUCA.
Open Access
Stabilization and disturbance rejection for the beam equation
(IEEE, 2001) Morgül, Ö.
We consider a system described by the Euler-Bernoulli beam equation. For stabilization, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the controller is a marginally stable positive real function which may contain poles on the imaginary axis. We then give various asymptotical and exponential stability results. We also consider the disturbance rejection problem.