I.S. Sadek

Maximum principle for optimal boundary control of vibrating structures with applications to beams

A maximum principle is derived for open-loop boundary control of one dimensional structures undergoing transverse vibrations. The optimal control law is obtained using a maximum principle and the applicability of the results to the boundary control of vibrating beams is demonstrated. The method of solution involves the transformation of the original problem into one with homogeneous boundary conditions for a general set of boundary forces and torques. An adjoint variable is introduced and used in the formulation of a Hamiltonian function which in turn leads to the derivation of the maximum principle. The effectiveness of the proposed control mechanism is illustrated numerically and it is shown that the implementation of the optimal boundary control using one force actuator can lead to substantial decrease in the dynamic response of a vibrating beam.

Optimal design and control of a cross-ply laminate for maximum frequency and minimum dynamic response

Abstract Optimal thickness design and optimal closed-loop and open-loop distributed control functions are determined for a symmetric, cross-ply laminate. The design/control problem is formulated as a multiobjective optimization problem by taking a performance index which comprises a weighted sum of the design and control objectives and a penalty functional of the control force. The design objective is the maximization of the fundamental frequency. The control objective is the minimization of the dynamic response of the plate, which is expressed in terms of the energy of the structure. The design/control problem is solved using two different approaches, namely closed-loop (feedback) control and the open-loop control. In the former approach, displacement and velocity feedback controls are employed. In the latter approach, the open-loop control involves an unknown control function which is determined optimally using a maximum principle. The design variables are determined by direct minimization of the design and control objectives. Numerical results are presented for a rectangular laminate made of an advanced composite material. Comparisons are given for controlled and uncontrolled laminates as well as for optimally designed and non-optimal laminates.

Necessary and sufficient conditions for the optimal control of distributed parameter systems subject to integral constraints

Abstract A class of optimal control problems for a damped distributed parameter system governed by a system of partial differential equations with side contraints(equality and/or inequality) is considered. Optimal control problems in structural mechanics are often formulated in this framework. A maximum principle is shown to be a necessary condition for the control to be optimal. Under additional convexity assumptions on state variables, the maximum principle is shown to be sufficient for optimality.

Optimal control of a Timoshenko beam by distributed forces

The present paper considers the problem of optimally controlling the deflections and/or velocities of a damped Timoshenko beam subject to various types of boundary conditions by means of a distributed applied force and moment. An analytic solution is obtained by employing a maximum principle.

Maximum principle for the optimal control of a hyperbolic equation in one space dimension, part 2: application

The optimal open-loop control of a beam subject to initial disturbances is studied by means of a maximum principle developed for hyperbolic partial differential equations in one space dimension. The cost functional representing the dynamic response of the beam is taken as quadratic in the displacement and its space and time derivatives. The objective of the control is to minimize a performance index consisting of the cost functional and a penalty term involving the control function. Application of the maximum principle leads to boundary-value problems for hyperbolic partial differential equations subject to initial and terminal conditions. The explicit solution of this system is obtained yielding the expressions for the state and optimal control functions. The behavior of the controlled and uncontrolled beam is studied numerically, and the effectiveness of the proposed control is illustrated.

Maximum principle for the optimal control of a hyperbolic equation in one space dimension, part 1: theory

A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression.

Structural control to minimize the dynamic response of Mindlin—Timoshenko plates

Abstract The problem of damping out the vibrations of a thick plate is solved using the optimal control theory of distributed parameter systems. The plate is modelled as a Mindlin- Timoshenko plate to include shear effects and may exhibit viscous damping. The dynamic response of the structure comprises the displacement and velocity components which are combined with the amount of force expended in controlling the motion in a multiobjective cost functional. This functional is minimized with respect to distributed controls. A maximum principle is formulated to relate the control forces to adjoint variables, the use of which leads to the explicit solution of the title problem. The control over the plate is exercised by distributed moment and transverse forces which are in practice applied by means of torque and force actuators.

SIMULTANEOUS DESIGN/CONTROL OPTIMIZATION OF SYMMETRIC CROSS-PLY LAMINATES OF HYBRID CONSTRUCTION

Abstract Optimal level of hybridization and optimal closed-loop and open-loop control functions are determined for a symmetric, cross-ply laminate of hybrid construction. The objectives of the optimization problem are to maximize the fundamental frequency (design objective) and to minimize the dynamic response to external disturbances (control objective) with minimum expenditure of control energy. The design/control problem is formulated as a multiobjective optimization problem by employing a performance index which combines the design and control objectives in a weighted sum. The control energy is limited by taking a quadratic functional of the control force as a penalty term in this performance index. The plate is constructed as a sandwich hybrid laminate with outer layers of a high-stiffness material. Hybridization refers to the relative amounts of high and low stiffness fibers. Comparative numerical results are given for hybrid and non-hybrid laminates which indicate that although the hybrid laminate ...

Optimal control of time-delay systems with distributed parameters

An optimal control problem with a prescribed performance index for parabolic systems with time delays is investigated. A necessary condition for optimality is formulated and proved in the form of a maximum principle. Under additional conditions, the maximum principle gives sufficient conditions for optimality. It is also shown that the optimal control is unique. As an illustration of the theoretical consideration, an analytic solution is obtained for a time-delayed diffusion system.

Dynamically loaded beam subject to time-delayed active control

Feedback control is emptoyed to present excessive vibrations of slender and flexible structures working under dynamic loads. As the control by a feedback mechanism has the ability to deal with unexpected system disturbances, its application to various structural elements has steadily increased. Indeed, the results for the active control of structural elements include: wings [I], plates [2], bridges [3,4], and beams [5,6,7].