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A maximum principle is derived for the optimal control of a beam with Kelvin–Voigt damping subject to an external excitation with the control exercised by means of piezoelectric patch actuators. The objective functional is defined as a weighted quadratic functional of the displacement and velocity which is to be minimized at a given terminal time. A penalty term is also part of the objective functional defined as the control voltage used in the control process. The maximum principle makes use of a Hamiltonian defined in terms of an adjoint variable and the control function. The optimal control problem is expressed as a coupled system of partial differential equations in terms of state, adjoint and control variables subject to boundary, initial and terminal conditions. The solution is obtained by expanding the state and adjoint variables in terms of eigenfunctions and determining the optimal control using the maximum principle. Numerical examples are given to demonstrate the applicability and the efficiency of the proposed method. [ABSTRACT FROM PUBLISHER]

Time delays exist in active control systems. Although a time delay in most cases is small, it results in the actuator applying energy to the control system with a time delay. This might cause instability of the dynamic systems and degrade the performance of the control systems. The objective of the present work is to propose a feasible methodology that can achieve good control performance for a dynamic beam structure system by time-delayed boundary torque actuators. The control problem is to determine optimal time-delayed boundary control by minimizing a given performance criterion. The performance criterion is specified as a weighted quadratic functional of the dynamic responses of the beam which is to be minimized at a specified terminal time using continuous time-delayed torque actuators. The modal expansion approach is used to convert the optimal control problem of a distributed parameter system into the optimal control problem of a linear lumped parameter system (LPS). A direct state-control parameterization approach is proposed where wavelets are employed to solve time-delayed LPS. The operational matrices of integration and delay are utilized to reduce the solution of linear time-varying delayed systems to the solution of algebraic equations. A numerical simulation is presented to illustrate the efficiency of the proposed control algorithm. [ABSTRACT FROM PUBLISHER]

Abstract: Active vibration control in a flexible beam is investigated by taking into account the time-delay which may occur in voltage controlled piezoelectric actuators. The voltage applied to a piezoelectric patch actuator to suppress the dynamic responses of the beam at a specified terminal time is computed by solving an optimal control problem. The quadratic performance index function consists of related potential energy and kinetic energy at the terminal time as well as control effort as a penalty term. The solution method is based on a combination of spatial modal expansion and direct approximation approaches. The spatial modal expansion approach is used to transfer the optimal control problem of distributed parameter system (DPS) into the optimal control problem of a linear time-invariant lumped parameter system (LPS). A direct state-control parametrization approach is formulated where Legendre wavelets are employed to solve time delayed LPS. The operational matrices of integration and delay are utilized to reduce the solution of linear time-varying delayed systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the applicability and the efficiency of the proposed method and the numerical results indicate the effectiveness of the proposed control mechanism. [Copyright &y& Elsevier]

Abstract: Active control of a vibrating beam using piezoelectric patch actuators is considered. The specific structure to be studied is an Euler–Bernoulli beam with piezoelectric actuators bonded to the top and bottom surfaces of the beam. The equation of motion includes Heaviside functions and their derivatives due to finite size piezo patches which provide the control force to damp out vibrations. Optimal control theory is formulated with the objective function specified as a weighted quadratic functional of the dynamic responses of the beam which is to be minimized at a specified terminal time. The expenditure of the control forces is included in the objective function as a penalty term. The optimal control law is derived using a maximum principle developed by Sadek et al. . The maximum principle involves a Hamiltonian expressed in terms of an adjoint variable with the state and adjoint variables linked by terminal conditions leading to a boundary-initial-terminal value problem. The explicit solution of the problem is developed using eigenfunction expansions of the state and adjoint variables. The numerical results are given to assess the effectiveness and the capabilities of piezo actuation to damp out the vibrations. [Copyright &y& Elsevier]

Abstract: A class of optimal control problems for hyperbolic systems in two-dimensional space is considered. An approach is proposed to damp the undesirable vibrations in the structures by pointwise moving force actuators extending over the spatial region occupied by the structure. A class of performance indices is introduced that includes functions of the state variable, its first and second-order space derivatives and first-order time derivative evaluated at a preassigned terminal time, and a suitable penalty term involving the control forces. A maximum principle is given for such general scanning control problem that facilitates the determination of the unique optimal control. A solution method is developed for the active vibration control of plates of general shape. The implementation of the method is presented and the effectiveness of a single moving force actuator is investigated and compared to a single fixed force actuator by a specific numerical example. [Copyright &y& Elsevier]

Abstract: Optimal control theory is formulated and applied to damp out the vibrations of micro-beams where the control action is implemented using piezoceramic actuators. The use of piezoceramic actuators such as PZT in vibration control is preferable because of their large bandwidth, their mechanical simplicity and their mechanical power to produce controlling forces. The objective function is specified as a weighted quadratic functional of the dynamic responses of the micro-beam which is to be minimized at a specified terminal time using continuous piezoelectric actuators. The expenditure of the control forces is included in the objective function as a penalty term. The optimal control law for the micro-beam is derived using a maximum principle developed by Sloss et al. [J.M. Sloss, J.C. Bruch Jr., I.S. Sadek, S. Adali, Maximum principle for optimal boundary control of vibrating structures with applications to beams, Dynamics and Control: An International Journal 8 (1998) 355–375; J.M. Sloss, I.S. Sadek, J.C. Bruch Jr., S. Adali, Optimal control of structural dynamic systems in one space dimension using a maximum principle, Journal of Vibration and Control 11 (2005) 245–261] for one-dimensional structures where the control functions appear in the boundary conditions in the form of moments. The derived maximum principle involves a Hamiltonian expressed in terms of an adjoint variable as well as admissible control functions. The state and adjoint variables are linked by terminal conditions leading to a boundary-initial-terminal value problem. The explicit solution of the problem is developed for the micro-beam using eigenfunction expansions of the state and adjoint variables. The numerical results are given to assess the effectiveness and the capabilities of piezo actuation by means of moments to damp out the vibration of the micro-beam with a minimum level of voltage applied on the piezo actuators. [Copyright &y& Elsevier]

Abstract: Optimal induced strain actuator sizes for an L-shaped laminated frame are determined under deterministic and uncertain loads with the objective of minimizing the tip deflection. The bending moments generated by the induced strain actuators control the deflection in order to minimize the maximum deflection. The frame is elastically restrained at the tip point and is subjected to a tip load which is specified for the deterministic case and lies in an uncertainty domain in the uncertain case. The specific uncertain loading studied in the present paper involves a load of given magnitude but of unknown direction which should be determined to induce the highest tip deflection caused by the least favorable loading. The effects of the elastic restraints and actuator locations on the deflection behaviour are investigated. It is shown that the optimal actuator size depends on the magnitude of the moments provided by the actuators. Moreover, it is observed that a deterministic design under an uncertain load leads to excessive deflections indicating the necessity of a robust design. [Copyright &y& Elsevier]