Finite deformations of thin shells with piezoelectric sensors and actuators

We treat coupled electromechanical problem of finite deformations of piezoelectric shells with the help of the direct approach. A shell is considered as a material surface with mechanical degrees of freedom of particles and with an additional field variable, namely electric potential on the electrodes. This results both in the nonlinear system of equations of piezoelectric shells and in the appropriate numerical scheme. Application of the direct approach is preceded with the three-dimensional asymptotic analysis of a linear electromechanical problem for a non-homogeneous piezoelectric plate, which provides the constitutive relations for the nonlinear theory. As a sample problem, we present finite element analysis of deformation and local buckling of cylindrical panel, equipped with piezoelectric sensors. The latter influence the mechanical behavior and produce signals, which can be interpreted in terms of structural entities. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite volume analysis of adaptive beams with piezoelectric sensors and actuators

This paper presents a finite volume (FV) formulation for the free vibration analysis and active vibration control of the smart beams with piezoelectric sensors and actuators. The governing equations based on Timoshenko beam theory are discretized using the finite volume method. For the purpose of forced vibration control of beam structures, the negative velocity feedback controller is designed for the single-input, single-output system. To achieve the best effect, the piezoelectric sensors and actuators are coupled with the host structure in different positions and then the performance of the designed control system is evaluated for each position. In the test examples, first the shear locking free feature of the present formulation is demonstrated. This has been performed by doing static and natural frequency analysis of some reference models. Then, the capability of the proposed method for the prediction of uncontrolled forced vibration response and active vibration control of a beam structure is studied.     

Optimal actuators placements for the active control of flexible structures

A scheme for the optimal spatial placement of a limited number of sensors and actuators under a minimum energy requirement for the active control of flexible structures is proposed. The method is based on the interpretation of the functional relationship (transfer matrix/conrol influence matrix) between the actuators and modes of the structural system. It is shown that, from the form of the matrix, the controllability and observability of the system with respect to differing locations of the sensors and actuators can be established. The algorithm presented circumvents prevailing problems encountered in contemporary optimal control applications. In particular, and in order to enhance the results presented in this paper, numerical simulation for a prismatic beam subjected to horizontal random wind loads and a simply supported square plate modelled as a single degree of freedom system are given to illustrate the placement strategy.     

Global optimization approach to nonlinear optimal control

To determine the optimum in nonlinear optimal control problems, it is proposed to convert the continuous problems into a form suitable for nonlinear programming (NLP). Since the resulting finite-dimensional NLP problems can present multiple local optima, a global optimization approach is developed where random starting conditions are improved by using special line searches. The efficiency, speed, and reliability of the proposed approach is examined by using two examples.Financial support from the Natural Science and Engineering Research Council under Grant A-3515 as well as an Ontario Graduate Scholarship are gratefully acknowledged. All the computations were done with the facilities of the University of Toronto Computer Centre and the Ontario Centre for Large Scale Computations.     

Numerical methods for solving applied optimal control problems

For an optimal control problem with state constraints, an iterative solution method is described based on reduction to a finite-dimensional problem, followed by applying a successive linearization algorithm with the use of an augmented Lagrangian. The efficiency of taking into account state constraints in optimal control computation is illustrated by numerically solving several application problems.     

Parametric optimization of a four-link close-chain manipulator with active and passive actuators

We investigate the problem of optimization of motion laws and design parameters of a four-link manipulator with a closed-chain kinematic structure. The manipulator performs cyclic transfer operations in a horizontal plane under the action of active and passive (springs and dampers) actuators. As a minimization criterion, we take a quadratic (with respect to control moments of forces) functional. An algorithm is proposed for constructing a suboptimal solution of the formulated problem based on parametrization of the generalized coordinates of the manipulator with a family of given functions and on the use of numerical procedures of mathematical programming.     

An applied cell mapping method for optimal control problems

From the application point of view, a series of modifications are proposed for the cell mapping method discussed in Ref. 1 for the optimal control analysis of dynamical systems. The cell order around the target set is rearranged. A set of common discriminate principles is used for the selection of the optimal one among competing control strategies of the same cost. Inequality constraints of the system are taken into account. The number of elements in the set of allowable time intervals is not prescribed, but left open. These modifications seem to make the cell mapping method more efficient for analyzing feedback systems and for obtaining their global optimal control information. The algorithms presented in this paper could broaden the application of the cell mapping approach of Ref. 1 to a wider class of engineering problems.     

A new algorithm for finding numerical solutions of optimal feedback control

Bing Sun Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People's Republic of China and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa Email: bzguo{at}iss.ac.cn Received on March 15, 2007; Revision received October 17, 2007. A new algorithm for finding numerical solutions of optimal feedbackcontrol based on dynamic programming is developed. The algorithmis based on two observations: (1) the value function of theoptimal control problem considered is the viscosity solutionof the associated Hamilton–Jacobi–Bellman (HJB)equation and (2) the appearance of the gradient of the valuefunction in the HJB equation is in the form of directional derivative.The algorithm proposes a discretization method for seeking optimalcontrol–trajectory pairs based on a finite-differencescheme in time through solving the HJB equation and state equation.We apply the algorithm to a simple optimal control problem,which can be solved analytically. The consistence of the numericalsolution obtained to its analytical counterpart indicates theeffectiveness of the algorithm.     

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.     

On homotopy analysis method applied to linear optimal control problems

In this paper, the homotopy analysis method (HAM) is employed to solve the linear optimal control problems (OCPs), which have a quadratic performance index. The study examines the application of the homotopy analysis method in obtaining the solution of equations that have previously been obtained using the Pontryagin’s maximum principle (PMP). The HAM approach is also applied in obtaining the solution of the matrix Riccati equation. Numerical results are presented for several test examples involving scalar and 2nd-order systems to demonstrate the applicability and efficiency of the method.     

Influence of boundary conditions and temperature of dissipative heating on active damping of forced axisymmetric resonant bending vibrations of circular viscoelastic plates by piezoelectric sensors and actuators

The problem of forced monoharmonic, axisymmetric, bending vibrations and dissipative heating of circular viscoelastic plates with piezoelectric sensors and actuators is considered. We describe the viscoelastic behavior of a passive (without the piezoeffect) and a piezoactive material according to the concept of complex modules depending on temperature. The nonlinear coupled problem of electrothermoviscoelasticity is solved by numerical methods. The influence of boundary conditions and temperature of dissipative heating on the active damping of forced resonant vibrations of circular viscoelastic plates using piezoelectric sensors and actuators is investigated.     

Hybrid functions approach for nonlinear constrained optimal control problems

In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrix of integration is introduced. This matrix is then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

An explicit procedure for discretizing continuous,optimal control problems

An explicit procedure for obtaining discrete approximations to general, nonlinear, fixed-time, continuous, optimal control problems with no intermediate trajectory constraints is presented. It is proved that, if the associated system of differential equations is linear in the control variable, then the optimal solutions of these approximationsconverge to extremals of the original continuous problem.     

Successive approximation procedure for steady-state optimal control of bilinear systems

The optimum regulation problem of a bilinear system with a quadratic performance criterion is obtained in terms of a sequence of algebraic Lyapunov equations. The results are based on the method of successive approximations. The proof of convergence of the proposed scheme is given and the design procedure is illustrated by two examples.     

An extragradient method for finding the saddle point in an optimal control problem

We propose an extragradient method for finding the saddle point of a convex-concave functional defined on solutions of controlled systems of linear ordinary differential equations. We prove the convergence of the method.     

Regularized extragradient method for finding a saddle point in an optimal control problem

We study n-dimensional cubical pseudomanifolds and their cellular mappings. In particular, we consider a discrete n-cube and all of its (n ? 1)-faces. Then, there exist either one or two or four faces of the cube each of which is mapped onto one face.     

Multilevel optimization for PDAE-constrained optimal control problems with pointwise constraints on control and state

We have developed a fully adaptive optimization environment suitable to solve complex optimal control problems restricted by partial differential algebraic equations (PDAEs) and pointwise constraints on the control [1, 2]. This contribution is devoted to the inclusion of pointwise constraints on the state within the optimization environment. To this end we first give a brief introduction into the architecture of the environment and the inclusion of pointwise constraints on the state by Moreau-Yosida regularization. Then, we test the new tool by applying it to an optimal boundary control problem for the cooling of hot glass down to room temperature, modeled by radiative heat transfer and semi-transparent boundary conditions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explicit and numerical methods for finding optimal therapeutics

Modeling is often applied to pharmacologic problems. But there are not many approaches to solve the associated control problems allowing to calculate optimal therapeutics. In this paper we propose some simple methods (numerical and analytical) giving an explicit (or calculated) optimal therapeutic. Applications to concrete drugs was done. An extension to nonlinear systems is given.