Controlled Duffing Oscillator Research Articles

Reliability control of nonlinear stochastic dynamical system based on discrete data

Stochastic optimal control based on reliability plays a crucial role in mitigating structural failure and ensuring safe operation. For general stochastic vibration systems, a data-driven method for reliability-based optimal control is proposed based on random state data. Firstly, the feedback control is split into conservative and dissipative components in coherence with physical intuition, and each component is expanded upon using pre-selected basic functions, resulting in a modified system with undetermined coefficients. Then, from discrete random samples, the expressions for the probability densities of the first-passage time and the stationary response can be identified, explicitly including system and excitation parameters, as well as initial and safety boundary conditions. Finally, a performance index, combining system reliability and control cost, is constructed and can be explicitly expressed in terms of the undetermined coefficients of the control forces. The optimal control problem for determining the optimal control forces is reformulated as an optimization problem for determining the coefficients that minimize the performance index. This unconstrained optimization problem can be easily solved. Three typical nonlinear stochastic systems, a controlled Duffing oscillator, a controlled nonlinear hysteretic system, and a controlled 2-DOF nonlinear dynamical system, are given to illustrate the validity and accuracy of the proposed data-driven method. Additionally, the control performance of the optimal control strategy in improving the system reliability is also discussed.

A numerical method for solving optimal control problems via Legendre polynomials

PurposeThe purpose of this paper is to propose an iterative Legendre technique to deal with a continuous optimal control problem (OCP).Design/methodology/approachFor the system in the considered problem, the control variable is a function of the state variables and their derivatives. State variables in the problem are approximated by Legendre expansions as functions of time t. A constant matrix is given to express the derivatives of state variables. Therefore, control variables can be described as functions of time t. After that, the OCP is converted to an unconstrained optimization problem whose decision variables are the unknown coefficients in the Legendre expansions.FindingsThe convergence of the proposed algorithm is proved. Experimental results, which contain the controlled Duffing oscillator problem demonstrate that the proposed technique is faster than existing methods.Originality/valueExperimental results, which contained the controlled Duffing oscillator problem demonstrate that the proposed technique can be faster while securing exactness.

The mechanism of switching combination synchronization for three distinct nonautonomous systems under sinusoidal constraints

In this paper, the mechanism of switching combination synchronization the controlled Duffing oscillator with Van der Pol system and Pendulum system is studied. Based on the theory of discontinuous dynamical systems, the analytical conditions for chaos synchronization of three different systems are obtained under sinusoidal constraints. With these conditions, the control parameter maps and the synchronization invariant sets are derived, respectively. Numerical illustrations for partial and full combination synchronization among three systems are carried out. The circuits of switching synchronization are used for the validation of the numerical simulations.

Non-linear stochastic optimal control of acceleration parametrically excited systems

Acceleration parametrical excitations have not been taken into account due to the lack of physical significance in macroscopic structures. The explosive development of microtechnology and nanotechnology, however, motivates the investigation of the acceleration parametrically excited systems. The adsorption and desorption effects dramatically change the mass of nano-sized structures, which significantly reduces the precision of nanoscale sensors or can be reasonably utilised to detect molecular mass. This manuscript proposes a non-linear stochastic optimal control strategy for stochastic systems with acceleration parametric excitation based on stochastic averaging of energy envelope and stochastic dynamic programming principle. System acceleration is approximately expressed as a function of system displacement in a short time range under the conditions of light damping and weak excitations, and the acceleration parametrically excited system is shown to be equivalent to a constructed system with an additional displacement parametric excitation term. Then, the controlled system is converted into a partially averaged Itô equation with respect to the total system energy through stochastic averaging of energy envelope, and the optimal control strategy for the averaged system is derived from solving the associated dynamic programming equation. Numerical results for a controlled Duffing oscillator indicate the efficacy of the proposed control strategy.

A Numerical Approach for Solving Optimal Control Problems Using the Boubaker Polynomials Expansion Scheme

In this paper, we present a computational method for solving optimal control problems and the controlled Duffing oscillator. This method is based on state parametrization. In fact, the state variable is approximated by Boubaker polynomials with unknown coefficients. The equation of motion, performance index and boundary conditions are converted into some algebraic equations. Thus, an optimal control problem converts to a optimization problem, which can then be solved easily. By this method, the numerical value of the performance index is obtained. Also, the control and state variables can be approximated as functions of time. Convergence of the algorithms is proved. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.

A numerical solution of the nonlinear controlled Duffing oscillator by radial basis functions

In this research, a new numerical method is applied to investigate the nonlinear controlled Duffing oscillator. This method is based on the radial basis functions (RBFs) to approximate the solution of the optimal control problem by using the collocation method. We apply Legendre–Gauss–Lobatto points for RBFs center nodes in order to use the numerical integration method more easily; then the method of Lagrange multipliers is used to obtain the optimum of the problems. For this purpose different applications of RBFs are used. The differential and integral expressions which arise in the dynamic systems, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Application of Chebyshev polynomials to derive efficient algorithms for the solution of optimal control problems

In this paper, new and efficient algorithms for solving optimal control problems and the controlled Duffing oscillator are presented. The solution is based on state parameterization, such that the state variable can be considered as a linear combination of Chebyshev polynomials with unknown coefficients. First, an optimization problem in (n+1)-dimensional space is changed into a one-dimensional optimization problem, which can then be solved easily. By these algorithms, the control and state variables can be approximated as a function of time. Convergence of the algorithms is proved and some illustrative examples are presented to show the efficiency and reliability of the presented method.

THE MECHANISM OF A CONTROLLED PENDULUM SYNCHRONIZING WITH PERIODIC MOTIONS IN A PERIODICALLY FORCED, DAMPED DUFFING OSCILLATOR

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in the Duffing oscillator are developed using the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domain is obtained. The partial and full synchronizations of the controlled pendulum with periodic motions in the Duffing oscillator are discussed. The control parameter map for the synchronization is achieved from the analytical conditions, and numerical illustrations of the partial and full synchronizations are carried out to illustrate the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. Because the periodically forced, damped Duffing oscillator possesses periodic and chaotic motions, further investigation on the controlled pendulum synchronizing with complicated periodic and chaotic motions in the Duffing oscillator will be accomplished in sequel.

Sinusoidal synchronization of a Duffing oscillator with a chaotic pendulum

In this Letter, analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator are presented. From the analytical conditions, the invariant domain of such sinusoidal synchronization is determined, and the control parameter map of the synchronicity is achieved. Under specific parameters, numerical illustrations of the partial and full sinusoidal synchronizations of the controlled Duffing oscillator with the pendulum are carried out for a better understanding of such synchronization under specific function constraints. The methodology presented in this Letter is applicable to synchronizations with any specific function constraints.

The chaotic synchronization of a controlled pendulum with a periodically forced, damped Duffing oscillator

In this paper, the chaotic synchronization of the Duffing oscillator and controlled pendulum is investigated. From the analytical conditions developed in [1], the partial and full synchronizations of the controlled pendulum with chaotic motions in the Duffing oscillator are discussed. Compared with the periodic synchronization, in the chaotic synchronization, switching points for appearance and vanishing of the partial synchronization are chaotic. The control parameter map for the synchronization is developed from the analytical conditions, and the partial and full synchronizations are illustrated to show the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. For a better understanding of synchronization characteristics between two different dynamical systems, effects with other parameters will be discussed later.

Numerical solution of the controlled Duffing oscillator by semi-orthogonal spline wavelets

This paper presents a numerical method for solving the controlled Duffing oscillator. The method can be extended to nonlinear calculus of variations and optimal control problems. The method is based upon compactly supported linear semi-orthogonal B-spline wavelets. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

BIFURCATION ANALYSIS OF A DELAYED DYNAMIC SYSTEM VIA METHOD OF MULTIPLE SCALES AND SHOOTING TECHNIQUE

This paper presents a detailed study on the bifurcation of a controlled Duffing oscillator with a time delay involved in the feedback loop. The first objective is to determine the bifurcating periodic motions and to obtain the global diagrams of local bifurcations of periodic motions with respect to time delay. In order to determine the bifurcation point, an analysis on the stability switches of the trivial equilibrium is first performed for all possible parametric combinations. Then, by means of the method of multiple scales, an analysis on the local bifurcation of periodic motions is given. The static bifurcation diagrams on the amplitude-delay plane exhibit two kinds of local bifurcations of periodic motions, namely the saddle-node bifurcation and the pitchfork bifurcation, indicating a sudden emergence of two periodic motions with different stability and a Hopf bifurcation, respectively, in the sense of dynamic bifurcation. The second objective is to develop a shooting technique to locate both stable and unstable periodic motions of autonomous delay differential equations such that the periodic motions and their stability predicted using the method of multiple scales could be verified. The efficacy of the shooting scheme is well illustrated by some examples via phase trajectory and time history. It is shown that periodic motions located by the shooting method agree very well with those predicted on the bifurcation diagrams. Finally, the paper presents some interesting features of this simple, but dynamics-rich system.

GLOBAL DYNAMICS OF A DUFFING OSCILLATOR WITH DELAYED DISPLACEMENT FEEDBACK

This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.

NUMERICAL SOLUTION OF THE CONTROLLED DUFFING OSCILLATOR BY THE INTERPOLATING SCALING FUNCTIONS

This paper present a numerical method for solving the controlled Duffing oscillator. The method can be extended to nonlinear calculus of variations and optimal control problems. The method is based upon interpolating scaling functions. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A Chebyshev expansion method for solving nonlinear optimal control problems

This paper presents a numerical technique for solving the controlled Duffing oscillator. The control and state variables are approximated by Chebyshev series. A technique is provided for approximating the system dynamics, boundary conditions, and performance index. This technique is based on using an explicit formula for the Chebyshev polynomials in terms of arbitrary order of their derivatives. The system dynamics and the performance index are converted into some algebraic equations. Then the optimal control problem is reduced to constrained optimization problem. Results and comparisons are given at the end of this paper.

Numerical solution of the controlled Duffing oscillator by hybrid functions

A numerical method for solving the controlled Duffing oscillator is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions which consists of block-pulse functions plus Legendre polynomials are given. The method is based upon expanding various time functions in the system as their truncated hybrid functions. The operational matrix of product is introduced. This matrix together with the operational matrix of integration are utilized to reduce the solution of controlled Duffing oscillator to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A Chebyshev spectral method for the solution of nonlinear optimal control problems

This paper presents a spectral method of solving the controlled Duffing oscillator. The method is based upon constructing the Mth degree interpolation polynomials, using Chebyshevs nodes, to approximate the state and the control vectors. The differential and integral expressions that arise from the system dynamics and the performance index are converted into some algebraic equations. The optimum condition is obtained by applying the method of constrained extremum.

On the optimal spectral Chebyshev solution of a controlled nonlinear dynamical system

In a recent paper we considered the numerical solution of the controlled Duffing oscillator: minimize J = 1/2∫ -T 0 U 2 (τ)dτ, subject to X(τ) + w 2 X(τ) + ∈X 2 (τ) = U(τ) (- T ≤ τ ≤ 0), where T is known, with X(- T) = x 0 , X(0) = 0, by the pseudospectral Legendre method, which shows that in order to maintain spectral accuracy the grids on which a physical problem is to be solved must also be obtained by spectrally accurate techniques. This paper presents an alternative spectrally accurate computational method of solving the nonlinear controlled Duffing oscillator. The method is based upon constructing the Mth-degree interpolation polynomials, using Chebyshev nodes, to approximate the state and the control vectors. The differential and integral expressions which arise from the system dynamics and the performance index are converted into an algebraic nonlinear programming problem. The results of computer-simulation studies compare favourably with optimal solutions obtained by closed-form analysis and/or by other numerical schemes.

A cell-averaging Chebyshev spectral method for the controlled Duffing oscillator

In order to maintain spectral accuracy, the grids on which a physical problem is to be solved must also be obtained by spectrally accurate techniques. This paper presents a spectral method of solving the controlled Duffing oscillator. The method is based upon constructing the Mth-degree interpolation polynomials, using Chebyshev nodes, to approximate the state and the control vectors. The differential and integral expressions which arise from the system dynamics and the performance index are converted into an algebraic nonlinear programming problem. The results of computer simulation studies compare favorably to optimal solutions obtained by closed-form analysis and/or by other numerical schemes.

A Chebyshev approximation for solving optimal control problems

This paper presents a numerical solution for solving optimal control problems, and the controlled Duffing oscillator. A new Chebyshev spectral procedure is introduced. Control variables and state variables are approximated by Chebyshev series. Then the system dynamics is transformed into systems of algebraic equations. The optimal control problem is reduced to a constrained optimization problem. Results and comparisons are given at the end of the paper.