Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control

Analytical derivations and numerical calculations are employed to gain insight into the parametric resonance of a stochastically driven van der Pol oscillator with delayed feedback. This model is the prototype of a self-excited system operating with a combination of narrow-band noise excitation and two time delayed feedback control. A slow dynamical system describing the amplitude and phase of resonance, as well as the lowest-order approximate solution of this oscillator is firstly obtained by the technique of multiple scales. Then the explicit asymptotic formula for the largest Lyapunov exponent is derived. The influences of system parameters, such as magnitude of random excitation, tuning frequency, gains of feedback and time delays, on the almost-sure stability of the steady-state trivial solution are discussed under the direction of the signal of largest Lyanupov exponent. The non-trivial steady-state solution of mean square response of this system is studied by moment method. The results reveal the phenomenon of multiple solutions and time delays induced stabilization or unstabilization, moreover, an appropriate modulation between the two time delays in feedback control may be acted as a simple and efficient switch to adjust control performance from the viewpoint of vibration control. Finally, theoretical analysis turns to a validation through numerical calculations, and good agreements can be found between the numerical results and the analytical ones.     

Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method

Hopf bifurcation control in nonlinear stochastic dynamical system with nonlinear random feedback method is studied in this paper. Firstly, orthogonal polynomial approximation is applied to reduce the controlled stochastic nonlinear dynamical system with nonlinear random controller to the deterministic equivalent system, solvable by suitable numerical methods. Then, Hopf bifurcation control with nonlinear random feedback controller is discussed in detail. Numerical simulations show that the method provided in this paper is not only available to control the stochastic Hopf bifurcation in nonlinear stochastic dynamical system, but is also superior to the deterministic nonlinear feedback controller.     

Stabilization of the rotating disk-beam system with a delay term in boundary feedback

In this article, the stabilization problem of a rotating disk-beam system is addressed. It is assumed that the flexible beam is free at one end, whereas the other end is attached to the center of the rotating disk whose angular velocity is time-varying. The proposed feedback law consists of a torque control which acts on the disk, whereas a delayed boundary force control is exerted at the free end of the beam. Thereafter, it is proved that the presence of such controls in the system guarantees the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk as well as the feedback gain in the delay term. This result is illustrated by numerical examples.     

Attitude control of a rigid spacecraft with one variable-speed control moment gyro

Nonlinear controllability and attitude stabilization are studied for the underactuated nonholonomic dynamics of a rigid spacecraft with one variable-speed control moment gyro(VSCMG), which supplies only two internal torques.Nonlinear controllability theory is used to show that the dynamics are locally controllable from the equilibrium point and thus can be asymptotically stabilized to the equilibrium point via time-invariant piecewise continuous feedback laws or time-periodic continuous feedback laws. Specifically,when the total angular momentum of the spacecraft-VSCMG system is zero, any orientation can be a controllable equilibrium attitude. In this case, the attitude stabilization problem is addressed by designing a kinematic stabilizing law, which is implemented through a nonlinear proportional and derivative controller, using the generalized dynamic inverse(GDI)method. The steady-state instability inherent in the GDI controller is elegantly avoided by appropriately choosing control gains. In order to obtain the command gimbal rate and wheel acceleration from control torques, a simple steering logic is constructed to accommodate the requirements of attitude stabilization and singularity avoidance of the VSCMG. Illustrative numerical examples verify the efcacy of the proposed control strategy.     

受周期和白噪声激励的分段线性系统的吸引域与离出问题研究

离出行为是随机非线性系统的重要现象之一,而离出问题是除随机动力系统理论以外考察随机非线性系统随机稳定性的另一种重要的方法.分段线性系统是一个经典的非线性动力学模型,受随机激励后成为随机系统,但并不是严格的随机动力系统,因而此时随机动力系统理论也不适用.为了研究同时受周期和白噪声激励的分段线性系统,首先使用Poincaré截面模拟其在无噪声时确定性的动力学行为,然后使用Monte Carlo模拟对其在白噪声激励下的离出行为进行了数值仿真分析.其次,为了考察离出问题中的重要参数,系统的平均首次通过时间（mean first-passage time,MFPT）,使用van der Pol变换,随机平均法,奇异摄动法和射线方法进行了量化计算.通过对理论结果与模拟结果的对比分析,得到结论:当系统吸引子对应的吸引域边界出现碎片化时,理论结果与模拟结果的误差极大;而当吸引域边界足够光滑的以后,理论结果与模拟结果才会相当吻合.      

RESEARCH FOR ATTRACTING REGION AND EXIT PROBLEM OF A PIECEWISE LINEAR SYSTEM UNDER PERIODIC AND WHITE NOISE EXCITATIONS

离出行为是随机非线性系统的重要现象之一,而离出问题是除随机动力系统理论以外考察随机非线性系统随机稳定性的另一种重要的方法.分段线性系统是一个经典的非线性动力学模型,受随机激励后成为随机系统,但并不是严格的随机动力系统,因而此时随机动力系统理论也不适用.为了研究同时受周期和白噪声激励的分段线性系统,首先使用Poincaré截面模拟其在无噪声时确定性的动力学行为,然后使用Monte Carlo模拟对其在白噪声激励下的离出行为进行了数值仿真分析.其次,为了考察离出问题中的重要参数,系统的平均首次通过时间（mean first-passage time,MFPT）,使用van der Pol变换,随机平均法,奇异摄动法和射线方法进行了量化计算.通过对理论结果与模拟结果的对比分析,得到结论：当系统吸引子对应的吸引域边界出现碎片化时,理论结果与模拟结果的误差极大;而当吸引域边界足够光滑的以后,理论结果与模拟结果才会相当吻合.     

On the efficiency of a linear controller under additional geometric constraints

One heuristic approach to taking into account geometric constraints on the controls in stabilization problems is to use controls obtained by truncation (at the constraint values) of a control signal linear in the phase variables. With the introduction of the truncated control, the originally linear system becomes substantially nonlinear, which complicates the analysis. In numerous papers, the phase plane method was used to analyze the control defined as the sign of a control signal linear in the phase variables. In [1, 2], the asymptotic stability of linear dynamical systems with nonlinear controls of special type different from that considered below was studied. The problem of stabilization of a mechanical system by a geometrically constrained control was considered in [3]. The asymptotic stability of an arbitrary linear system with a truncated control was studied in [4], where some estimates for the attraction domain of the trivial solution of the system were obtained and necessary and sufficient conditions under which this domain can be made arbitrarily large were given. In the present paper, we solve the problem of ensuring the asymptotic stability of amechanical system with arbitrarily many degrees of freedom and with componentwise geometric constraints on the control.     

Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation

The principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation is studied by using the method of multiple scales and numerical simulations. The first-order approximations of the solution, together with the modulation equations of both amplitude and phase, are derived. The effects of the frequency detuning, the deterministic amplitude, the intensity of the random excitation and the time delay on the dynamical behaviors, such as stability and bifurcation, are studied through the largest Lyapunov exponent. Moreover, the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time delay can broaden the stable region of the trivial steady-state solution and enhance the control performance. The theoretical results are well verified through numerical simulations.     

Control strategy of optimal deployment for spacecraft solar array system with initial state uncertainty

A control strategy combining feedforward control and feedback control is presented for the optimal deployment of a spacecraft solar array system with the initial state uncertainty. A dynamic equation of the spacecraft solar array system is established under the assumption that the initial linear momentum and angular momentum of the system are zero. In the design of feedforward control, the dissipation energy of each revolute joint is selected as the performance index of the system. A Legendre pseudospectral method (LPM) is used to transform the optimal control problem into a nonlinear programming problem. Then, a sequential quadratic programming algorithm is used to solve the nonlinear programming problem and offline generate the optimal reference trajectory of the system. In the design of feedback control, the dynamic equation is linearized along the reference trajectory in the presence of initial state errors. A trajectory tracking problem is converted to a two-point boundary value problem based on Pontryagin’s minimum principle. The LPM is used to discretize the two-point boundary value problem and transform it into a set of linear algebraic equations which can be easily calculated. Then, the closed-loop state feedback control law is designed based on the resulting optimal feedback control and achieves good performance in real time. Numerical simulations demonstrate the feasibility and effectiveness of the proposed control strategy.     

Robust stabilization the Korteweg–de Vries–Burgers equation by boundary control

The problem of robust global stabilization by nonlinear boundary feedback control for the Korteweg–de Vries–Burgers equation on the domain [0,1] is considered. The main purpose of this paper is to derive nonlinear robust boundary control laws which make the system robustly globally asymptotically stable in spite of uncertainty in the system parameters. Furthermore, we show that the proposed boundary controllers guarantee L ₂-robust exponential stability, L _∞-robust asymptotic stability and boundedness in terms of both L ₂ and L _∞.     

Active vibration control of a dynamical system via negative linear velocity feedback

In this paper, a negative velocity feedback is added to a dynamical system which is represented by second-order nonlinear differential equations having quadratic coupling, quadratic, and cubic nonlinearities. The system describes the vibration of the system subjected to multi-parametric excitation forces. The method of multiple scale perturbation technique is applied to obtain the response equation near the simultaneous internal and super-harmonic resonance case of this system. The stability to the system is investigated applying frequency response equations. The numerical solution and the effects of some parameters on the vibrating system are investigated and reported. The simulation results are achieved using MATLAB 7.0 program. A comparison is made with the available published work.     

First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r(1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system. A partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi-partially integrable Hamiltonian system. In the present paper, the procedures for studying the first-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems are proposed. First, the stochastic averaging method for quasi-partially integrable Hamiltonian systems is briefly reviewed. Then, based on the averaged Itô equations, a backward Kolmogorov equation governing the conditional reliability function, a set of generalized Pontryagin equations governing the conditional moments of first-passage time and their boundary and initial conditions are established. After that, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and of maximization of mean first-passage time are formulated. The relationship between the backward Kolmogorov equation and the dynamical programming equation for reliability maximization, and that between the Pontryagin equation and the dynamical programming equation for maximization of mean first-passage time are discussed. Finally, an example is worked out to illustrate the proposed procedures and the effectiveness of feedback control in reducing first-passage failure.     

Observer-based synchronization of memristive systems with multiple networked input and output delays

This paper investigates the synchronization problem of memristive systems with multiple networked input and output delays via observer-based control. A memristive system is set up, and the fuzzy method has been employed to linearize the dynamical system of the memristive system; the networked input and output delays are considered in the synchronization problem of this system. A truncated predictor feedback approach is employed to design the observers. Under certain restrictions, a class of finite-dimensional observer-based output feedback controllers is designed. A numerical example is carried out to demonstrate the effectiveness of the proposed methods.     

Stabilization of stochastic cycles and chaos suppression for nonlinear discrete-time systems

In this paper we consider a nonlinear discrete-time control system with regular and chaotic dynamics forced by stochastic disturbances. The problem addressed is the design of the feedback regulator which stabilizes a limit cycle of the closed-loop deterministic system and synthesizes a required dispersion of random states for the corresponding stochastic system. To solve this problem, we propose a new method based on the stochastic sensitivity function technique. This function approximates a dispersion of random states distributed around deterministic cycle. Explicit formulas for the intercoupling between stochastic sensitivity function and considered system parameters are worked out. The problem of the design of the required stochastic sensitivity function for cycles by feedback regulators is solved. Coefficients of the feedback regulator are constructed and corresponding attainability sets are described. The effectiveness of the proposed approach is demonstrated on the stochastic Verhulst model. It is shown that constructed regulators provide a low level of sensitivity and suppress chaotic oscillations.     

Second Order Adaptive Boundary Conditions for Exterior Flow Problems: Non-Symmetric Stationary Flows in Two Dimensions

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct – by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude. Peter Wittwer: Supported in part by the Fonds National Suisse.     

时滞状态正反馈在振动控制中的新特征

控制理论中广泛采用负反馈,而正反馈的应用不多, 一个重要原因是正反馈将系统的变化放大而使系统的稳定性变差. 如果反馈环节具有时滞, 那么正反馈未必使系统稳定性变差. 本文以线性振动系统为例, 采用稳定性切换方法和利用确定时滞系统稳定性的最大实部特征根, 详细研究了时滞状态正反馈在镇定系统不稳定运动和改善系统稳定性方面的作用. 我们发现,时滞位移正反馈明显优于时滞位移负反馈, 表现为: (1). 正反馈控制可以用较小的时滞去镇定不稳定运动和改善系统稳定性; (2). 正反馈控制可容许的时滞范围很大, 而负反馈控制的可容许时滞范围很小; (3). 正反馈对应的闭环系统的最大实部特征根的实部的极小值可显著小于负反馈对应的闭环系统的最大实部特征根的实部的极小值, 因而在相同的初始条件下, 正反馈作用下的闭环系统比之负反馈作用下的闭环系统可以更快地稳定到平衡点. 我们还发现, 对时滞速度反馈与时滞加速度反馈来说, 负反馈优于正反馈; 而对相同的反馈增益, 时滞位移正反馈优于时滞速度正反馈和时滞加速度正反馈. 关键字镇定,振动控制,时滞正反馈, 稳定性切换, 特征根      

ew Features of Positive Time-Delayed Feedbacks in Vibration Control

控制理论中广泛采用负反馈,而正反馈的应用不多, 一个重要原因是正反馈将系统的变化放大而使系统的稳定性变差. 如果反馈环节具有时滞, 那么正反馈未必使系统稳定性变差. 本文以线性振动系统为例, 采用稳定性切换方法和利用确定时滞系统稳定性的最大实部特征根, 详细研究了时滞状态正反馈在镇定系统不稳定运动和改善系统稳定性方面的作用. 我们发现,时滞位移正反馈明显优于时滞位移负反馈, 表现为: (1). 正反馈控制可以用较小的时滞去镇定不稳定运动和改善系统稳定性; (2). 正反馈控制可容许的时滞范围很大, 而负反馈控制的可容许时滞范围很小; (3). 正反馈对应的闭环系统的最大实部特征根的实部的极小值可显著小于负反馈对应的闭环系统的最大实部特征根的实部的极小值, 因而在相同的初始条件下, 正反馈作用下的闭环系统比之负反馈作用下的闭环系统可以更快地稳定到平衡点. 我们还发现, 对时滞速度反馈与时滞加速度反馈来说, 负反馈优于正反馈; 而对相同的反馈增益, 时滞位移正反馈优于时滞速度正反馈和时滞加速度正反馈. 关键字镇定,振动控制,时滞正反馈, 稳定性切换, 特征根     

利用时滞反馈控制自参数振动系统的振动

主要研究采用时滞状态反馈控制自参数动力吸振器减振系统中主系统的振动问题.系统在简谐激励作用下,采用多尺度方法得到了自参数动力吸振器减振系统中饱和控制的范围.当系统处于饱和控制时,引入时滞状态反馈控制主系统的振动.主要分析了反馈增益系数和时滞两控制参数对主系统振动的影响.结果表明,存在反馈增益系数和时滞的调节区域能够减小主系统的振动.对某一反馈增益系数,可以在某段区间内调节时滞以减小主系统的振动.在时滞的调节区间内存在一个时滞的``最大减振点',能够在该反馈增益系数下最大程度地减小主系统的振动.研究还表明,随着反馈增益系数的不断增大,时滞在``最大减振点'时系统的减振能力也不断提高.通过合理的选择反馈增益系数和时滞两参数,主系统的振动几乎可以完全消除.      

Finite-time stabilization of uncertain non-autonomous chaotic gyroscopes with nonlinear inputs

Gyroscopes are one of the most interesting and everlasting nonlinear nonautonomous dynamical systems that exhibit very complex dynamical behavior such as chaos.In this paper,the problem of robust stabi...     

LMI-based stabilization of a class of fractional-order chaotic systems

Based on the theory of stabilization of fractional-order LTI interval systems, a simple controller for stabilization of a class of fractional-order chaotic systems is proposed in this paper. We consider the structure of the chaotic systems as fractional-order LTI interval systems due to the limited amplitude of chaotic trajectories. We introduce a simple feedback controller for the interval system and then, based on a recently established theorem for stabilization of interval systems, we reach to a linear matrix inequality (LMI) problem. Solving the LMI yields an appropriate decoupling feedback control law which suffices to bring the chaotic trajectories to the origin. Several illustrative examples are given which show the effectiveness of the method.