哈密顿系统正则变换在时变最优控制中的应用

利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.      

Numerical solutions of linear quadratic control for time-varying systems via symplectic conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented. It gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation. The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.     

Numerical solutions of linear quadratic control for time-varying systems via symplectic par conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented.It gives a uniform way to solve the linear quadratic control (LQ control) problems for linear time-varying systems accurately and efficiently, whose key points are solutions of differential Riccati equation (DRE) with variable coefficients and the state feedback equation.The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.     

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.     

Recursive identification of discrete-time nonlinear cascade systems with time-varying output hysteresis

Nonlinear Dynamics - A recursive identification method for discrete-time nonlinear cascade systems with a linear dynamic system followed by a time-varying output hysteresis is presented. The...     

Delay-dependent stability analysis and \mathcal {H}_{\infty } control for LPV systems with parameter-varying state delays

This paper develops the stability analysis and delay-dependent \(\mathcal {H}_{\infty }\) control synthesis for linear parameter-varying (LPV) systems with time-varying state delays. On the basis of the Finsler’s lemma, sufficient conditions on \(\mathcal {H}_{\infty }\) performance analysis are formulated in terms of parameterized linear matrix inequalities. The interesting annihilator matrix is constituted by time-varying parameters of LPV systems to reduce the conservatism. A numerical example is presented to confirm the efficiency of the proposed method.     

Linear feedback controller design method for time-delay chaotic systems

Based on the stability theory of time-delay systems, this paper discusses chaos control and synchronization problem of time-delay chaotic systems. Through the combining of a new theorem and the characters of the chaotic system, we have designed a linear controller to realize chaos control and synchronization for Lorenz system with time-varying lags. Finally, numerical simulations are provided to verify the effectiveness and feasibility of the developed method.     

Leader-following formation control for second-order multiagent systems with time-varying delay and nonlinear dynamics

In this paper, the leader-following formation control problem for second-order multiagent systems with time-varying delay and nonlinear dynamics is considered. Two different cases of coupling topologies, fixed topology and switching topology, are analyzed. Based on the Lyapunov theory combined with the linear matrix inequality (LMI) method, sufficient conditions in terms of LMIs are given to ensure the multiagent systems can reach and maintain the desired formation. The simulation results are provided to demonstrate the effectiveness of the obtained theory results.     

Secure communication based on chaotic synchronization via?interval?time-varying delay feedback control

In this paper, a synchronization method of Lur??e systems for chaotic secure communication systems with interval time-varying delay feedback control is proposed. To increase communication security, the transmitted message is encrypted with the techniques of N-shift cipher and public key. Based on Lyapunov method and linear matrix inequality (LMI) formulation, new delay-dependent synchronization criteria are established to not only guarantee stable synchronization of both transmitter and receiver systems but also recover the transmitted original signal at the receiver. Throughout a numerical example, the validity and superiority of the proposed method are shown.     

Dynamic response of a single-mesh gear system with periodic mesh stiffness and backlash nonlinearity under uncertainty

Parametric uncertainties play a critical role in the response predictions of a gear system. However, accurately determining the effects of the uncertainty propagation in nonlinear time-varying models of gear systems is awkward and difficult. This paper improves the interval harmonic balance method (IHBM) to solve the dynamic problems of gear systems with backlash nonlinearity and time-varying mesh stiffness under uncertainties. To deal with the nonlinear problem including the fold points and uncertainties, the IHBM is improved by introducing the pseudo-arc length method in combination with the Chebyshev inclusion function. The proposed approach is demonstrated using a single-mesh gear system model, including the parametrically varying mesh stiffness and the gear backlash nonlinearity, excited by the transmission error. The results of the improved IHBM are compared with those obtained from the scanning method. Effects of parameter uncertainties on its dynamic behavior are also discussed in detail. From various numerical examples, it is shown that the results are consistent meanwhile the computational cost is significantly reduced. Furthermore, the proposed approach could be effectively applied for sensitivity analysis of the system response to parameter variations.     

Discontinuous Lyapunov functional approach to synchronization of time-delay neural networks using sampled-data

This paper investigates the synchronization problem of neural networks with time-varying delay under sampled-data control in the presence of a constant input delay. Based on the extended Wirtinger inequality, a discontinuous Lyapunov functional is introduced, which makes full use of the sawtooth structure characteristic of sampling input delay. A simple and less conservative synchronization criterion is given to ensure the master systems synchronize with the slave systems by using the linear matrix inequality (LMI) approach. The design method of the desired sampled-data controller is also proposed. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Constrained predictive synchronization of discrete-time chaotic Lur’e systems with time-varying delayed feedback control

This paper presents a predictive synchronization method for discrete-time chaotic Lur’e systems with input constraints by using time-varying delayed feedback control. Based on the model predictive control scheme, a delay-dependent stabilization criterion is derived for the synchronization of chaotic systems that is represented by Lur’e systems with input constraints. By constructing a suitable Lyapunov–Krasovskii functional and combining with a reciprocally convex combination technique, a delay-dependent stabilization condition for synchronization is obtained via linear matrix inequality (LMI) formulation. The control inputs are obtained by solving a min-max problem subject to cost monotonicity, which is expressed in terms of LMIs. The effectiveness of the proposed method will be verified throughout a numerical example.     

Delay-dependent robust stability and H_∞ analysis of stochastic systems with time-varying delay

This paper investigates the robust stochastic stability and H∞ analysis for stochastic systems with time-varying delay and Markovian jump. By using the freeweighting matrix technique, i.e., He's technique, and a stochastic Lyapunov-Krasovskii functional, new delay-dependent criteria in terms of linear matrix inequalities are derived for the the robust stochastic stability and the H∞ disturbance attenuation. Three numerical examples are given. The results show that the proposed method is efficient and much less conservative than the existing results in the literature.     

轴向移动系统的参数振动问题研究进展

对近二十年来轴向移动系统(移动弦，移动梁和移动带等)的参数振动研究进展进行了详细的评述，特别关注了轴向张紧力和移动速度随时间改变时轴向移动系统的参数振动特性和稳定性等问题。文章首先讨论了所研究问题的控制方程。然后详细说明了目前研究中人们较为关注的几个重点问题，如参数激励的形式，求解方法和所研究的问题等。接着在其后的两节中，分别评述了在张紧力和移动速度随时间变化时，轴向移动弦和轴向移动梁的振动问题近年来的研究进展，详细、深入讨论了模型的类型、张紧力和轴向移动速度随时间变化的形式以及在研究中使用的解题方法和系统的振动特性(振动响应、固有频率和动态稳定性)等；最后给出了在此领域今后研究中应关注的问题。     

Full- and reduced-order synchronization of a class of time-varying systems containing uncertainties

Investigation on chaos synchronization of autonomous dynamical systems has been largely reported in the literature. However, synchronization of time-varying, or nonautonomous, uncertain dynamical systems has received less attention. The present contribution addresses full- and reduced-order synchronization of a class of nonlinear time-varying chaotic systems containing uncertain parameters. A unified framework is established for both the full-order synchronization between two completely identical time-varying uncertain systems and the reduced-order synchronization between two strictly different time-varying uncertain systems. The synchronization is successfully achieved by adjusting the determined algorithms for the estimates of unknown parameters and the linear feedback gain, which is rigorously proved by means of the Lyapunov stability theorem for nonautonomous differential equations together with Barbalat’s lemma. Moreover, the synchronization result is robust against the disturbance of noise. We illustrate the applicability for full-order synchronization using two identical parametrically driven pendulum oscillators and for reduced-order synchronization using the parametrically driven second-order pendulum oscillator and an additionally driven third-order Rossler oscillator.     

Sufficient conditions of the various stabilities of the linear time-varying delayed differential equations

Due to the appearance and the study of the ornithopter and flexible-wing micro air vehicles, etc., the time-varying systems become more and more important and ubiquitous in the study of the mechanics. In this letter, the sufficient conditions of the uniform asymptotic stability are first presented for the delayed time-varying linear differential equations with any time delay by employing the Dini derivative, Lozinskii measure and the generalized scalar Halanay delayed differential inequality. They are especially based on the estimation of the arbitrary solutions but not the fundamental solution matrix since their solutions' space is infinite-dimensional. Then some sufficient conditions of the stability, asymptotic stability and uniform asymptotic stability of the delayed time-varying linear system with a sufficiently small time delay are reported by employing Taylor expansion and Dini derivative. It implies that these stabilities can be guaranteed by the Lozinskii measure of the matrix composing of the time delay and the coefficient matrices of the system.     

Delay-range-dependent stability criteria for delayed discrete-time Lur’e system with sector-bounded nonlinearities

This paper is devoted to the absolute and robust stability for uncertain discrete-time Lur’e systems with interval time-varying delays and sector-bounded nonlinearities. Both the cases with time-invariant and time-varying nonlinearities are considered. By dividing the variation interval of the time delays into some subintervals, some new delay-range-dependent robust stability criteria are derived in the form of linear matrix inequalities (LMIs) via a modified Lyapunov-Krasovskii functional (LKF) approach. The criteria are less conservative than some existing results. Finally, some numerical examples are presented to show the effectiveness of the proposed approach.     

A Re-Configurable Test Apparatus for Complex Nonlinear Dynamic Systems

This paper reports the results of an analytical and experimental study to design, analyze, construct, test, and evaluate a re-configurable test-bed to allow the convenient performance of sophisticated experiments in a laboratory setting for investigating a broad category of important nonlinear, time-varying phenomena that are widely encountered in the applied mechanics field. The essential elements of the test apparatus include an electro-dynamic exciter that drives an oscillating mass whose restoring force is easily adjustable from one resembling a linear SDOF mass-spring-damper system with constant coefficients, to one that can represent systems with nonlinear elastic forces, to one that models systems possessing hysteretic properties with precisely-controlled time-varying characteristics. The apparatus design is economical to fabricate, convenient to manipulate, and it provides results that can be accurately replicated under repeated test combinations of system parameters and dynamic loads. The utility of the apparatus to provide an investigational tool for studying the dynamic response of systems with time-varying dry-friction forces that give rise to hysteretic phenomena is demonstrated. Sophisticated on-line system-identification techniques are used to estimate the parameters of reduced-order models that capture the dominant features of the physical model. It is shown that the apparatus under discussion is a useful research tool for investigators conducting studies in physics-based models of generic nonlinear dynamic systems.     

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.     

Adaptive fuzzy synchronization for a class of chaotic systems with unknown nonlinearities and disturbances

By incorporating a time-varying parameter into T-S fuzzy logic systems with nonlinear consequents (T-S-FLS-NRC), the synchronization of driver-response chaotic systems with unknown nonlinearities and disturbances is synthesized via state feedback controllers and updated adaptive laws. During designing process of synchronization, only three common parameters are needed to be adjusted automatically, and the number of adaptive laws is not related with the number of IF-THEN rules. Meanwhile, T-S-FLS-NRC is employed to approximate the unknown nonlinearities for the master and slave systems. The general form and high approximate capacity of T-S-FLS-NRC is useful to obtain fewer fuzzy rules than other fuzzy logic systems such as Mamdani or T-S fuzzy logic system with linear consequents. The synchronization method in this paper cannot only significantly reduce the on-line computational burden, but also can synthesize the fuzzy rules with high interpretability by means of intuition inferences. Finally, a numerical example is used to show the validity of the proposed synchronization method.