Asymptotic Lyapunov stability with probability one of quasi-integrable Hamiltonian systems with delayed feedback control

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and nonresonant Hamiltonian systems with time-delayed feedback control subject to multiplicative (parametric) excitation of Gaussian white noise is studied. First, the time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay and the system is converted into ordinary quasi-integrable and nonresonant Hamiltonian system. Then, the averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems and the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained approximately by letting the largest Lyapunov exponent to be negative. An example is worked out in detail to illustrate the above mentioned procedure and its validity and to show the effect of the time delay in feedback control on the largest Lyapunov exponent and the stability of the system.     

Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach

The Hamilton-Jacobi-Bellman (HJB) equation corresponding to constrained control is formulated using a suitable nonquadratic functional. It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS). The value function of this HJB equation is solved for by solving for a sequence of cost functions satisfying a sequence of Lyapunov equations (LE). A neural network is used to approximate the cost function associated with each LE using the method of least-squares on a well-defined region of attraction of an initial stabilizing controller. As the order of the neural network is increased, the least-squares solution of the HJB equation converges uniformly to the exact solution of the inherently nonlinear HJB equation associated with the saturating control inputs. The result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line.     

Stabilisation of time-varying linear systems via Lyapunov differential equations

This article studies stabilisation problem for time-varying linear systems via state feedback. Two types of controllers are designed by utilising solutions to Lyapunov differential equations. The first type of feedback controllers involves the unique positive-definite solution to a parametric Lyapunov differential equation, which can be solved when either the state transition matrix of the open-loop system is exactly known, or the future information of the system matrices are accessible in advance. Different from the first class of controllers which may be difficult to implement in practice, the second type of controllers can be easily implemented by solving a state-dependent Lyapunov differential equation with a given positive-definite initial condition. In both cases, explicit conditions are obtained to guarantee the exponentially asymptotic stability of the associated closed-loop systems. Numerical examples show the effectiveness of the proposed approaches.     

Application of generalized Zubov's method to power system stability†

The generalized form of Zubov's partial differential equation (Szego 1962) is written using the dynamical equations of a power system. Then a Lyapunov function which satisfies the partial differential equation is obtained by using a transformation of state variables. The V function obtained is used to describe the region of stability. The application of the method to the power system stability problem is illustrated by considering a synchronous generator connected to an infinite bus, Three examples, using three different models for the synchronous machine, are given.     

Chebyshev series approximation for solving differential–algebraic equations (DAEs)

The numerical solution of differential–algebraic equations (DAEs) using the Chebyshev series approximation is considered in this article. Two different problems are solved using the Chebyshev series approximation and the solutions are compared with the exact solutions. First, we calculate the power series of a given equation system and then transform it into Chebyshev series form, which gives an arbitrary order for solving the DAE numerically.     

Stochastic minimax control for stabilizing uncertain quasi- integrable Hamiltonian systems

A procedure for designing feedback control to asymptotically stabilize, with probability one, quasi-integrable Hamiltonian systems with bounded uncertain parametric disturbances is proposed. First, the partially averaged Itô stochastic differential equations are derived from given system by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Second, the Hamilton-Jacobi-Issacs (HJI) equation for the ergodic control problem of the averaged system and a performance index with undetermined cost function is established based on the principle of optimality. This equation is then solved to yield the worst disturbances and the associated optimal controls. Third, the asymptotic Lyapunov stability with probability one of the optimally controlled system with worst disturbances is analyzed by evaluating the maximal Lyapunov exponent of the fully averaged Itô equations. Finally, the cost function and feedback control are determined by the requirement of stabilizing the worst-disturbed system. A simple example is worked out to illustrate the application of the proposed procedure and the effects of optimal control on stabilizing the uncertain system.     

STABILITY CONDITION OF DISTRIBUTED DELAY SYSTEMS BASED ON AN ANALYTIC SOLUTION TO LYAPUNOV FUNCTIONAL EQUATIONS

An analytic solution to Lyapunov functional equations for distributed delay systems is derived. The analytic solution is computed using a matrix exponential function, while conventional computation has been relied on numerical approximations. Based on the analytic solution, a necessary and sufficient stability condition for distributed delay systems with unknown but bounded constant delay is proposed.     

New-type stability theorem for stochastic functional differential equations with application to SFDSs with distributed delays

In this paper, a new-type stability theorem for stochastic functional differential equations (SFDEs) is established, which is not a direct copy of the basic stability theorem for deterministic functional differential equations (DFDEs). By the new-type stability theorem, one can use the most simple Lyapunov functions and employ the equations repeatedly to deal with the delayed terms encountered conveniently and to carry out stability criteria for the equations. Based on the theorem, a practical stability theorem in accordance with the Lyapunov function method is also established, and then the asymptotic stability of SFDEs with distributed delays in the diffusive terms is investigated and a stability criterion for SFDSs is obtained, which is described by algebraic matrix equations. Finally, an example is given to illustrate the effectiveness of our method and results.     

Automatic determination of dynamical symmetries of ordinary differential equations

We describe the application of a computer program CRACKSTAR for the exact analytic solution of overdetermined systems of differential equations which result in determining point-, contact- and dynamical symmetries of ordinary differential equations. Examples are discussed.     

Stability and control of nonlinear systems described by retarded functional equations: a review of recent results

This paper reports on recent results in a series of the work of the authors on the stability and nonlinear control for general dynamical systems described by retarded functional differential and difference equations. Both internal and external stability properties are studied. The corresponding Lyapunov and Razuminkhin characterizations for input-to-state and input-to-output stabilities are proposed. Necessary and sufficient Lyapunov-like conditions are derived for robust nonlinear stabilization. In particular, an explicit controller design procedure is developed for a new class of nonlinear time-delay systems. Lastly, sufficient assumptions, including a small-gain condition, are presented for guaranteeing the input-to-output stability of coupled systems comprised of retarded functional differential and difference equations.     

Robust Control Using Interval Analysis

The synthesis procedure of a control law that guarantees properties of robust stability with respect to structured parameter perturbations is proposed. The solution of the considered problem is based on the Razumikhin's method for functional differential equations generalized for parameter perturbation systems with time delay. The extension is obtained by using interval Lyapunov functions. The robust control law is represented through a solution of an interval matrix Riccati type equation.     

Families of Lyapunov functions for nonlinear systems in criticalcases

Lyapunov functions are constructed for nonlinear systems of ordinary differential equations whose linearized system at an equalized point possesses either a simple zero eigenvalue or a complex conjugate pair of simple, pure imaginary eigenvalues. The construction is explicit, and yields parameterized families of Lyapunov functions for such systems. In the case of a zero eigenvalue, the Lyapunov functions contain quadratic and cubic terms in the state. Quartic terms appear as well for the case of a pair of pure imaginary eigenvalues. Predictions of local asymptotic stability using these Lyapunov functions are shown to coincide with those of pertinent bifurcation-theoretic calculations. The development of the paper is carried out using elementary properties of multilinear functions. The Lyapunov function families thus obtained are amenable to symbolic computer coding     

Bounds in the Lyapunov matrix differential equation

Upper and lower bounds for the trace of the solution of the Lyapunov matrix differential equation are derived. It is shown that they are obtained as solutions to simple scalar differential equations. As a special case, the bounds for the stationary solution give ones for the solution to the Lyapunov algebraic equation.     

Global Error Minimization method for solving strongly nonlinear oscillator differential equations

A modified variational approach called Global Error Minimization (GEM) method is developed for obtaining an approximate closed-form analytical solution for nonlinear oscillator differential equations. The proposed method converts the nonlinear differential equation to an equivalent minimization problem. A trial solution is selected with unknown parameters. Next, the GEM method is used to solve the minimization problem and to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations (ODEs). This approach is simple, accurate and straightforward to use in identifying the solution. To illustrate the effectiveness and convenience of the suggested procedure, a cubic Duffing equation with strong nonlinearity is considered. Comparisons are made between results obtained by the proposed GEM method, the exact solution and results from five recently published methods for addressing Duffing oscillators. The maximal relative error for the frequency obtained by the GEM method compared with the exact solution is 0.0004%, which indicates the remarkable precision of the GEM method.     

求解任意波数的三维Helmholtz方程

提出通过Adomian分解法求解任意波数的三维Helmholtz方程。通过Adomian分解法可以把三维Helmholtz微分方程转换成递归代数公式,并进一步把其边界条件转换成适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后通过数值计算以验证Adomian分解法求解任意波数下三维Helmholtz方程的有效性。数值计算结果表明：Adomian分解法的计算结果非常接近精确解,并且该方法在大波数情况下还具有良好的收敛性。     

关于非綫性控制系統的稳定性问題

本文利用李亚普诺夫方法研究非线性控制系统的全局稳定性、大范围稳定和其吸引区的寻求以及一个品质问题.对非线性元件采用了分段线性的模型,从而使得到的条件中也包含有非线性特性的参数.     

一类无限维随机关联大系统的全局指数稳定性

基于箱体理论,利用向量函数法,研究了一类无限维随机非线性关联大系统的全局指数稳定性.通过分析相应的随机微分不等式的稳定性,得到了该类大系统全局指数稳定的一个判据.该判据利用随机大系统的系数矩阵以及与大系统关联的Lyapunov矩阵方程的解构造判定条件来判定大系统的全局指数稳定性,计算简便,便于应用.     

Limit cycle construction using Liapunov functions

This paper deals with a generalization of the method of Zubov [2] for the construction of Liapunov functionsV(x)useful in estimating the location of stability boundaries. For a systemdot{x}=f(x), V(x)is taken as the solution of(nablaV)' f(x)=-h(x)g(V)whereh(x)is positive semi-definite and not identically zero on a non-trivial trajectory andg(V)exhibits the significant behavior of the system. For a second order system having (with time reversed) an unstable limit cycle analytic in a parameter ε, a suitableg(V)would beg(V) = V(1-V)dotVsatisfying the above partial differential equation may be developed as a power series in e and the position of the limit cycle can be estimated fromV = 1. As an example of the procedure, the method is applied to van der Pol's equation and the position of the limit cycle is estimated to order ε².     

Stability analysis of hybrid composite dynamical systems: Descriptions involving operators and differential equations

We address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Fig. 1, consisting of a block (usually the plant) which is described by an operatorLand of a finite-dimensional block described by a system of ordinary differential equations (usually the controller). We establish results for the well-posedness, attractivity, asymptotic stability, uniform boundedness, asymptotic stability in the large, and exponential stability in the large for such systems. The hypotheses of these results are phrased in terms of the I/O properties ofLand in terms of the Lyapunov stability properties of the subsystem described by the indicated ordinary differential equations. The applicability of our results is demonstrated by means of general specific examples (involving C0-semigroups, partial differential equations, or integral equations which determineL).     

Accelerating technique for implicit finite difference schemes for two dimensional parabolic partial differential equations

In this paper we define a new accurate fast implicit method for the finite difference solution of the two dimensional parabolic partial differential equations with first level condition, which may be obtained by any other method. The stability region is discussed. The suggested method is considered as an accelerating technique for the implicit finite difference scheme, which is used to find the first level condition. The obtained results are compared with some famous finite difference schemes and it is in satisfactory agreement with the exact solution.