Tchebychev acceleration technique for large scale nonsymmetric matrices

Summary The acceleration by Tchebychev iteration for solving nonsymmetric eigenvalue problems is dicussed. A simple algorithm is derived to obtain the optimal ellipse which passes through two eigenvalues in a complex plane relative to a reference complex eigenvalue. New criteria are established to identify the optimal ellipse of the eigenspectrum. The algorithm is fast, reliable and does not require a search for all possible ellipses which enclose the spectrum. The procedure is applicable to nonsymmetric linear systems as well.     

Extremum principles for optimal control in normed linear spaces

Extremum principles intended for use in optimal control are derived in the form of necessary conditions and sufficient conditions, formulated in general normed linear spaces. The method of application is illustrated by several examples involving optimal control problems, mathematical programming problems, lumped-parameter systems, and distributed-parameter systems. The basic theorems provide a unified approach which is applicable to a wide variety of problems in open-loop optimal control.     

非零势能的耗散力学控制系统的位形能控性

在拉格朗日力学控制系统的仿射联络框架下,基于Sussmann对有限维流形上一般仿射非线性控制系统的能控性讨论,将简单力学控制系统短时间局部位形能控的一个可计算的充分条件推广到迷向耗散的系统上,并给出系统是平衡点能控的一个充分条件,其中,系统的拉格朗日函数为动能减势能A·D2在问题的讨论中,系统的能控李代数的向量场李括号运算,以及与系统位形流形的Levi-Civita联络相关的对称积起了重要作用．尽管势能项会使系统的位形能控性讨论复杂化,但Liouville向量场又简化了系统的能控李代数计算．     

Identification of heat transfer dynamics for non-modal analysis of thermoacoustic stability

A systematic approach for non-modal stability analysis of thermoacoustic systems with a localized heat source is proposed. The response of the heat source to flow perturbations is obtained from unsteady computational fluid dynamics combined with correlation-based linear system identification. A model for the complete thermoacoustic system is formulated with a Galerkin expansion technique, where the heat source is included as an acoustically compact element. The eigenvalues of the resulting system are obtained from discretization of the solution operator, the maximum growth factor is estimated from the pseudospectra using Kreiss’ theorem.The approach is illustrated with a simple Rijke tube configuration. Results obtained with a simple “baseline” model for the heat source dynamics based on King’s law - widely used in hot wire anemometry - are compared against the more advanced treatment developed here. Analysis of pseudospectra diagrams shows that the choice of the heat source model does influence the sensitivity of eigenvalues to perturbations and hence the non-normal behavior. The maximum growth factor for the system with the heat source model based on King’s law is more sensitive to changes in the heat source location than the CFD-based heat source model.     

Eigenstructure assignment for linear parameter-varying systems with applications

The aim of this paper is to show that a recently proposed technique for eigenstructure assignment of linear time-invariant systems can be extended to solve the corresponding eigenstructure assignment problem for linear parameter-varying systems, whose state-space matrices depend on a set of time-varying parameters that are bounded and available online. In particular, the design of eigenstructure assignment is performed without requiring any conditions on the closed-loop eigenvalues, and provides a simple, complete and analytical parametric approach as well as the most degrees of design freedom for the eigenstructure assignment problem of linear parameter-varying systems. A parameter-varying attitude control system of refueling spacecraft in-orbit is used to demonstrate the usefulness and practicality of the proposed approach.     

Algebraic Multiplicities Arising from Static Feedback Control Systems of Parabolic Type

In stabilization studies of linear parabolic control systems, a successful approach is a scheme employing dynamic compensators in the feedback loop. An essential reason is the fact that both sensors and actuators cannot be designed freely, especially in the case of boundary observation/boundary feedback. Most fundamental in this scheme is a simple stabilization result under the static feedback control scheme. In this scheme, little attention has been paid to how to assign new eigenvalues of the feedback system. In this article, we show a new feature of pole assignment that shows some choices of new eigenvalues cause a deterioration of the stability property. An algebraic growth rate is added to the feedback system in such a choice.     

单输入多时滞离散系统的线性二次调节问题

研究了单输入多时滞的离散时间系统的线性二次调节问题(LQR问题),给出了求解最优控制输入序列的一种简单有效而又新颖的方法.将该动态的离散时滞系统的LQR最优控制问题最终转化成了一个静态的、不带时滞的数学规划模型——带等式线性约束的严格凸二次规划问题,并利用两种方法解这个二次规划问题,均成功地导出了系统的最优控制输入序列.仿真结果验证了我们的方法的正确有效性.     

Partial Differential Equation for Evolution of Star-Shaped Reachability Domains of Differential Inclusions

The problem of reachability for differential inclusions is an active topic in the recent control theory. Its solution provides an insight into the dynamics of an investigated system and also enables one to design synthesizing control strategies under a given optimality criterion. The primary results on reachability were mostly applicable to convex sets, whose dynamics is described through that of their support functions. Those results were further extended to the viability problem and some types of nonlinear systems. However, non-convex sets can arise even in simple bilinear systems. Hence, the issue of nonconvexity in reachability problems requires a more detailed investigation. The present article follows an alternative approach for this cause. It deals with star-shaped reachability sets, describing the evolution of these sets in terms of radial (Minkowski gauge) functions. The derived partial differential equation is then modified to cope with additional state constraints due to on-line measurement observations. Finally, the last section is on designing optimal closed-loop control strategies using radial functions.     

General Approach to Integrating Invertible Dynamical Systems Defined by Transformations from the Cremona group Cr(P k n ) of Birational Transformations

A general approach is developed for integrating an invertible dynamical system defined by the composition of two involutions, i.e., a nonlinear one which is a standard Cremona transformation, and a linear one. By the Noether theorem, the integration of these systems is the foundation for integrating a broad class of Cremona dynamical systems. We obtain a functional equation for invariant homogeneous polynomials and sufficient conditions for the algebraic integrability of the systems under consideration. It is proved that Siegel's linearization theorem is applicable if the eigenvalues of the map at a fixed point are algebraic numbers.     

Compensation spectrale et taux de décroissance optimal de l'énergie de systèmes partiellement amortis

We study the stabilization of systems of two equations, for which only one equation is damped by a feedback control. We show that a well chosen control can compensate the real parts of the eigenvalues of the system, therefore, giving the optimal polynomial energy decay rate of the system for smooth initial data. To cite this article: P.  Loreti, B. Rao, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Optimal m-stage Runge-Kutta schemes for steady-state solutions of hyperbolic equations

A numerical scheme is developed to find optimal parameters and time step of m-stage Runge-Kutta (RK) schemes for accelerating the convergence to -steady-state solutions of hyperbolic equations. These optimal RK schemes can be applied to a spatial discretization over nonuniform grids such as Chebyshev spectral discretization. For each m given either a set of all eigenvalues or a geometric closure of all eigenvalues of the discretization matrix, a specially structured nonlinear minimax problem is formulated to find the optimal parameters and time step. It will be shown that each local solution of the minimax problem is also a global solution and therefore the obtained m-stage RK scheme is optimal. A numerical scheme based on a modified version of the projected Lagrangian method is designed to solve the nonlinear minimax problem. The scheme is generally applicable to any stage number m. Applications in solving nonsymmetric systems of linear equations are also discussed. © 1993 John Wiley & Sons, Inc.     

Flight control system synthesis using eigenstructure assignment

The central topic of this paper is the establishment of an efficient practical synthesis procedure for modern flight control systems. Unlike the classical design methodology (Bode plots, Nichols plot, etc.) and optimal control techniques, the present approach provides the designer a direct approach for the synthesis of desired control laws. Although the setting is the now familiar state space, the actual design is performed relative to classical specifications (i.e., modes and mode distribution) by placing closed-loop eigenvalues and eigenvectors at some desired locations (regions) within the state space. The new method can handle output feedback configurations, subject to controller structural constraints. Complete theoretical background and a realistic numerical example are provided.     

Design of optimal observers with specified eigenvalues via shifted Legendre polynomials

The quadratic performance measure of estimation errors in approximated by using the Legendre polynomial approach for the design of optimal observers with specified distinct and multiple eigenvalues. This method is simple as compared with other design techniques of optimal observers. One example is illustrated, and only a small number (m=6) of shifted Legendre series are needed to produce a much better result than that obtained by the convenient block-pulse function.     

Numerical detection and study of singularities in solutions of differential equations

New simple and robust methods are proposed for detecting singularities, such as poles, logarithmic poles, and mixed singularities, in systems of ordinary differential equations. The methods produce characteristics of these singularities with an a posteriori asymptotically precise error estimate. They are applicable in the case of an arbitrary parametrization of integral curves, including one in terms of the arc length, which is optimal for stiff and ill-conditioned problems. Following this approach, blowup solutions can be detected for a broad class of important nonlinear partial differential equations, since they are reducible by the method of lines to systems of ordinary differential equations of huge orders. The simplicity and reliability of the approach are superior to those of previously known methods.     

粘性阻尼线性振动系统的复模态特征值问题的一种新的矩阵摄动分析解法

本文提出了对粘性阻尼线性振动系统的复模态二次广义特征值问题进行高效近似求解的一种新的矩阵摄动分析方法,即先将阻尼矩阵分解为比例阻尼部分和非比例阻尼部分之和,并求得系统的比例阻尼实模态特征解;然后以此为初始值,将阻尼矩阵的非比例部分作为对其比例部分的小量修改,利用摄动分析方法简捷地得到系统的复模态特征值问题的近似解.这一新方法适用于振系阻尼分布不十分偏离比例阻尼情况的问题,因此对大阻尼(非过阻尼)振动系统也有效.这是它优于以前提出的基于无阻尼实模态特征解的类似摄动分析方法的重要特点.文中建立了复模态特征值和特征向量的二阶摄动解式,并通过算例证实了其有效性.此外还讨论了利用比例阻尼假定估计阻尼系统固有振动的复特征值的可行性.     

On the general theory of exact controllability for skew symmetric operators

We introduce a general formalism for linear evolution equations with skew adjoint operators. We make explicit the controllability operator as an expansion with respect to eigenfunctions. Using the fact that the eigenvalues are purely imaginary, we give sufficient controllability conditions. This approach is convenient for studying the asymptotic behaviour of the optimal control.     

On the Order Reduction Approach for Singularly Perturbed Optimal Control Systems

The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to be zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed problem with a small parameter. In this paper we establish that if relaxed controls are allowed then the answer to the question whether or not this method is valid depends essentially on one simple parameter: the dimension of the fast variable, denoted n. More specifically, if n=1 then the order reduction method is indeed applicable, while if n>1 then the set of singularly perturbed optimal control systems for which it is not applicable is dense (in the L norm).     

Numerical diagnostics of solution blowup in differential equations

New simple and robust methods have been proposed for detecting poles, logarithmic poles, and mixed-type singularities in systems of ordinary differential equations. The methods produce characteristics of these singularities with a posteriori asymptotically precise error estimates. This approach is applicable to an arbitrary parametrization of integral curves, including the arc length parametrization, which is optimal for stiff and ill-conditioned problems. The method can be used to detect solution blowup for a broad class of important nonlinear partial differential equations, since they can be reduced to huge-order systems of ordinary differential equations by applying the method of lines. The method is superior in robustness and simplicity to previously known methods.     

Asymptotics of the optimal time in a time-optimal control problem with a small parameter

A time-optimal control problem for a singularly perturbed linear autonomous system is considered. The main difference between this case and the case of systems with fast and slow variables studied earlier is that the eigenvalues of the matrix at the fast variables do not satisfy the standard requirement of negativity of the real part. We obtain and justify a complete power asymptotic expansion in the sense of Erdélyi of the optimal time and optimal control in a small parameter at the derivatives in the equations of the system.     

A simple method of chaos control for a class of chaotic discrete-time systems

In this paper, a simple method is proposed for chaos control for a class of discrete-time chaotic systems. The proposed method is built upon the state feedback control and the characteristic of ergodicity of chaos. The feedback gain matrix of the controller is designed using a simple criterion, so that control parameters can be selected via the pole placement technique of linear control theory. The new controller has a feature that it only uses the state variable for control and does not require the target equilibrium point in the feedback path. Moreover, the proposed control method cannot only overcome the so-called “odd eigenvalues number limitation” of delayed feedback control, but also control the chaotic systems to the specified equilibrium points. The effectiveness of the proposed method is demonstrated by a two-dimensional discrete-time chaotic system.