一类离散双线性系统可控性研究

在工程中, 系统离散化前后可控性是否一致是一个重要问题. 近来, Elliott就一类双线性系统, 首次构造出了一个连续时不可控、但经离散化后可控的二阶反例. 鉴于离散化后的系统可控性无法利用已有方法进行判断, 本文给出了一个二阶离散双线性系统可控性的充分条件, 从而在更一般情形下得到了一类连续时系统不可控, 离散化后可控的反例, 深化了对双线性系统可控性的认识. 进而, 本文证明了对于该类系统, 当其阶数大于2时, 可控性反例不再存在.     

双线性系统可控性综述

双线性系统是一类特殊的非线性系统,广泛存在于现实世界中,如工程、经济、生物、生态等领域,被认为是最接近于线性系统的非线性系统．对双线性系统的研究已历经了近半个世纪. 作为系统最基本的属性,双线性系统可控性的研究一直以来是热点和难点.本文分别对连续双线性系统可控性和离散双线性系统可控性进行讨论, 综述了双线性系统可控性的研究.特别地,报告了近来对离散双线性系统可控性研究的新成果.最后,例举了一些可控的双线性系统例子.     

离散广义双线性系统的稳定控制

针对离散广义双线性系统,研究了稳定控制存在的条件,给出了相应的控制方案。所采用的方法是利用广义系统分解,选择适当的控制,运用Lyapunov方法证明线性状态控制可以使闹环系统在原点附近渐近稳定。对于这一类离散广义双线性系统有关理论的证明,可以看作是正常离散广义双线性系统的拓广。给出了一个可行性的应用算例,同时仿真结果也说明了该方法的有效性所得稳定性的判据对于选择合适的控制方案具有实际指导意义。     

离散双线性系统的稳定控制分析

对于一般离散双线性系统 ,本文提出了系统在线性反馈下的稳定性判据和最小吸引域 ,并给出了一种简单的控制方案 ,保证闭环系统在原点的一致渐近稳定性 .仿真结果证明了该方法的有效性 .所得稳定性判据对于选择合适的控制方案具有重要的指导意义 .     

对数字仿真方法的研究

本文提出并证明了一个定理,指出单步离散化数字仿真方法只能达到二阶精度。考虑到数字仿真离散化方法中现有方法是二阶的情况,提出了一类两步四阶精度的方法。在时域内对提出的方法的稳定性进行了分析,提出了稳定性的约束条件,给出了绝对稳定区间,最后给出了计算实例结果。     

对数字仿真方法的研究

本文提出并证明了一个定理,指出单步离散化数字仿真方法只能达到二阶精度。考虑到数字仿真离散化方法中现有方法是二阶的情况,提出了一类两步四阶精度的方法。在时域内对提出的方法的稳定性提出了分析,提出了稳定性的约束条件,给出了绝对稳定区间。最后给出了计算实例结果。     

一类双线性随机系统的辨识

本文对于一类具有规范形式的双线性、多变量、离散时间随机系统给出了确定结构指标和参数估计的递推方法及实现算法。首先导出了状态方程规范形等价的输入输出方程的一般形式,由此得到 Popov 规范形所对应的输入输出方程,然后借助于一、二阶相关     

快速离散化双线性插值算法

双线性插值算法在数字图像处理中有广泛的应用.它具有比最近邻点法更好的连续性,但计算速度慢.为提高其计算速度,提出了离散化双线性插值算法.把像素分割成子像素,用双线性插值函数计算子像素中心坐标处的像素值,可以得到每一个子像素的卷积函数C,用C来代替双线性插值函数.理论分析表明离散化算法减少了加法和乘法的浮点运算次数.通过图像旋转实验表明该算法与双线性插值算法相比计算速度有较大的提高,连续性略有损失,图像质量基本相同.     

高超声速飞行器基于特征模型的自适应姿控系统稳定性分析

文中研究了高超声速飞行器基于特征模型的全系数自适应姿态控制系统的稳定性问题,提出了一种通过计算有限个矩阵导出1范数或无穷范数来分析系统稳定性的方法,克服了特征模型时变和参数辨识给稳定性分析造成的困难.首先利用泰勒展开,获得高超飞行器姿态动力学输入输出描述的离散化方程.该组方程具有二阶特征模型的形式,可以分析出系统处于一个工作点附近时特征模型参数的取值范围.当使用线性反馈控制器,并引入一定保守性的情况下,可以将闭环系统看作一类由区间时变参数决定的时变对象.对于这样的一类对象,文中给出了其指数稳定的充要条件.最后利用前述结果,给出了判断高超声速飞行器基于特征模型的自适应控制系统稳定的充分条件,并通过一个算例来说明该方法的应用.     

一种分析和设计Sm ith 预估器的新方法及其应用

针对积分、一阶和二阶稳定与不稳定带长时滞的过程,给出了Smith预估器的分析和设计新方法,使其设定值的响应从负载干扰响应中分离出来,并用参数辨识的方法解决了在实际中难以得到过程精确的数学模型和控制器离散化的问题。模拟仿真和离散化系统仿真都达到了满意的效果,表明该方法是工程中很有前途的一种应用方法。     

On Near‐controllability of Discrete‐time Upper‐triangular Bilinear Systems with Applications to Controllability

Near‐controllability is defined for those systems that are uncontrollable but have a large controllable region. It is a property of nonlinear control systems introduced recently, and it has been demonstrated on two classes of discrete‐time bilinear systems. This paper studies near‐controllability of discrete‐time upper‐triangular bilinear systems, which are uncontrollable and are more general than the two classes mentioned. A necessary and sufficient condition for the systems in dimension two to be nearly controllable is presented, which covers the existing results. For the systems with high dimensions, necessary conditions and sufficient conditions of near‐controllability are provided, which generalize the existing results. In particular, the obtained near‐controllability results are applied to controllability of discrete‐time bilinear systems. An example also is given to demonstrate the effectiveness, which shows that the controllability problems of discrete‐time bilinear systems can be solved by near‐controllability.     

On controllability of a class of discrete‐time homogeneous bilinear systems with solvable controls

This paper studies the controllability of a class of discrete‐time homogeneous bilinear systems. Necessary and sufficient conditions for the controllability are obtained. In particular, an algorithm for computing the controls, which achieve given states transition, for the controllable systems is given with examples. Furthermore, the necessary and sufficient conditions are applied to present four cases of the change of the controllability for Euler discretization, which show that Euler discretization may and may not change the controllability of nonlinear systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Controllability of second-order infinite-dimensional systems

In the paper, the approximate controllability of linear abstract second-order infinite-dimensional dynamical systems is considered. It is proved using the frequency-domain method, that approximate controllability of second-order system can be verified by the approximate controllability conditions for the corresponding simplified first-order system. General results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure. Some special cases are also considered. Moreover, remarks and comments on the relationships between different concepts of controllability are given. The paper extends earlier results on approximate controllability of second-order abstract dynamical systems.     

The reachable set of a linear endogenous switching system

In this work, switching systems are named endogenous when their switching pattern is controllable. Linear endogenous switching systems can be considered as a particular class of bilinear control systems. The key idea is that both types of systems are equivalent to polysystems, i.e. to systems whose flow is piecewise smooth. The reachable set of a linear endogenous switching system can be studied consequently. The main result is that, in general, it has the structure of a semigroup, even when the Lie algebra rank condition is satisfied since the logic inputs cannot reverse the direction of the flow. The adaptation of existing controllability criteria for bilinear systems is straightforward.     

Implementation of multi-rate model reference adaptive control for continuous systems via bilinear transformation based on block-pulse functions

The discretization or approximation techniques for continuous systems using the well-known delta operator and the bilinear transformation based on block-pulse functions or the trapezoidal rule are discussed. Then implementation techniques of multi-rate indirect model reference adaptive control for continuous systems purely using digital computers arc described. The scheme is composed of three components: a general recursive least-squares type parameter estimator, a continuous plant model and a controller designed in continuous-time domain. To reduce the computational burden, the algorithm is implemented in a multi-rate manner with a small sampling interval of the system signals and a relatively large parameter estimation interval. Comparisons of the discretization methods for the adaptive system using block-pulse functions, the trapezoidal integrating rule and the well-known delta operator are discussed through simulation study. It is shown that the block-pulse function method is the most effective one.     

On controllability of discrete-time bilinear systems by near-controllability

In this paper, controllability of discrete-time bilinear systems is studied. By applying a recent result on near-controllability, a new sufficient condition for controllability of the systems is presented, where controllability is proved by approximation with near-controllability. The new condition is algebraically verifiable and is hence easy to apply compared with a classical result on controllability of discrete-time bilinear systems, which can be effective even when the classical result does not work. Furthermore, the control inputs to achieve the transition of the systems between any given pair of states are approximately computable according to near-controllability. Therefore, near-controllability can be used to not only better characterize the system properties, but also prove controllability with computable control inputs. The new condition is then generalized to derive similar results on controllability and near-controllability of the systems. Finally, examples are given to illustrate the results of this paper.     

Simultaneous control of a rod equation and a simple Schrödinger equation

We show that a corollary of a local approximate controllability result for the bilinear rod equation in [1] is that the controls which “steer” the modes of the rod equation also move the modes of a controlled Schrödinger equation. Specifying a target point for the Schrödinger system restricts but does not determine the outcome of the controlled rod equation. The local result is a special case of a general local result for hyperbolic systems, which is used in [1] to obtain a global approximate controllability result. After modifying the latter, we obtain a global result for the two systems.     

Frequency Interval Cross Gramians for Linear and Bilinear Systems

In many control engineering problems, it is desired to analyze the systems at particular frequency intervals of interest. This paper focuses on the development of frequency interval cross gramians for both linear and bilinear systems. New generalized Sylvester equations for calculating the frequency interval cross gramians are derived in order to be used to obtain information regarding controllability and observability within a single matrix. The advantage of the proposed method is that it is computationally more efficient compared to existing gramian‐based techniques since only half of the number of equations need to be solved in order to obtain information regarding the controllability and observability of a system compared to existing techniques. Numerical examples are provided to demonstrate the computational efficiency of the proposed method which uses frequency interval cross gramians relative to existing methods.