同时镇定问题的一个数值解法*

本文运用多项式稳定性充分判据，将不同阶次线性系统的同时镇定问题化成非线性不等式组的求解，然后提出求解不等式组的拟牛顿下山数值法，并应用于同时镇定问题中，算例表明该方法的有效性。     

线性系统的同时镇定问题

线性系统的同时镇定问题,是系统与控制理论中的基本问题,有着广泛的理论意义和应用价值.本文介绍了线性系统同时镇定问题的研究现状和最新进展.首先回顾了同时镇定问题的研究内容、基本方法及相关结果.其次,从理论求解和控制器设计的角度对线性系统同时镇定研究中著名的"香槟问题","比利时巧克力问题"和"威士忌问题"进行了分析探讨,并基于不等式型定理机器证明理论,给出了线性系统同时镇定问题的相关工作.最后指出了对于同时镇定多个线性系统方面的若干研究方向.     

严格正则多输入多输出对象同时镇定

使用逆LQ方法讨论了r个严格正则多输入多输出对象的同时镇定问题,基于矩阵 不等式方法得到了静态输出反馈可同时镇定的充要条件,本文证明,r个对象静态输出反馈同 时镇定等价于r个耦合LQ控制问题的解.然后,基于迭代线性矩阵不等式技术给出了一种 迭代求解方法,并给出了算例.     

一类含未建模动态的奇异时滞系统的鲁棒镇定问题

利用线性矩阵不等式的方法,研究了一类同时含有参数摄动和未建模动态的线性奇异时滞系统的鲁棒镇定问题.得到了系统可鲁棒镇定的一个充分条件,用线性矩阵不等式的方法,将控制器的求解问题转化为受限线性矩阵不等式的求解问题,并给出了受限线性矩阵不等式的具体解法．最后举例说明了所提出方法的正确性．     

基于LMI的仿射型模糊控制系统的稳定性研究

基于凸优化方法，分析了仿射型模糊控制系统的稳定性与镇定问题。首先以LMI的形式给出了仿射型模糊系统稳定的充分条件；然后基于这一条件讨论了闭环系统的镇定问题。得出的镇定条件是BMI，使用轮换极小法将这一BMI问题转化为2个LMI问题，从而可以用MATLAB中的LMI优化工具箱进行求解。仿真例子说明了所提方法的有效性。     

一类结构不确定性的离散时滞大系统的分散镇定

研究了一类离散时滞大系统的时滞相关鲁棒分散镇定问题, 系统的不确定与标称系统具有结构相似性. 通过构造性的差分格式, 将时滞相关鲁棒分散镇定控制器的存在条件转化为一列线性矩阵不等式的可行解问题. 提出了控制时滞及控制器的时滞鲁棒性概念, 在获得分散镇定控制器的同时给出了鲁棒时滞上界的求解, 因此比现有文献的结果更具有实际应用价值.     

基于迭代线性矩阵不等式的奇异摄动系统同时镇定

研究了采用一个线性状态反馈控制器镇定多个线性奇异摄动系统的问题.同时镇定条 件可以表达为一组矩阵不等式条件,所得条件与摄动参数无关,从而有效地回避了病态问题.采 用基于快慢分解的两步法可以得到镇定控制器增益和相应的Lyapunov函数.而在每一步需要利 用迭代线性矩阵不等式技术求解相应的双线性矩阵不等式,相关定理研究了算法的收敛性.本文 所得结论可同时适用于标准与非标准奇异摄动系统.文末给出了相应的仿真算例.     

基于互质分解的同时镇定控制器参数化

本文基于稳定互质分解介绍了所有分散镇定控制器的参数化方法，给出一种同时镇定控制器的参数化方法，并得到同时镇定问题的解的参数化结果，即由一广义对象的控制器的分散和同时结构约束导出了参数需满足的一系列二次方程的约束条件。     

同时镇定：逆LQ方法

本文使用逆ＬＱ方法了同时镇定问题，首先得到了ｒ个正则ＭＩＭＯ对象可静态输出反馈同时定的充要条件，然后给出了状态反馈问题同时镇定的一种解，这些条件均以一组耦合Ｒｉｃｃａｔｉ方程和Ｒｉｃｃａｔｉ不等式的形式给出，本语文证明，任一同时镇定反馈增益均可描述为一组具有合适交叉项的耦合ＬＱ控制问题的解。     

同阶线性系统在Kharitonov空间的同时镇定

运用Kharitonov定理给出一个期望的闭环特征区间多项式,然后将同阶线性系统 的同时镇定问题化成一组线性不等式的求解,并给出求解该不等式组的线性规划法,讨论了 三种解的意义和性质.算例表明该方法的有效性.     

Finite‐time robust simultaneous stabilization of a set of nonlinear time‐delay systems

This paper investigates the finite‐time robust simultaneous stabilization problem of a set of nonlinear time‐delay systems with general forms and proposes some new simultaneous stabilization results. First, by developing an equivalent form and applying augmented technique, this paper obtains an augmented equivalent form of the original systems. Secondly, based on the equivalent form, we study finite‐time simultaneous stabilization problem and present some new stabilization results by constructing some suitable Lyapunov functionals. Thirdly, using the simultaneous stabilization results obtained, this paper investigates the finite‐time robust simultaneous stabilization problem for the set systems and proposes a delay‐dependent robust simultaneous stabilization result. Finally, the study of an illustrative example shows that the results obtained by this paper work well in the finite‐time robust simultaneous stabilization the set systems. It is shown that, by using the method in this paper, the developed conditions do not contain delay terms, which can avoid solving nonlinear mixed matrix inequalities and reduce effectively computational burden in studying nonlinear time‐delay systems.     

一类非光滑优化及其在控制系统稳定化中的应用

研究一类来自控制系统稳定化中的非光滑优化问题．考虑Lyapunov函数是非光滑的,特别是有限个光滑函数的极大值函数．建立了相应的非光滑优化模型,进一步导出了这类非光滑优化的KKT系统,然后基于非线性互补函数将KKT系统转化成一个非光滑方程组,最后分别用广义牛顿法和光滑化牛顿法求解此非光滑方程组。使得此类稳定化设计可以具体实现．     

A lagrange approach to set-theoretic control synthesis

A Lagrange approach to solving the nonlinear constrained optimization problem arising in the set-theoretic control problem is described. By introducing matrix Lagrange multipliers, the problem is reduced to that of solving a set of nonlinear simultaneous matrix equations, one of which is the familiar matrix Riccati equation frequently encountered in linear-quadratic control theory. The structural similarities and differences between set-theoretic and linear-quadratic control methods are identified. The results obtained from the set-theoretic control approach are compared with those obtained from the linear-quadratic control approach.     

A functional equations approach to nonlinear discrete-time feedback stabilization through pole-placement

The present work proposes a new approach to the nonlinear discrete-time feedback stabilization problem with pole-placement. The problem's formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived. Using tools from functional equations theory, one can prove that the solution to the above system of nonlinear functional equations is locally analytic, and an easily programmable series solution method can be developed. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete-time state feedback control law that are both computed through the solution of the system of nonlinear functional equations, the feedback stabilization with pole-placement design objective can be attained under rather general conditions. The key idea of the proposed single-step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach. However, since the proposed method does not involve an external reference input, it cannot meet other control objectives such as trajectory tracking and model matching.     

Acceleration-constrained time-optimal control inndimensions

The minimum-time acceleration of a generalized point mass from an initial position and velocity to the origin inndimensions is solved by transforming the problem to an equivalent problem in two dimensions and analytically integrating the system differential equations. Computation of the optimal control is thereby reduced to the solution of five simultaneous nonlinear equations. A numerical continuation method is presented for solving these equations by starting at the known solution of a related single-dimensional problem and progressing incrementally to the desired solution. The problem and solution method are illustrated by a numerical example.     

Simulation package for small-scale systems

A simulation package for microcomputers has been developed which is applicable to small- and medium-size problems in the educational and industrial environment. The package consists of: an ordinary differential equations simulator capable of solving up to 12 simultaneous first-order ordinary differential equations; a simultaneous linear and nonlinear equations solver capable of solving systems containing up to 12 linear and nonlinear equations; an extended polynomial/nonlinear curve fitting program capable of fitting a variety of equations to a set of data points of Y_i versus X_i; and an extended multiple-linear-regression program which can fit mathematical models containing up to five linear parameters to experimental data.The paper describes this simulation package, demonstrates some typical applications, and summarizes recent experience in using the package in an educational environment.     

Semi‐Global Robust Output Regulation for A Class of Singular Nonlinear Systems with Unknown Algebraic Equations

This paper considers semi‐global robust output regulation problem for a class of singular nonlinear systems whose algebraic equations are not precisely known. Since the algebraic equations are not known, the output regulation problem of singular nonlinear systems cannot be solved by directly reducing the singular nonlinear system into a normal nonlinear system. Based on internal model principle, we convert the robust output regulation problem of singular nonlinear systems into a robust stabilization problem of an augmented singular nonlinear system. The augmented singular nonlinear system is also with unknown algebraic equations. However, without transforming the singular nonlinear system into a normal nonlinear system, it is shown that the augmented singular nonlinear system can be semi‐globally stabilized by a high gain output feedback control law under some reasonable assumptions. Moreover, the semi‐global stabilization control law of the augmented singular nonlinear systems also solves the semi‐global robust output regulation problem of the original singular nonlinear system.     

Simultaneous stabilization for a set of Boolean control networks

This paper investigates simultaneous stabilization of a collection of Boolean control networks (BCNs) by using the semi-tensor product method, and presents a number of new results. First, an algebraic expression of the BCNs is obtained by the semi-tensor product, based on which some necessary and sufficient conditions are presented to solve the simultaneous stabilization problem by a free control sequence, a state-feedback control, and an output-feedback control, respectively. Second, using the column stacking form of matrices, a new procedure is established to design both state-feedback and output-feedback controllers for the simultaneous stabilization problem. The study of two illustrative examples shows that the new results obtained in this paper are very effective in solving simultaneous stabilization of a collection of BCNs.     

Neural-network-based approximate output regulation of discrete-time nonlinear systems.

The existing approaches to the discrete-time nonlinear output regulation problem rely on the offline solution of a set of mixed nonlinear functional equations known as discrete regulator equations. For complex nonlinear systems, it is difficult to solve the discrete regulator equations even approximately. Moreover, for systems with uncertainty, these approaches cannot offer a reliable solution. By combining the approximation capability of the feedforward neural networks (NNs) with an online parameter optimization mechanism, we develop an approach to solving the discrete nonlinear output regulation problem without solving the discrete regulator equations explicitly. The approach of this paper can be viewed as a discrete counterpart of our previous paper on approximately solving the continuous-time nonlinear output regulation problem.     

离散化微分模型方程的反应器网络综合

构造基于全混流和交叉流反应器的反应器网络超结构,并建立优化模型,优化模型由复杂的高度非线性的微分／代数方程所构成。本文采用有限元正交配置法离散化微分方程的策略来简化超结构数学模型,将离散化所得的代数方程组与其它约柬条件一起,作为反应器网络超结构的数学模型,然后运用数学软件优化求解。实例研究表明,优化结果与文献相一致或优于文献,表明离散化法求解含有微分模型方程的反应器网络综合问题是有效的。