非线性约束下非完整系统的平衡稳定性

Kozlov将Liapunov第一方法推广到非线性力学系统,用来解决保守和耗散力场中,运动力学系统平衡位置的不稳定性.文中讨论的系统运动限于理想的非线性非完整约束.将势能和约束函数展开为Maclaurin级数,对其第一非平凡多项式的阶,确定了相互间关系的5种情况,并对生成的非线性非完整约束方程进行了分析.将3种线性齐次约束下的非完整系统平衡位置的不稳定定理(Kozlov,1986),推广到非线性非完整约束.另外两种情况下的新定理,也是将Kozlov(1994)的结果,拓展到非线性约束下的非完整系统.     

肌型血管生物数学模型的非线性状态反馈同步

用非线性状态反馈方法研究肌型血管生物数学模型的同步问题,利用驱动系统与响应系统的误差信号,通过施加反馈控制,设计一个单变量非线性反馈控制器,使响应系统跟踪驱动系统,即误差系统渐近稳定到平衡点,从而实现了两个混沌系统在单变量驱动的情况下,消耗更少的能量达到同步,从理论上验证了处于痉挛状态下的血管运动可以与正常血管运动同步,为有效防治和治疗心肌梗塞等疾病提供一定的理论依据.仿真结果表明了此方法的有效性.     

粘弹性板混沌振动的输出变量反馈线性化控制

研究了粘弹性板混沌振动的控制问题·　应用非线性系统精确线性化控制理论导出了一类非仿射控制系统的非线性反馈控制律·　建立了描述材料非线性的粘弹性板运动的数学模型并利用Ｃａｌｅｒｋｉｎ 方法进行简化·　采用相空间曲线和频率谱密度函数说明了在特定参数条件下系统将出现混沌振动,并以位移为输出变量将混沌振动控制为给定的周期运动·     

两自由度非线性振动系统周期运动及其稳定性研究

运用Liapunov函数方法,对一类两自由度非线性振动系统周期运动及其稳定性进行了研究,得到了存在唯一渐近稳定的周期解的充分条件.     

用精确线性化控制Lorenz混沌

本文采用状态空间精确线性化方法研究Ｒａｙｌｅｉｇｈ数可控的Ｌｏｒｅｎｚ系统中混沌的控制问题·在证明系统可以精确线性化的基础上，借助非线性反馈构造出变换关系使原系统转化为线性可控系统而实现控制，并给出了控制算例     

非线性振动系统周期解的数值分析

用直接数值积分法求非线性振动系统的周期解,求解时对初始条件进行迭代,使它与终点条件相一致.积分时间区间(即周期)或运动方程中的某些参数,也可在迭代过程中随同变化,积分方法是变步长的. 用这种“打靶”法求周期解,所需计算工作量相对较少.其中误差主要来源于数值积分,故不难估计并控制它足够小.这种方法可处理各种类型的振动问题,如单自由度和多自由度系统的自由无阻尼振动、强迫振动、自激振动和参数振动等等;也能求得不稳定解和那些对参数变动十分敏感的解.解的稳定性根据相关的周期系数微分方程来研究.求共振曲线或其他振动特性曲线时,利用插值方法并自动调节步长来定出迭代始值. 为了阐明这种方法的通用性,计算了若干例子.非线性的描述可用解析函数或任何其他形式,例如分段线性函数.文中还就所得周期解指出了非线性振动的一些值得注意的性质.部分计算结果与已有的近似解或实验结果作了比较.     

路面方向扰动下重型汽车的横向稳定性及分岔、混沌运动

针对三轴重型汽车建立了二自由度非线性人-车-路闭环模型,考虑驾驶员控制和路面方向扰动,推导了系统动力学方程．在运用Hopf分岔理论进行分析的基础上,以临界车速为评价指标,通过数值模拟研究了轴距、预瞄距离、载重量、驾驶员控制时滞和轮胎侧偏刚度对转向稳定性的影响,并确定了转向系统的数值稳定范围．另外,还通过分岔图、时程曲线、相轨线、功率谱、Poincaré图和Lyapunov指数研究了不同车速下汽车的非线性动力学响应．结果表明,随着车速的增加汽车可能发生周期运动、拟周期运动及混沌运动,汽车的横向稳定性与车辆和驾驶员参数密切相关.     

一类时变仿射非线性系统的动态状态反馈线性化的条件

本文用微分几何方法讨论了一类时变仿射非线性系统（即ｎ＝ｍ＋１）的动态状态反馈线性化与系统的线性近似的关系，得出了系统可动态状态反馈线性化的条件。     

具有控制输入约束的轮式移动机器人轨迹跟踪最优保性能控制

针对考虑模型不确定性因素影响及控制输入约束的轮式移动机器人的轨迹跟踪问题,文章提出了一种具有控制输入约束的轨迹跟踪最优保性能控制器.依据实际机器人与虚拟机器人的相对位姿关系建立轨迹跟踪误差动态模型.采用平衡点线性化方法将非线性形式的轨迹跟踪误差动态模型转换为线性形式的轨迹跟踪误差动态模型.通过线性矩阵不等式处理方法,给出了具有鲁棒性能上界的最优保性能控制器的存在条件及设计方法.仿真及实验结果表明,文章所提控制器能较好抑制模型不确定性的影响,提高移动机器人的轨迹跟踪性能和鲁棒性.     

关于非线性控制系统的线性化问题

本文首先研究了一般形式的非线性控制系统,得到了可以用状态反馈线性化时应具有的形式,即系统方程与控制规律一般条件下都应是对控制变量是线性的。然后研究了用状态变换进行线性化的问题,得到了变换应满足的条件。最后讨论了同时用状态变换与反馈变换对非线性控制系统进行线性化以及综合问题。     

Robust Stability and Stabilization of a Class of Nonlinear Itô-Type Stochastic Systems via Linear Matrix Inequalities

This article presents a new approach to robust quadratic stabilization of nonlinear stochastic systems. The linear rate vector of a stochastic system is perturbed by a nonlinear function, and this nonlinear function satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of stochastic systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The new formulation provides a suitable setting for robust stabilization of nonlinear stochastic systems where the underlying deterministic systems satisfy the generalized matching conditions. Our sufficient conditions are written in matrix forms, which are determined by solving linear matrix inequalities (LMIs), which have significant computational advantage over any other existing techniques. Examples are given to demonstrate the results.     

离散交通流模型的反馈线性化与拥堵控制

在自动化高速公路环境下,提出一种改进的宏观离散交通流模型密度控制方法.利用反馈线性化方法,将宏观离散交通流模型转换为一般容易处理的线性系统模型,简化了密度控制器的设计.利用线性系统中具有输入变换的跟踪反馈控制方法,对线性化后的系统模型设计控制律.通过控制该线性系统的状态变量,间接稳定离散交通流模型中的交通流密度,达到对道路交通流拥堵的控制.同时给出设计方法和步骤,仿真实例说明了方法的实用性.     

Template-Based Stabilization of Relative Equilibria in Systems with Continuous Symmetry

We present an approach to the design of feedback control laws that stabilize relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. We present two different approaches of control design. The first approach assumes that the closed loop system is affine in the control and that the actuation is equivariant. We derive feedback laws for the reduced system that minimize a quadratic cost function. The second approach is more general; here the actuation need not be equivariant, but the actuators can be translated in the symmetry direction. The controller resulting from this approach leaves the dynamics associated with the symmetry variable unchanged. Both approaches are simple to implement, as they use standard tools available from linear control theory. We illustrate the approaches on three examples: a rotationally invariant planar ODE, an inverted pendulum on a cart, and the Kuramoto-Sivashinsky equation with periodic boundary conditions.     

Modal exact linearization of a class of second-order switched nonlinear systems

This paper considers the problem of stabilizing single-input affine switched nonlinear systems. The main idea is to transform a switched nonlinear system to an equivalent controllable switched linear system. First, we define the notion of modal state feedback linearization. Then, we develop a set of conditions for modal state feedback linearizability of a certain class of second order switched nonlinear systems. Considering two special structures, easily verifiable conditions are proposed for the existence of suitable state transformations for modal feedback linearization. The results are constructive. Finally, the method is illustrated with two examples, including a Continuous Stirred Tank Reactor (CSTR) to demonstrate the applicability of the proposed approach.     

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.     

Linearization of affine systems based on control-dependent changes of independent variable

To transform single-input affine systems into linear control systems, we suggest to use control-dependent changes of independent variable. We show that the use of such changes of variables in conjunction with feedback linearization enables one to linearize systems to which known linearization methods do not apply. We prove that a linearizing change of independent variable can be found by solving a system of partial differential equations. The approach developed in the paper is applied to the construction of solutions of terminal problems.     

Invariants for feedback equivalence and Cauchy characteristic multifoliations of nonlinear control systems

Linearization of a nonlinear feedback control system under nonlinear feedback is treated as a problem of equivalence-under the Lie pseudogroup of feedback transformations-of distributions on the product manifold of the state and control variables. The new feature of this paper is that it introduces the Cauchy characteristic sub-distributions of these distributions and their derived distributions. These Cauchy characteristic distributions are involutive and nested, hence define a Multifoliate Structure. A necessary condition for feedback equivalence of two nonlinear control systems is that these multifoliations be transformed under the feedback pseudogroup. For linear systems, this Cauchy characteristic multifoliate structuee is readily computed in terms of the (A, B)-matrix that defines the linear system. Assuming that the conditions for local feedback linearization are satisfied, the existence of a global feedback linearizing transformation is dependent on computing an element of the first cohomology group of the space with coefficients in the sheaf of groupoid of infinitesimal feedback automorphisms of the linear system. The theorem quoted above about the Cauchy characteristic multifoliations provides some information about this groupoid. It is computed explicitly and directly for control systems with one- or two-state dimensions. Finally, these Cauchy characteristic sub-distributions must inevitably play a role in the numerical or symbolic computational analysis of the Hunt-Su partial differential equations for the feedback-linearizing transformation.Senior Research Associate of the National Research Council at the Ames Research Center of NASA.     

Stabilization of stochastic singular nonlinear hybrid systems

This paper deals with the control of the class of singular nonlinear stochastic hybrid systems. Under some appropriate assumptions, results on stochastic stability and stochastic stabilization are developed. Two state feedback controllers (linear and nonlinear) that stochastically stabilize the class of systems we are considering are designed. LMI sufficient conditions are developed to compute the gains of these controllers.     

The explicit synthesis of stabilizing (time optimal) feedback controls for the attitude control of a rotating satelite

The attitude stabilization problem for a spinning satellite controlled by two small jets may be modelled as a four-dimensional, nonlinear control system, linear in the controls. The recent feedback linearization theorem of Hunt and Su may be applied to transform this system, via state feedback and a local coordinate change, to a pair of uncoupled, two-dimensional, linear systems. Feedback controls for the problem of time optimal transfer to the origin for these linear systems are explicitly calculated and then transformed to give explicit feedback controls for time optimal stabilization in the original nonlinear problem. The theory is illustrated by sample calculations.     

A new approach to the incorporation of servo constraints in multibody dynamics

A new index reduction approach is developed to solve the servo constraint problems [2] in the inverse dynamics simulation of underactuated mechanical systems. The servo constraint problem of underactuated systems is governed by differential algebraic equations (DAEs) with high index. The underlying equations of motion contain both holonomic constraints and servo constraints in which desired outputs (specified in time) are described in terms of state variables. The realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Since the (differentiation) index of the DAEs is often higher than three for underactuated systems, in which the number of degrees of freedom is greater than the control outputs/inputs, we propose a new index reduction method [1] which makes possible the stable numerical integration of the DAEs. We apply the proposed method to differentially flat systems, such as cranes [1,4,5], and non-flat underactuated systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)