Controllability of product systems

For nonlinear continuous-time control systems we introduce the notion of controllability with selectable time, which is a little bit stronger than the usual controllability notion. For a large class of nonlinear systems on compact manifolds, controllability with selectable time is equivalent to controllability together with strong accessibility. It turns out that the controllability of products of independent control systems is related to this new controllability notion. The main result states that a smooth control affine product system is controllable if and only if all factors are controllable and at most one factor is not controllable with selectable time, provided the accessibility algebra of the product system has full rank.     

On the controllability of a nonlinear control system

This paper considers the problem of controllability of a class of nonlinear systems. Sufficient conditions are given for a nonlinear system to be locally controllable and globally completely controllable. In the case of a linear time-invariant system, these conditions are also necessary. Furthermore, our result reveals that, for a nonlinear control system, there is a relationship among controllability, stability and optimality.     

Homomorphisms and controllability of parallel composite systems

The concept of a homorphism is introduced to study the local weak controllability of parallel composite analytic systems. If two systems are simultaneously homomorphic to a nontrivial system, then their parallel composition is not locally weakly controllable. The converse, however, is not true. It is this negative aspect that shows the complexity of nonlinear systems in contrast with the linear ones.     

Output feedback design and controllable cascade connections of nonlinear systems

For linear control systems necessary and sufficient conditions have been established for transforming a linear multiinput multioutput controllable system to a single input controllable system by use of linear output feedback. This problem is studied here for nonlinear systems affine in control, extending previous work of the authors dealing with the more restricted case of state feedback. The results obtained are subsequently applied to analyze controllability of cascade connections of nonlinear systems. A notable consequence of the development presented here, is that for linear systems the above questions can be answered under milder conditions than those required by the linear theory, provided that nonlinear output feedback is allowed.     

Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearisation

The concepts of local controllability and observability of nonlinear discrete-time systems with the Caputo-, Riemann–Liouville- and Grünwald–Letnikov-type h-difference fractional order operators are studied. The Implicit Function Theorem is used in order to show that the nonlinear systems are locally observable or controllable in a finite number of steps if their linear approximations are observable or controllable, respectively, in the same number of steps.     

Controllability of Mechanical Systems in the Class of Controls Bounded Together with Their Derivatives

A series of studies concerned with controllability of the nonlinear dynamic mechanical systems such as manipulators, aircraft, or water-craft was continued. For the class of smooth controls which comprises swing-bounded controls with bounded derivatives, a criterion for controllability of mechanical systems was established in addition to the criterion for controllability in the class of bounded controls that was established earlier. The controllability criterion (necessary and sufficient conditions) was formulated for a set of mechanical systems, rather than for a particular system as usual. Stated differently, conditions for controllability of any system from the set were established. The elements of this set are represented by the systems obeying the Lagrange equations of the second kind and having also bounded inertial characteristics and generalized forces. The established controllability conditions have an obvious physical sense. For example, for a manipulation robot to be controllable, it is required that the control forces dominate over any other generalized forces such as weight or environmental resistance.Domination is necessary both in the amplitude of forces and rate of their variation, that is, the derivative.     

On the relation between local controllability and stabilizabilityfor a class of nonlinear systems

The problem of local stabilizability of locally controllable nonlinear systems is considered. It is well known that, contrary to the linear case, local controllability does not necessarily imply stabilizability. A class of nonlinear systems for which local controllability implies local asymptotic stabilizability using continuous static-state feedback is described, as for this class of systems the well-known Hermes controllability condition is necessary and sufficient for local controllability     

一类齐次离散双线性系统可控性分析

在系统分析中,可控性是系统的一个重要特性.在工程实际操作中,往往需要对一个连续系统进行离散化处理,人们希望系统在离散化后能保留原系统的重要系统特征,比如可控性.对于线性系统,我们有成熟的判断方法.然而,对于非线性系统则无统一的判别方法.Elliott在2005年给出了一个二阶双线性系统经过离散化后,可控性发生变化的例子.它表明一个系统在离散化前后,它的可控性可能会发生改变.本文旨在给出一类二阶离散化双线性系统可控性充分条件,并和已有结果作比较,表明本文结果更具有一般性.另外,本文对于3阶及以上的这类系统可控性做出了不可控的判断.     

感应电动机系统的能控性与能观测性分析

分析了感应电动机系统的能控性与能观测性。结果表明系统是弱能控与弱能观的。而且,采用非线性状态反馈解耦控制和非线性坐标变换后的局部闭环解耦线性系统是能控与能观的。     

对称非线性控制系统的能控性

本文采用微分几何方法,研究了具有对称结构的非线性控制系统的能控性。本文证明:若对称非线性控制系统在某点能控,那么该系统在此点所属的轨道上的所有点均能控。     

Controllability for single-input mechanical control systems with dissipation

Within the affine connection framework of Lagrangian control systems , based on the results of Sussmann on small- time locally controllability of single- input affine nonlinear control systems , the controllability results for mechanical control systems with single- input are extended to the case of the systems with isotropic damping ,where the Lagrangian is the kinetic energy associated with a Riemannian metric. A sufficient condition of negative small- time locally controllability for the system is obtained. Then ,it is demonstrated that such systems are small- time locally configuration controllable if and only if the dimension of the configuration manifold is one. Finally , two examples are given to illustrate the results. Lie bracketting of vector fields and the symmetric product show the advantages in the discussion.     

Controllability for single-input mechanical control systems with dissipation

Within the affane connection framework of Lagrangian control systems, based on the results of Sussmann on small-time locally controllability of single-input affine nonlinear control systems, the controllability results for mechanical control systems with single-input are extended to the case of the systems with isotropic damping, where the Lagrangian is the kinetic energy associated with a Riemannian metric, A sufficient condition of negative small-time locally controllability for the system is obtained.Then,it is demonstrated that such systems are small-time locally configuration controllable if and only if the dimemion of the configuration manifold is one. Finally, two examples are given to illustrate the results. Lie bracketting of vector fields and the symmetric product show the advantages in the discussion.     

DC-DC变换器在恒功率负载下的能控性

带有恒功率负载的DC-DC变换器是切换非线性系统.以Boost变换器为例,本文建立了其切换系统模型;它包括三个模态.模态I,II,III分别对应于开关器件导通、关断、电感电流断续时的电路拓扑.因为电容电压是非负的,所以本文对平面拓扑学的最新结果加以修正,导出了半平面仿射非线性系统全局能控的充分必要条件——判别函数在任意非零控制曲线上变号.据此证明了模态II的全局能控性.由于模态II为运转中的必经阶段,故切换系统能控.研究方法也适用于Buck等变换器的能控性分析.     

Approximate controllability of fractional stochastic evolution equations

A class of dynamic control systems described by nonlinear fractional stochastic differential equations in Hilbert spaces is considered. Using fixed point technique, fractional calculations, stochastic analysis technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations is formulated and proved. In particular, we discuss the approximate controllability of nonlinear fractional stochastic control system under the assumptions that the corresponding linear system is approximately controllable. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result. Finally as a remark, the compactness of semigroup is not assumed and subsequently the conditions are obtained for exact controllability result.     

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.     

Exact and approximate feedback linearization without the linear controllability assumption

The feedback linearization problem of nonlinear control systems has been solved in the literature under the assumption that the nonlinear system is linearly controllable. In this paper, the assumption of linear controllability is removed; necessary and sufficient conditions are given through Lie orbital symmetries, thus giving a geometric characterization of the problem in the analytic case. Both the exact and the approximate linearization problems are considered in the analytic case.     

Spacecraft attitude control and stabilization: Applications of geometric control theory to rigid body models

In this paper, we study the attitude control problem for spacecraft with gas jet or momentum exchange actuators, using the recent nonlinear geometric control theory. We give necessary and sufficient conditions for controllability of the system in the case that the gas jet actuators yield one, two, or three independent torques. In the case of momentum exchange devices, controllability is studied with three independent actuators, and controllability is shown to be impossible with fewer devices. The former conditions with gas jet actuators are presented in three equivalent ways, and an equivalence is established with an earlier condition by Baillieul. The local controllability problem is also studied in the case of gas jet actuators yielding two independent torques. Using these results, an algorithm stabilizing the controllable system around an equilibrium state and trajectory is outlined, as proposed by Hermes. In the situations considered, however, the linearized systems are not controllable.     

Practical stabilization of driftless systems on Lie groups: the transverse function approach

A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system's controllability. Its outcome is the practical stabilization of any trajectory, i.e., not necessarily a solution of the control system, in the state-space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given.     

高阶系统方法—Ⅱ.能控性与全驱性

本文首先简述了基于状态空间模型的一阶动态系统的能控性进展, 指出了一阶系统方法中卡尔曼能控性体系的一些问题.然后证明了线性定常系统能控的充要条件是它能化成一个高阶全驱系统, 同时还在一定程度上将这一结果推广到非线性系统的情形.基于这一发现, 本文定义了一般动态系统的完全能控性, 明确其意义在于存在控制律使得闭环系统为一线性定常的高阶系统, 并且可以任意配置闭环特征多项式的系数矩阵, 同时还指出其多方面相关结论.     

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.