基于径向基函数网络的非线性离散时间系统的自适应控制

对于一类离散时间的非线性系统x(k+1)=f(x(k))+u(k)+d(k),当系统中的非 线性函数f(x(k))满足线性增长条件时,首先证明了{x(k)}落入一紧集中,然后根据高斯径 向基函数网络的逼近性质,给出了自适应控制器的设计方法.利用李亚普诺夫稳定性理论,证 明了控制算法是全局稳定的,跟踪误差收敛于零的某一领域中.     

关于《一类非线性时滞系统的稳定化控制器设计研究》一定理的讨论

文献[1]将精确线性化的方法应用到一类时滞非线性系统稳定化控制器设计中,提出一种新的稳定化控制器设计方法,其思路是可取的.但是我认为其中的定理1有误,下面提出一家之言与作者商讨.考虑单输入非线性时滞系统x=f(x) g(x)u(t-τ),(1)其中x∈Rn,u∈R,f(.),g(.)为C∞非线性向量场;τ为时滞,f(0)=0.同时引入线性时滞系统w=Aw bu(t-τ),(2)其中w=[w1…wn]T∈Rn为新的状态变量A=01…0?0……0,　b=0?1.(3)　　原定理为定理1.对于非线性时滞系统(1),通过微分同胚变换w=T(x)将其转化成线性时滞系统(3)的充分必要条件是(i)rankM(x)=n,其中M(…     

微分几何方法与非线性控制系统（4）

7 非线性控制系统的几何理论 7.1 非线性控制系统的几何描述 定义在R~n上的一般非线性系统可以用微分方程描述如下: x=f(x,4) (7.1a) y=h(x) (7.1b)这里,状态x∈R~n;控制u∈R~m;输出y∈R~r,等式(7.1a)称为状态方程,(7.1b)称为输出方程。为了便于使用微分几何的工具,假设f对x和u都是光滑或解析函数,同样,h(x)也是C~∞     

弱化缓冲算子性质研究

在灰色系统缓冲算子公理体系下,证明了下列结果:若d是一弱化缓冲算子,则x(k)d是由x(k)···x(n)所构成的表达式.f为严格单调递增函数,g为其反函数.在d中,用f(x(k))替换x(k)d,对得到的表达式用函数g加以作用,最后的表达式记为e.若d为弱化缓冲算子,则e也为弱化缓冲算子.     

对偶函数的研究

首次明确定义了反映二元域GF(2)上序列之间平移跨距关系的对偶函数,研究了基于二元域GF(2)上多项式f(x)对偶函数的存在性和若干性质,证明了当且仅当二元域GF(2)上多项式f(x)为本原多项式时才存在相应多项式下的对偶函数,给出了便于对偶函数计算的3条性质.最后作为对偶函数的一个应用,证明了m序列状态与共轭状态跨距之分布的均匀性.     

一般非线性微分方程边值问题的六阶三对角差分法

在文中介绍了一种满足边界条件y(a)=A和y(b)=B的一般二阶非线性微分方程y″=f(x,y,y′)的六阶三对角差分法,并证明了本方法是六阶收敛的。     

一族迭代格式的Kanforovich型收敛定理和最佳渐近误差

本文建立了由King提出一族迭代格式[1]■的Kantorovich型收敛定理,并在下列两个条件(ⅰ) Kantorovich型收敛定理对参数的限制:0≤β≤2; (ⅱ) 一类函数:f(x)≡{f(x)|f″(x)~2-f′(x)f″(x)≥0}下证明了Ostrowski-Traub迭代格式[2],[3]■具有最小的渐近误差。     

应用无源性分析时变非线性系统的稳定性

应用无源性分析研究时变非线性系统的稳定性.通过引进一个非线性复合微分算子dI(x)=k(x)s和一个时变非线性复合微分算子dLI(x,t)=k(x,t)sk(x,t),讨论了基于无源系统理论的时变非线性系统的稳定性分析.这里s=d/dt为普通的微分算子,x为所研究系统的状态变量.应用复合微分算子,构造出了一类严格无源的时变非线性系统,进一步给出了相应系统的渐近稳定条件.     

简单前馈结构控制系统的镇定

讨论前馈结构系统 x1=Ax1+g(x2, u), x2 =f(x2, u)的反馈镇定, 应用前向设计方法给出了当系统 x2=f(x2, u)可以反馈镇定的时候, 整个系统反馈镇定的条件. 在设计中实现了输入_状态增益的最终线性, 以及状态的收敛性. 文章在此基础上探讨了应用复杂前馈结构系统的可能性.     

具有线性结构的弹性函数非线性度的新上界

讨论了具有线性结构的弹性函数的两个指标:沃什谱和非线性度,得到了具有线性结构的布尔函数的一些性质.利用沃尔什变换和汉明重量的方法,发现了:如果V是n元布尔函数,f(x)的线性结构,那么得到f(x)的沃尔什变换在为零这一事实,同时得到了一个布尔函数没有k(k≥0)维线性结构的充分条件.最后,利用以上结果推出了具有线性结构的弹性函数的非线性度的上界表达式.     

Invertibility of multivariable nonlinear control systems

This paper gives sufficient conditions for the invertibility of nonlinear control systems of the formdot{x}= A(x)+ u_{1}B_{1}(x) +... + u_{m} B_{m}(x);y=c(x), where the state space is a real analytic manifold. These conditions are also necessary in the case of single-input nonlinear systems and multivariable time-invariant linear systems. For invertible systems we construct nonlinear inverse systems.     

On nonlinear reinforcement schemes

This paper deals with the analysis of nonlinear reinforcement schemes for learning automata. The learning automaton is connected in feedback loop to a random environment. The correction term of the action probability vector depends on a nonlinear function φ(x). Results concerning the convergence, the convergence rate, and the effect of the function φ(x) are stated. A comparison between the convergence rate of nonlinear and linear reinforcement schemes is presented     

On multi-input chattering-free second-order sliding mode control

A solution to the problem of eliminating the chattering effect, which is always associated with practical implementations of variable structure control, is presented with reference to a class of uncertain multi-input nonlinear systems. The solution procedure relies on the application of an original control approach capable of enforcing a second-order sliding mode (i.e., a sliding regime on a surface s[x(t)]=0 in the system state space, with s˙[x(t)] identically equal to zero, a regime enforced by a control signal depending on s[x(t)], but directly acting only on s¨[x(t)]). Such an approach, in its original formulation, only applies to single-input nonlinear systems with particular types of uncertainties. In the present paper, its validity is extended to multi-input nonlinear systems characterized by uncertainties of more general nature, covering a wide class of real processes     

H∞-control of discrete-time nonlinear systems

This paper presents an explicit solution to the problem of disturbance attenuation with internal stability via full information feedback, state feedback, and dynamic output feedback, respectively, for discrete-time nonlinear systems. The H_∞-control theory is first developed for affine systems and then extended to general nonlinear systems based on the concepts of dissipation inequality, differential game, and LaSalle's invariance principle in discrete time. A substantial difficulty that V(A(x)+B(x)u+E(x)w) [respectively, V(f(x,u,w))] is no longer quadratic in [_w^u] arising in the case of discrete-time nonlinear systems has been surmounted in the paper. In the case of a linear system, we show how the results reduce to the well-known ones recently proposed in the literature     

Lyapunov function of a fourth-order system

The Lyapunov functionVand its time derivativedot{V}are expressed in matrix form byx'Sxandx'Tx, respectively, whereSandTcontain elements which involve the state variables, andx'is the transpose ofx. A given fourth-order nonlinear system is characterized bydot{x}=A(x)x, whereA(x)contains nonlinear elements. Simanov's problem is extended to a fourth-order system whose nonlinearity is a constrained function of two state variables.     

A new method for the control of discrete nonlinear dynamic systems using neural networks

A new controller design method for nonaffine nonlinear dynamic systems is presented in this paper. An identified neural network model of the nonlinear plant is used in the proposed method. The method is based on a new control law that is developed for any discrete deterministic time-invariant nonlinear dynamic system in a subregion Psi(x), of an asymptotically stable equilibrium point of the plant. The performance of the control law is not necessarily dependent on the distance between the current state of the plant and the equilibrium state if the nonlinear dynamic system satisfies some mild requirements in Psi(x). The control law is simple to implement and is based on a novel linearization of the input-output model of the plant at each instant in time. It can be used to control both minimum phase and nonminimum phase nonaffine nonlinear plants. Extensive empirical studies have confirmed that the control law can be used to control a relatively general class of highly nonlinear multiinput-multioutput (MIMO) plants.     

The existence and uniqueness of Volterra series for nonlinear systems

Given an input-output map described by a nonlinear control systemdot{x}=f(x,u)and nonlinear outputy=h(x), we present a simple straightforward means for obtaining a series representation of the outputy(t)in terms of the inputu(t). When the control enters linearly,dot{x} =f(x)+ ug(x), the method yields the existence of a Volterra series representation. The proof is constructive and explicitly exhibits the kernels. It depends on standard mathematical tools such as the Fundamental Theorem of Calculus and the Cauchy estimates for the Taylor series coefficients of analytic functions. In addition, the uniqueness of Volterra series representations is discussed.     

Losslessness, feedback equivalence, and the global stabilization ofdiscrete-time nonlinear systems

In this paper a necessary and sufficient condition for a nonlinear system of the form Σ, given by x(k+1)=f(x(k))+g(x(k))u(k), y(k)=h(x(k))+J(x(k))u(k), to be lossless is given, and it is shown that a lossless system can be globally asymptotically stabilized by output feedback if and only if the system is zero-state observable. Then, we investigate conditions under which Σ can be rendered lossless via smooth state feedback. In particular, we show that this is possible if and only if the system in question has relative degree {0,...,0} and has lossless zero dynamics. Under suitable controllability-like rank conditions, we prove that nonlinear systems having relative degree {0,...,0} and lossless zero dynamics can be globally stabilized by smooth state feedback. As a consequence, we obtain sufficient conditions for a class of cascaded systems to be globally stabilizable. The global stabilization problem of the nonlinear system Σ without output is also investigated in this paper by means of feedback equivalence. Some of the results are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear     

导出变系数非线性动力学系统拉格朗日函数的两种方法

利用从运动微分方程出发和从第一积分出发导出拉格朗日函数的两种直接方法,构造变系数非线性动力学系统¨x+b(x)x~2+c(x)x=0的拉格朗日函数和c(x)=0特殊情况的拉格朗日函数族.另外,讨论了这种非保守系统广义能量守恒的物理意义.     

非线性切换系统的集合稳定性分析

将输出对状态稳定和状态模可观测等概念进行推广.提出了输出对V(x)稳定、小时间y(z)可观测、大时间V(x)可观测的定义.利用上述定义,给出了非线性切换系统的V(x)零值集稳定的充分条件.分别利用统一Lyapunov函数和多Lyapunov函数证明了所提出的结论,并详细讨论了输出对V(x)稳定与输出对状态稳定及其他相关定义之间的关系.数值例子验证了所提出结论的正确性.