Remarks on nonlinear planar control systems which are linearizable by feedback

The possibility that certain nonlinear systems can be linearized by the nonlinear feedback group has recently attracted a great deal of attention in the literature. At least for mechanical systems, the apparent ubiquity of linearizable systems is often accounted for by the claim that scalar control systems defined on R² are generically linearizable. Since there are several possible interpretations of the term ‘generic’ and since, until now, no rigorous proof of genericity for any use of the term has been offered, I thought it worthwhile to investigate in what sense, if any, this ‘folklore theorem’ is valid. Contrary to the prevailing intuition, linearizable systems fail to be dense in any topology for which evaluation of vector fields at points is a continous operation. This includes, for example, both the weak and strong C^∞-topologies.What, then, are the global properties of the class of linearizable systems? In this note, it is also shown that this class has no interior points in any reasonable, complete topology making feedback operations continuous. In particular, linearizable systems in R² are far from being structurally stable in the weak C^∞-topology. In the other hand, one of the main results presented here is that globally linearizable systems are open in the strong, or Whitney, topology. Thus, globally linearizable planar systems are structurally stable. If the well-known local conditions actually implied global linearizability, this result would be trivial. Recent work of Boothby shows, however, that global linearization is far more subtle. For this reason, in the course of the proof it is necessary to modify the existing global linearization criteria. In particular, a ‘nonvanishing’ criterion for irjectivity of smooth maps on R″, sharpening the so-called ‘ratio condition’, is presented. This result appears to be of independent interest and leads to improvements of the global linearization condition in R″, due to Hunt, Su and Meyer, and of the global non-linear observability conditions in R″, due to Kou, Elliott and Tarn.     

Global output feedback stabilisation for a class of upper triangular stochastic nonlinear systems

This paper investigates the problem of global output feedback stabilisation for a class of upper triangular stochastic nonlinear systems which are neither necessarily feedback linearisable nor affine in the control input. Based on the adding of a power integrator technique and homogeneous domination approach, an output feedback controller is explicitly constructed to ensure that the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability. A simulation example is provided to demonstrate the effectiveness of the design scheme.     

基于调节函数的一类三角结构非线性系统的自适应滑模控制

针对一类三角结构的非线性系统,基于状态参考自适应控制算法和滑模控制技术,研究了其在非匹配未知参数和不确定性干扰下的跟踪控制问题,提出了自适应滑模控制策略,实现了不确定非线性系统的鲁棒输出跟踪.与一般自适应控制相比,允许系统存在非参数化的不确定性和未知扰动,增强了控制系统鲁棒性.仿真算例证明了理论研究成果的正确性和可行性.     

On the optimal control of nonlinear systems

The Lie algebra of tensors on a Hilbert space is used to obtain optimal controls for a class of nonlinear systems.     

一类高阶非线性不确定系统自适应稳定控制

研究了一类高阶非线性不确定性系统的自适应稳定控制设计问题.因该系统的非线性程度高,其控制系数不等同、符号已知、但数值未知,故在此之前其稳定控制设计问题没有得到解决.本文应用自适应技术,结合设计参数的适当选取,从而得到了设计该类非线性系统状态反馈稳定控制器的新方法,并基于反推技术,给出了稳定控制器的设计步骤.所设计的状态反馈控制器使得闭环系统的状态全局渐近收敛于零,其余闭环信号一致有界.最后通过一个仿真例子说明了控制设计方法的有效性.     

Receding horizon control of nonlinear systems without differentiability of the optimal value function

In this paper, we show that receding horizon control, when applied to a class of nonlinear systems, has stabilizing properties. Our assumptions are necessarily fairly strong because they guarantee implicity that a smooth stabilizing feedback can be constructed. This is an extension of our previous results which require the finite horizon value function to be continuously differentiable and therefore applies to a more restricted class of system.Continuous differentiability of the value function is replaced by local Lipschitz continuity; Dini derivatives and a generalization of the Invariance Principle are employed to establish stability.     

-bounded robust control of nonlinear cascade systems

In this paper, we consider the L^∞-bounded robust control problem for a class of nonlinear cascade systems with disturbances. Sufficient conditions are provided under which a hard bound is imposed on the system performance measure. The backstepping approach is used for controller design. A practical example is provided to illustrate the method.     

一类非线性系统的无超调控制

针对一类严格反馈非线性系统,利用后推法设计一种无超调跟踪控制律.对于阶数小于4的对象,给出了控制律参数应满足的充要条件;对于阶数更高的对象,给出了求解一个充分条件的方法.该控制律适用于零与非零初始条件,且参考输出不限制为阶跃信号.通过两个数值仿真例子验证了所提出控制律的有效性.     

Observer-based control of a class of time-delay nonlinear systems having triangular structure

Time-delay systems constitute a special class of dynamical systems that are frequently present in many fields of engineering. It has been shown in the literature that the existence of a stabilizing observer-based controller is related to delay-dependent conditions that are generally satisfied for a small time delay. Motivating works towards reducing the conservatism of the results are among the on-going research topics especially when partial-state measurements are imposed. This paper investigates the problem of observer-based stabilization of a class of time-delay nonlinear systems written in triangular form. First, we show that a delay nonlinear observer is globally convergent under the global Lipschitz condition of the system nonlinearity. Then, it is shown that a parameterized linear feedback that uses the observer states can stabilize the system whatever the size of the delay. An illustrative example is provided to approve the theoretical results.     

On the convergence of nonlinear optimal control using pseudospectral methods for feedback linearizable systems

We consider the optimal control of feedback linearizable dynamic systems subject to mixed state and control constraints. In contrast to the existing results, the optimal controller addressed in this paper is allowed to be discontinuous. This generalization requires a substantial modification to the existing convergence analysis in terms of both the framework as well as the notion of convergence around points of discontinuity. Although the nonlinear system is assumed to be feedback linearizable, the optimal control does not necessarily linearize the dynamics. Such problems frequently arise in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. We prove that a sequence of solutions obtained using the Legendre pseudospectral method converges to the optimal solution of the continuous‐time problem under mild conditions. Published in 2007 by John Wiley & Sons, Ltd.     

一类非线性系统的奇异点克服控制

针对一类非线性系统, 研究存在奇异点时的跟踪控制问题. 在采用反馈线性化方法将对象转换成标准型后, 构造线性补偿器并结合期望轨迹的高阶导数构成伪控制量. 通过引入梯度动力学方法求解控制律, 以克服在控制过程中遇到的奇异点问题. 通过稳定性分析验证了闭环系统的稳定性和跟踪误差的收敛性. 仿真结果表明, 此类控制器具有良好的控制性能, 并且能有效克服奇异点问题.

     

Decentralised adaptive control for a class of stochastic switched interconnected nonlinear systems

In this paper, we focus on the problem of adaptive stabilisation for a class of interconnected uncertain switched stochastic nonlinear systems. Classical adaptive and backstepping technique are employed for control synthesis. Instead of estimating the switching parameters directly, we design the adaptive controller based on the estimations of bounds on switching time-varying parameters. A smooth function is introduced to deal with the difficulties caused by unknown interactions and tuning function approach is used to circumvent the overparameter problem. It is shown that under the action of the proposed controller all the signals of the overall closed-loop systems are globally uniformly bounded in probability under arbitrary switching. Simulation results are presented to illustrate the effectiveness of the proposed approach.     

一类非线性多时滞系统的干扰解耦控制

考虑了一类MIMO非线性多时滞系统的干扰解耦控制问题.在一定条件下,通过构造非线性状态反馈控制规律,确保闭环系统的输出不受干扰影响而且是时滞无关的.基于微分几何理论,给出该状态反馈控制规律存在的充分必要条件.仿真验证了所得结果的有效性.     

Output transient trajectory shaping control for a class of nonlinear systems

This paper presents a transient trajectory shaping (TTS) control method for the SISO strict feedback nonlinear systems. The TTS control refers to explicitly constraining the system output tracking error transient trajectories within predesigned time‐varying boundaries while they are converging to equilibrium. By this method, the boundaries of system output transient trajectories can be designed a priori according to the system transient control performance requirements in both symmetric and asymmetric ways. With a class of time‐varying boundary functions, control laws can be devised by utilizing the enhanced differential unbounded function techniques. Such control laws can ascertain that the system output tracking errors travel within their respectively predesigned time‐varying boundaries while converging to origin. To handle the control input exaggeration issue in TTS, input constraint control techniques are proposed to effectively reduce the required control input magnitude for second‐order systems. A numerical example is utilized to show the effectiveness of the proposed TTS control methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class of finite-dimensional, linear, single-input (u), single-output (y) systems (A,b,c) is universally stabilized by the feedback strategy u = Λ(λ)y, λ = y², where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class ^k of nonlinear control systems (a,b,c), with positively homogeneous (of degree k 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ(λ)/vby/vb^k−1y, assures ^k-universal stabilization. More generally, the strategy u = Λ(λ)exp(/vby/vb)y, assures -universal stabilization, where = _{k 1} ^k.     

On the controllability of nonlinear systems with symmetry

Sufficient and necessary conditions are given for the controllability of nonlinear systems possesing symmetry.     

Observer-based fuzzy adaptive fault control for a class of MIMO nonlinear systems

In this paper, the fault-tolerant control (FTC) problem is investigated for a class of multi-input multiple output nonlinear systems with time-varying delays, and an active FTC method is proposed. The controlled system contains unknown nonlinear functions, unknown control gain functions and actuator faults, which integrates time-varying bias and gain faults. Then, fuzzy logic systems are used to approximate the unknown nonlinear functions and unknown control gain functions, fuzzy adaptive observers are used for fault detection and isolation. Further, based on the obtained information, an accommodation method is proposed for compensating the actuator faults. It is shown that all the variables of the closed-loop system are semi-globally uniformly bounded, the tracking error converges to an arbitrary small neighbourhood of the origin. A simulation is given to demonstrate the effectiveness of the proposed approach.     

一类非线性系统镇定问题的分层控制方法

通过利用分层控制设计方法,研究了一类仿射非线性系统的镇定问题.为了镇定这类系统,分层方法构建了一个简单的抽象系统,然后根据抽象系统的控制律求得原始系统的控制律.最后,一个四维仿射非线性系统的仿真例子说明了这种方法的有效性.     

Robust adaptive control of a class of nonlinear uncertain systems

A smooth robust dynamic feedback controller is constructed, and the problem of robust H∞ almost disturbance attenuation with internal stability is solved for high-order nonlinear systems with parameter uncertainties. Finally, illustrative example and simulation results demonstrate the effectiveness of the proposed method.