Asymptotic feedback stabilization: A sufficient condition for the existence of control Lyapunov functions

In this paper we study the feedback stabilization problem for a wide class of nonlinear systems that are affine in the control. We offer sufficient conditions for the existence of ‘Control Lyapunov functions’ that according to [3,23] and [28–30] guarantee stabilization at a specified equilibrium by means of a feedback law, which is smooth except possibly at the equilibrium. We note that the results of the paper present a local nature.     

Asymptotic stabilization with locally semiconcave control Lyapunov functions on general manifolds

Asymptotic stabilization on noncontractible manifolds is a difficult control problem. If a configuration space is not a contractible manifold, we need to design a time-varying or discontinuous state feedback control for asymptotic stabilization at the desired equilibrium.     

Constructive stabilization for quadratic input nonlinear systems

In this paper stabilization of nonlinear systems with quadratic multi-input is considered. With the help of control Lyapunov function (CLF), a constructive parameterization of controls that globally asymptotically stabilize the system is proposed. Two different cases are considered. Firstly, under certain regularity assumptions, the feasible control set is parameterized, and continuous feedback stabilizing controls are designed. Then for the general case, piecewise continuous stabilizing controls are proposed. The design procedure can also be used to verify whether a candidate CLF is indeed a CLF. Several illustrative examples are presented as well.     

Backstepping control of flexible joint manipulator based on hyperbolic tangent function with control input and rate constraints

In this paper, we investigate the trajectory tracking problems of the link angle and angle speed of the flexible joint manipulator model based on external disturbance, the control input and rate constraints. The controller of the flexible joint manipulator model is designed using the backstepping control scheme. To achieve this objective, the smooth hyperbolic tangent function is used to solve the problems of control input and rate constraints, and the stability is proved using Lyapunov function in the design procedure of the backstepping control scheme. Finally, the effectiveness of the proposed backstepping controller is verified by numerical simulation.     

基于多输入时滞脉冲模型的采样数据控制系统的随机镇定

针对具有非均匀采样的采样数据控制系统,把区间内连续分布的采样间隔序列描述为多元独立同分布随机过程,把闭环系统转化为多输入时滞脉冲模型,通过构造合适的非连续Lyapunov泛函,以及合理地利用在所有采样间隔内输入时滞的时间导数等于1的条件,结合自由权矩阵方法推导了基于LMIs的全局均方渐近稳定性条件,在此基础上运用调节因子法和锥补线性化方法,把控制器设计转化为具有LMI约束的非线性优化问题,并给出了基于LMIs的迭代求解算法.数值实例和实验表明了所得理论结果的优越性和有效性.     

基于鲁棒控制Lyapunov 函数的非线性预测控制

针对一类约束不确定性非线性仿射系统,提出一种可保证闭环系统鲁棒镇定的非线性模型预测控制算法.利用鲁棒控制Lyapunov函数得到改进的Sontag公式,并以此为基础,构造一种计算有效的单自由度鲁棒预测控制器.以Matlab语言为仿真工具,对一开环不稳定振荡器进行了仿真研究,结果表明,利用该控制算法得到的闭环系统不仅渐近稳定于原点,而且所得控制量和系统状态都满足系统约束,从而验证了控制算法的有效性.     

Global stabilization of nonlinear cascaded systems with a Lyapunov function in superposition form

The global stabilization problem of nonlinear cascaded systems has been well studied in literature. In particular, a Lyapunov function in superposition form has been explicitly constructed for the closed-loop system in a recent paper provided the nonlinearities are polynomial. This paper removes this polynomial assumption and gives a more general result. For this purpose, a special version of changing supply function technique is utilized which preserves the superposition form of supply functions during the “changing” procedure.     

Control Lyapunov functions for adaptive nonlinear stabilization

For the problem of stabilization of nonlinear systems linear in unknown constant parameters, we introduce the concept of an adaptive control Lyapunov function (aclf) and use Sontag's constructive proof of Artstein's theorem to design an adaptive controller. In this framework the problem of adaptive stabilization of a nonlinear system is reduced to the problem of nonadaptive stabilization of a modified system. To illustrate the construction of aclf's we give an adaptive backstepping lemma which recovers our earlier design.     

Global robust stabilization of nonlinear systems subject to input constraints

Our main purpose in this paper is to further address the global stabilization problem for affine systems by means of bounded feedback control functions, taking into account a large class of control value sets: p, r ‐weighted balls ??^m _r (p), with 1<p?∞, defined via p, r ‐weighted gauge functions. Observe that p=∞ is allowed, so that m‐dimensional r ‐hyperboxes ??^m _r (∞)?[?r₁^?,r₁⁺]×???×[?r_m^?,r_m⁺], r_j^±>0 are also considered. Working along the line of Artstein–Sontag's approach, we construct an explicit formula for a one‐parameterized family of continuous feedback controls taking values in ?? _r ^m(p) that globally asymptotically stabilize an affine system, provided an appropriate control Lyapunov function is known. The designed family of controls is suboptimal with respect to the robust stability margin for uncertain systems. The problem of achieving disturbance attenuation for persistent disturbances is also considered. Copyright © 2002 John Wiley & Sons, Ltd.     

具有输入未建模动态的纯反馈非线性系统自适应控制

对一类具有状态和输入未建模动态且控制增益符号未知的纯反馈非线性系统,利用非线性变换、改进的动态面控制方法以及Nussbaum函数性质,提出两种自适应动态面控制方案.利用正则化信号来约束输入未建模动态,从而有效地抑制其产生的扰动.通过引入动态信号,有效地处理了由状态未建模动态引起的动态不确定性.通过在总的李雅普诺夫函数中引入非负正则化信号,并利用稳定性分析中引入的紧集,证明了闭环控制系统是半全局一致终结有界的.数值仿真验证了所提方案的有效性.     

Decentralized measurement feedback stabilization of large-scale systems via control vector Lyapunov functions

This paper studies the problem of decentralized measurement feedback stabilization of nonlinear interconnected systems. As a natural extension of the recent development on control vector Lyapunov functions, the notion of output control vector Lyapunov function (OCVLF) is introduced for investigating decentralized measurement feedback stabilization problems. Sufficient conditions on (local) stabilizability are discussed which are based on the proposed notion of OCVLF. It is shown that a decentralized controller for a nonlinear interconnected system can be constructed using these conditions under an additional vector dissipation-like condition. To illustrate the proposed method, two examples are given.     

Continuous simultaneous stabilization of single‐input nonlinear stochastic systems

This paper investigates the simultaneous stabilization of a collection of continuous single‐input non‐linear stochastic systems, with coefficients that are not necessarily locally Lipschitz. A sufficient condition for the existence of a continuous simultaneously stabilizing feedback control is proposed — it is based on the generalized stochastic Lyapunov theorem and on the technique of stochastic control Lyapunov functions. This condition is also necessary, provided that the system's coefficients satisfy some regularity conditions. Moreover, the proposed feedback can be chosen to be bounded under the assumption that appropriate control Lyapunov functions are known. All the proposed simultaneously stabilizing state feedback controllers are explicitly constructed. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed approach.     

Robust global stabilization of linear systems with input saturation via gain scheduling

The problem of robust global stabilization of linear systems subject to input saturation and input‐additive uncertainties is revisited in this paper. By taking advantages of the recently developed parametric Lyapunov equation‐based low gain feedback design method and an existing dynamic gain scheduling technique, a new gain scheduling controller is proposed to solve the problem. In comparison with the existing ?₂‐type gain scheduling controller, which requires the online solution of a state‐dependent nonlinear optimization problem and a state‐dependent ?₂ algebraic Riccati equation (ARE), all the parameters in the proposed controller are determined a priori. In the absence of the input‐additive uncertainties, the proposed controller also partially recovers Teel's ?_∞‐type scheduling approach by solving the problem of global stabilization of linear systems with actuator saturation. The ?_∞‐type scheduling approach achieves robustness not only with non‐input‐additive uncertainties but also requires the closed‐form solution to an ?_∞ ARE. Thus, the proposed scheduling method also addresses the implementation issues of the ?_∞‐type scheduling approach in the absence of non‐input‐additive uncertainties. Copyright © 2009 John Wiley & Sons, Ltd.     

Simultaneous stabilization of a collection of single‐input nonlinear systems based on CLFs

This paper considers the simultaneous stabilization problem of a collection of single‐input nonlinear systems. Based on the technique of control Lyapunov functions (CLFs), a sufficient condition for the existence of a simultaneously stabilizing state feedback controller is proposed. It is shown that a collection of feedback linearizable systems in canonical form can be simultaneously globally asymptotically stabilized by a single state feedback controller. Moveover, for a set of three‐order chaotic dynamical systems, the simultaneous stabilization problem is considered and a similar result is derived. All the proposed simultaneously stabilizing state feedback controllers are explicitly constructed. Numerical examples are provided to illustrate the effectiveness of the proposed schemes. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

基于障碍Lyapunov函数的未知控制方向非线性系统的约束鲁棒输出调节

本文研究一类不确定非线性系统的约束鲁棒输出调节问题. 在具有未知控制输入方向的情况下, 要求被控线性系统的受控输出在趋近于零的同时, 被限制在给定的范围内. 文章结合障碍Lyapunov函数和Nussbaum增益技术, 进行反馈控制器的设计. 通过MATLAB仿真验证了控制算法的合理性. 同时, 与基于二次型Lyapunov函数设计的仿真结果进行对比, 表明了控制算法的有效性     

基于障碍Lyapunov 函数的输出有界全局收敛鲁棒控制

为了克服基于障碍Lyapunov函数反演控制方法在输出约束控制中约束变量收敛区间小和要求模型精确已知的缺陷,提出一种输出有界且全局收敛的鲁棒控制策略。将收敛区间扩展到全局,并使输出保持在约束区间内;同时消除了由未知干扰和模型不确定性引起的稳态误差,放宽了对指令信号连续可导的限制。应用于电动舵机位置伺服控制的仿真结果表明,舵面能够全局稳定且摆幅有界,稳态无静差,动态响应快,抗扰动能力强。     

基于分段Lyapunov 函数的Hammerstein-Wiener 非线性预测控制

针对输入和输出受约束的Hammerstein-Wiener型非线性系统,建立T-S模糊模型,并提出一种基于分段Lyapunov函数的非线性预测控制算法.通过构造分段二次Lyapunov函数,分析非线性系统的稳定性,降低普通二次Lyapunov函数的保守性;通过离线设计分段反馈控制律,在线实施符合条件的反馈控制律,极大程度地提高了在线计算效率.仿真结果验证了该方法的有效性.     

Lyapunov stabilization of the nonlinear control systems via the neural networks

In this paper, we have used a neural network to obtain an approximate solution to the value function of the HJB (Hamilton–Jacobi–Bellman) equation. Then, we have used it to stabilize the affine control nonlinear systems. The requisite control input is generated as the output of a neural network, which is trained off-line. We have designed two various neural networks, in which learning algorithm in the first one it is the steepest descent method, and in the second one is in its unconstrained optimization method. The proposed methods are compared with the traditional methods, and sometimes our methods are proved to be more efficient than the traditional methods. Numerical examples indicate the effectiveness of the proposed neural networks.     

Semiglobal stabilization of saturated linear systems via multiple parametric Lyapunov equations

This paper investigates the problem of semiglobal stabilization with guaranteed flexible pole placement for saturated linear systems. To retain the advantages of the parametric Lyapunov equation, matrix‐partitioning idea is used to derive a new pole shift lemma. Starting from system matrix transformations, a recursive algorithm is proposed to shift every eigenvalue of a linear system separately without mode decomposition in each step. A new method introducing various parameters to every Lyapunov equation in each step is presented. As an application, the semiglobal stabilization with guaranteed flexible pole placement for saturated linear systems can be achieved by this method. Finally, its effectiveness and advantages are demonstrated via a simulation example. Copyright © 2013 John Wiley & Sons, Ltd.