Multilayer Minimum Projection Method with Singular Point Assignment for Nonsmooth Control Lyapunov Function Design

Control Lyapunov function (CLF) design on a manifold is a difficult problem in control theory. To address this problem, we have proposed the multilayer minimum projection method. The method requires CLFs on different manifolds from the manifold where the control problem is defined. In this paper, we relax the requirement by desingularization of the functions on the manifolds. The paper focuses on the problem of desingularization in the multilayer minimum projection method. We show that the functions on other manifolds need not be CLFs by consideration of desingularization. Moreover, we propose a CLF design method by singular point assignment based on the advantage of desingularization. The method enables us to merge local CLFs into the global CLF. This paper proposes two CLF design methods: desingularization and singular point assignment. A CLF design example is provided for each method; the advantages of the proposed methods are confirmed by those two examples.     

一类非光滑优化及其在控制系统稳定化中的应用

研究一类来自控制系统稳定化中的非光滑优化问题．考虑Lyapunov函数是非光滑的,特别是有限个光滑函数的极大值函数．建立了相应的非光滑优化模型,进一步导出了这类非光滑优化的KKT系统,然后基于非线性互补函数将KKT系统转化成一个非光滑方程组,最后分别用广义牛顿法和光滑化牛顿法求解此非光滑方程组。使得此类稳定化设计可以具体实现．     

Feedback stabilization for a class of discontinuous systems driven by integrator

This paper investigates the feedback stabilization problem for a class of discontinuous systems which is characterized by Filippov differential inclusion. Lyapunov-based backstepping design method is generalized with nonsmooth Lyapunov functions to solve the control problem. A set-valued time derivative is introduced first for nonsmooth function along discontinuous vector fields, which enables us to perform Lyapunov-based design with nondifferentiable Lyapunov function. Conditions for designing a virtual control law which is shown nondifferentiable in general in the recursive design problem are proposed. Finally, as a special case, piecewise linear system is discussed to demonstrate the application of the presented design approach.     

Stabilization of nonlinear systems via the center manifold approach

This paper considers the problem of the stabilization of affine nonlinear control systems. First, we assume that the systems under investigation are of the generalized Byrnes–Isidori normal form. A new way to approximate the center manifold is proposed, which can reduce the error degree of the center manifold approximation. A new matrix product, called the semi-tensor product, is introduced to obtain the approximation of the center manifold. Then the Lyapunov function with homogeneous derivative (LFHD) is used to design a stable center manifold by state feedback control. Finally, the method is applied to general affine nonlinear control systems.     

Multiple robust H∞ controller design using the nonsmooth optimization method

This paper proposes a systematic technique to design multiple robust H_∞ controllers. The proposed technique achieves a desired robust performance objective, which is impossible to achieve with a single robust controller, by dividing the uncertainty set into several subsets and by designing a robust controller to each subset. To achieve this goal with a small number of divisions of the uncertainty set, an optimization problem is formulated. Since the cost function of this optimization problem is not a smooth function, a numerical nonsmooth optimization algorithm is proposed to solve this problem. This method avoids the use of Lyapunov variables, and therefore it leads to a moderate size optimization problem. A numerical example shows that the proposed multiple robust control method can improve the closed‐loop performance when a single robust controller cannot achieve satisfactory performance. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive fuzzy switched control design for uncertain nonholonomic systems with input nonsmooth constraint

In this paper, a fuzzy adaptive switched control approach is proposed for a class of uncertain nonholonomic chained systems with input nonsmooth constraint. In the control design, an auxiliary dynamic system is designed to address the input nonsmooth constraint, and an adaptive switched control strategy is constructed to overcome the uncontrollability problem associated with x₀(t₀) = 0. By using fuzzy logic systems to tackle unknown nonlinear functions, a fuzzy adaptive control approach is explored based on the adaptive backstepping technique. By constructing the combination approximation technique and using Young's inequality scaling technique, the number of the online learning parameters is reduced to n and the ‘explosion of complexity’ problem is avoid. It is proved that the proposed method can guarantee that all variables of the closed-loop system converge to a small neighbourhood of zero. Two simulation examples are provided to illustrate the effectiveness of the proposed control approach.     

The Exponential Stability for a Class of Hybrid Systems

The exponential stability with a nonsmooth Lyapunov function for a class of hybrid systems is studied in this paper. First, a sufficient condition is derived that has to be satisfied by the feedback control for the hybrid systems. Then, for the special case where the Lyapunov function involved is a kind of nonsmooth function, the maximum of finitely many smooth functions (for short, max‐type function), the stability of a hybrid system is considered, and a convenient criterion to determine the stability of the system is established. Finally, a numerical method of determining the control input value is developed.     

三阶系统族的共同二次Lyapunov函数

研究了寻找三阶系统族的共同二次Lyapunov函数问题.针对给定的稳定的三阶系统提出了寻找二次Lyapunov函数集的方法,然后获得了三阶系统族具有共同二次Lyapunov函数的充分条件.该充分条件易于构造,从而具有较强的工程实用性.文中实例验证了所得结果的有效性.     

基于l1-非凸非光滑函数的图像恢复方法

针对有整齐边界且加性噪声满足均匀、椒盐、拉普拉斯等非高斯分布的图像,根据逐步拟合及变量转换思想,提出一种基于l1-非凸、非光滑函数的逐步非凸图像恢复方法。根据原始非凸、非光滑函数建立一组拟合函数,获得原始问题的优良初始值,利用变量转换解决l1-非凸、非光滑函数的非可微性问题。实验结果表明,该方法恢复的图像效果较好,峰值信噪比较高。     

Manifold Ranking using Hessian Energy

In recent years, learning on manifolds has attracted much attention in the academia community. The idea that the distribution of real-life data forms a low dimensional manifold embedded in the ambient space works quite well in practice, with applications such as ranking, dimensionality reduction, semi-supervised learning and clustering. This paper focuses on ranking on manifolds. Traditional manifold ranking methods try to learn a ranking function that varies smoothly along the data manifold by using a Laplacian regularizer. However, the Laplacian regularization suffers from the issue that the solution is biased towards constant functions. In this work, we propose using second-order Hessian energy as regularization for manifold ranking. Hessian energy overcomes the above issue by only penalizing accelerated variation of the ranking function along the geodesics of the data manifold. We also develop a manifold ranking framework for general graphs/hypergraphs for which we do not have an original feature space (i.e. the ambient space). We evaluate our ranking method on the COREL image dataset and a rich media dataset crawled from Last.fm. The experimental results indicate that our manifold ranking method is effective and outperforms traditional graph Laplacian based ranking method.     

Stabilization of uncertain chained form systems within finite settling time

This note considers finite time stabilization of uncertain chained form systems. The objective is to design a nonsmooth state feedback law such that the controlled chained form system is both Lyapunov stable and finite-time convergent within any given settling time. We propose a novel switching control strategy with help of homogeneity, time-rescaling, and Lyapunov-based method. Also, the simulation results show the effectiveness of the proposed control design approach.     

Immersion and invariance adaptive control of linear multivariable systems

We show in this paper that it is possible to globally adaptively stabilize linear multivariable systems with reduced prior knowledge of the high-frequency gain. In particular, we relax the restrictive (nongeneric) symmetry condition usually required to solve this problem. Instrumental for the establishment of our result is the use of the new immersion and invariance approach to adaptive control recently proposed in the literature. The controllers obtained with this technique are not certainty equivalent—though smooth and without projections or overparameterizations—and the resulting Lyapunov functions contain cross-terms between the plant states and the parameter errors.     

基于量化依赖Lyapunov函数的有界丢包网络控制系统的保成本控制

研究了一类具有有界丢包的网络控制系统（Networked control systems，NCSs）的保成本控制问题，提出了一种包含量化反馈的网络控制系统数学模型，该模型将系统的镇定问题转化为镇定一系列子系统的鲁棒控制问题.在对网络控制系统的分析中，区别于常用的二次型Lyapunov函数，本文采用了一种新的且能够降低保守性的量化依赖Lyapunov函数方法.基于本文的Lyapunov函数，得到了充分考虑丢包过程的保成本控制器的设计方法.仿真算例验证了所给出方法的有效性.     

Robust stabilization of a class of uncertain system via block decomposition and VSC

In this paper, a block decomposition procedure for sliding mode control of a class of nonlinear systems with matched and unmatched uncertainties, is proposed. Based on the nonlinear block control principle, a sliding manifold design problem is divided into a number of sub‐problems of lower dimension which can be solved independently. As a result, the nominal parts of the sliding mode dynamics is linearized. A discontinuous feedback is then used to compensate the matched uncertainty. Finally, a step‐by‐step Lyapunov technique and a high gain approach is applied to obtain hierarchical fast motions on the sliding manifolds and to achieve the robustness property of the closed‐loop system motion with respect to unmatched uncertainty. Copyright © 2002 John Wiley & Sons, Ltd.     

Immersion and invariance adaptive control of a class of nonlinear systems with unknown control direction

The goal of this article is to extend the adaptive control problem of parametric strict feedback form nonlinear systems, using immersion and invariance to the case of unknown, possibly, time-varying control direction. The idea is to immerse a target system in ? ^n?1, which is stabilised through the design of virtual controllers, into an extended system in ? ^n+p . The designed controller takes advantage of the well-known Nussbaum functions to deal with the unknown sign of input multiplier and is designed through the manifold dynamics belonging to ? ^p+1. The effectiveness of the proposed method is shown through a simulation example and is also compared to the classical adaptive backstepping approach with an unknown control direction.     

视频人脸识别中判别性联合多流形分析

将基于视频的人脸识别转换为图像集识别问题,并提出两种流形来表示每个图像集:一种是类间流形,表示每个图像集的平均脸信息;另一种是类内流形,表示每个图像集的所有原始图像的信息.类间流形针对图像集之间的区别提取整体判别信息,作用是选出几个与待识别图像集较为相似的候选图像集.类内流形则考虑图像集内各原始图像之间的关系,负责从候选图像集中找出最为相似的一个.不同于现有的非线性流形方法中每幅图像对应流形中的一个点,采用分片技术学习两种流形的投影矩阵,每个分片对应流形中的一个点,所学到的特征更具有判别性,进而使流形边界更加清晰,同时解决了传统非线性流形方法中的角度偏差和不充分采样问题.还提出了与分片技术相匹配的流形之间的距离度量方法.最后在几个广为研究的数据集上进行了实验,结果表明:新方法的识别准确率高,尤其适用于不受控环境下的视频识别,而且不受视频段长短的影响.     

A New Projection-Based Neural Network for Constrained Variational Inequalities

This paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations. Five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of Lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function. The proposed neural network includes an existing model, and can be applied to solve some nonmonotone and nonsmooth problems. The validity and transient behavior of the proposed neural network are demonstrated by some numerical examples.      

Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties

This paper considers the stabilization problem of a one-dimensional unstable heat conduction system (rod) modeled by a parabolic partial differential equation (PDE), powered with a Dirichlet type actuator from one of the boundaries. By applying the Volterra integral transformation, a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties. The associated Lyapunov function is used for designing an infinite-dimensional sliding manifold, on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances. It is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero. A continuous control law satisfying the reaching condition is obtained by passing a discontinuous (signum) signal through an integrator.     

Practical Formation Control of Multiple Unicycle-Type Mobile Robots with Limited Sensing Ranges

This paper presents a design of cooperative controllers that force a group of N unicycle-type mobile robots with limited sensing ranges to perform a desired tight formation and that guarantee no collisions between any robots in the group. The desired formation can be stabilized at any reference trajectories with bounded time derivatives. The formation control design is based on several nonlinear coordinate changes, the transverse function approach, the backstepping technique, the Lyapunov direct method, and smooth or p −times differentiable step functions. These functions are introduced and incorporated into novel potential functions to solve the collision avoidance problem without the need of switchings despite of the robots’ limited sensing ranges. The proposed formation control system is applied to solve a gradient climbing problem.     

Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems

A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output zeroing manifold and its friend. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to higher order dynamics. This is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one. We also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective. Our construction does not invoke certainty equivalence, nor requires a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. Finally, it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form, yielding new stabilizing control laws. We illustrate the method with several academic and practical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application.