Design of sliding observers for Lipschitz nonlinear system using a new time-averaged Lyapunov function

This paper provides a procedure for designing a sliding mode observer for a nonlinear system in the presence of Gaussian input disturbances and sensor noises. The paper first proposes a novel candidate for Lyapunov stability, termed a Time-averaged Lyapunov (TAL) function. The TAL averages the Lyapunov analysis over a small finite time interval, allowing for intuitive analysis of noises and disturbance. The paper then provides the necessary and sufficient condition for the design of the sliding observer gainsusing the TAL in the form of Linear Matrix Inequality (LMI). The LMIs can then be explicitly solved offline using commercial LMI solvers. The paper compares the LMIs for the two observer designs to demonstrate the design of the sliding mode observer using TAL can greatly enhance the scope of observer design for nonlinear systems.     

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.     

基于输出反馈和滑模控制的一类二阶非线性系统有限时间镇定方法

本文研究了一类具有不确定非线性动力学和未知外部扰动的二阶非线性系统的全局有限时间输出镇定问 题. 首先, 提出了一种全局状态反馈有限时间控制器, 实现了二阶非线性系统的有限时间镇定. 为了解决只有系统输 出可用这种更有挑战性的情况, 采用了一种新颖的设计思想, 即非分离原理. 构造了一个有限时间收敛的状态观测 器来估计未知状态. 在此观测器的基础上, 提出了一种基于输出的有限时间复合控制器. 基于李雅普诺夫方法, 证明 了整个闭环系统的全局有限时间稳定性. 仿真结果表明了理论的有效性.     

Robust stabilization of nonlinear systems with pointwisenorm-bounded uncertainties: a control Lyapunov function approach

The authors give a necessary and sufficient condition for globally stabilizing a nonlinear system, robustly with respect to unstructured uncertainties Φ˜(u,x,t), norm-bounded for each fixed x and u. This condition requires one to find a smooth, proper, and positive definite solution V(x) of a suitable partial differential inequality depending only on the system data. A procedure, based on the knowledge of V(x), is outlined for constructing almost smooth robustly stabilizing controllers. Our approach, based on Lyapunov functions, generalizes previous results for linear uncertain systems and establishes a precise connection between robust stabilization, on one hand, and H_∞ -control sector conditions and input-to-state stabilization on the other     

Constructive Lyapunov stabilization of nonlinear cascade systems

We present a global stabilization procedure for nonlinear cascade and feedforward systems which extends the existing stabilization results. Our main tool is the construction of a Lyapunov function for a class of (globally stable) uncontrolled cascade systems. This construction serves as a basis for a recursive controller design for cascade and feedforward systems. We give conditions for continuous differentiability of the Lyapunov function and the resulting control law and propose methods for their exact and approximate computation     

基于Lyapunov方法和快速终端滑模的轨迹跟踪控制

针对移动机器人的运动学模型,提出一种具有全局渐近稳定性的跟踪控制器。该跟踪控制器的设计分为两部分：第一部分是采用全局快速终端滑动模态的思想设计了角速度的控制律,用来渐近镇定移动机器人跟踪的前向角误差;第二部分是采用Lyapunov方法设计了线速度的控制律,用来渐近镇定移动机器人跟踪的平面坐标误差。采用Lyapunov稳定性定理,证明了移动机器人在满足这些控制律条件下,实现了对参考轨迹的全局渐近跟踪。实验结果表明移动机器人能够有效地跟踪期望轨迹,有利于在实际应用中推广。     

Asymptotic stabilization with group-wise sparse input based on control Lyapunov function approach

This study proposes a novel stabilizing controller for nonlinear systems using group-wise sparse inputs. The input variables are divided into several groups. In the situations when the input constraints can be ignored, one input becomes active for each group at each moment. Our method improves energy efficiency, as sparse input vectors often reduce the standby power of inactive actuators. Large-scale systems, such as those consisting of multiple subsystems, often require the manipulation of multiple inputs simultaneously to be controlled. Our method can be applied to such systems due to the group-wise sparsity of the inputs. The proposed controller is based on the control Lyapunov function approach and includes Sontag's universal formula as a special case. The controllers designed in our method have best-effort property, which means even when a restriction for the decreasing rate of the Lyapunov function cannot be fulfilled, the controller minimizes the time derivative of the Lyapunov function within the input constraint. The effectiveness of the proposed method can be confirmed through simulations.     

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 vector function method in the motion stabilisation problem for nonholonomic mobile robot

In this paper we propose a sampled-data control law in the stabilisation problem of nonstationary motion of nonholonomic mobile robot. We assume that the robot moves on a horizontal surface without slipping. The dynamical model of a mobile robot is considered. The robot has one front free wheel and two rear wheels which are controlled by two independent electric motors. We assume that the controls are piecewise constant signals. Controller design relies on the backstepping procedure with the use of Lyapunov vector-function method. Theoretical considerations are verified by numerical simulation.     

Constructing quasi-invariant control systems based on the Lyapunov function method for nonlinear plants with an incomplete mathematical model

The problem of constructing feedback control systems for one-dimensional plants with incomplete mathematical models and known structures is studied. A quadratic form decomposable into linear multipliers used as the Lyapunov function helps decompose the control into two components viz. the Utkin-Drazhenovich equivalent control and the stabilizing control. A condition used to find the stabilizing control is given. An equation involving the observation error is obtained. The equation is used to derive the estimation formula for the nonlinearity and external perturbation. The high estimation accuracy and the feasible unlimited increase of the gain coefficients of the controller and observer without breaking the system??s stability ensure the high performance and tracking accuracy of the reference trajectory. Model stabilization problems are solved in Matlab/Simulink.     

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.     

Switching fuzzy controller design based on switching Lyapunov function for a class of nonlinear systems

This paper presents a switching fuzzy controller design for a class of nonlinear systems. A switching fuzzy model is employed to represent the dynamics of a nonlinear system. In our previous papers, we proposed the switching fuzzy model and a switching Lyapunov function and derived stability conditions for open-loop systems. In this paper, we design a switching fuzzy controller. We firstly show that switching fuzzy controller design conditions based on the switching Lyapunov function are given in terms of bilinear matrix inequalities, which is difficult to design the controller numerically. Then, we propose a new controller design approach utilizing an augmented system. By introducing the augmented system which consists of the switching fuzzy model and a stable linear system, the controller design conditions based on the switching Lyapunov function are given in terms of linear matrix inequalities (LMIs). Therefore, we can effectively design the switching fuzzy controller via LMI-based approach. A design example illustrates the utility of this approach. Moreover, we show that the approach proposed in this paper is available in the research area of piecewise linear control.     

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.     

Deep learning controller for nonlinear system based on Lyapunov stability criterion

Neural Computing and Applications - For the current paper, the technique of feed-forward neural network deep learning controller (FFNNDLC) for the nonlinear systems is proposed. The FFNNDLC...     

On the stabilization of nonlinear systems via input-dependent sliding surfaces

This article proposes the use of sliding regimes, defined on a suitable input-dependent sliding surface, as a means of making more robust any model-based smooth feedback control scheme designed for the stabilization of a general nonlinear plant. The approach naturally produces a, robust, ‘outer loop’ redundant discontinuous feedback scheme, with several advantageous properties regarding insensitivity to external perturbation signals, modelling errors and sudden failure of the smooth portion of the feedback loop.     

A new Lyapunov design approach for nonlinear systems based on Zubov's method

The present research work proposes a new nonlinear controller synthesis approach that is based on the methodological principles of Lyapunov design. In particular, it relies on a short-horizon model-based prediction and optimization of the rate of “energy dissipation” of the system, as it is realized through the time derivative of an appropriately selected Lyapunov function. The latter is computed by solving Zubov's partial differential equation based on the system's drift vector field. A nonlinear state feedback control law with two adjustable parameters is derived as the solution of an optimization problem that is formulated on the basis of the aforementioned Lyapunov function and closed-loop performance characteristics. A set of system-theoretic properties of the proposed control law are examined as well. Finally, the proposed Lyapunov design method is evaluated in a chemical reactor example which exhibits nonminimum-phase behaviour.     

Linear stabilization of nonlinear systems program motion

The problem of nonlinear systems control in presence of indeterminacy is considered. It is proposed to stabilize the program motion by strong linear feedback. Such a control law is proved to ensure the tracking of the program motion with any given accuracy under control restrictions. Results may be adopted to robot-manipulator control.     

基于结构Lyapunov矩阵的静态输出反馈镇定

线性时不变系统的静态输出反馈控制可行性等价于两个耦合的线性矩阵不等式解的存在性问题,这导致了一个非线性最优化问题,是无法直接求解的．针对线性时不变系统(LTI),深入研究这两个矩阵不等式的关系,通过构造一个结构Lyapunov矩阵,给出了一个问题有解的充分条件,并在此基础上提出一个静态输出反馈镇定算法．利用线性矩阵不等式(LMI)方法,可直接求解出相应的输出反馈增益．数值实例证明了该方法的有效性．     

Enhanced backstepping sliding mode controller for motion tracking of a nonlinear 2-DOF piezo-actuated micromanipulation system

Microsystem Technologies - In this paper a robust backstepping sliding mode controller is developed for tracking control of 2-DOF piezo-actuated micromanipulation system. The control approach is...