A constructive solution for stabilization via immersion and invariance: The cart and pendulum system

Immersion and Invariance (I&I) is the method to design asymptotically stabilizing control laws for nonlinear systems that was proposed in [Astolfi, A., & Ortega, R. (2003). Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems. IEEE Transactions on Automatic Control, 48, 590-606]. The key steps of I&I are (i) the definition of a target dynamics, whose order is strictly smaller than the order of the system to be controlled; (ii) the construction of an invariant manifold such that the restriction of the system dynamics to this manifold coincides with the target dynamics; (iii) the design of a control law that renders the manifold attractive and ensures that all signals are bounded. The second step requires the solution of a partial differential equation (PDE) that may be difficult to obtain. In this short note we use the classical cart and pendulum system to show that by interlacing the first and second steps, and invoking physical considerations, it is possible to obviate the solution of the PDE. To underscore the generality of the proposed variation of I&I, we show that it is also applicable to a class of n-dimensional systems that contain, as a particular case, the cart and pendulum system.     

Algebraic separation in realizing a linear state feedback control law by means of an adaptive observer

Based on a novel adaptive observer, which does not require signal boundedness in its stability proof, an algebraic separation property of linear state feedback control and adaptive state observation is established. This means, whenever a linear, stabilizing state feedback control law is realized with the state replaced by the state estimate of the given stable adaptive observer, then the resulting nonlinear control system is also globally asymptotically Lyapunov stable with respect to the initial state and parameter observation error of the adaptive observer. In particular, no assumptions on the system dynamics nor on the speed of the adaptation are made.     

New class of control laws for robotic manipulators Part 1. Non–adaptive case

A new class of exponentially stabilizing control laws for joint level control of robot arms is introduced. It has recently been recognized that the non-linear dynamics associated with robotic manipulators have certain inherent passivity properties. More specifically, the derivation of the robotic dynamic equations from Hamilton's principle gives rise to natural Lyapunov functions for control design based on total energy considerations. Through a slight modification of the energy Lyapunov function and the use of a convenient lemma to handle third–order terms in the Lyapunov function derivatives, closed–loop exponential stability for both the set point and tracking control problem is demonstrated. In one new design, the nonlinear terms are decoupled from real-time measurements which completely removes the requirement for on–line computation of non–linear terms in the controller implementation. In general, the new class of control laws offers alternatives to the more conventional computed torque method, providing trade–offs between computation and convergence properties. Furthermore, these control laws have the unique feature that they can be adapted in a very simple fashion to achieve asymptotically stable adaptive control.     

Two results for adaptive output feedback stabilization of nonlinear systems

The problem of (adaptive) stabilization by means of output feedback of a class of nonlinear systems is addressed and solved. The proposed method relies on the asymptotic reconstruction of a stabilizing state feedback control law, does not require stable zero dynamics nor the construction of a Lyapunov function for the closed loop system, and treats in a unified way unknown parameters and unmeasured states. The applicability of the proposed method is discussed via theoretical examples. Finally, it is shown that the proposed method yields a solution to the problem of output feedback regulation for a DC-to-DC power converter and the efficacy of the resulting controller is verified via experiments.     

Robust adaptive control for a class of uncertain non-affine nonlinear systems using affine-type neural networks

A robust adaptive control is proposed for a class of single-input single-output non-affine nonlinear systems. In order to approximate the unknown nonlinear function, a novel affine-type neural network is used, and then to compensate the approximation error and external disturbance a robust control term is employed. By Lyapunov stability analysis for the closed-loop system, it is proved that tracking errors asymptotically converge to zero. Moreover, an observer is designed to estimate the system states because all the states may not be available for measurements. Furthermore, the adaptation laws of neural networks and the robust controller are given out based on the Lyapunov stability theory. Finally, two simulation examples are presented to demonstrate the effectiveness of the proposed control method.     

漂浮基空间机器人自适应RBF 网络终端滑模控制

主要研究漂浮基空间机器人对工作空间连续轨迹跟踪控制问题．针对系统动力学模型中非线性项未知，以及参数不确定性和外界扰动无法估计的情况，提出了基于自适应RBF网络终端滑模控制方法．该方法结合了非线性滑动流形与径向基函数特性，利用自适应RBF网络在线学习系统中的不确定性，使得无需精确的动力学模型亦能保证系统在有限时间内快速稳定．根据Lyapunov方法设计的自适应增益保证闭环控制系统具有全局稳定性，并且有效抑制抖振现象．针对6关节空间机器人的轨迹跟踪控制仿真表明，提出的自适应RBF网络终端滑模控制方法能够基于不完整动力学模型实现高精度轨迹跟踪，且误差在有限时间内快速收敛，系统抖振也得到了有效抑制．     

Adaptive Mittag‐Leffler Stabilization of a Class of Fractional Order Uncertain Nonlinear Systems

This paper deals with the stabilization of a class of commensurate fractional order uncertain nonlinear systems. The fractional order system concerned is of the strict‐feedback form with uncertain nonlinearity. An adaptive control scheme combined with fractional order update laws is proposed by extending classical backstepping control to fractional order backstepping scheme. The asymptotic stability of the closed‐loop system is guaranteed under the construction of fractional Lyapunov functions in the sense of generalized Mittag‐Leffler stability. The fractional order nonlinear system investigated can be stabilized asymptotically globally in presence of arbitrary uncertainty. Finally illustrative examples and numerical simulations are performed to verify the effectiveness of the proposed control scheme.     

含有非线性不确定参数的电液系统滑模自适应控制

针对含有非线性不确定参数的电液控制系统, 提出了一种滑模自适应控制方法. 该控制方法主要是为了解决由于初始控制容积的不确定性而引起的, 非线性不确定参数自适应律设计的难题. 其主要特点为, 通过定义一个新型的特Lyapunov 函数, 进而构建系统的自适应控制器及参数自适应律, 并结合滑模控制方法及一种简单的鲁棒设计方法, 给出整个电液系统的滑模自适应控制器, 及所有不确定参数的自适应律. 试验结果表明, 采用该控制方法能够取得良好的性能, 尤其可以补偿非线性不确定参数对系统的影响.     

Stabilizing controllers design for switched nonlinear systems in strict-feedback form

This paper considers the stabilization problem for a class of switched nonlinear systems under arbitrary switching. Based on the backstepping method and the control Lyapunov function approach, it is shown that, under a simultaneous domination assumption, a switched nonlinear system in strict-feedback form can be globally uniformly asymptotically stabilized by a continuous state feedback controller. A universal formula for constructing stabilizing feedback laws is presented. One example is included for verifying the obtained results.     

A Detailed Study of Adaptive Control of Chaotic Systems with Unknown Parameters

In this paper, we study the control of chaotic systems with unknown parameters. A stable adaptive control scheme is used to guarantee that the parameter estimator converges to stabilizing values such that the controlled chaotic system asymptotically approaches a reference point. A Lyapunov function approach is used to prove a global result which guarantees the stability of both controlled chaotic system and the adaptive parameter estimator. The center manifold theorem is used to prove the stability of the adaptive parameter estimator.To demonstrate the usefulness of this adaptive control of chaotic systems, computer simulation results are provided. We use Chua's circuit with cubic nonlinearity in our simulations. We provide the simulation results of control of Chua's circuit with 6 unknown parameters.     

一类非线性系统有限时间流形吸引的浸入与不变控制

本文对一类非线性系统,提出了一种设计渐近稳定控制律的有效方法.其中,通过更新系统浸入与不变流形理论的应用方法,流形的吸引坐标可以在有限时间内收敛到平衡点.为了得到闭环系统的稳定性,增广系统的各个信号被证明是有界的.本文得出的一个重要成果是流形吸引有限时间的计算方法.此外,在施加了有限时间流形吸引控制器之后,流形对外部有界未知扰动具有不敏感性.最后利用车摆系统来论述所提出的控制方法的设计步骤,以及通过仿真验证控制器的性能.     

Robust adaptive prescribed performance control for a class of nonlinear pure‐feedback systems

An adaptive prescribed performance control design procedure for a class of nonlinear pure‐feedback systems with both unknown vector parameters and unmodeled dynamics is presented. The unmodeled dynamics lie within some bounded functions, which are assumed to be partially known. A state transformation and an auxiliary system are proposed to avoid using the cumbersome formula to handle the nonaffine structure. Simultaneously, a parameter‐type Lyapunov function and L function are designed to ensure the prescribed performance of the pure‐feedback system. As illustrated by examples, the proposed adaptive prescribed performance control scheme is shown to guarantee global uniform ultimate boundedness. At the same time, this method not only guarantees the prescribed performance of the system but also makes the tracking error asymptotically close to a certain value or stable.     

Smooth patchy control Lyapunov functions

A smooth patchy control Lyapunov function for a nonlinear system consists of an ordered family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy certain further increase or decrease conditions. We prove that such a control Lyapunov function exists for any asymptotically controllable nonlinear system. We also show a construction, based on such a control Lyapunov function, of a stabilizing hybrid feedback that is robust to measurement noise.     

Adaptive tracking H-infinity control for switched nonlinear systems with unknown control gain sign

This paper addresses the adaptive tracking control scheme for switched nonlinear systems with unknown control gain sign. The approach relaxes the hypothesis that the upper bound of function control gain is known constant and the bounds of external disturbance and approximation errors of neural networks are known. RBF neural networks (NNs) are used to approximate unknown functions and an H-infinity controller is introduced to enhance robustness. The adaptive updating laws and the admissible switching signals have been derived from switched multiple Lyapunov function method. It’s proved that the resulting closed loop system is asymptotically Lyapunov stable such that the output tracking error performance and H-infinity disturbance attenuation level are well obtained. Finally, a simulation example of Forced Duffing systems is given to illustrate the effectiveness of the proposed control scheme and improve significantly the transient performance.     

连续时间混沌系统的参数自适应控制

研究了连续时间混沌系统的参数自适应控制,提出了关于多重参数混沌非线性系统 的参数自适应控制的新方法.考虑系统参数是线性形式的自适应控制,利用Lyapunov方法 证明了参数控制方程是全局渐近稳定的.研究结果表明该控制方法是分析混沌参数白适应控 制的一个十分有效的方法.     

连续时间混沌系统的参数自适应控制1)

研究了连续时间混沌系统的参数自适应控制,提出了关于多重参数混沌非线性系统的参数自适应控制的新方法.考虑系统参数是线性形式的自适应控制,利用Lyapunov方法证明了参数控制方程是全局渐近稳定的.研究结果表明该控制方法是分析混沌参数自适应控制的一个十分有效的方法.     

Distributed adaptive control for synchronization of unknown nonlinear networked systems

This paper is concerned with synchronization of distributed node dynamics to a prescribed target or control node dynamics. A design method is presented for adaptive synchronization controllers for distributed systems having non-identical unknown nonlinear dynamics, and for a target dynamics to be tracked that is also nonlinear and unknown. The development is for strongly connected digraph communication structures. A Lyapunov technique is presented for designing a robust adaptive synchronization control protocol. The proper selection of the Lyapunov function is the key to ensuring that the resulting control laws thus found are implementable in a distributed fashion. Lyapunov functions are defined in terms of a local neighborhood tracking synchronization error and the Frobenius norm. The resulting protocol consists of a linear protocol and a nonlinear control term with adaptive update law at each node. Singular value analysis is used. It is shown that the singular values of certain key matrices are intimately related to structural properties of the graph.     

Adaptive state-feedback stabilization for a class of high-order nonlinear uncertain systems

For high-order nonlinear uncertain systems, there have been a lot of investigations under a strong assumption that the lower bounds of the unknown control coefficients should be exactly known. In this paper, this assumption is removed and a unified approach is developed to systematically construct a state-feedback adaptive stabilizing control law for a class of high-order nonlinear uncertain systems with unknown control coefficients. By using the method of the so-called adding a power integrator merging with adaptive technique, a recursive design procedure is provided to achieve a smooth adaptive state-feedback control law, which guarantees that the closed-loop system is globally uniformly stable while the original system states globally asymptotically converge to zero. Finally, a simulation example is given to illustrate the correctness of the theoretical results.     

Discrete-time certainty equivalence output feedback: allowing discontinuous control laws including those from model predictive control

We present certainty equivalence output feedback results for discrete-time nonlinear systems that employ possibly discontinuous control laws in the feedback loop. Coupling assumptions of nominal robustness with uniform observability or detectability assumptions, we assert nominally robust stability for output feedback closed loops. We further show that model predictive control (MPC) can be used to generate a feedback control law that is robustly globally asymptotically stabilizing when used in a certainty equivalence output feedback closed loop. Allowing for discontinuous feedback control laws is important for systems employing MPC, since the method can, and sometimes necessarily does, result in discontinuous control laws.     

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.