Asymptotic Stabilization of the Inverted Equilibrium Manifold of the 3-D Pendulum Using Non-Smooth Feedback

The 3-D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. In , almost global stabilization of the inverted equilibrium manifold was studied using a smooth globally defined feedback. Here, we study the problem of almost global stabilization of the inverted equilibrium manifold using non-smooth feedback of angular velocity and a reduced attitude vector of the 3-D pendulum. The importance of the non-smooth feedback is that the almost global domain of attraction is a geometrically simple set that excludes the hanging attitude manifold. Unlike the closed-loop for a 3-D pendulum with a smooth controller, the closed-loop designed in this paper does not exhibit a performance constraint. These new results are based on Lyapunov analysis of the non-smooth closed-loop 3-D pendulum.      

Asymptotic stabilization of the hanging equilibrium manifold of the 3D pendulum

The 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. The 3D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. This paper shows that a controller based on angular velocity feedback can be used to asymptotically stabilize the hanging equilibrium manifold of the 3D pendulum. Lyapunov analysis and nonlinear geometric methods are used to assess the global closed‐loop properties. We explicitly construct compact sets that lie in the domain of attraction of the hanging equilibrium of the closed‐loop. Finally, this controller is shown to achieve almost global asymptotic stability of the hanging equilibrium manifold. An invariant manifold of the closed‐loop that converges to the inverted equilibrium manifold is identified. Copyright © 2007 John Wiley & Sons, Ltd.     

Stabilization of a 3D axially symmetric pendulum

Stabilizing controllers are developed for a 3D pendulum assuming that the pendulum has a single axis of symmetry and that the center of mass lies on the axis of symmetry. This assumption allows development of a reduced model that forms the basis for controller design and global closed-loop analysis; this reduced model is parameterized by the constant angular velocity component of the 3D pendulum about its axis of symmetry. Several different controllers are proposed. Controllers based on angular velocity feedback only, asymptotically stabilize the hanging equilibrium. Then controllers are introduced, based on angular velocity and reduced attitude feedback, that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium. These problems can be viewed as stabilization of a Lagrange top. Finally, if the angular velocity about the axis of symmetry is assumed to be zero, controllers are introduced, based on angular velocity and reduced attitude feedback, that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium. This problem can be viewed as stabilization of a spherical pendulum.     

基于能量反馈的圆轨倒立摆摆起控制

基于拉格朗日方程建立圆轨倒立摆单摆模型,根据能量反馈控制理论,设计圆轨倒立摆摆起控制器,从而使摆杆从稳定平衡点摆起到不稳定平衡点并稳定不倒。采用具有流水线指令结构CIP-51控制器内核的C8051单片机实现控制算法。实验结果表明,采用能量反馈控制器可以对具有复杂非线性、强耦合、自然不稳定特性的倒立摆系统实现摆起控制,摆起效果良好。同时也为其他非线性多变量系统的控制提供了有效方法。     

基于滑模控制的3D刚体摆姿态稳定性

研究3D刚体摆姿态稳定性的滑模控制问题.3D刚体摆由一个刚体绕一固定且无摩擦的支点旋转,刚体受到恒重力作用且具有三个转动自由度.针对3D刚体摆平衡位置处的姿态稳定控制问题,设计了滑模控制器并分析了角速度和姿态的渐进稳定性.由Lyapunov直接法找出了各个滑模系数取值的充分条件,并通过数值仿真实验验证了滑模控制方法的有效性.     

Synthesis of upper-triangular non-linear systems with marginally unstable free dynamics using state-dependent saturation

In this paper we present sufficient conditions under which a fairly large class of single-input non-linear systems including feedforward systems and the well-known ball-and-beam model, are globally asymptotically and locally exponentially stabilizable by smooth state feedback. A nested saturation controller with state-dependent saturation levels is constructed explicitly, using a novel design approach which combines the nested saturation strategy for marginally unstable linear systems subject to input saturation, with the small feedback design technique, developed for global asymptotic stabilization of general non-affine systems with marginally stable free dynamics. The power of the state-dependent saturation design method is demonstrated by solving a number of non-linear control problems, particularly, the global stabilization problem of a class of two-dimensional non-linear systems and the ball-and-beam system.     

A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon

We show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium. This result is directly applicable to mechanical systems having rotational degrees of freedom. In particular, the result applies to the attitude motion of a rigid body. In light of this result, we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability.     

Passivity, feedback equivalence, and the global stabilization ofminimum phase nonlinear systems

Conditions under which a nonlinear system can be rendered passive via smooth state feedback are derived. It is shown that, as in the case of linear systems, this is possible if and only if the system in question has relative degree one and is weakly minimum phase. It is proven that weakly minimum phase nonlinear systems with relative degree one can be globally asymptotically stabilized by smooth state feedback, provided that suitable controllability-like rank conditions are satisfied. This result incorporates and extends a number of stabilization schemes recently proposed for global asymptotic stabilization of certain classes of nonlinear systems     

Stability improvement of nonlinear systems by feedback

This paper is concerned with the property of stabilizing a nonlinear system to a specified equilibrium point arbitrarily fast by appropriate smooth feedback. Two closely related forms of this property are explored. One refers to the asymptotic transfer to the critical point with exponential decay, and it is shown that the systems possessing the above property are precisely those whose corresponding linearization is controllable. The second stabilizability form defined here amounts to driving the system back to an arbitrarily small neighborhood of the given point arbitrarily fast by smooth feedback. A global stabilization result is also included, finding application in Hamiltonian systems. Special emphasis is given to the bilinear case.     

Smooth output feedback stabilization for a class of nonlinear systems with time‐varying powers

This paper investigates the global asymptotic stabilization problem for a class of nonlinear systems with time‐varying powers. First, adding a power integrator technique is revamped to design a smooth state feedback controller, which is implementable with only upper and lower bounds of the time‐varying powers. When the system state is not fully available and the time‐varying power is exactly known, a smooth output feedback controller constituted by a state feedback and a nonlinear state observer is constructed to globally stabilize the system. Copyright © 2017 John Wiley & Sons, Ltd.     

A nonlinear position and attitude observer on SE(3) using landmark measurements

This paper addresses the problem of position and attitude estimation, based on landmark readings and velocity measurements. A derivation of a nonlinear observer on SE(3) is presented, using a Lyapunov function conveniently expressed as a function of the difference between the estimated and the measured landmark coordinates. The resulting feedback laws are explicit functions of the landmark measurements and velocity readings, exploiting the sensor information directly in the observer. The proposed observer yields almost global asymptotic stabilization of the position and attitude errors and exponential convergence in any closed ball inside the region of attraction. Also, it is shown that the asymptotic convergence of the estimation error trajectories is shaped by the landmark geometry and observer design parameters. The problem of non-ideal velocity readings is also considered, and the observer is augmented to compensate for bias in the angular and linear velocity measurements. The resulting position, attitude, and bias estimation errors are shown to converge exponentially fast to the desired equilibrium points, for bounded initial estimation errors. Simulation results are presented to illustrate the stability and convergence properties of the observer.     

A nonlinear hybrid controller for swinging-up and stabilizing the Furuta pendulum

The conventional switching strategy for solving the inverted pendulum control problem is based on two steps: swinging-up and stabilization. In this note, first, a new strategy for swinging the Furuta pendulum up towards the desired upright position is designed using the Speed-Gradient method, which uses only directly measured coordinates. Then, a nonlinear controller, based on the Forwarding approach, stabilizes the upright position. As a new contribution the latter leads to a nonlinear stabilizer around the upright position, whose Lyapunov function yields a larger size estimation of the domain of attraction than the one obtained with the traditional linear approach. This estimation allows us to use it in a global switching strategy in the practical implementation and guarantees almost-global asymptotic stability of the equilibrium. Successful experimental results are reported with the available laboratory Furuta pendulum.     

Global stabilization of the unstable Reaction-Wheel Pendulum

The paper deals with the problem of the Reaction Wheel Pendulum stabilization about unstable (inverted) position for arbitrary initial conditions. Considered mechanical system consists of a physical pendulum with a symmetric disk attached to the end of the pendulum, which is free to spin about an axis parallel to the axis of rotation of the pendulum. The disk is actuated by a DC-motor. The coupling torque generated by the angular acceleration of the disk is used to control of the pendulum. The switching control law is proposed to swinging up the pendulum and balancing it about the inverted position. The nonlinear swinging up control law is proposed ensuring global stabilization of the pendulum about inverted position. The Energy-based Speed-gradient (EBSG) control scheme is used to designing the swinging-up controller. The modification of the EBSG method is proposed to ensure attainability of the inverted position of the pendulum for all initial states of the system. The balance controller is designed on the basis of the Variable Structure Control with forced sliding mode. Numerical simulation results are presented showing achievement of the posed control goal by means of the control action of small magnitude.     

Velocity‐free attitude stabilization with inertial vector measurements

This paper deals with the attitude stabilization problem of a rigid body, where neither the angular velocity nor the attitude is used in the feedback; only body‐referenced vector measurements are needed. The proposed control scheme is based on an angular velocity observer‐like system relying solely on vector measurements. The proposed controller ensures almost global asymptotic stability and provides some interesting performance properties through an appropriate tuning of the control gains. The performance and effectiveness of the proposed control scheme are illustrated via simulation results where the control gains are adjusted using a nonlinear optimization. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

基于四元数的航天器姿态非线性控制器

针对由四元数描述的航天器姿态控制系统进行了非线性反馈控制器设计.首先,得到一类含角速度测量的反馈控制律.该控制律不仅可以保证闭环系统是全局渐近稳定的,而且保证了2个期望的平衡位置都是稳定的.然后,利用基于四元数的滤波器取代角速度测量,得到一类无需角速度测量的反馈控制律.该控制律同样保证了闭环系统的全局渐近稳定性和2个稳定的期望平衡位置.值得注意的是,2个稳定的期望平衡位置,将促使由任意初始姿态出发的轨迹都就近收敛至任意期望平衡位置.最后,数值仿真展示了所得反馈控制律的优势.     

基于受控拉格朗日函数的垂直起降飞机控制器设计

本文将单欠驱动度力学系统基于受控拉格朗日函数(CL)的控制器设计技术应用到具有输入耦合的垂直起降(PVTOL)飞机的控制问题中. 在显式陀螺力的匹配条件下, 得到的光滑反馈控制律保证了系统几乎全局渐近稳定. 与现有同类方法所得到的结果相比, 该匹配控制器形式更简单而收敛性能得到保持.     

Robust attitude stabilization of spacecraft using nonlinearquaternion feedback

This paper considers the problem of three-axis attitude stabilization of a rigid spacecraft. A nonlinear control law which uses the feedback of the unit quaternion and the measured angular velocities is proposed and is shown to provide global asymptotic stability. The control law does not require the knowledge of the system parameters and is, therefore, robust to modeling errors. The significance of the control law is that it can be used for large-angle maneuvers with guaranteed stability     

A continuous feedback approach to global strong stabilization ofnonlinear systems

We present a continuous feedback approach to the problem of global strong stabilization, for genuinely nonlinear systems that may not be stabilized, even locally, by any smooth feedback. We describe conditions under which it is possible to prove, while no smooth controllers exist, the existence of continuous state feedback control laws that achieve global strong stability (GSS) in the sense of Kurzweil (1956). The proof is constructive and carried out by developing a machinery, which combines the theory of homogeneous systems with the idea of adding a power integrator, for the explicit construction of globally stabilizing continuous controllers. We then illustrate, by means of examples, how this machinery can be used to overcome the topological obstruction caused by smooth feedback, and hence resulting in new solutions to a variety of open control problems, including global stabilization of an underactuated unstable two degree of freedom mechanical system     

Asymptotic stabilization of minimum phase nonlinear systems

How a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown. More precisely, conditions under which, for every compact set of initial states, it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described. The theory includes the development of globally defined transformations of the system equations to their global normal form