A time-optimal control synthesis for a nonlinear pendulum

A time-optimal control synthesis is obtained that steers a nonlinear pendulum to the bottom stable equilibrium position. The solution is based on the maximum principle and involves the analytical investigation in combination with numerical calculations. As a result, for different values of the maximum admissible control torque, switching curves and dispersal curves (or separation curves), which confine domains in the phase space that correspond to different values of the bang-bang optimal control, are constructed. The investigated patterns of synthesis reveal the mechanism of generation of numerous breaks on switching curves for small control torques.     

Finding the principal bifurcation value of the maximum control torque in the problem of optimal control synthesis for a pendulum

The principal bifurcation value in the problem of synthesis of a time-optimal control for damping the oscillations of a nonlinear pendulum is found. A one-link pendulum controlled by a torque of forces produced at the suspension point is considered. The magnitude that bounds the modulus of the control torque is the only essential parameter of the investigated system. It is supposed that an admissible control may exceed the moment of the gravitational force. By the principal bifurcation value, we mean the value of the maximum admissible control torque for which a trajectory with two switches of relay optimal control arises in the synthesis pattern. This publication is a logical complement to papers [1–3], where the mentioned synthesis patterns were obtained numerically with the use of the maximum principle. In contrast to [1–3], we formulate a system of explicit relations of integral type for the numerical construction of the boundaries of domains of the phase plane inside which the trajectories with two switches start. Based on these relations, an analytical upper estimate of the principal bifurcation value is obtained. This value is also found numerically. In a special coordinate system, the synthesis patterns are constructed that illustrate the specific features of the optimal control for large angular velocities of pendulum rotations.     

Time-optimal control of an inverted pendulum in the feedback form

A time-optimal feedback control is synthesized that steers a nonlinear pendulum to the top unstable equilibrium position. The solution is based on the maximum principle and involves an analytical investigation in combination with numerical computations. As a result, for a number of values of the maximum admissible control torque, the switching curves and dispersal curves that delimit the domains in the phase space that correspond to different values of the bang-bang optimal control are constructed.     

Brachistochrone for a rigid body sliding down a curve

Extremality conditions in the brachistochrone problem for a perfectly rigid body sliding without friction down an unknown (to be determined) curve in a vertical plane are found. In this case, the body has to track the tangent to the trajectory. According to the principle of constraint release, the reaction of the support creates a torque used as control. For dimensionless Appell’s equations with a single similarity coefficient, the standard problem of the fastest descent from a given initial point to a given terminal point assuming that the initial velocity is zero is formulated. The Okhotsimskii-Pontryagin method is used to analyze the differential of the objective function. Necessary optimality conditions are found, and a formula for the optimal control that does not involve adjoint variables is derived from them. Properties of the optimal trajectories are investigated analytically both in the general case and for the limiting (zero and infinite) values of the similarity coefficient. It is found that cycloid-shaped brachistochrones occur as the similarity coefficient tends to infinity. For some values of the similarity coefficient, numerical results are presented that demonstrate the shape of the corresponding brachistochrones and the optimal time of motion. The results are compared with those obtained by solving the classical brachistochrone problem.     

On the problem of optimal spacecraft attitude control

The problem of the optimal control of a spacecraft reorientation from an arbitrary initial position into a prescribed final angular position is studied. For optimization, we use a generalized integral index characterizing the complexity of the rotation trajectory from the viewpoint of the “distance covered,” which is the generalized rotation angle that takes into account the different weights of the spacecraft axes in the sense of expenditures (of fuel, time, or another irreplaceable resource) needed to rotate the spacecraft by the same angle. An analytical solution of this problem is obtained. Two versions of the optimal spacecraft slew maneuver problem (using the shortest trajectory) are considered—the quickest maneuver and a maneuver in the prescribed time. The optimal control problem is solved for several types of constraints on the control variables. The time of starting the deceleration is determined based on the actual motion parameters (mismatch angle and angular velocity) using the terminal control principles (based on the angular position and angular velocity measurements). An example and simulation results of the spacecraft dynamics under the optimal control are presented, which demonstrate the practical usefulness of the proposed control algorithms.     

Control of giant swing motion of a two-link horizontal bar gymnastic robot

We investigated a control method to realize the three different types of free giant swing motions produced by a two-link horizontal bar gymnastic robot. By evaluating the eigenvalues of the transitional error matrix on the Poincare plane, it was found that the stable giant swing motions could be obtained by a proposed configuration control, in which the actuated joint torque is controlled such that the measured state variables follow the reference configuration with respect to the angular position of the passive joint. We also demonstrated that the two types of stable giant swing motions could be accomplished by the configuration method.     

分区四元数姿态控制

提出了一种基于分区控制策略的四元数姿态控制律. 其基本思想是基于姿态四元数误差分区设计目标角速度, 由此将问题降阶为一个角速度跟踪问题; 基于不同的角速度跟踪误差, 设计了切换类型的抗干扰姿态控制律. 该控制策略可以使得姿态快速收敛, 并且在合适的参数选择条件之下还能同时满足控制力矩的饱和约束. 通过综合相平面和Lyapunov函数的分析方法严格证明了闭环系统全局收敛的性质. 最后, 通过数值仿真验证了本文提出的控制方案的有效性.     

On optimal spacecraft damping

The problem of spacecraft damping (damping of initial angular velocity to zero) for a minimal time is studied. Two variants of formulation of the optimization problem are considered; these variants differ in the form of constraints on the control torque. Analytical solution to the formulated problem is obtained in the closed form and numerical expressions for synthesis of optimal angular velocity control program are given. Similar problem of time-optimal angular acceleration of the spacecraft to the given value is also solved. Procedure for determination of the control torque at the initial time instant for the problem of acceleration of the spacecraft to the required angular velocity is presented. Numerical example of solution of the problems of buildup and damping of spacecraft rotation velocity for a minimal time is given.     

行为受限下Acrobot起摆平衡控制研究

Acrobot是一种典型的二自由度欠驱动机械系统,针对实际中可能存在的杆二摆角及驱动力矩受限的问题,研究了行为受限下系统的起摆与平衡控制问题.将整个过程分为起摆和平衡2个阶段,首先利用垂直向上不稳定平衡点处近似线性模型,设计了基于LQR方法的平衡控制器,并以减小平衡控制过程力矩为目标讨论了切换姿态的优化问题;起摆控制采用了基于参考路径的能量泵入法,提出了一个改进的参考路径,采用部分反馈线性化方法实现对参考路径的跟踪,可以保证摆角在一定范围内摆动的情况下系统机械能不断增加,使起摆结束时具有更优的切换姿态,从而更容易切换至平衡控制阶段,同时使得平衡控制过程更加平稳,所需力矩更小.最后给出了实物实验结果,证明了所提控制策略的有效性.     

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.     

射频平面阵列目标位置控制系统的研究

阵列式射频仿真技术开始用于论证大型设备在复杂电磁环境下的效能评估试验,常规单一试验中心的球面阵列已难以适应此类试验,迫切需要构造一种新型阵列系统来完成新的试验,为此提出采用射频平面阵列式目标位置控制方案.设计并制作了平面阵列结构,论述了目标位置控制系统的阵列馈电方案以及软件模块,重点阐述了精位控制算法和粗位控制原理.提出四元组合成方案实现精位控制,并推导了四元组合成位置的角闪烁方程.最后在微波暗室内测试了典型频段下该系统的辐射信号合成角位置精度,结果表明该阵列系统在X波段的角位置模拟精度可达1 mrad,能够用于大型设备的效能评估试验.     

Quasi-Static Motion of a Two-Link System along a Horizontal Plane

Quasi-static motion of a two-link system along a rough horizontal plane is considered. Dry friction acts between the links of the system and the plane. During the motion, the configuration of the linkage changes, whereas the points of contact of the linkage and the plane remain the same. The control torque applied at the joint of the linkage is chosen so as to provide equilibrium condition for each of the links. It is shown that the quasi-static motion of the two-link system is uncontrollable and the uniqueness of this motion is proved for given initial position of the linkage. The trajectories of the vertices are found, depending on the parameters of the system.     

Optimal bang-bang control of linear stochastic systems with a small noise parameter

This study considers the problem of determining optimal feedback control laws for linear stochastic systems with amplitude-constrained control inputs. Two basic performance indices are considered, average time and average integral quadratic form. The optimization interval is random and defined as the first time a trajectory reaches the terminal regionR. The plant is modeled as a stochastic differential equation with an additive Wiener noise disturbance. The variance parameter of the Wiener noise process is assumed to be suitably small. A singular perturbation technique is presented for the solution of the stochastic optimization equations (second-order partial differential equation). A method for generating switching curves for the resulting optimal bang-bang control system is then developed. The results are applied to various problems associated with a second-order purely inertial system with additive noise at the control input. This problem is typical of satellite attitude control problems.     

自旋卫星喷气姿态控制动力学

本文讨论了自旋卫星向任意目标姿态机动的动力学问题.文中导出了一组受一串脉冲控 制力矩作用的卫星姿态运动的解析表达式.这些公式描述了角动量、章动角和姿态相位角与 控制力矩、控制次数之间的变化规律,可以直接应用于卫星姿控系统和姿态确定设计. 必须指出的是:关于将卫星自旋轴控制到垂直于轨道平面或将自旋轴控制到指向太阳, 都是本文所得结果的两个特例.     

Terminal synthesis of orbital orientation for a spacecraft

This paper presents an algorithm for the terminal synthesis of the orbital orientation for a spacecraft. The algorithm is based on solving the following problems: the analytical determination of the program values for the angular velocity vector components in an orbital coordinate system and the stabilization of the program values for the turn rate vector components and the desired terminal angular position of a spacecraft. The program values of the angular velocity vector components are determined analytically using the method previously proposed by the authors for solving a boundary value problem based on the parameter identification of a discrete model with the use of modal control. The simulation results demonstrating the effectiveness of the proposed algorithm, as well as the high accuracy level of the steady-state control, are presented.     

Neural network learning from demonstration and epipolar geometry for visual control of a nonholonomic mobile robot

The control of a robot system using camera information is a challenging task regarding unpredictable conditions, such as feature point mismatch and changing scene illumination. This paper presents a solution for the visual control of a nonholonomic mobile robot in demanding real world circumstances based on machine learning techniques. A novel intelligent approach for mobile robots using neural networks (NNs), learning from demonstration (LfD) framework, and epipolar geometry between two views is proposed and evaluated in a series of experiments. A direct mapping from the image space to the actuator command is conducted using two phases. In an offline phase, NN–LfD approach is employed in order to relate the feature position in the image plane with the angular velocity for lateral motion correction. An online phase refers to a switching vision based scheme between the epipole based linear velocity controller and NN–LfD based angular velocity controller, which selection depends on the feature distance from the pre-defined interest area in the image. In total, 18 architectures and 6 learning algorithms are tested in order to find optimal solution for robot control. The best training outcomes for each learning algorithms are then employed in real time so as to discover optimal NN configuration for robot orientation correction. Experiments conducted on a nonholonomic mobile robot in a structured indoor environment confirm an excellent performance with respect to the system robustness and positioning accuracy in the desired location.     

Analytic Controlling Reorientation of a Spacecraft Using a Combined Criterion of Optimality

We consider the problem of optimally controlling the reorientation of a spacecraft (SC) from an arbitrary initial angular state into a given final angular position. We study the case when the minimized functional joins, in the given proportion, the time spent and the integral of the squared modulus of the angular momentum on the reorientation of a SC. The problem is solved in a kinematic setting. We consider two versions of the problem of the optimal rotation of a SC, with bounded and unbounded control. Using the necessary optimality conditions in the form of the Pontryagin maximum principle and the quaternion method for solving control problems on the motion of spacecrafts, we obtain an analytical solution of the posed problem. The solution of the problem is based on the quaternionic differential equation relating the angular momentum vector of a SC with the orientation quaternion of the related coordinate system. We present formalized equations and give computational expressions for constructing the optimal control program. We state the control law as an explicit dependence of the control variables on the phase coordinates. Using the transversality condition as a necessary optimality condition, we determine the maximal value of the modulus of the angular momentum for the optimal motion. For a dynamically symmetric SC, the problem of reorientation in space is solved completely: we obtain the dependences for the optimal law of the change of the angular momentum vector as explicit time functions. We give the results of the mathematical modeling of the motion for optimal control which demonstrate the practical realizability of designed algorithm for controlling the spatial orientation of a SC.     

Optimal attitude control of two pivotally connected bodies in the supportless phase of motion

The motion of two pivotally connected bodies in the supportless phase (weightlessness) is investigated. The minimization and maximization of the slew time from a given initial position to the given terminal position is solved. It is assumed that the angular momentum of the system about its center of mass is distinct from zero. This problem is a simple model of attitude control of a hopping machine in the supportless phase of the hop.     

Quasi-optimal deceleration of rotations of a rigid body with a moving mass in a resistive medium

The problem of time quasi-optimal deceleration of the rotations of a dynamically symmetric rigid body is studied. It is assumed that the body contains a point mass connected to it with a strong viscoelastic element (damper). The body is acted on by a small linear resistance torque of the medium that is proportional to the angular momentum and a small control torque bounded by an ellipsoidal domain. An approximate synthesis of control is proposed, and an asymptotic solution based on a procedure of averaging the precession motion over the phase is obtained; numerical integration is performed. The main properties of the quasi-optimal motion are determined.     

Adaptive output feedback control of synchronous motors

This paper solves the problem of position tracking for a permanent magnet synchronous motor when the angular velocity is not measured and all the parameters of the dynamic equations are unknown. The mathematical model is expressed in the reference fixed to the stator. The adaptation algorithm gives a degree of robustness to the control scheme since it enables one to identify the motor's parameters under the hypothesis of persistent excitation, which is always satisfied when the unknown parameters are only the stator resistance and the load torque. The identification is exponential in these cases. Moreover, it is shown that the closed-loop trajectory corresponding to the nominal parameter values is not influenced by variations of stator resistance and load torque. The performances of the proposed controller are compared with those of a controller which solves the tracking problem with an observer for the angular velocity, designed on the basis of the motor's nominal parameters.