Dynamic shape control for a flexible beam

In this paper we discuss the dynamic shape control for a flexible beam such that even under unknown step disturbances, the shape of the beam will track the desired shape after a certain regulation time. In order to treat the case of point controls and point observations, we formulate the flexible beam by a second-order partial differential equation on a Hilbert space H^?1(0, 1). We apply the theory of multivariable tuning regulators to the dynamic shape control problem. Two kinds of methods will be proposed for the determination of an integral feedback gain matrix M. Simulation results will show the efficiency of the theory developed in this paper.     

On the optimal design of elastic beams

In this paper we investigate the optimal dynamics of simply supported nonlinearly elastic beams with rectangular cross-sections. We consider the elastic beam under the assumption of time-dependent intensive transverse loading. The state of the beam is described by a system of partial differential equations of the fourth order. We deal with the problem of choosing the optimal shape for the beam. The optimal shape is determined in such a way that the deflection of the nonlinearly elastic beam for any given time is minimal. The problem of choosing the optimal shape is formulated as an optimal control problem. To solve the obtained problem effectively, we use the optimality principle of Bellman (Bellman and Dreyfus 1962; Bryson and Ho 1975) and the penalty function method (Polyak 1987). We present a constructive algorithm for the optimal design of nonlinearly elastic beams. Some simple examples of the implementation of the proposed numerical algorithm are given.     

Optimal distributed real-time scheduling of flexible manufacturing networks modeled as hybrid dynamical systems

The paper considers a class of flexible manufacturing networks. We employ hybrid dynamical systems to model such networks. The main and new achievement of the paper is that we propose a distributed implementable in real time scheduling rule such that the corresponding closed-loop system is stable and optimal. In stable systems the processes converge to periodic ones. The paper gives computing relations for the determination of the parameters of the periodic processes. These are very much suitable for planning purposes. On this basis—and this, we consider, is also a new, significant result—optimal arrival (demand) rates determination method is proposed. Quality characteristics are outlined. Field of application of hybrid dynamical approach for FMS scheduling is analyzed. The results open perspectives for MRP level task planning. Example and simulation results are presented.     

Optimal control of a single-link flexible manipulator

This paper deals with an exact state space dynamic model for manipulators with flexible links. We use the Bernoulli-Euler beam equations to derive a frequency domain matrix transfer function. This transfer function is then used to compute the Laplace transform of the state vector as a function of the lateral position along a single link manipulator. The problem of optimal end point control of the beam is then addressed. A sixth-order state space model is derived for the manipulator and the controller is based on this model. Several control laws are studied for this model. Next, the manipulator is modeled as eighth order but the control law based on the sixth-order model is retained. We then estimate the six states from the output of the eighth-order model and feed these states back to the controller to derive the control torque used to drive the manipulator. A filter is introduced to compensate for spillover. The results are very satisfactory, and are illustrated by simulated case studies.     

Optimal control computation for nonlinear systems with state-dependent stopping criteria

In this paper, we consider a challenging optimal control problem in which the terminal time is determined by a stopping criterion. This stopping criterion is defined by a smooth surface in the state space; when the state trajectory hits this surface, the governing dynamic system stops. By restricting the controls to piecewise constant functions, we derive a finite-dimensional approximation of the optimal control problem. We then develop an efficient computational method, based on nonlinear programming, for solving the approximate problem. We conclude the paper with four numerical examples.     

Semantic approximation of data stream joins

We consider the problem of approximating sliding window joins over data streams in a data stream processing system with limited resources. In our model, we deal with resource constraints by shedding load in the form of dropping tuples from the data streams. We make two main contributions. First, we define the problem space by discussing architectural models for data stream join processing and surveying suitable measures for the quality of an approximation of a set-valued query result. Second, we examine in detail a large part of this problem space. More precisely, we consider the number of generated result tuples as the quality measure and we propose optimal offline and fast online algorithms for it. In a thorough experimental study with synthetic and real data, we show the efficacy of our solutions.     

Perturbation of strongly and polynomially stable Riesz-spectral operators

In this paper we consider bounded and relatively bounded finite rank perturbations of a Riesz-spectral operator generating a polynomially stable semigroup of linear operators on a Hilbert space. We concentrate on a commonly encountered situation where the spectrum of the unperturbed operator is contained in the open left half-plane of the complex plane and approaches the imaginary axis asymptotically. We present conditions on the perturbing operator such that the spectrum of the perturbed operator is contained in the open left half-plane of the complex plane and additional conditions for the strong and polynomial stabilities of the perturbed semigroup. We consider two applications of the perturbation results. In the first example we apply the results to the perturbation of a polynomially stabilized one-dimensional wave equation. In the second example we consider the perturbation of a closed-loop system consisting of a distributed parameter system and an observer-based feedback controller solving the robust output regulation problem related to an infinite-dimensional signal generator.     

Adaptive hybrid control of manipulators on uncertain flexible objects

This paper presents an adaptive hybrid control approach for a robot manipulator to interact with its flexible object. Because of its flexibility, the object dynamics influence the robot's control system, and since it is usually a distributed parameter system, the object dynamics as seen from the robot change when the robot moves. The problem becomes further complicated such that it is difficult to decompose the robot's position and contact force control loops. In this paper, we approximate the object's distributed parameter model into a lumped 'position state-varying' model. Then, by using the well-known nonlinear feedback compensation, we decompose the robot's control space into a position control subspace and object torque control subspace. We design the optimal state feedback for the position control loop and control the robot's contact force through controlling the resultant torque of the object. We use the model-reference simple adaptive control strategy to control the torque control loop. We also study the problem on how to select a reasonable reference model for this control loop. Experiments of a PUMA robot interacting with an aluminum beam show the effectiveness of our approach.     

中心刚体-柔性梁应变反馈多目标优化控制

本文针对柔性结构在轨组装任务背景,将自动飞船和待运送部件分别简化为中心刚体和柔性梁附件,研究自由漂浮中心刚体-柔性梁系统的动力学与控制问题.首先基于小变形和低转速假设,将梁用假设模态法离散,采用Lagrange方程导出系统动力学方程.继而设计一种带有应变反馈的PD控制律用于完成中心刚体的状态镇定以及柔性梁振动抑制.此外,本文还使用遗传算法对控制器中的参数进行多目标优化.最后,通过数值算例验证了所设计控制器的有效性.     

Feedback control of decoupled bending and torsional vibrations of flexible beams

In this article, we consider a flexible beam with a rigid body, of which the bending vibration and the torsional vibration are decoupled. Therefore we need two control motors to suppress the vibrations. Each set of dynamic equations is treated in the form of an appropriate Hilbert space. A stabilizing feedback control law of each rotation motor will be established on the basis of modal analysis. A set of experiments has been carried out, and the results demonstrate the effectiveness of the dynamic model and the proposed control law. © 1994 John Wiley & Sons, Inc.     

Distributed control of a class of flexible mechanical systems with global constraint

In this paper, the constrained problem is investigated for both flexible string model and Euler–Bernoulli beam model with the tip payload, based on an infinite dimensional generalisation of a distributed control method. The control objectives are to develop the control law so that the motion of flexible mechanical systems can track a desired reference signal, and ensure that the string or beam remain in a constrained space. We prove that, with the proposed control, the tracking error is exponentially stable without violation of the constraint. The proof of convergence is based on an Integral-Barrier Lyapunov Function (IBLF), and extensive simulations are provided to illustrate the performance of the control system.     

Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin–Voigt damping

We consider the beam equation coupled by a transmission condition with a wave equation in an elastic beam. The beam has clamped boundary conditions and the wave equation has Dirichlet boundary conditions. The damping which is locally distributed acts through one of the equations only; its effect is transmitted to the other equation through the coupling. First, we consider the case where the dissipation acts through the beam equation. We show that in this case the coupled system is polynomially stable by using a recent result of Borichev and Tomilov on polynomial decay characterisation of bounded semigroups, we provide precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this system is also polynomially stable and we provide precise polynomial decay estimates for its energy. Finally, we show the lack of uniform exponential decay of solutions for both models.     

Optimisation of a Simulated-Annealing-based Heuristic for Single Row Machine Layout Problem by Genetic Algorithm

We discuss a procedure to determine the optimal set of parameters relevant to heuristics based on the Simulated Annealing technique, an algorithm which is widely applied to combinatorial problems in the field of manufacturing systems. We consider the search for the best set as a second optimisation problem that we solve by a Genetic Algorithm. The performance of our approach is tested in the particular case of backtracking minimisation in a single row machine layout problem for flexible manufacturing systems.     

Optimal reservation policy for two queues in tandem

In this paper, we consider the optimal reservation problem for two multi-server loss queues in tandem, which is usually used to model network and communication systems. In previous work [European J. Oper. Res. 21 (1985) 399-409; Ph.D. diss., 1995; IEEE Trans. Automatic Control 42 (1997) 1017-1023], under appropriate conditions, the optimal reservation policy that maximizes the expected total discounted reward over an infinite horizon is shown to be a switching curve in two-dimensional state space. However, some counterintuitive examples for variation do exist in the numerical experiments. We therefore discuss and analyze the variation in these switching curves with the number of customers in the system. We propose two sufficient conditions under which the counterintuitive situation will not occur.     

Stability and robustness analysis of boundary control of a flexible space structure

We consider the question of the robustness of boundary control for a simplified SCOLE (spacecraft control laboratory experiment) model. A continuum model consisting of a long flexible mast joining two rigid bodies, one of which represents the space shuttle orbiter, the other an antenna reflector, is developed. The dynamics of the system are represented by a coupled set of ordinary and partial differential equations, and formulated as a wave equation in a Hilbert space by using a semigroup approach. For a continuum, distributed controls have often been proposed to stabilize the perturbed system. However, distributed controls for the continuous system are not practical to implement. Hence, for easy implementation we employ the boundary controls, as a special class of localized controls, that are control forces or torques applied only at the boundary. In this paper it is shown that the flexible system subject to a certain class of perturbations remains stabilizable by the same boundary feedback control as that for the nominal system. The results of numerical simulations for the simplified model are also presented to demonstrate the effective strong stabilization and robustness of the boundary controller with respect to the various perturbations.     

柔性臂振动系统角速度反馈控制的最优指数衰减率问题

讨论柔性臂的端点角速度反馈控制问题。通过对系统特征值和特征函数的渐近表示式的进一步研究,用特征扰动的Payley-Wiener稳定性理论,证明了该系统的最优指数衰减率可由系统的谱来确定。     

Dynamics and Noncollocated Model‐Free Position Control for a Space Robot with Multi‐Link Flexible Manipulators

In this paper, both the dynamics and noncollocated model‐free position control (NMPC) for a space robot with multi‐link flexible manipulators are developed. Using assumed modes approach to describe the flexible deformation, the dynamic model of the flexible space robotic system is derived with Lagrangian method to represent the system dynamic behaviors. Based on Lyapunov's direct method, the robust model‐free position control with noncollocated feedback is designed for position regulation of the space robot and vibration suppression of the flexible manipulators. The closed‐loop stability of the space robotic system can be guaranteed and the guideline of choosing noncollocated feedback is analyzed. The proposed control is easily implementable for flexible space robot with both uncertain complicated dynamic model and unknown system parameters, and all the control signals can be measured by sensors directly or obtained by a backward difference algorithm. Numerical simulations on a two‐link flexible space robot are provided to demonstrate the effectiveness of the proposed control.     

Optimal Control of Underactuated Nonholonomic Mechanical Systems

In this paper, we use an affine connection formulation to study an optimal control problem for a class of nonholonomic, underactuated mechanical systems. In particular, we aim to minimize the norm-squared of the control input to move the system from an initial to a terminal state. We consider systems evolving on general manifolds. The class of nonholonomic systems we study in this paper includes, in particular, wheeled-type vehicles, which are important for many robotic locomotion systems. The two special aspects of this optimal control problem are the nonholonomic constraints and underactuation. Nonholonomic constraints restrict the evolution of the system to a distribution on the manifold. The nonholonomic connection is used to express the constrained equations of motion. Many robotic systems are underactuated since control inputs are usually applied through the robot's internal configuration space only. While we do not consider symmetries with respect to group actions in this paper, the fact that the system is underactuated is taken into account in our problem formulation. This allows one to compute reaction forces due to any inputs applied in directions orthogonal to the constraint distribution. We illustrate our ideas by considering a simple example on a three-dimensional manifold, including obstacle avoidance using the method of navigation functions.     

Inventory models for multi-warehouse systems under fixed and flexible space leasing contracts

In this paper, we analyze a practical situation in which an inventory manager is faced with several options to store excess stocks whenever the storage capacity of his/her warehouse is insufficient. The manager can choose from either storage space providers through fixed long term or flexible leasing contracts, or the manager can acquire the extra required space from the spot market. We formulate this inventory problem with multiple storage facilities as a nonlinear program and show that it has a global optimal solution. We then provide closed-form solutions for the optimal ordering quantity and leased spot market space depending on the value of the unconstrained economic order quantity. In addition, we develop some structural properties for the optimal ordering policy and include several examples to illustrate the formulated models.     

Stochastic system controller synthesis for reachability specifications encoded by random sets

We consider a reach–avoid specification for a stochastic hybrid dynamical system defined as reaching a goal set at some finite time, while avoiding an unsafe set at all previous times. In contrast with earlier works which consider the target and avoid sets as deterministic, we consider these sets to be probabilistic. An optimal control policy is derived which maximizes the reach–avoid probability. Special structure on the stochastic sets is exploited to make the computation tractable for large space dimensions.