Fastest recovery from feedback loss: Bounded overshoot

The problem of designing robust optimal controllers to reduce in minimal time operating errors that had accumulated during a period of feedback loss is revisited, with the objective of imposing a constraint on the maximal overshoot of the controlled system. It is shown that robust optimal controllers that satisfy this constraint exist under rather broad conditions. It is also shown that optimal performance can be closely approximated by bang–bang controllers – controllers that are relatively easy to design and implement.     

Optimal robust control of nonlinear time-delay systems: Maintaining low operating errors during feedback outages

The problem of maintaining low operating errors during feedback outages is considered for a class of nonlinear systems with time-delays in the input channel. It is shown that there are optimal controllers that keep operating errors below a specified bound for the longest time possible. Furthermore, it is shown that optimal performance can be approximated as closely as desired by using bang-bang controllers – controllers that are relatively easy to calculate and implement.     

Fastest recovery after feedback disruption: nonlinear delay-differential systems

The problem of reducing operating errors after a feedback disruption is considered for a class of nonlinear delay-differential systems. It is shown that there are optimal controllers that reduce such errors in minimal time, once feedback has been restored. It is further shown that optimal performance can be approximated by bang-bang controllers – controllers that are easy to design and implement.     

Bang–bang control model with optimistic value criterion for uncertain switched systems

This paper studies an optimal control problem for uncertain switched linear systems with subsystems perturbed by uncertainty. A model for this problem is investigated with optimistic value criterion. The goal is to jointly design a deterministic switching law and a continuous feedback to optimize an uncertain objective function. A two-stage algorithm is applied to handle such model. In the first stage, the maximum value of the objective function and the bang–bang control are obtained under fixed switching instants, and in the second stage, GA and PSO algorithm are used to get the optimal switching instants, respectively. An example is shown to validate the method.     

Robust Optimal Control of Nonlinear Systems With System Disturbance During Feedback Disruption

In this paper, a robust optimal control problem of nonlinear systems with system disturbance during feedback disruption is considered. This is an extended work of previous time‐delay optimal control results, by adding external disturbance in the considered system. It is shown that there exists an optimal input signal which keeps the performance error within the specified bound for the longest time. Then, it is shown that such an optimal input signal can be approximated by an implementable bang‐bang input signal in terms of control performance. Two examples are given for illustration.     

Robust adaptive feedforward control and achievable tracking for systems with time delays

A feedback/feedforward controller architecture is developed that characterises the achievable reference tracking of real time inputs for both minimum phase and non-minimum phase systems with time delays, when there are no modelling errors or external disturbances. This characterisation is obtained by factoring the plant into its minimum phase, non-minimum phase, and time delay components, which are used to design two feedforward controllers that inject signals into two points of the feedback loop. Design constraints are provided that determine both the types of signals that may be achieved, and the feedforward controllers that will generate that output. Of course, in practice, both modelling errors and external disturbances will be present. In this case, we develop robust analysis tools that both guide the feedback controller design process, and provide rigorous robust tracking performance that guarantees for the overall resulting closed-loop system. Robust methods for designing the feedforward controllers are presented, and numerical examples are provided. The performance of this architecture depends strongly on the choice of design parameters, and the accuracy of the plant model used. Hence, the use of adaptation methods is also considered, and it is shown that they can readily be employed to improve the performance of this control methodology.     

Optimality and stability in a class of bang–bang controlled biochemical reaction systems

This paper deals with the optimization of biochemical reaction systems of rank one. Two optimization problems are solved: the problem of optimal operation for maximum productivity in steady state and the problem of the start-up to the optimal steady state. Application of Pontryagin's maximum principle shows that the controller is of the bang–bang type, with no singular intervals. The determination of the optimal switching surface involves the solution of a two point boundary value problem. Solving such a difficult problem is avoided by choosing candidate switching surfaces on a heuristic basis. This study shows that switching on the stability boundary of the nominal operating point corresponding to the maximum dilution rate is the best choice. Here the value of the cost index is minimum amongst the various switching surfaces considered and the stability boundary satisfies the conditions imposed on a candidate switching surface for proper operation. Simple, robust algorithms are formulated for accurately estimating the system's stability boundary. The obtained results display the influence of feedback control on the stability of the set point. The bang–bang controller substantially increases the set point's region of attraction in state space as compared to the uncontrolled bioreactor.     

Distributed tracking control of networked chained systems

This paper considers distributed tracking control of multiple nonholonomic chained systems using neighbours’ information. With the aid of the cascade structure of each system and properties of persistently excited signals, distributed state feedback tracking controllers and distributed output feedback tracking controllers are proposed such that the tracking errors exponentially converge to zero. To show applications of the proposed results, formation control of wheeled mobile robots is considered. Distributed controllers are obtained with the aid of the proposed theorems. Simulation results verify the effectiveness of the proposed results.     

Periodic sampling: maximising the sampling period

The process of periodic sampling is investigated for a class of nonlinear systems. The objective is to achieve the longest sampling period that is compatible with a specified error bound. It is shown that there are robust optimal controllers that achieve this objective. It is also shown that the performance of such optimal controllers can be approximated by bang-bang controllers – controllers that are relatively easy to design and implement.     

基于微分对策的最优状态观测器和最优状态反馈控制器的设计

研究了线性系统基于二次型指标的最优状态观测器和最优状态反馈控制器的设计问题．将观测状态的状态反馈和状态误差的输出反馈分别作为两个对局方,应用微分对策理论研究了系统的最优控制问题．给出了最优状态观测器和基于状态观测器的最优状态反馈控制的存在性条件．将系统的最优状态观测器和最优控制器的设计问题转化为一对Riccati方程的求解问题．研究表明最优状态观测器在一般情况下不存在．并进一步研究了基于状态观测器的次优控制问题,给出了基于LMI的优化算法．     

Modified bang‐bang controller for maximal and minimal time optimal control problems

Recently, there have been a series of results regarding two time optimal control problems for a class of linear and nonlinear systems ‐ one is to keep the system states within certain bound for the longest time during feedback disruption and the other is to derive the system states to near the origin as fast as possible after feedback recovery, both under bounded control inputs. These are called maximal and minimal time optimal control problems, respectively. In the existing results, a bang‐bang controller has been commonly suggested as the actual implementation of the optimal controller. In this paper, we suggest a modified version of the bang‐bang controller which can also serve as an approximate optimal controller. Our proposed controller provides the (near) optimal performance with (i) possible reduction of a number of switchings; (ii) possible reduction of control input magnitude.     

Online optimal control of nonlinear discrete-time systems using approximate dynamic programming

In this paper,the optimal control of a class of general affine nonlinear discrete-time(DT) systems is undertaken by solving the Hamilton Jacobi-Bellman(HJB) equation online and forward in time.The proposed approach,referred normally as adaptive or approximate dynamic programming(ADP),uses online approximators(OLAs) to solve the infinite horizon optimal regulation and tracking control problems for affine nonlinear DT systems in the presence of unknown internal dynamics.Both the regulation and tracking contro...     

Reliability-based robust Pareto design of linear state feedback controllers using a multi-objective uniform-diversity genetic algorithm (MUGA)

In this paper, fuzzy threshold values, instead of crisp threshold values, have been used for optimal reliability-based multi-objective Pareto design of robust state feedback controllers for a single inverted pendulum having parameters with probabilistic uncertainties. The objective functions that have been considered are, namely, the normalized summation of rising time and overshoot of cart (S_RO–C) and the normalized summation of rising time and overshoot of pendulum (S_RO–P) in the deterministic approach. Accordingly, the probabilities of failure of those objective functions are also considered in the reliability-based design optimization (RBDO) approach. A new multi-objective uniform-diversity genetic algorithm (MUGA) is presented and used for Pareto optimum design of linear state feedback controllers for single inverted pendulum problem. In this way, Pareto front of optimum controllers is first obtained for the nominal deterministic single inverted pendulum using the conflicting objective functions in time domain. Such Pareto front is then obtained for single inverted pendulum having probabilistic uncertainties in its parameters using the statistical moments of those objective functions through a Monte Carlo simulation (MCS) approach. It is shown that multi-objective reliability-based Pareto optimization of the robust state feedback controllers using MUGA with fuzzy threshold values includes those that may be obtained by various crisp threshold values of probability of failures and, thus, remove the difficulty of selecting suitable crisp values. Besides, the multi-objective Pareto optimization of such robust feedback controllers using MUGA unveils some very important and informative trade-offs among those objective functions. Consequently, some optimum robust state feedback controllers can be compromisingly chosen from the Pareto frontiers.     

On the use of positive feedback for improved torque control

This paper considers the torque control problem for robots with flexible joints driven by electrical actuators. It is shown that the achievable closed-loop tracking bandwidth using PI torque controllers may be limited due to transmission zeros introduced by the load dynamics. This limitation is overcome by using positive feedback from the load motion in unison with PI torque controllers. The positive feedback is given in terms of load velocity, acceleration and jerk. Stability conditions for designing decentralized PI torque controllers are derived in terms of Routh-Hurwitz criteria. Disturbance rejection properties of the closed system are characterized and an analysis is carried out investigating the use of approximate positive feedback by omitting acceleration and/or jerk signals. The results of this paper are illustrated for a two DoF (degrees of freedom) system. Experimental results for a one DoF system are also included.     

Optimal control during feedback failure

The problem of controlling a perturbed open loop system so as to keep its performance errors within bounds is considered. The objective is to maximise the time during which performance errors remain below a prescribed ceiling, while the controlled system's parameters are within a specified neighbourhood of their nominal values. It is shown that there is an optimal open loop controller that achieves this objective. Conditions under which the optimal controller generates a bang-bang control input signal are characterised. In general, it is shown that the performance of the optimal controller can always be approximated by a bang-bang signal.     

输入随机的不可靠柔性制造系统的反馈控制

对于具有随机输入和随机需求的一类不可靠柔性制造系统,利用转移率一致化技术和随机动态规划方法,给出了输入率和服务率分配的最优反馈控制策略,指出系统的最优控制具有bang-bang形式的天关结构,数值例子验证了文中的结果。     

Universal controllers in model matching optimal control problems for unknown external signals

Linear-quadratic model matching optimal control problems are studied for unknown external signals and noise in the control plant. Given natural assumptions for two practically significant classes of external signals (stochastic with an unknown, rapidly decreasing spectral density and polyharmonic with the known spectrum and unknown amplitudes), controllers universal in the sense that they generate an optimal or approximately optimal process in the problem for any signal from the stated class are proved to exist. Linear controllers are shown to exist among them, with the class of linear universal controllers described exhaustively.     

Optimal control of saturable linear plants for minimum integral-square-error and similar performance criteria†

The paper deals with the problem of minimizing integral–square–error and similar performance indices for linear single–variable plants with saturating control input. It is shown that the optimal control can be obtained by treating it as the limit as υ→∞ of the Sequence of controls which minimize the performance index on the class of all bang–bang controls containing not more than υ switches. These optimal υ switch controls are shown to satisfy a modified maximum principle.The method enables the computation of optimal controls from any given initial plant state. The particular problems of minimizing integral–modulus–error for a two–integrator plant and integral–square–error for a three–integrator plant are worked out in detail so as to illu3trata the techniques involved.     

面向反馈运动控制器的多目标求解

目的 基于物理模拟的人体运动生成方法由于能够合成符合自然规律的运动片段,可实时响应环境的变化,且生成的物理运动不是机械性的重复,因此是近年来计算机动画和虚拟现实领域中最活跃的研究方向之一。然而人体物理模型具有高维、非线性及关节间强耦合性等特点,求解人体物理运动十分困难。反馈控制器常用于人体物理运动控制,求解时通常需要对多个目标函数加权求和,然而权重的设置需多次试验,烦杂耗时。针对运动控制器求解困难的问题,本文提出了一种面向反馈运动控制器的多目标求解方法。方法 首先,对运动数据进行预处理并提取关键帧求解初始控制器,并设计一种改进的反馈控制机制;在此基础上,种群父代个体变异产生子代,采用禁选区域预筛选策略去除不满足约束的个体,并通过重采样获取新解;然后,通过物理仿真获得多目标适应度值,采用区域密度多层取优选取分布均匀的优秀个体作为下一代父代,并通过基于剪枝的多阶段物理求解算法决定是否进入下一阶段优化;经过多次迭代后获得物理控制器,从而生成具有反馈的人体物理运动。结果 针对提出的方法,本文针对多个测试函数和物理运动分别进行实验：在测试函数实验中,本文分别采用经典的测试函数进行实验对比,在相同的迭代次数下,相比之前算法,本文算法中满足约束的优秀个体命中率更高,反转世距离更小,且最优解集的分布更加均匀;物理运动生成实验中,分别针对走路、跑步和翻滚等运动进行物理运动生成,与之前算法进行对比,本文算法可以更早地完成收敛,同时目标函数值更小,表明生成的运动效果更好。结论 本文提出的进化求解方法可以生成不同运动的控制器,该控制器不仅可以生成物理运动,而且还具备外力干扰下保持平衡的能力,解决了运动控制器求解中多目标权重设置困难、优化时间长的问题;除此之外,本文算法还对具有约束的多目标问题具有较好的求解效果。     

LQ bumpless transfer between two tracking controllers

In this article, a linear quadratic (LQ) scheme is proposed for bumpless transfer between two tracking controllers. The controllers have integral actions which are used to eliminate the steady-state tracking errors. A compensator is designed using the LQ technique to minimise the differences between two sets of signals, the differences between two control signals as well as the differences between the two input signals which drive the two controllers. Under such circumstances, complete information of the system is taken into account and utilised for achieving satisfactory control. It is shown that the ideal bumpless transfer is achievable with the proposed control scheme. Moreover, the results are extended to discrete-time systems. Finally, simulation results are given to demonstrate the effectiveness of the proposed control scheme.