Robust optimal control: low-error operation for the longest time

The problem of maximising the time during which an open-loop system can operate without exceeding a specified error bound is considered for linear systems that are subject to uncertainties about their parameters and their initial conditions, and whose operation is hampered by disturbance signals. The objective is to characterise an optimal input signal that keeps performance errors within specified bounds for the longest time. It is shown that such an input signal exists, and that it can be approximated by a bang-bang input signal without significantly affecting performance.     

Robust optimal control during feedback disruption for nonlinear systems controlled by an observer-based output feedback controller

In this paper, a robust optimal control problem during feedback disruption is considered for a class of nonlinear systems which have been controlled by an observer-based output feedback controller. It is shown that during feedback disruption, there exists an optimal control input which keeps both system states and observer errors within a specified bound for the longest time. Then, it is shown that such an optimal control input can be practically implemented by using a bang-bang control input in terms of control performance. One numerical and one practical examples are given for clear illustration.     

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.     

Optimal control of a solar collector loop using a distributed-lumped model

This paper considers the optimal control of a solar collector loop described by a bilinear distributed parameter model for the collector fluid temperature and a bilinear lumped parameter model for the storage fluid temperature. The objective is to control the collector fluid velocity so as to maximize the net energy collected over a fixed time period. Necessary conditions for optimality, given by a set of equations whose solution yields the optimal control, are derived. It is shown that the optimal control is an open-loop, bang-bang control which depends on two terms: a measurable quantity which depends on the state of the collector fluid, and a quantity which depends on a future knowledge of the weather data. It is also shown that for the case in which only two switches occur during the period of operation, the optimal control depends only on the temperature difference across the collector. Thus, one can construct a feedback on/off controller for the system provided that it is known a priori that only two switches will occur during the time interval under consideration.     

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.     

For model-based control design, closed-loop identification gives better performance

We compare open loop versus closed loop identification when the identified model is used for control design, and when the system itself belongs to the model class, so that only variance errors are relevant. Our measure of controller performance (which is used as our design criterion for identification) is the variance of the error between the output of the ideal closed loop system (with the ideal controller) and that of the actual closed loop system (with the controller computed from the identified model). Under those conditions, we show that, when the controller is a smooth function of the input-output dynamics and the disturbance spectrum, the best controller performance is achieved by performing the identification in closed loop with an operating controller that we characterize. For minimum variance and model reference control design criteria, we show that this ‘optimal operating controller for identification’ is the ideal controller. This then leads to a suboptimal but feasible iterative scheme.     

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.     

Hierarchical optimal force-position control of complex manufacturing processes

A hierarchical optimal controller is developed in this paper to regulate the machining force and axis positions, simultaneously, in a micro end milling process. The process is divided into two levels of decision making. The bottom level includes the measurable states, which in this work comprises the axis positions. The top level includes the higher order objectives, which can be derived from the bottom level objectives by an aggregation relationship. In this work, the top level's objective is to regulate the machining force. A series of simulations were conducted in which the weighting between the top and the bottom level objectives is adjusted within the feasible range. The results demonstrated that excellent tracking of both axis positions and machining force are achieved during the steady state regardless of the weighting. However, the transient performance of the system could be systematically shaped to achieve better performance of either objective. For the purpose of comparison a decentralized optimal controller was constructed and simulated for the feasible range of controller weights. When the axis position errors were weighted heavily, both controllers were able to regulate the axis errors well, while the hierarchical controller had smaller machining force errors. When the machining force errors were weighted heavily, although the machining force error decreased for the decentralized controller the axis position errors increased significantly. However, with heavy machining force weighting, the hierarchical controller was able to manipulate the axial errors in a way that while the machining force error was reduced, the contour error (i.e., smallest deviation from the tool tip to the desired contour) remained small.     

Energy-Efficient Optimal Guaranteed Cost Intermittent-Switch Control of a Direct Expansion Air Conditioning System

To improve the energy efficiency of a direct expansion air conditioning (DX A/C) system while guaranteeing occupancy comfort, a hierarchical controller for a DX A/C system with uncertain parameters is proposed. The control strategy consists of an open loop optimization controller and a closed-loop guaranteed cost periodically intermittent-switch controller (GCPISC). The error dynamics system of the closed-loop control is modelled based on the GCPISC principle. The difference, compared to the previous DX A/C system control methods, is that the controller designed in this paper performs control at discrete times. For the ease of designing the controller, a series of matrix inequalities are derived to be the sufficient conditions of the lower-layer closed-loop GCPISC controller. In this way, the DX A/C system output is derived to follow the optimal references obtained through the upper-layer open loop controller in exponential time, and the energy efficiency of the system is improved. Moreover, a static optimization problem is addressed for obtaining an optimal GCPISC law to ensure a minimum upper bound on the DX A/C system performance considering energy efficiency and output tracking error. The advantages of the designed hierarchical controller for a DX A/C system with uncertain parameters are demonstrated through some simulation results.      

Design of Bang-bang Controller Based on a Fuzzy-Neuro Approach: Application to a Heating System

A fuzzy-neuro approach for the design of bang-bang controller is presented in this paper. The approach has been used with success for the time optimal bang-bang control of a heating system. The improved bang-bang controller suppresses the oscillations often observed at the output of an on-off controller. A fuzzy system is used for the implementation of the on-off control. An extension of the fuzzy control is provided by an equivalent neural network of the fuzzy system. A test application, that of a house heating with a two-state furnace, is developed and evaluated with standard hysteresis switching, fuzzy control, and fuzzy-neuro control.     

The practical implementation of time-optimal control for robotic manipulators

This paper presents experimental results for time-optimal control of robotic manipulators along specified paths. The implementation of time-optimal control represents several unique problems: (1) the control is generally discontinuous (bang-bang), (2) actuator dynamics are usually ignored in order to reduce the system order, and (3) the optimal control leaves no control authority to compensate for tracking errors caused by unmodeled dynamic and the delays introduced by the on-line feedback controller. To overcome these difficulties, we compensate for motor dynamics using a simplified friction model, and account for the dynamics of the feedback controller using trajectory preshaping. Implemented for the UCLA Direct Drive Arm, this is shown to drastically reduce the tracking errors compared to the errors obtained with no preshaping and no compensation for motor dynamics. The experimental results demonstrate the merit of time optimal control for reducing motion time as well as for increasing tracking accuracy.     

Nonlinear stochastic receding horizon control: stability,robustness and Monte Carlo methods for control approximation

This work considers the stability of nonlinear stochastic receding horizon control when the optimal controller is only computed approximately. A number of general classes of controller approximation error are analysed including deterministic and probabilistic errors and even controller sample and hold errors. In each case, it is shown that the controller approximation errors do not accumulate (even over an infinite time frame) and the process converges exponentially fast to a small neighbourhood of the origin. In addition to this analysis, an approximation method for receding horizon optimal control is proposed based on Monte Carlo simulation. This method is derived via the Feynman–Kac formula which gives a stochastic interpretation for the solution of a Hamilton–Jacobi–Bellman equation associated with the true optimal controller. It is shown, and it is a prime motivation for this study, that this particular controller approximation method practically stabilises the underlying nonlinear process.     

A novel fractional order fuzzy PID controller and its optimal time domain tuning based on integral performance indices

A novel fractional order (FO) fuzzy Proportional-Integral-Derivative (PID) controller has been proposed in this paper which works on the closed loop error and its fractional derivative as the input and has a fractional integrator in its output. The fractional order differ-integrations in the proposed fuzzy logic controller (FLC) are kept as design variables along with the input–output scaling factors (SF) and are optimized with Genetic Algorithm (GA) while minimizing several integral error indices along with the control signal as the objective function. Simulations studies are carried out to control a delayed nonlinear process and an open loop unstable process with time delay. The closed loop performances and controller efforts in each case are compared with conventional PID, fuzzy PID and PI^λD^μ controller subjected to different integral performance indices. Simulation results show that the proposed fractional order fuzzy PID controller outperforms the others in most cases.     

Fastest recovery after feedback disruption

The problem of quickly reducing operating errors during recovery from a feedback disruption is considered. The objective is to design controllers that reduce operating errors as quickly as possible, once feedback has been restored. It is shown that robust optimal feedback controllers that achieve this objective do exist. Furthermore, it is shown that the performance of optimal controllers can be approximated as closely as desired by controllers that generate bang–bang input signals for the controlled system. Controllers that generate bang–bang signals are relatively easy to derive and implement, since bang–bang signals are characterised by a finite list of scalars – their switching times.     

Continuous bounded controllers for active control of structures

This paper considers nonlinear controller under bounded capacity of the actuators. The controller is first derived based on bang-bang type controllers. In order to avoid high-speed switching effect, a continuous function is used to replace signum function in bang-bang controller. Although a variety of continuous and bounded functions are available, a hyperbolic tangent is chosen as the candidate of the controller. The proposed control law is then applied to three- and six-storey buildings utilising active bracing systems. Different earthquake excitations with various intensities are used in the simulation to show the potential benefit of the proposed control law. Numerical results show that in contrast to bang-bang type controllers that utilise maximum capacity of the actuator, the proposed controller is adaptive in the sense that the maximum capacity of the actuator is utilised when only needed. In addition, the proposed controller can avoid the high-speed switching effects presence in the bang-bang controllers if the weighting variables are chosen appropriately.     

基于PID和LQR的四旋翼无人机控制系统研究

为提高四旋翼无人机的飞行稳定性、无人飞行器控制系统的鲁棒性和控制精度,以建立的四旋翼无人机飞行控制系统模型为基础,采用现代控制理论与传统控制论相结合的方法,针对姿态角速率、姿态角分别设计内环LQR(线性二次型调节器)控制器,及外环PID控制的双回路闲环控制器.充分利用PID控制器易于掌握且对模型要求精度低、LQR控制器能改善内回路的动态特性和稳态性能的特点,完成四旋翼无人机的飞行控制.通过实验遴选该双闭环控制器相关参数并进行优化,实验结果表明所设计的双回路控制器控制性能指标良好.     

非线性广义最小方差控制律综述

如何设计简单的控制策略对复杂非线性系统进行控制是控制界还未解决的难题．非线性广义最小方差控制律的提出使得非线性控制器的设计可以基于更为一般的非线性模型，并且控制器易于实现．整个系统包含时滞环节，稳定的非线性输入子系统和一个可以用多项式或者状态空间描述的子系统．通过最小化由误差加权项、状态加权项和输入加权项组成的信号的方差得到优化控制器．在系统为开环稳定的情况下，可用史密斯预估器进行控制．本文首先介绍了非线性广义最小方差控制的发展进程，然后综述了基于状态空间和多项式描述的系统的非线性广义最小方差控制器的设计以及其现状和今后的发展方向．     

高精度磁场式时栅传感器激励信号对测量误差的影响分析及系统设计

为提高磁场式时栅传感器测量精度,本文从理论上推导分析了时栅传感器激励信号源幅值和相位不一致产生的谐波成分对时栅传感器测量精度的影响,提出了一种基于DDS原理并采用完整闭环调节的高性能时栅激励信号源设计方案。以FPGA为微处理器,通过编程分频系统时钟,设置频率、相位控制字对DDS输出的信号频率、相位进行调节,使用增益控制器配合相位累加器实现相位到幅值精确转换。搭建了信号调理电路和信号反馈电路,通过实时对比反馈控制,解决了系统电路阻抗不匹配及干扰导致的激励信号相位不正交性和幅值不一致性的问题。实验结果表明：本文所设计的激励信号源输出信号幅值相对误差只有0.4%,正交性相对误差只有0.05%,并且采用该激励信号源,磁场式时栅传感器测角原始误差从±103.4"降低到了±20.3",有效抑制由于激励信号源幅值不一致和相位不正交带来的谐波误差。经修正后对极内角位移测量误差只有±1.3",整周角位移测量精度达到±2",满足高精度位移测量要求。     

Generalized bang-bang control for feedforward constrained regulation

In the behavioral framework for continuous-time linear scalar systems, simple sufficient conditions for the solution of the minimum-time rest-to-rest feedforward constrained control problem are provided. The investigation of the time-optimal input-output pair reveals that the input or the output saturates on the assigned constraints at all times except for a set of zero measure. The resulting optimal input is composed of sequences of bang-bang functions and linear combinations of the modes associated to the zero dynamics. This signal behavior constitutes a generalized bang-bang control that can be fruitfully exploited for feedforward constrained regulation. Using discretization, an arbitrarily good approximation of the optimal generalized bang-bang control is found by solving a sequence of linear programming problems. Numerical examples are included.