Revisited LQ output-feedback control: minimax controller for a set of initial states

A minimax linear quadratic (LQ) output-feedback controller is introduced which minimises the maximal value of the performance index over all initial states belonging to some set separated out a priori. If the set is an ellipsoid or a polygon, such controllers are synthesised in terms of linear matrix inequalities (LMIs). In particular case when a size of this set tends to zero tightening to a point, the minimax LQ controller approaches the optimal LQ output-feedback controller for the given initial state, while in another extreme case when this size tends to infinity, we have the worst-case LQ output-feedback controller. Numerical results for an inverted pendulum are presented.     

Optimal Decoupling Control Method and Its Application to a Ball Mill Coal-pulverizing System

The conventional optimal tracking control method cannot realize decoupling control of linear systems with a strong coupling property. To solve this problem, in this paper, an optimal decoupling control method is proposed, which can simultaneously provide optimal performance. The optimal decoupling controller is composed of an inner-loop decoupling controller and an outer-loop optimal tracking controller. First, by introducing one virtual control variable, the original differential equation on state is converted to a generalized system on output. Then, by introducing the other virtual control variable, and viewing the coupling terms as the measurable disturbances, the generalized system is open-loop decoupled. Finally, for the decoupled system, the optimal tracking control method is used. It is proved that the decoupling control is optimal for a certain performance index. Simulations on a ball mill coal-pulverizing system are conducted. The results show the effectiveness and superiority of the proposed method as compared with the conventional optimal quadratic tracking (LQT) control method.      

Optimal controller tuning for nonlinear processes

The present work proposes a systematic methodology for the optimal selection of controller parameters in the sense of minimizing a performance index, which is a quadratic function of the tracking error and the control effort. The performance index is calculated explicitly as an algebraic function of the controller parameters by solving Zubov's partial differential equation (PDE). Standard optimization techniques are then employed for the calculation of the optimal values of the controller parameters. The solution of Zubov's PDE is also used to estimate the closed-loop stability region for the chosen values of the controller parameters. The proposed approach is finally illustrated in a chemical reactor control problem.     

在连续过程的离散最优控制中采样周期和加权系数的选择

本文提出了在连续过程的离散最优控制中选择采样周期和加权系数的一个方法。这个选择基于:1)线性的时间离散控制器;2)控制变量的范围受到限制;3)根据对系统的实际要求给定输出量的初始偏移。系统的性能则采用时间连续的二次型输出偏移损失函数以及类似的控制损失函数来表示。对不同的采样周期,利用标准的LQ方法来计算控制器,并通过选择控制损失项的加权系数,以使得对于给定的初始偏移,允许的控制范围得到充分利用。最后的采样周期则是权衡了偏移损失和控制损失的一个折衷选择。本文介绍的方法尤其适用于需要尽可能长的采样周期的情况。     

一种基于人工免疫原理的最优模糊神经网络控制器

提出了一种基于人工免疫原理的最优RBF模糊神经网络控制器设计方案．首先给出了控制器结构，其次将免疫进化算法用于控制器参数的优化，设计了一种满足二次型性能指标的最优RBF模糊神经网络控制器．将该控制器用于控制实际倒立摆系统，并采用状态变量合成方法以大大减少模糊规则的数目，实验结果验证了该控制器的有效性．     

基于H2与H∞的"动中通"伺服系统控制器设计

本文以"动中通"伺服系统为对象, 研究了一种基于H_2和H_{/infty}指标的控制器设计方案. 在反馈控制器方面, 将满足LQ最优准则的扩展状态反馈与状态观测设计整合成一个H_2设计问题, 给出理论推导与设计方法; 在前馈控制器方面, 将扰动抑制H_{/infty}指标设计转换为模型匹配设计, 并给出有效求解方法. 在控制器参数整定过程中, 考虑主谐振模态、不确定性等对系统的∞影响, 给出了一种兼顾动态性能和鲁棒稳定性的整定方法. 该设计方案从反馈到前馈均着重考虑扰动抑制能力, 并且在实际系统上取得了较好的效果, 对扰动抑制能力要求较高的控制器设计有一定指导作用.     

一类工业过程运行反馈优化控制方法

为了克服流程工业运行优化中控制回路闭环系统的动态误差对运行优化性能的影响,本文针 对一类工业过程提出了使运行指标实际值与目标值偏差和控制回路输出与设定值跟踪误差的二次性能 指标极小化的运行优化反馈控制方法. 该方法由运行层设定值反馈控制和回路控制层设定值跟踪控制组成,其中设定值反馈控制采用基于LMI的 模型预测控制,回路控制采用衰减率可调的带有积分项的状态反馈调节律. 本文给出了保证运行优化反馈控制闭环系统渐近稳定的充分条件,并开展了浮选过程运行优化反馈控制仿 真实验,实验结果表明所提方法的有效性.     

具有不完全微分的Fuzzy-GA PID控制器及其在智能仿生人工腿中的应用

以模糊推理和遗传算法为基础，提出了一种新的具有不完全微分的最优PID控制器的设计方 法．该控制器由离线和在线两部分组成．在离线部分，以系统响应的超调量、上升时间以及 调整时间为性能指标，利用遗传算法搜索出一组最优的PID参数Kp*、Ti* 和 Td*，作为在线部分调整的初始值．在在线部分，一个专用的PID参数优化程序以离 线部分获得Kp*、Ti* 和Td*为基础，根据系统当前的误差e和误差变化率e ，通过一个模糊推理系统在线调整系统瞬态响应的PID参数，以确保系统的响 应具有最优的动态和稳态性能．该控制器已被用来控制由作者设计的智能仿生人工腿中的执 行电机．计算机仿真结果表明，该控制器具有良好的控制性能和鲁棒性能．     

Adaptive control utilizing optimally sensitive design techniques†

In the recent literature a control problem in which both the initial state vector as well as a vector of constant plant parameters ore unknown ha3 been treated utilizing sensitivity techniques referred to as optimally sensitive control. In this paper the concepts of optimally sensitive control as developed by Kokotovic, Perkins, Cruz and others are extended to the problem in which the state dynamics contain a vector of stochastic inputs which can be represented as Martingale processes. The resulting optimally sensitive system is shown to be an effective and realistic adaptive controller for systems containing unknown time-varying parameters. A numerical example is presented to demonstrate the effectiveness of the resulting control system at identifying and adapting to the levels of the unknown time-varying inputs.     

Fuel gas blending benchmark for economic performance evaluation of advanced control and state estimation

A simulation of a fuel gas blending process and its measurement system is proposed as a benchmark test case for advanced control and state estimation. The simulation represents an industrial facility and employs a well-established software environment. The objective is to maintain four controlled variables within specified bounds while minimizing an economic performance index. The controlled variables are the fuel gas pressure and three measures of gas quality. Six feed gas flow rates may be adjusted to achieve the objective. Each has a limited availability.The benchmark consists of three reproducible scenarios, each a 46-h period during which 23 discrete upsets occur and the feed gas compositions vary gradually with time. A benchmark multi-loop feedforward–feedback structure is described, tested, and compared to an estimate of optimal performance. The operating cost provided by the benchmark controller is from 1.19 to 1.71 times higher than the estimated minimum.Readers are challenged to download the simulation model, benchmark controller and estimated optimal performance from the URL given in this paper, and to devise case studies of advanced state estimation and control strategies to better the proposed benchmark controller.     

Optimal linear-quadratic control: From matrix equations to linear matrix inequalities

The results of the present authors’ studies on the design of optimal linear-quadratic control based on the linear matrix inequalities were summarized. Uniform consideration was given both to the nondegenerate and degenerate problems of optimal control by state and measured output, the so-called orthogonality condition being either satisfied or not. In the problem of control by the measured output, the matrix of parameters of the optimal feedback depends substantially on the initial state of the plant, which makes this control law practically impossible. The need for passing from the integral functional as the system performance index to the maximal value of the ratio of this functional to the square of the initial state norm over all initial states and the corresponding passage from the optimal control law to the minimax linear-quadratic law were substantiated for this case.     

Finite horizon optimal control of discrete-time nonlinear systems with unfixed initial state using adaptive dynamic programming

In this paper, we aim to solve the finite horizon optimal control problem for a class of discrete-time nonlinear systems with unfixed initial state using adaptive dynamic programming (ADP) approach. A new ε-optimal control algorithm based on the iterative ADP approach is proposed which makes the performance index function converge iteratively to the greatest lower bound of all performance indices within an error according to ε within finite time. The optimal number of control steps can also be obtained by the proposed ε-optimal control algorithm for the situation where the initial state of the system is unfixed. Neural networks are used to approximate the performance index function and compute the optimal control policy, respectively, for facilitating the implementation of the ε-optimal control algorithm. Finally, a simulation example is given to show the results of the proposed method.     

Implications of the degree of controllability of controlled plants in the sense of LQR optimal control

In this paper, the relationship between the degree of controllability (DOC) of controlled plants and the corresponding quadratic optimal performance index in LQR control is investigated for the electro-hydraulic synchronising servo control systems and wind turbine systems, respectively. It is shown that for these two types of systems, the higher the DOC of a controlled plant is, the better the quadratic optimal performance index is. It implies that in some LQR controller designs, the measure of the DOC of a controlled plant can be used as an index for the optimisation of adjustable plant parameters, by which the plant can be controlled more effectively.     

参数未知的离散系统Q-学习优化状态估计与控制

控制系统的应用中存在状态不能直接测量或测量成本高的实际问题,给模型参数未知的系统完全利用状态数据学习最优控制器带来挑战性难题.为解决这一问题,首先构建具有状态观测器且系统矩阵中存在未知参数的离散线性增广系统,定义性能优化指标;然后基于分离定理、动态规划以及Q-学习方法,给出一种具有未知模型参数的非策略Q-学习算法,并设计近似最优观测器,得到完全利用可测量的系统输出和控制输入数据的非策略Q-学习算法,实现基于观测器状态反馈的系统优化控制策略,该算法的优点在于不要求系统模型参数全部已知,不要求系统状态直接可测,利用可测量数据实现指定性能指标的优化;最后,通过仿真实验验证所提出方法的有效性.     

Quantized stabilization of linear systems: complexity versus performance

Quantized feedback control has been receiving much attention in the control community in the past few years. Quantization is indeed a natural way to take into consideration in the control design the complexity constraints of the controller as well as the communication constraints in the information exchange between the controller and the plant. In this paper, we analyze the stabilization problem for discrete time linear systems with multidimensional state and one-dimensional input using quantized feedbacks with a memory structure, focusing on the tradeoff between complexity and performance. A quantized controller with memory is a dynamical system with a state space, a state updating map and an output map. The quantized controller complexity is modeled by means of three indexes. The first index L coincides with the number of the controller states. The second index is the number M of the possible values that the state updating map of the controller can take at each time. The third index is the number N of the possible values that the output map of the controller can take at each time. The index N corresponds also to the number of the possible control values that the controller can choose at each time. In this paper, the performance index is chosen to be the time T needed to shrink the state of the plant from a starting set to a target set. Finally, the contraction rate C, namely the ratio between the volumes of the starting and target sets, is introduced. We evaluate the relations between these parameters for various quantized stabilizers, with and without memory, and we make some comparisons. Then, we prove a number of results showing the intrinsic limitations of the quantized control. In particular, we show that, in order to obtain a control strategy which yields arbitrarily small values of T/lnC (requirement which can be interpreted as a weak form of the pole assignability property), we need to have that LN/lnC is big enough.     

Application of stochastic optimal control to a hydrofoil boat problem

In this paper the solution of a stochastic optimal control problem described by linear equations of motion and a nonquadratic performance index is presented. The theory is then applied to the dynamics of a single-foil and a hydrofoil boat flying on rough water. The random disturbances caused by sea waves are represented as the response of an auxiliary system to a white noise input. The control objective is formulated as an integral performance index containing a quadratic acceleration term and a nonquadratic term of the submergence deviation of the foil from calm water submergence. The stochastic version of the maximum principle is used in the formulation of a feedback control law. The Riccati equations and the feedback gains associated with a nonquadratic performance index are non-linear functions of the state and auxiliary state variables. These equations are integrated forward with the state equations for the steady-state solution of the problem. The controller for a nonquadratic performance index contains computing elements which perform the integration of the Riccati equations to generate the instantaneous values of the feedback gains. The effect of a nonquadratic penalty on the submergence deviation and the effect of a nonquadratic control penalty on the response of the system are investigated. A comparison between an optimal nonlinear control law and a suboptimal linear control law is presented.     

Sensitivity of the Performance of Optimal Linear Control Systems to Parameter Variations†

The sensitivity of the index of performance to parameter variations in optimal control systems is examined in this paper. It is shown, in the case of linear optimal systems with quadratic performance criteria, that the value of the performance index, after the system parameters have deviated from a nominal set of values, is still given by a symmetric positive definite quadratic form of the initial state. The matrix of this quadratic form is governed by a special case of the matrix Riccati equation. It is shown also, that similar results hold for the performance index sensitivity function.

Because the sensitivity problem closely parallels the original optimization problem, the computational techniques used in the design of the optimal system may be reapplied in the sensitivity analysis.     

基于定量反馈理论的时滞系统自抗扰控制参数整定

工业过程对象普遍存在时滞、模型参数不确定性和外部扰动多等特点,传统Smith预估控制方法难以设计出满足期望性能的鲁棒控制器.针对模型参数不确定性和外部扰动,本文采用自抗扰控制技术进行估计和补偿.针对系统存在时滞的特点,本文提出改进Smith预估器结构,提升扩张状态观测器对于扰动估计的实时性.在此基础上,本文以一阶时滞系统为例提出了控制器参数整定方法.首先根据最优参数选取准则确定预估器模型,然后在等效模型框架下采用定量反馈理论整定自抗扰控制器参数,确保控制系统达到预期性能指标.在仿真实验中,将所提出方法与几种常见时滞系统控制方法进行比较,通过设定值跟踪、抗扰及蒙特卡罗实验验证了所提出方法具有良好抗扰能力与鲁棒性.     

Parallel Control for Optimal Tracking via Adaptive Dynamic Programming

This paper studies the problem of optimal parallel tracking control for continuous-time general nonlinear systems. Unlike existing optimal state feedback control, the control input of the optimal parallel control is introduced into the feedback system. However, due to the introduction of control input into the feedback system, the optimal state feedback control methods can not be applied directly. To address this problem, an augmented system and an augmented performance index function are proposed firstly. Thus, the general nonlinear system is transformed into an affine nonlinear system. The difference between the optimal parallel control and the optimal state feedback control is analyzed theoretically. It is proven that the optimal parallel control with the augmented performance index function can be seen as the suboptimal state feedback control with the traditional performance index function. Moreover, an adaptive dynamic programming (ADP) technique is utilized to implement the optimal parallel tracking control using a critic neural network (NN) to approximate the value function online. The stability analysis of the closed-loop system is performed using the Lyapunov theory, and the tracking error and NN weights errors are uniformly ultimately bounded (UUB). Also, the optimal parallel controller guarantees the continuity of the control input under the circumstance that there are finite jump discontinuities in the reference signals. Finally, the effectiveness of the developed optimal parallel control method is verified in two cases.      

Fuzzy Optimal Control for Bilinear Stochastic Systems with Fuzzy Parameters

In this paper we consider the control problem for a class of partially observed bilinear stochastic systems with fuzzy parameters. Using Takagi–Sugeno fuzzy model, the problem is described by three sets of fuzzy stochastic differential equations: one for the state process, one for the observed process and one for the controller which is assumed to be driven by the observed process. With this formulation, the original stochastic control problem can be treated as a deterministic identification problem in which the controller parameters and the corresponding membership functions are the unknowns. Using a suitable performance index, we have developed a set of necessary conditions for determining the parameters of the controller and the corresponding membership functions. Finally, some numerical simulations are presented to illustrate the effectiveness of the proposed fuzzy control scheme.