Time-optimal control strategy for saturating linear discrete systems

An easily implemented strategy for obtaining the required minimal-time and time-optimal control sequence for a given initial state of an nth-order linear discrete system with constraints on the input is described. This strategy uses the boundary faces of the reachable set for the determination of the minimal time. The time-optimal control sequence is achieved by dividing the reachable set into linear and saturated parts.     

A note on the existence and optimal control for mixed Volterra–Fredholm-type integrodifferential dispersion system of third order

The optimal control problem consists of a performance index subject to a set of differential equations that describes the path of the control and state variables. The main aim of this article is to prove the existence and uniqueness of a mild solution, optimal control, and time-optimal control of a mixed Volterra–Fredholm-type third-order dispersion system. By applying the strongly continuous semigroup theory and the Banach fixed-point theorem, we prove the existence and uniqueness of the considered system. The optimal control results are proved by using Mazur's lemma, Gronwall's inequality, and the minimizing sequence technique. The discussion on the time-optimal control of the third-order dispersion system is also presented.     

Robust mixed time-optimal control for third-order systems with model uncertainty

In this paper, a robust near time-optimal controller for some third-order uncertain systems with constrained input is presented. The response is usually composed of two parts. At first, the states move towards the goal state from arbitrary initial conditions by pure time-optimal control (bang-bang) and then there is sliding along a predetermined surface within an ellipsoid. The sliding surface is chosen in such a way that the integral of the absolute value of the system error is minimized. Using a combined control algorithm, robustness and stability of the system is guaranteed even with parameter uncertainty and disturbance. The error convergence rate is also improved compared with pure time-optimal control. Simulation results demonstrate the advanced performance of the proposed control algorithm.     

Particle swarm optimization for time-optimal control design

In this paper,a particle swarm optimization(PSO)based method is proposed to obtain the time-optimal bang-bang control law for both linear and nonlinear systems.By introducing a penalty function,the method can be modified to deal with systems with constraints.Compared with existing computational methods,the proposed method can be implemented in a straightforward manner.The convergent solutions can be achieved by selecting suitable PSO parameters regardless of the initial guess of the switching times.A double integrator and a third-order nonlinear system are used to demonstrate the effectiveness and robustness of the proposed method.The method is applied to obtain the time-optimal control law for a high performance linear motion positioning system.The results show the practicality of the proposed algorithm.     

Time-optimal control and high-gain linear state feedback

It is known that the fastest possible response of a linear system under input constraints is typically achieved using extreme control efforts at all times, or bang-bang control. However, closed-loop minimum-time control is very difficult to implement except for a few simple second-order systems, since a closed-form solution for the time-optimal switching hypersurface usually cannot be found for high-order systems. In this paper the connection between linear state feedback and time-optimal control is explored. It is shown that for a class of systems an equivalent time-optimal, linear, switching hypersurface exists corresponding to an initial condition. Based on these switching planes, a high-gain linear state feedback law can be used to achieve minimum-time control on systems of order higher than two, provided that the feedback coefficients are adjusted according to the initial conditions at the start of each operation. Three different cases are investigated in detail: a second-order DC servo drive, a third-order servo system including actuator dynamics, and a fourth-order flexible mechanism. Since the proposed algorithm resembles in form a sliding-mode controller, the similarities and differences between the two control algorithms are also discussed.     

Robust near time-optimal control

A robust control algorithm for a realization of nearly time-optimal control of double integrator plants is presented. The technique involves the combination of traditional bang-bang time-optimal control with the methods of sliding-mode control. The result is a nonlinear feedback scheme for maintaining a desired functional relationship among dynamic variables, in this case imitating the idealized dynamics of time-optimal control. The approach is nearly time optimal rather than exactly time optimal, since the bang-bang control components are restricted to values below full actuator saturation, thus reserving some actuator effort for compensation of disturbances and model imperfections. The algorithm blends smoothly into a linear controller near a stable goal location     

Time-optimal control algorithm for two-time-scale discrete system

A linear, shift-invariant discrete system exhibiting two-time-scale (slow and fast) character is addressed. An iterative algorithm based on linear programming for the computation of time-optimal control is presented by decoupling the original system into slow and fast subsystems. It is found that the computation time and storage capacity for the computation of time-optimal control of the slow and fast subsystems is less than that required for the original system. A numerical example is provided to illustrate the proposed algorithm.     

Sequential synthesis of the time-optimal control in real time

The paper considers the sequential synthesis method for time-optimal control of linear systems. The method is based on piecewise constant finite controls that ensure approximate solutions for time-optimal control problems. A sequence of finite controls is thereafter transformed into the optimal control. The appropriate computations are reduced to a sequence of linear algebraic equation problems and the integration of a matrix differential equation over the intervals of control switching points and final point change. It is proven that the sequence of finite controls converge to the optimal control. The sliding mode conditions are obtained, as well as the control structure modifications for the motions on switching manifolds. The initial approximations reducing considerably computational complexity are considered. The computational algorithm, together with the modelling and numerical results, are presented.     

非线性时滞系统次优控制的逐次逼近法

对状态变量含有时滞的非线性系统的次优控制问题进行了研究，提出了一种次优控制的逐次逼近设计方法．针对由最优控制理论导出的既含有时滞项又含有超前项的非线性两点边值问题，构造了其解序列一致收敛于原问题最优解的非齐次线性两点边值问题序列．从而将两点边值问题解序列的有限次迭代结果作为系统的次优控制律．仿真结果表明了所提出方法的有效性．     

基于改进遗传算法的时间最优控制问题求解

对原有遗传算法的不足进行分析 ,提出改进的遗传算法。对于高维、高精度问题 ,改进算法相对原算法可节省大量存储空间和解码时间。提出的选择算子仅与父代的大小顺序有关 ,既可避免原算法对适应值必须为正的限制 ,又可避免算法过早收敛到局部解。证明了新算法的全局收敛性 ,并对新的选择算子进行了性能分析。将改进的遗传算法引入受约束时间最优控制问题的求解 ,获得了令人满意的结果     

Robust model predictive control with guaranteed setpoint tracking

In this paper a novel robust model predictive control (RMPC) algorithm is proposed, which is guaranteed to stabilize any linear time-varying system in a given convex uncertainty region while respecting state and input constraints. Moreover, unlike most existing RMPC algorithms, the proposed algorithm is guaranteed to remove steady-state offset in the controlled variables for setpoints (possibly) different from the origin when the system is unknown linear time-invariant. The controller uses a dual-mode paradigm (linear control law plus free control moves to reach an appropriate invariant region), and the key step is the design of a robust linear state feedback controller with integral action and the construction of an appropriate polyhedral invariant region in which this controller is guaranteed to satisfy the process constraints. The proposed algorithm is efficient since the on-line implementation only requires one to solve a convex quadratic program with a number of decision variables that scale linearly with the control horizon. The main features of the new control algorithm are illustrated through an example of the temperature control of an open-loop unstable continuous stirred tank reactor.     

鲁棒模型预测控制在变风量空调系统中的应用

在变风量空调系统中二次泵压差控制可以有效地减少空调能耗,为克服二次泵模型的不确定性,提高二次泵变频调速控制的响应速度和精度,采用基于线性矩阵不等式的鲁棒预测控制策略。算法分为离线和在线两个部分,离线时首先用传统算法得出目标函数上界,以此为已知量重新优化得到一系列较大的渐近稳定的不变椭圆集。在线时,每个采样周期用三个相邻的椭圆集优化来对状态变量进行精确定位,并给出控制量。给出在线优化的理论证明。通过和传统算法的仿真比较,表明该算法的有效性。二次泵压差控制的实验表明该算法可得到较大的可行域,系统响应快,控制效果好。     

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.     

LP-based velocity profile generation for robotic manipulators

In this paper we examine the minimum-time velocity profile generation problem which belongs to the second stage of the decoupled robot motion planning. The time-optimal profile generation problem can be translated to a convex optimal control task through a nonlinear change of variables. When the constraints of the problem have special structure, the time-optimal solution can be obtained by linear programming (LP). In this special case, the velocity of the robot along the path is maximised, instead of time minimising. The benefit of the LP solution is the lower computational time. Validation of the LP algorithm is also presented based on simulation results.     

近似时间最优的舵机多模位置控制策略

高超声速飞行器要求其伺服作动系统具有高动态高精度的特性,针对传统控制方法难以兼顾系统动态和精度的难题,本文设计了一种近似时间最优的舵机多模位置控制策略.首先建立了以一阶惯性环节串联积分器表征舵机输入输出特性的特征模型.然后以相平面为分析工具,给出了其近似时间最优控制的切换区;在切换区外以快速性为目标而采用bang-bang最优控制,在切换区内以避免振荡和超调为目标而采用bang-bang次优控制;为提高稳态性能,在小误差时时采用线性控制.实验表明,该方法响应快速,避免了振荡和超调,很好地满足了对伺服作动系统高动态高精度的要求.     

An algorithm for non-linear space—Time nuclear reactor control

A novel multivariable control algorithm for non-linear space-time nuclear reactor dynamics is proposed in this paper. The multivariable control algorithm is based on a mathematical model of the nuclear reactor which includes: a single energy group of neutrons, delayed neutron precursors, iodine, xenon and thermal-hydraulic feedback. The multivariable control algorithm is composed of non-linear time-varying feedforward and feedback control signals, a reference model of the nuclear reactor and a dynamic observer. The non-linear proportional plus integral feedback controller forces the nuclear reactor to follow the response of the reference model. The dynamic observer estimates the unmeasurable state variables. The feedforward and feedback control signals are determined in a novel approach by specifying the form of the closed-loop response of the neutron density variables. By virtue of the multivariable control algorithm the closed-loop differential equations are linear and time-varying. A linear stability analysis for base-load and load-cycle operation indicates that the closed-loop system is stable provided that the thermal-hydraulic subsystem is inherently stable. The simulated dynamic response indicates that the multivariable control algorithm provides excellent response characteristics.     

非线性系统近似最优PD动态补偿控制

本文研究了一类基于动态补偿的非线性系统的近似最优PD控制的问题.用微分方程的逐次逼近理论将非线性系统的最优控制问题转化为求解线性非齐次两点边值序列问题,并提供了从时域最优状态反馈到频域最优PD控制器参数的优化方法,从而获取系统最优的动态补偿网络,设计出最优PD整定参数,给出其实现算法.最后仿真示例将所提出的方法与传统的线性二次型调节器（LQR）逐次逼近方法相比较,表明该方法具有良好的动态性能和鲁棒性.     

带正弦干扰的线性时滞系统的次优控制

研究状态变量含有时滞的线性系统在正弦干扰下的前馈反馈次优减振控制问题，首先构造其解收敛于原时滞系统的无时滞系统序列；然后将时滞系统的最优控制问题化为无时滞系统的最优控制序列问题；最后通过截取最优控制序列解的有限项，得到系统的前馈反馈次优控制律，仿真结果表明，该方法抑制正弦干扰的鲁棒性优于经典反馈最优控制。     

求解量子系统时间最优控制问题的同伦算法研究

针对数值求解量子系统时间最优控制问题中反复调用梯度算法导致计算量大的问题,本文提出一类同伦算法用以快速求解量子系统的时间最优控制问题.与已有算法不同,这一算法通过引入同伦变量在减小终端时间的方向上搜索最优解.在这一算法中,可通过自由函数构造保真度函数对控制变量的梯度方向,也可通过方向函数引导算法的搜索方向,以加快算法的搜索速度.本文将这一算法用于求解量子系统态转移和门变换的时间最优控制问题.仿真结果表明这一算法的有效性.     

A time-optimal adaptive control system via adaptive switching hypersurface

This paper presents a technique for synthesizing a time-optimal adaptive control system. The technique consists of combining off-line memorization with simple on-line calculations to determine the control signal. The adaptive switching hypersurface consists of images of switching points of each plant in the extended state space (the present and past state variables). The approximation is obtained by linearizing the projection of the adaptive switching hypersurface in a subspace, and finally storing the parameters of the hyperplanes thus obtained in memory. The on-line calculation corresponds to a simple function generator. The function generator consists of the memory containing the values of parameters of the hyperplanes and a linear interpolator. The input of the function generator is the present and past state variables, and its final output is+Kor-K. The technique is applied to the pitch axis adaptive regulator control of the F-101B aircraft in various flight conditions, and simulations of the system by the hybrid computer are obtained.