Time-Optimal Boundary Control for Systems Defined by a Fractional Order Diffusion Equation

We consider the optimal control problem for a system defined by a one-dimensional diffusion equation with a fractional time derivative. We consider the case when the controls occur only in the boundary conditions. The optimal control problem is posed as the problem of transferring an object from the initial state to a given final state in minimal possible time with a restriction on the norm of the controls. We assume that admissible controls belong to the class of functions L_∞[0, T ]. The optimal control problem is reduced to an infinite-dimensional problem of moments. We also consider the approximation of the problem constructed on the basis of approximating the exact solution of the diffusion equation and leading to a finitedimensional problem of moments. We study an example of boundary control computation and dependencies of the control time and the form of how temporal dependencies in the control dependent on the fractional derivative index.     

Symbolic elimination in dynamic optimization based on block-triangular ordering

We consider dynamic optimization problems for systems described by differential-algebraic equations (DAEs). Such problems are usually solved by discretizing the full DAE. We propose techniques to symbolically eliminate many of the algebraic variables in a preprocessing step before discretization. These techniques are inspired by the causalization and tearing techniques often used when solving DAE initial value problems. Since sparsity is crucial for some dynamic optimization methods, we also propose a novel approach to preserving sparsity during this procedure. The proposed methods have been implemented in the open-source JModelica.org platform. We evaluate the performance of the methods on a suite of optimal control problems solved using direct collocation. We consider both computational time and probability of solving the problem in a timely manner. We demonstrate that the proposed methods often are an order of magnitude faster than the standard way of discretizing the full DAE, and also significantly increase probability of successful convergence.     

An observer for non-linear differential-algebraic systems

In this paper, we consider design of observers for non-linear models containing both dynamic and algebraic equations, so-called differential-algebraic equations (DAE), of index 1. The observer is formulated as a DAE that, by construction, has index 1. The main results of the paper include conditions that ensure local stability of the observer error dynamics. Design methodology is presented and illustrated using a small simulation study.     

桥式吊车系统的伪谱最优控制设计

对于桥式吊车系统的最优控制问题,根据实际的工况要求,性能指标有时不一定是标准的二次形式.同时,在实际的控制问题中,状态和控制输入往往会受到一些边界条件和路径过程中的约束.针对这一问题,本文应用Chebyshev伪谱优化算法来处理,它可以处理状态和控制约束的非线性最优化问题以及一个非标准的目标函数.首先对桥式吊车系统模型进行一系列的坐标变换,将其转变为上三角系统形式的误差模型.然后将桥式吊车最优控制问题转化成具有一系列代数约束的参数优化问题,即非线性规划问题.通过求解离散化后的参数优化问题,得到桥式吊车的最优控制律.本文还给出了Chebyshev伪谱最优解的可行性和一致性分析.最后,在仿真研究中验证该控制器的有效性.     

Time-optimal flatness based control of a gantry crane

This contribution deals with the flatness based control of a gantry crane, where the control objective is to transfer the load from an initial rest position to a final rest position in a minimal transition time. It is well-known that the type of crane model we consider is a differentially flat system, and that the position of the load is a flat output. We exploit this property both for the design of a tracking control as well as for planning time-optimal reference trajectories for the load. We discuss the design of the tracking control in detail, and show in particular how a standard approach which can be found in the literature can be modified systematically such that instead of measurements of certain time derivatives of the flat output we can use measurements of the state of the system. We also present a new approach for the design of time-optimal reference trajectories. In order to solve the resulting nonlinear optimization problem numerically, we use a primal-dual interior point method. Finally, we conclude with measurement results that stem from an implementation on a laboratory model.     

面向性能仿真的非连续微分代数模型的指标分析

针对物理系统性能仿真形成的非连续高指标微分代数模型,提出一种基于加权二部图的指标转换方法．该方法将微分代数系统表示为加权二部图,基于二部图匹配算法可以判定微分代数系统是否为高指标系统．对于高指标系统,采用文中方法可以找出需要求导的最小结构奇异方程子集,以便将高指标系统转换为低指标形式．最后,针对定结构与变结构的非连续微分代数模型给出了相应的指标分析策略．文中策略与相关算法已在基于Modelica语言的建模仿真平台MWorks上实现．     

最优预见伺服系统与最优预见FF补偿系统的统一处理

最优预见伺服系统一般与基本最优伺服系统共用同一个二次型性能指标函数设 计预见前馈补偿项,其设计的着眼点是进一步减小性能指标函数.当控制系统的基本反馈部 分不是采用最优控制方法设计时,历史上采用另一个性能指标函数设计预见前馈补偿,并把 所得系统称为预见FF(前馈)补偿系统.这里把两种设计方法统一起来处理后,不仅最优预见 伺服系统与最优预见FF补偿系统都仅仅是特例,而且给设计者扩大了选择的余地.最后给 出了数值仿真,把这种设计方法与最优伺服系统、最优预见伺服系统进行了比较.     

基于最优控制的行车系统消摆控制

针对行车系统运动的特点,根据理论力学理论,采用拉格朗日方程建立了行车系统的动力学模型。该模型完整地描述了行车系统的水平方向、垂直方向与前后方向的三维运动以及由这些运动引起的负载摆动,并在该非线性模型基础上给出了近似条件下的行车系统的线性化模型。根据专家经验,借助最优控制理论,提出一种防摆控制方法。该方法将消摆和运行控制结合起来考虑,在运行过程中不需实时跟踪测量负载摆角、摆速,仅通过控制行车各阶段的运行时间或距离即可实现行车停车与负载消摆的同步控制。仿真结果表明该方法简单合理,易于工程实现。     

机载液压舵机伺服系统优化控制研究与仿真

研究机载液压舵机伺服系统液位精确控制问题。飞行器在空中起降高度发生大幅变化时导致气压不稳,液压受到气压不稳的影响发生不规律性和非线性波动。传统液压舵机伺服控制方法中,以PID算法为基础,控制参数灵活,不能很好地处理液压异常变化带来的液位零漂、滞后造成的影响,造成液位控制精度不高的问题。为此提出了参数动态调整的机载液压舵机伺服系统液位优化控制算法,通过建立参数动态模型,运用动态模型使得液压参数可以根据实际情况变化,保证了干扰条件下的控制精度。仿真结果表明,优化控制方法能够有效解决气压干扰下液位的控制精度,为系统优化提供思路。     

一类微处理机控制最优自适应转台伺服系统

针对存在干摩擦和负载转动惯量不固定的微处理机控制转台伺服系统,提出一种离散最 优自适应控制综合方法.实际系统运行结果表明:用此方法综合的系统优于线性最优伺服系统.     

Exact controllability of a Rayleigh beam with a single boundary control

We prove exact boundary controllability for the Rayleigh beam equation ${\varphi_{tt} -\alpha\varphi_{ttxx} + A\varphi_{xxxx} = 0, 0 < x < l, t > 0}$ with a single boundary control active at one end of the beam. We consider all combinations of clamped and hinged boundary conditions with the control applied to either the moment ${\varphi_{xx}(l, t)}$ or the rotation angle ${\varphi_{x}(l, t)}$ at an end of the beam. In each case, exact controllability is obtained on the space of optimal regularity for L ²(0, T) controls for ${T > 2l\sqrt{\frac{\alpha}{A}}}$ . In certain cases, e.g., the clamped case, the optimal regularity space involves a quotient in the velocity component. In other cases, where the regularity for the observed problem is below the energy level, a quotient space may arise in solutions of the observed problem.     

Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems

We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions can we assure optimal controls are bounded? This question is related to one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial in closing the gap between the conditions arising in existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the latter problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover optimal control problems with general nonlinear dynamics.     

Singular Controls in the Classical Sense for the Optimal Control Problem with Nonlocal Boundary Conditions

We consider an optimal control problem in which the states of the system are determined by the control systems of ordinary differential equations with two-point boundary conditions. Admissible controls are selected from the class of bounded measurable functions with values in an open set. We calculate the increment of the functional formula of the second order. We use variations of the control to derive the necessary optimality conditions for singular controls in the classical sense.     

基于输入整形技术的吊车系统防摆控制

针对传统吊车防摆闭环控制方法存在的控制器设计相对复杂、控制效果不理想、不易实现等问题,提出了一种基于经典控制理论与前馈输入整形技术相结合的控制技术。首先,对吊车系统进行建模并简化数学模型,采用经典控制理论设计吊车的伺服控制系统。其次,在吊车伺服系统的基础上,引入输入整形技术设计吊车系统的ZV输入整形器。最后,通过MATLAB仿真验证所提出控制技术的有效性。     

Optimal control of mobile sources for heat conductivity processes

We consider a problem of optimal control of the mobile sources for heat conductivity processes, described by parabolic equation and systems of ordinary differential equations. Sufficient conditions for Frechet differentiability of the performance criterion and an expression for its gradient were determined. The necessary optimality condition was established in the form of the integral principles of maximum. The theoretical conclusions are illustrated by the solution of a numerical example.     

Natural killer or T-lymphocyte cells: Which is the best immune therapeutic agent for cancer? An optimal control approach

Therapeutic vaccines are being developed as a promising new approach to treatment for cancer patients. There are still many unanswered questions about which kind of therapeutic vaccines are the best for the cancer treatments? In this paper we consider a mathematical model, in the form of a system of ordinary differential equations (ODE), this system is an example from a class of mathematical models for immunotherapy of the tumor that were derived from a biologically validated model by Lisette G. de Pillis. The problem how to schedule a variable amount of which vaccines to achieve a maximum reduction in the primary cancer volume is consider as an optimal control problem and it is shown that optimal control is quadratic with 0 denoting a trajectory corresponding to no treatment and 1 a trajectory with treatment at maximum dose along that all therapeutics are being exhausted. The ODE system dynamics characterized by locating equilibrium points and stability properties are determined by using appropriate Lyapunov functions. Especially we attend a parametric sensitivity analysis, which indicates the dependency of the optimal solution with respect to disturbances in model parameters.     

Index reduction by minimal extension for the inverse dynamics simulation of cranes

The present work deals with the inverse dynamics simulation of underactuated mechanical systems relying on servo constraints. The servo-constraint problem of discrete mechanical systems is governed by differential–algebraic equations (DAEs) with high index. We propose a new index reduction approach, which makes possible the stable numerical integration of the DAEs. The new method is developed in the framework of a specific crane formulation and facilitates a reduction from index five to index three and even to index one. Particular attention is placed on the special case in which the reduced index-1 formulation is purely algebraic. In this case the system at hand can be classified as differentially flat system. Both redundant coordinates and minimal coordinates can be employed within the newly developed approach. The success of the proposed method is demonstrated with two representative numerical examples.     

The Control Problem for Stage-by-Stage Changing Linear Systems of Loaded Differential Equations

We consider the control problem for stage-by-stage changing linear differential equations and the optimal control problem with a quality criterion defined for the entire time interval. We formulate necessary and sufficient conditions for complete controllability and the existence of a program control and motion. We construct an explicit form of the control action for the control problem and propose a method for solving the optimal control problem. We give a solution of the control problem for a specific loaded system.