Mixed type symmetric duality for multiobjective variational problems with cone constraints

In this study, a pair of multiobjective variational mixed symmetric dual programs involving cone constraints is presented and usual duality results are established using the notion of generalized ‐convexity. Self‐duality is discussed under additional condition of skew symmetry, and a time‐independence (static) version of the problems is also incorporated.     

Duality and symmetry in constrained estimation and control problems

The Lagrangian dual of a constrained linear estimation problem is derived where the resulting dual system is a linear optimal control problem with projected cost variables. The symmetry between constrained estimation and optimal control is revealed when the primal problem is expressed in an equivalent form involving projected variables.     

Singular optimal control problems: On the order of a singular arc

The higher-order tests for optimality of singular arcs in optimal control problems, as well as the characterization of the continuity and smoothness properties of the optimal control at points, called junctions, joining nonsingular and singular arcs of the control, depend upon the orders of the arcs involved. To data, definitions of order have been rigorously given only for terminal unconstrained control problems or by means of a normality assumption in the order cases. We give here a new definition of the local order of a singular arc which unifies those already given in the literature. One advantage of the proposed definition is that, unlike the previous ones, it remains valid even when there are terminal constraints. One example will illustrate the fact that such a basic concept is essential to solve the very difficult synthesis problem as soon as the dimension of the state space is greater than two.     

Sliding mode control design via reduced order model approach

This paper presents a design of continuous-time sliding mode control for the higher order systems via reduced order model.It is shown that a continuous-time sliding mode control designed for the reduced order model gives similar performance for the higher order system.The method is illustrated by numerical examples.The paper also introduces a technique for design of a sliding surface such that the system satisfies a cost-optimality condition when on the sliding surface.     

一种高阶滑模控制算法的改进及应用

针对一类高阶滑模算法收敛速度慢、控制作用不连续等不足,以二自由度轮式机器人为控制对象设计了改进算法．首先,引入虚拟控制项以增加系统相对阶的形式进行高阶滑模算法改进,得出了一种新的高阶滑模控制律．与原算法相比,系统收敛速度快、控制作用平滑连续．然后,为解决二阶超螺旋算法仅适用于系统相对阶为1的问题,以跟踪误差的高阶导数建立滑动量,构造适用于高阶系统的二阶超螺旋算法．最后,通过数值仿真验证了改进算法的有效性．     

大型空间飞行器的高阶滑模姿态控制律设计

针对大型空间飞行器的大角度姿态控制问题,考虑航天器惯量矩阵中的不确定性和外部扰动力矩,应用高阶滑模控制方法设计了姿态跟踪控制律.采用的二阶滑模控制方法改善了系统针对不确定性及外部扰动的鲁棒性,并减弱了振颤现象.针对所设计的控制器进行了仿真验证,并与一阶滑模控制进行了对比,仿真结果表明了所提出方法的有效性.     

Program separation and definitional higher order programming

We describe a program separation scheme based on the notions of form and content of an algorithm. The content of an algorithm consists of all local operations needed to compute the algorithm on the basis of a given definition. Separating an algorithm can thus be described as the process of finding definitions such that the algorithm is locally definable. The form of the algorithm then gives the global structure of the algorithm.

The scheme is given in a definitional context, and we show how this type of program separation can be used as a basis for definitional higher order programming.     

退化高阶抛物型分布参数系统的迭代学习控制

研究一类高阶分布参数系统的迭代学习控制问题,该类系统由退化高阶抛物型偏微分方程构成.根据系统所满足的性质,基于P型学习算法构建得到迭代学习控制器.利用压缩映射原理,证明该算法能使得系统的输出跟踪误差于L~2空间内沿迭代轴方向收敛于零.最后,仿真算例验证了算法的有效性.     

Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.     

On solving hybrid optimal control problems with higher index DAEs

The paper deals with hybrid optimal control problems described by higher index differential–algebraic equations (DAEs). We introduce a numerical procedure for solving these problems. The procedure has the following features: it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of systems equations by an implicit Runge–Kutta method; the consistent initialization procedure is applied whenever control functions jumps, or state variables transition occurs. The procedure can cope with hybrid optimal control problems which are defined by DAEs with the index not exceeding three. Our approach does not require differentiation of some system equations in order to transform higher index DAEs to the underlying ordinary differential equations (ODEs). The presented numerical examples show that the proposed approach can be used to solve efficiently hybrid optimal control problems with higher index DAEs.     

A computational method for solving two‐dimensional nonlinear variable‐order fractional optimal control problems

In this paper, a new class of two‐dimensional nonlinear variable‐order fractional optimal control problems (V‐OFOCPs) is introduced where the variable‐order fractional derivative is defined in the Caputo type. The general procedure for solving theses systems is expanding the state variable and the control variable based on the Legendre cardinal functions in the matrix form. Hence, we derive their operational matrix of derivative (OMD) and operational matrix of variable‐order fractional derivative (OMV‐OFD). More significantly, some properties of these basis functions are proved to be exploited in our approach. Using these achieved results, we simply expand the matrix form of the nonlinear performance index in terms of the Legendre cardinal functions and subsequently convert it to an algebraic equation. We emphasize that it is a valuable advantage of applying cardinal functions in approximation theory. Then, we implement the OMD and the OMV‐OFD of the Legendre cardinal functions to transform the variable‐order fractional dynamical system to a system of algebraic equations. Next, the method of constrained extremum is applied to adjoin the constraint equations including the given dynamical system and the initial‐boundary conditions to the performance index by a set of undetermined Lagrange multipliers. Finally, the necessary conditions of the optimality are derived as a system of nonlinear algebraic equations including the unknown coefficients of the state variable, the control variable and the Lagrange multipliers. The applicability and efficiency of the proposed approach are investigated through the various types of test problems.     

First order conditions for generalized variational problems over time scales

In this paper, we derive the weak Pontryagin principle for generalized optimal control problems over time scales. Three types of problems are considered, namely (i) the problems involving the values of the state endpoints in the Lagrangian and the dynamics, (ii) the problems with an integral of the state in the Lagrangian and the dynamics, and (iii) the isoperimetric problems. As special cases, we obtain the first order optimality conditions for the corresponding calculus of variations problems, which have been of a recent interest in the literature. Our method is based on transforming the generalized optimal control or calculus of variations problem into a traditional optimal control problem on time scales, to which the known weak Pontryagin principle can be applied.     

Active disturbance rejection control for nonlinear fractional‐order systems

This paper investigates active disturbance rejection control involving the fractional‐order tracking differentiator, the fractional‐order PID controller with compensation and the fractional‐order extended state observer for nonlinear fractional‐order systems. Firstly, the fractional‐order optimal‐time control scheme is studied to propose the fractional‐order tracking differentiator by the Hamilton function and fractional‐order optimal conditions. Secondly, the linear fractional‐order extend state observer is offered to acquire the estimated value of the sum of nonlinear functions and disturbances existing in the investigated nonlinear fractional‐order plant. For the disturbance existing in the feedback output, the effect of the disturbance is discussed to choose a reasonable parameter in fractional‐order extended state observer. Thirdly, by this observed value, the nonlinear fractional‐order plant is converted into a linear fractional‐order plant by adding the compensation in the controller. With the aid of real root boundary, complex root boundary, and imaginary boot boundary, the approximate stabilizing boundary with respect to the integral and differential coefficients is determined for the given proportional coefficient, integral order and differential order. By choosing the suitable parameters, the fractional‐order active disturbance rejection control scheme can deal with the unknown nonlinear functions and disturbances. Finally, the illustrative examples are given to verify the effectiveness of fractional‐order active disturbance rejection control scheme. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal discrete higher‐order sliding mode control of uncertain LTI systems with partial state information

The concept of discrete higher‐order sliding mode has received increased attention in the recent literature. This paper presents an optimal discrete higher‐order sliding mode control for an uncertain discrete LTI system using partial state information, which has been missing in literature. A new technique is proposed to design an optimal time‐varying higher‐order sliding surface and control input through the minimization of a quadratic performance index. Moreover, disturbance estimation technique is utilized to modify the control algorithm to reduce the width of the discrete higher‐order sliding mode band. The proposed algorithm is experimentally validated on a rectilinear plant. Copyright © 2017 John Wiley & Sons, Ltd.     

非匹配不确定高阶非线性系统的滑模控制新方法

对具有非匹配不确定的高阶非线性系统,采用改进的高阶滑模微分器获取已知状态的任意阶微分估计值,再以恰当阶次的状态微分估计值之差,得到非匹配不确定项及其微分的估计值,证明了其误差任意小.为避免奇异性和抖振,采用两种方案设计了滑模控制器,并设计鲁棒项提高系统鲁棒性.基于Lyapunov~论证明了系统稳定性.同现有其他方法相比,该方法具有适用范围更广、收敛速度快、控制精度高、运算量小、保守性低等优点.最后仿真证明了本文所有结论.     

Reaction wheel pendulum control using fourth‐order discontinuous integral algorithm

The fourth‐order model of the reaction wheel pendulum is considered and a fourth‐order discontinuous integral algorithm is used for stabilization and tracking of the system, using a continuous control signal. The states reach the origin or a reference signal in finite‐time, even in presence of uncertain control coefficient and a kind of matched and unmatched uncertainties/disturbances. A homogeneous Lyapunov function is designed to ensure local finite‐time stability of the system, which can be used for designing the controller gains. Simulations and experimental results illustrate the performance and advantages of the presented algorithm.     

Regularity of minimizers for higher order variational problems in one independent variable

This paper concerns problems in the calculus of variations in one independent variable, when the Lagrangian depends on derivates of the state trajectories up to order N. For first order problems (N = 1) it is well known that, under standard hypotheses of existence theory and a local boundedness condition on the Lagrangian, minimizers have uniformly bounded first derivatives. These properties are of interest, because they ensure validity of necessary conditions for analysing minimizers, such as the Euler–Lagrange equation, and give insights in appropriate descritization schemes for numerical solution. For Nth order problems one might expect, by analogy with the N = 1 case, that minimizers would have uniformly bounded Nth order derivatives. This is not the case in general, however, as illustrated by known counter examples. To guarantee boundedness of the Nth order derivatives it has been found necessary to introduce additional ‘integrability’ hypotheses on derivatives of the Lagrangian, evaluated along the minimizer. We show that the additional hypotheses, previously imposed to guarantee uniform boundedness of the highest order derivatives, can be significantly reduced. This paper improves in particular on recent work on the boundedness of the second order derivates for second order problems, based on an analysis specific to the N = 2 case.     

Reduced‐order design of high‐order sliding mode control system

To design an rth (r>2) order sliding mode control system, a sliding variable and its derivatives of up to (r ? 1) are in general required for the control implementation. This paper proposes a reduced‐order design algorithm using only the sliding variable and its derivatives of up to (r ? 2) as the extension of the second‐order asymptotic sliding mode control. For a linear time‐invariant continuous‐time system with disturbances, it is found that a high‐order sliding mode can be reached locally and asymptotically by a reduced‐order sliding mode control law if the sum of the system poles is less than the sum of the system zeros. The robust stability of the reduced‐order high‐order sliding mode control system, including the convergence to the high‐order sliding mode and the convergence to the origin is proved by two Lyapunov functions. Simulation results show the effectiveness of the proposed control algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust nonovershooting tracking control for fractional‐order systems

A robust tracking control is proposed for the fractional‐order systems (FOSs) to achieve a tracking response with no overshoot, even in the presence of a class of disturbances. The control proposed makes use of a newly designed integral sliding mode technique for FOSs, which is capable of rejecting the bounded disturbances acting through the input channel. The proposed integral sliding mode control design has two components: a nominal control component and a discontinuous control component. The overshoot in the system response is avoided by the nominal control designed with the use of Moore's eigenstructure assignment algorithm. The sliding mode technique is used for the design of discontinuous part of the control that imparts the desired robustness properties.     

Optimal higher order learning adaptive control approach for a class of SISO nonlinear systems

In this paper,an optimal higher order learning adaptive control approach is developed for a class of SISO nonlinear systems.This design is model-free and depends directly on pseudo-partial-derivatives derived on-line from the input and output information of the system.A novel weighted one-step-ahead control criterion function is proposed for the control law.The convergence analysis shows that the proposed control law can guarantee the convergence under the assumption that the desired output is a set point.Simulation examples are provided for nonlinear systems to illustrate the better performance of the higher order learning adaptive control.