Construction of the best ellipsoidal approximation of the attraction domain in stabilization problem for a wheeled robot

The synthesis control problem for the plane motion of a wheeled robot with constrained control resource is studied. The goal of the control is to bring the robot to an assigned curvilinear trajectory and to stabilize its motion along it. For a synthesized control law, the problem of finding the best in the sense of volume ellipsoidal approximation of the attraction domain of the target path is posed. To take into account constraints on the control, an approach based on methods of absolute stability theory is used, in the framework of which construction of an approximating ellipsoid reduces to solving a system of linear matrix inequalities. It is shown that the desired maximum-volume approximating ellipsoid can be found by solving a standard constrained optimization problem for a function of two variables.     

Attraction domain estimate for single-input affine systems with constrained control

Nonlinear single-input affine systems represented in a canonical (normal) form are considered. The control resource is assumed to be constrained. For a closed-loop system obtained by applying a linearizing feedback, the problem of finding an estimate of the attraction domain is set. A method for constructing an ellipsoidal estimate that is based on results of absolute stability theory is suggested. Construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.     

面向欠约束几何系统的一种同伦求解方法

针对几何约束系统的数值求解过程中，经常发生的数值不稳定性问题，构造了一种面向欠约束系统的同伦方法，并将其与现有的求解与分解方法有机地结合起来，提出了一种牛顿－同伦混合方法，在牛顿迭代失败的位置自动调用欠约束同伦法，既提高了几何约束求解器的效率，同时又保证了求解的效率。     

Convexity and convex approximations of discrete-time stochastic control problems with constraints

We investigate constrained optimal control problems for linear stochastic dynamical systems evolving in discrete time. We consider minimization of an expected value cost subject to probabilistic constraints. We study the convexity of a finite-horizon optimization problem in the case where the control policies are affine functions of the disturbance input. We propose an expectation-based method for the convex approximation of probabilistic constraints with polytopic constraint function, and a Linear Matrix Inequality (LMI) method for the convex approximation of probabilistic constraints with ellipsoidal constraint function. Finally, we introduce a class of convex expectation-type constraints that provide tractable approximations of the so-called integrated chance constraints. Performance of these methods and of existing convex approximation methods for probabilistic constraints is compared on a numerical example.     

Synthesis of a stabilizing feedback for a wheeled robot with constrained control resource

Stabilization of motion of a wheeled robot with constrained control resource by means of a continuous feedback linearizing the closed-loop system in a neighborhood of the target path is considered. We pose the problem of finding the feedback coefficients such that the phase portrait of the nonlinear closed-loop system is topologically equivalent to that of a linear system with a stable node, with the asymptotic rate of decrease of the deviation from the target path being as high as possible. On this family, we pose the problem of minimization of “overshooting” for arbitrary initial conditions. The solution of this optimization problem is proved to be a limit discontinuous control law. A hybrid control law is proposed that, on the one hand, ensures the desired properties of the phase portrait and minimal overshooting and, on the other hand, does not result in a chattering inherent in systems with discontinuous feedbacks.     

Semi-globally tracking control for uncertain non-affine self-restructuring systems with state constraints

It is still an open problem to achieve tracking control with state constraints if the initial states are out of the constrained region. This underlying problem becomes more challenging if the system suffers from modeling uncertainties and external disturbances. In this paper, we investigate such problem for a class of uncertain nonaffine self-restructuring systems subject to state constraints in the absence of any limitation on the initial values of states. In order to steer the original system states into the desired constraint region in finite time, certain auxiliary system is firstly constructed via introducing auxiliary variables. By resorting to some auxiliary nonlinear functions, the original system with state constraints is transformed into an unconstrained one in the sense that the constraint problem of the original system is equivalent to ensuring the stability of the transformed one. By following the backstepping design technique, an adaptive control scheme is developed that enables the states to enter into the constrained region for any initial point within any fixed time and then never violate it. All the closed loop signals are ensured to be semi-globally uniformly ultimately bounded. Finally, one numerical simulation example is presented to verify the theoretical analysis.     

计算机数控系统光滑时间最优轨迹规划

基于控制向量参数化(CVP)方法, 研究了计算机数控(CNC)系统光滑时间最优轨迹规划方法. 通过在规划问题中引入加加速度约束, 实现轨迹的光滑给进. 引入时间归一化因子, 将加加速度约束的时间最优轨迹规划问题转化为固定时间的一般性最优控制问题. 以路径参数对时间的三阶导数(伪加加速度)和终端时刻为优化变量, 并采用分段常数近似伪加加速度, 将最优控制问题转化为一般的非线性规划(NLP)问题进行求解. 针对加加速度、加速度等过程不等式约束, 引入约束凝聚函数, 将过程约束转化为终端时刻约束, 从而显著减少约束计算. 构造目标和约束函数的Hamiltonian函数, 利用伴随方法获得求解NLP问题所需的梯度.     

Newton-conjugate gradient （CG） augmented Lagrangian method for path constrained dynamic process optimization

In this paper, a Newton-conjugate gradient (CG) augmented Lagrangian method is proposed for solving the path constrained dynamic process optimization problems. The path constraints are simplified as a single final time constraint by using a novel constraint aggregation function. Then, a control vector parameterization (CVP) approach is applied to convert the constraints simplified dynamic optimization problem into a nonlinear programming (NLP) problem with inequality constraints. By constructing an augmented Lagrangian function, the inequality constraints are introduced into the augmented objective function, and a box constrained NLP problem is generated. Then, a linear search Newton-CG approach, also known as truncated Newton (TN) approach, is applied to solve the problem. By constructing the Hamiltonian functions of objective and constraint functions, two adjoint systems are generated to calculate the gradients which are needed in the process of NLP solution. Simulation examples demonstrate the effectiveness of the algorithm.     

Finite-horizon differential games for missile–target interception system using adaptive dynamic programming with input constraints

In this paper, the problem of intercepting a manoeuvring target within a fixed final time is posed in a non-linear constrained zero-sum differential game framework. The Nash equilibrium solution is found by solving the finite-horizon constrained differential game problem via adaptive dynamic programming technique. Besides, a suitable non-quadratic functional is utilised to encode the control constraints into a differential game problem. The single critic network with constant weights and time-varying activation functions is constructed to approximate the solution of associated time-varying Hamilton–Jacobi–Isaacs equation online. To properly satisfy the terminal constraint, an additional error term is incorporated in a novel weight-updating law such that the terminal constraint error is also minimised over time. By utilising Lyapunov's direct method, the closed-loop differential game system and the estimation weight error of the critic network are proved to be uniformly ultimately bounded. Finally, the effectiveness of the proposed method is demonstrated by using a simple non-linear system and a non-linear missile–target interception system, assuming first-order dynamics for the interceptor and target.     

基于动态规划的资源受限随机工序调度

为解决资源受限条件下的随机工序调度问题,该文提出一种基于离散随机动态系统描述的加工时间离散随机分布且同时具有不兼容和多种可更新资源约束的资源受限项目调度模型,使得在满足资源约束和工序约束的前提下,总的平均加工时间最短。该系统研究了动态规划算法求解该问题的方法。通过实例,验证了该方法的有效性和可行性。     

Synthesis of a stabilizing control for a wheeled robot following a curvilinear path

The synthesis control problem for the plane motion of a wheeled robot with constrained control resource is studied. The goal of the control is to bring the robot to an assigned curvilinear trajectory and to stabilize its motion along it. A new change of variable is suggested that reduces the problem of stabilizing robot’s motion to that of stabilizing the zero solution in the form that admits feedback linearization. A control law stabilizing robot’s motion along an arbitrary curvilinear target trajectory is synthesized. For a straight target path, the closed-loop system is shown to be asymptotically stable for any initial conditions except for the case where the initial direction of motion is perpendicular to the target path.     

Robust MPC for the constrained system with polytopic uncertainty

An improved method for synthesising the constrained robust model predictive controller is proposed in this study. It constructs a continuum of terminal constraint sets off-line, and achieves robust stability with a variable control horizon on-line from the very beginning and a time-varying terminal constraint set, by solving the min–max optimisation problem, which can be formulated as a linear matrix inequality problem. This algorithm not only dramatically reduces the on-line computation burden, but also guarantees the control performance by reserving at least one free control move in the whole process. Simulation results for the three-tank system with uncertain dynamic behaviour on flux coefficients are given.     

Control and simulation of a constrained dynamic two-link system

In this paper a solution to the attitude control problem of a two-link planar system, whose angular momentum about the centre of gravity is constant, is presented. Reference trajectories for simulating the motion of the constrained dynamic system are found by specifying the internal variables of the system and solving for the corresponding external variables. It is shown that the angular momentum constraint has quasi-holonomic character because it can be employed to reduce the dimensionality of the system by eliminating half a degree of freedom. A controller is designed for the reduced dynamic system and tested by digital computer simulation. It is shown that the same controller can be used for the unreduced system if the angular momentum constraint is a linear function of the states of the system. Such an approach, however, cannot be employed if the constraint is a non-linear function of the system state. In this case a controller must be designed which stabilizes both the system and the constraint equation. Computer simulations of the diving motion of a human are presented to illustrate the control approach.     

A linearizing feedback for stabilizing a car-like robot following a curvilinear path

The path following problem for a car-like robot is considered. The control goal is to bring the robot to a pre-assigned curvilinear path and to stabilize its motion along the path. A new canonical change of variables is suggested. It reduces the problem of stabilizing robot’s motion to that of stability of the zero solution of the transformed system in the form that admits feedback linearization. A new control law is synthesized that ensures linearity of the closed-loop system and stabilizes robot’s motion along a given target path if the initial conditions belong to a known region. Comparison of the new control law with two earlier obtained linearizing feedbacks known from the literature demonstrates its unquestionable advantages.     

Lagrangian duality applied to the vehicle routing problem with time windows

This paper considers the vehicle routing problem with time windows, where the service of each customer must start within a specified time interval. We consider the Lagrangian relaxation of the constraint set requiring that each customer must be served by exactly one vehicle yielding a constrained shortest path subproblem. We present a stabilized cutting-plane algorithm within the framework of linear programming for solving the associated Lagrangian dual problem. This algorithm creates easier constrained shortest path subproblems because less negative cycles are introduced and it leads to faster multiplier convergence due to a stabilization of the dual variables. We have embedded the stabilized cutting-plane algorithm in a branch-and-bound search and introduce strong valid inequalities at the master problem level by Lagrangian relaxation. The result is a Lagrangian branch-and-cut-and-price (LBCP) algorithm for the VRPTW. Making use of this acceleration strategy at the master problem level gives a significant speed-up compared to algorithms in the literature based on traditional column generation. We have solved two test problems introduced in 2001 by Gehring and Homberger with 400 and 1000 customers respectively, which to date are the largest problems ever solved to optimality. We have implemented the LBCP algorithm using the ABACUS open-source framework for solving mixed-integer linear-programs by branch, cut, and price.     

A new problem in control system design [and circuit design]

An optimisation problem arising in the design of circuits and control systems is formulated, and an outer approximation algorithm for solving it is presented. The problem is characterized by constraints that must be satisfied for all functions in a given subset of a function space. It is shown that an outer-approximation-type algorithm increases the number of variables and constraints at each iteration. A practical algorithm will require a constraint dropping scheme     

Canonical representation of the path following problem for wheeled robots

For the problem of stabilizing motion of an n-dimensional nonholonomic wheeled system along a prescribed path, the concept of a canonical representation of the equations of motion is introduced. The latter is defined to be a representation that can easily be reduced to a linear system in stabilizable variables by means of an appropriate nonlinear feedback. In the canonical representation, the path following problem is formulated as that of stabilizing the zero solution of an (n?1)-dimensional subsystem of the canonical system. It is shown that, by changing the independent variable, the construction of the canonical representation reduces to finding the normal form of a stationary affine system. The canonical representation is shown to be not unique and is determined by the choice of the independent variable. Three changes of variables known from the literature, which were earlier used for synthesis of stabilizing controls for wheeled robot models described by the third- and fourth-order systems of equations, are shown to be canonical ones and can be generalized to the n-dimensional case. Advantages and disadvantages of the linearizing control laws obtained by means of these changes of variables are discussed.     

约束非线性系统理想点多目标安全预测控制

考虑具有状态和控制约束的仿射非线性系统多目标安全控制问题,本文提出一种保证安全和稳定的多目标安全模型预测控制(MOSMPC)策略.首先通过理想点逼近方法解决多个控制目标的冲突问题.其次,利用控制李雅普诺夫障碍函数(CLBF)参数化局部控制律,并确定系统不安全域.在此基础上,构造非线性系统的参数化双模控制器,减少在线求解模型预测控制(MPC)优化问题的计算量.进一步,应用双模控制原理和CLBF约束,建立MOSMPC策略的递推可行性和闭环系统的渐近稳定性,并保证闭环系统状态避开不安全域.最后,以加热系统的多目标控制为例,验证了本文策略的有效性.     

蚁群算法在时延约束选播路由问题中的应用

在分析选播通信服务的基础上,研究了蚁群算法在网络选播路由问题中的应用,提出了一种基于蚁群系统原理,用于解决有时延约束的选播路由问题。算法采用调整最优解路径上的信息素和算法重启策略,较好地解决了蚁群算法易于陷入局部最优的问题。仿真实验结果表明,该算法是有效且切实可行的,它可以在满足延时约束的条件下迅速找到最优解。     

A global approach for the optimal path generation of redundant robot manipulators

In this article the optimal path generation of redundant robot manipulators is considered as an optimization problem, with given kinematics and subject to the robot requirements and a singularities avoidance constraint. This problem is formulated as a constrained continuous optimal control problem, which allows to consider joints and velocities constraints and/or manipulator dynamics. This approach is exemplified for a planar redundant manipulator and the resultant state constrained problem is solved by an efficient iterative numerical technique.