在逐点状态约束下一个四阶线性系统的时间最优控制

在逐点状态约束下,最优控制问题的求解是很困难的,已有的最大值原理和形态规划理论很难用来求解在逐点状态约束下最优控制问题.本文讨论逐点状态约束下一个四阶线性系统的时间最优控制问题.我们采用转换的方法给出了最优时间与最优控制的具体表达式.     

Optimal Nonlinear Income Taxation with a Two-Dimensional Population; A Computational Approach

We consider the optimal income tax problem when income differences are due to differences in abilities and in preferences between consumption and leisure among individuals. We model this problem as an optimal control problem and develop a numerical method for solving it. The method is based on the expansion of state and control variables in Lagrange series and on a spectral collocation method for approximating state equations. In this way the optimal control problem is reduced to a parameter optimization problem. The problem is difficult to solve, but we managed to do so with some limitations. On the basis of our calculations we conclude that the tax system in the two-dimensional case is more redistributive compared to that obtained from the one-dimensional model.     

On the convergence of nonlinear optimal control using pseudospectral methods for feedback linearizable systems

We consider the optimal control of feedback linearizable dynamic systems subject to mixed state and control constraints. In contrast to the existing results, the optimal controller addressed in this paper is allowed to be discontinuous. This generalization requires a substantial modification to the existing convergence analysis in terms of both the framework as well as the notion of convergence around points of discontinuity. Although the nonlinear system is assumed to be feedback linearizable, the optimal control does not necessarily linearize the dynamics. Such problems frequently arise in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. We prove that a sequence of solutions obtained using the Legendre pseudospectral method converges to the optimal solution of the continuous‐time problem under mild conditions. Published in 2007 by John Wiley & Sons, Ltd.     

Robust and optimal attitude control of spacecraft with disturbances

In this paper, a robust and optimal attitude control design that uses the Euler angles and angular velocities feedback is presented for regulation of spacecraft with disturbances. In the control design, it is assumed that the disturbance signal has the information of the system state. In addition, it is assumed that the disturbance signal tries to maximise the same performance index that the control input tries to minimise. After proposing a robust attitude control law that can stabilise the complete attitude motion of spacecraft with disturbances, the optimal attitude control problem of spacecraft is formulated as the optimal game-theoretic problem. Then it is shown that the proposed robust attitude control law is the optimal solution of the optimal game-theoretic problem. The stability of the closed-loop system for the proposed robust and optimal control law is proven by the LaSalle invariance principle. The theoretical results presented in this paper are illustrated by a numerical example.     

On constrained infinite-time linear quadratic optimal control with stochastic disturbances

In this work, we study the infinite-time linear quadratic optimal control problem for systems with stochastic disturbances and constrained inputs. A number of stochastic problem formulations under the full state information (FSI) structure are considered with a particular focus on the subject of feedback structure and its impact on certainty equivalence. In particular, we clarify results concerning the open-loop hard constrained, closed-loop statistically constrained, and closed-loop hard constrained cases. Extension to the infinite-time framework provides a vehicle for interpreting these controllers and indicates that the last of the three is of most interest to regulation type applications. Additionally, the partial state information problem is considered, and conditions are given for which a separated configuration consisting of the optimal estimator cascaded with the FSI optimal controller remains optimal.     

Necessary conditions for a class of optimal multiprocess with state constraints

In this article, we derive a maximum principle for a special class of free end time optimal control of multiprocesses involving a family of control systems acting in different regions defined by state constraints. We are mainly interested in problems with contiguous time intervals. The main feature of our maximum principle is that it covers the case where some of the regions considered may not be visited. This means that the intervals where the corresponding control systems are active may be reduced to a single point. The derivation of our maximum principle is done by reformulating the optimal multiprocess problem as an equivalent fixed time state constrained optimal control problem. This reformulated problem is also of interest since it provides the means to solve optimal multiprocess problems numerically via the direct method. We illustrate our findings with an example concerning the path planning of an autonomous underwater vehicle (AUV) using a simple kinematic model derived for simulations. We use simplified point‐mass model for the motion of an AUV in a horizontal plane and we assume that the ocean currents are known. We recast this problem as the multiprocess optimal control problem of interest and we study it via simulations presenting computational results partially validated by the maximum principle.     

Regional optimal control for a class of semilinear systems with distributed controls

This paper considers a regional optimal control problem for a class of semilinear infinite dimensional systems, with distributed controls. The existence of an optimal control is proven, and then this control is characterised for four cases of admissible controls. Theoretical results lead to a useful algorithm, that we illustrate by simulations of two heat equations.     

Optimal control for multi-stage and continuous-time linear singular systems

In this paper, optimal control problems for multi-stage and continuous-time linear singular systems are both considered. The singular systems are assumed to be regular and impulse-free. First, a recurrence equation is derived according to Bellman's principle of optimality in dynamic programming. Then, by applying the recurrence equation, bang-bang optimal controls for the control problems with linear objective functions subject to two types of multi-stage singular systems are obtained. Second, employing the principle of optimality, a equation of optimality for settling the optimal control problem subject to a class of continuous-time singular systems is proposed. The optimal control problem may become simpler through solving this equation of optimality. Two numerical examples and a dynamic input–output model are presented to show the effectiveness of the results obtained.     

Ensemble control of linear systems with parameter uncertainties

In this paper, we study the optimal control problem for a class of four-dimensional linear systems based on quaternionic and Fourier analysis. When the control is unconstrained, the optimal ensemble controller for this linear ensemble control systems is given in terms of prolate spheroidal wave functions. For the constrained convex optimisation problem of such systems, the quadratic programming is presented to obtain the optimal control laws. Simulations are given to verity the effectiveness of the proposed theory.     

A Stable Analytical Solution Method for Car-Like Robot Trajectory Tracking and Optimization

In this paper, the car-like robot kinematic model trajectory tracking and control problem is revisited by exploring an optimal analytical solution which guarantees the global exponential stability of the tracking error. The problem is formulated in the form of tracking error optimization in which the quadratic errors of the position, velocity, and acceleration are minimized subject to the rear-wheel car-like robot kinematic model. The input-output linearization technique is employed to transform the nonlinear problem into a linear formulation. By using the variational approach, the analytical solution is obtained, which is guaranteed to be globally exponentially stable and is also appropriate for real-time applications. The simulation results demonstrate the validity of the proposed mechanism in generating an optimal trajectory and control inputs by evaluating the proposed method in an eight-shape tracking scenario.     

Stabilization with discounted optimal control

We provide a condition under which infinite horizon discounted optimal control problems can be used in order to obtain stabilizing controls for nonlinear systems. The paper gives a mathematical analysis of the problem as well as an illustration by a numerical example.     

Discrete-time controller for stochastic nonlinear polynomial systems with Poisson noises

This paper presents a solution to the optimal control problem for discrete-time stochastic nonlinear polynomial systems confused with white Poisson noises subject to a quadratic criterion. The solution is obtained in the following way: a nonlinear optimal controller is first developed for polynomial systems, considering the state vector completely available for control design. Then, based on the solution of the state estimation problem for polynomial systems with white Poisson noises, the state estimate vector is used in the control law to obtain a closed-form solution. Performance of this controller is compared to that of the controller employing the extended Kalman filter and the linear-quadratic regulator and the controller designed for polynomial systems confused with white Gaussian noises.     

Optimal control problems with multiple characteristic time points in the objective and constraints

In this paper, we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points. These time points can be either fixed or variable. Using the control parametrization technique and a time scaling transformation, this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems. Each of these approximate problems can be viewed as a finite dimensional optimization problem. New gradient formulae for the cost and constraint functions are derived. With these gradient formulae, standard gradient-based optimization methods can be applied to solve each approximate optimal parameter selection problem. For illustration, two numerical examples are solved.     

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.      

On the optimal design of elastic beams

In this paper we investigate the optimal dynamics of simply supported nonlinearly elastic beams with rectangular cross-sections. We consider the elastic beam under the assumption of time-dependent intensive transverse loading. The state of the beam is described by a system of partial differential equations of the fourth order. We deal with the problem of choosing the optimal shape for the beam. The optimal shape is determined in such a way that the deflection of the nonlinearly elastic beam for any given time is minimal. The problem of choosing the optimal shape is formulated as an optimal control problem. To solve the obtained problem effectively, we use the optimality principle of Bellman (Bellman and Dreyfus 1962; Bryson and Ho 1975) and the penalty function method (Polyak 1987). We present a constructive algorithm for the optimal design of nonlinearly elastic beams. Some simple examples of the implementation of the proposed numerical algorithm are given.     

Parametric optimal control for uncertain linear quadratic models

In recent few decades, linear quadratic optimal control problems have achieved great improvements in theoretical and practical perspectives. For a linear quadratic optimal control problem, it is well known that the optimal feedback control is characterized by the solution of a Riccati differential equation, which cannot be solved exactly in many cases, and sometimes the optimal feedback control will be a complex time-oriented function. In this paper, we introduce a parametric optimal control problem of uncertain linear quadratic model and propose an approximation method to solve it for simplifying the expression of optimal control. A theorem is given to ensure the solvability of optimal parameter. Besides, the analytical expressions of optimal control and optimal value are derived by using the proposed approximation method. Finally, an inventory-promotion problem is dealt with to illustrate the efficiency of the results and the practicability of the model.     

Mathematical programs with complementarity constraints in the context of inverse optimal control for locomotion

In this paper an inverse optimal control problem in the form of a mathematical program with complementarity constraints (MPCC) is considered and numerical experiences are discussed. The inverse optimal control problem arises in the context of human navigation where the body is modelled as a dynamical system and it is assumed that the motions are optimally controlled with respect to an unknown cost function. The goal of the inversion is now to find a cost function within a given parametrized family of candidate cost functions such that the corresponding optimal motion minimizes the deviation from given data. MPCCs are known to be a challenging class of optimization problems typically violating all standard constraint qualifications (CQs). We show that under certain assumptions the resulting MPCC fulfills CQs for MPCCs being the basis for theory on MPCC optimality conditions and consequently for numerical solution techniques. Finally, numerical results are presented for the discretized inverse optimal control problem of locomotion using different solution techniques based on relaxation and lifting.     

基于二次型性能指标的混合动力汽车实时优化控制

为解决混合动力系统实时优化控制问题,本文提出了一种基于二次型性能指标最优的混合动力汽车功率分配优化方案.通过合理的假设和近似,建立了混合动力系统的线性模型,并利用二次型最优控制理论将混合动力最优控制问题转化为二次型最优调节问题进行求解,得到了一个结构简单的实时优化控制算法.5种道路工况下的仿真结果表明,本文提出的控制方法在未来道路工况未知的情况下能够实现混合动力系统的实时优化控制,且节油率与离线计算以燃油消耗最小为性能指标的全局最优控制的节油率相近.     

Multiple constrained rivaling actuators in the optimal control of miniaturized manipulators

Miniaturized robotic manipulators are a key elements in future high-precision telesurgery with artificial fingers and hands. A major challenge is the development of suitable drives for the micro-joints, because those small drives are either slow or can produce only low specific forces. A possible solution are joints driven by weak, but fast, and strong, but slow, actuators acting in parallel. This results in an optimal control problem of a multibody system subject to rivaling controls. Not only the control amplitude, but also the rate of change of the control is constrained for the respective actuator. In optimal control theory, derivatives of controls are forbidden by definition. The straightforward approach—defining the derivatives of the controls as the new control variables—results in a very sensitive differential-algebraic control problem with a complicated structure. A new approach is proposed: The original control problem is converted into a piecewise defined nonlinear multi-point boundary value problem for ordinary differential equations with nonconstant dimension. Based on the minimum principle, a strict and careful mathematical analysis of the coupling of the single parts of the boundary value problem leads to new interior point conditions at the junction points. As an example, three-dimensional energy optimal trajectories are calculated for a six-sectional branched manipulator. Stable, efficient and very accurate numerical solutions are obtained by the new procedure.     

Motion planning in uncertain environments with vision-like sensors

In this work we present a methodology for intelligent path planning in an uncertain environment using vision-like sensors, i.e., sensors that allow the sensing of the environment non-locally. Examples would include a mobile robot exploring an unknown terrain or a micro-UAV navigating in a cluttered urban environment. We show that the problem of path planning in an uncertain environment, under certain assumptions, can be posed as the adaptive optimal control of an uncertain Markov decision process, characterized by a known, control-dependent system, and an unknown, control-independent environment. The strategy for path planning then reduces to computing the control policy based on the current estimate of the environment, also known as the “certainty-equivalence” principle in the adaptive control literature. Our methodology allows the inclusion of vision-like sensors into the problem formulation, which, as empirical evidence suggests, accelerates the convergence of the planning algorithms. Further we show that the path planning and estimation problems, as formulated in this paper, possess special structure which can be exploited to significantly reduce the computational burden of the associated algorithms. We apply this methodology to the problem of path planning of a mobile rover in a completely unknown terrain.