Solving 1D Conservation Laws Using Pontryagin’s Minimum Principle

This paper discusses a connection between scalar convex conservation laws and Pontryagin’s minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar space-independent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the Lax–Oleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin’s minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both space-dependent and space-independent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with space-independent flux functions.     

A class of time-optimal velocity control laws for a spinning space vehicle

Several time-optimal velocity control laws are described for a spinning space vehicle. It is shown that several laws, rather than a unique law, result from the non-normality of the system. Given initial pitch and yaw body rates, a control system is defined that will drive these rates to zero in minimum time.     

Nonlinear feedback control with global stabilization

Hamilton-Jacobi-Bellman theory is shown to provide a unified framework for nonlinear feedback control laws for special classes of nonlinear systems. These classes include Jurdjevic-Quinn type systems, as well as minimum phase systems with relative degree {1, 1, ..., 1}. Several examples are given to illustrate these results. For the controlled Lorenz equation, results obtained by Vincent and Yu are extended. Next, for spacecraft angular velocity stabilization with two torque inputs, a family of nonlinear feedback control laws that globally asymptotically stabilize angular velocity is established. Special cases of this family of control laws include generalizations of the locally stabilizing control laws of Brockett and Aeyels to global stabilization as well as the globally stabilizing control laws of Irving and Crouch and Byrnes and Isidori. Finally, the results are applied to spacecraft angular velocity stabilization with only one torque input. These last results extend control laws given by Outbib and Sallet.     

Equivalence condition of optimal control problems†

Two optimal control problems which have the same optimal feedback control laws and identical minimum costs are said to be equivalent. The conditions for the equivalence of two optimal control problems have been presented by other authors, but it seems that the equivalence conditions given by other authors are not complete. In this paper, a necessary and sufficient condition for the equivalence classes is derived.     

Deadbeat control of linear multivariable generalized state-spacesystems

The classic idea of deadbeat control is extended to linear multivariable discrete-time generalized state-space systems using algebraic methods. The asymptotic properties of the linear quadratic regulator theory are used to obtain the classes of deadbeat controllers using stabilizing full semistate feedback. The solution is constructed from a `cheap control' problem. Both semistate and output deadbeat control laws are considered. The main design criteria are to drive the semistate and/or outputs of the system to zero in minimum time and that the closed-loop system be internally stable. Unique properties of these types of control laws are discussed. For semistate deadbeat control, all the (dynamic) poles including the ones at infinity are moved to the origin, whereas for output deadbeat, some of the finite transmission zeros are canceled. Numerically reliable algorithms are developed to solve both problems     

δ算子下的网络控制系统最优控制方法

研究网络控制系统的随机最优控制问题，提出了针对随机时延的网络控制系统最优控制律和二次型性能指标极小的控制律设计方案．在δ算子域内应用动态规划理论．设计网络控制系统的最优状态反馈和输出反馈控制律，得到的线性二次型高斯控制器可对系统中的随机长时延进行动态补偿．最后通过实例仿真验证了上述最优控制方案的可行性和有效性．     

Stabilization of Motions of the Descent Vehicle in the Upper Atmosphere

The paper was written in continuation of a series of papers on construction of a control law stabilizing a wide range of motions of the controlled plant. At that, the law in essence must be independent of the plant's dynamic characteristics and the environment. These multimodal (general-purpose) laws must generate the control signal with the minimum computational burden and in the minimum time. These laws have other important characteristics concerning stability of the closed-loop system, its robustness, and so on. The paper studied a controlled plant moving under the action of aerodynamic and other forces. The distinction of the formulated control problem is due to the deficiency of the control actions because it is assumed that the controlled plant has only two control engines. This is the case in the well-known problem of controlling the descent vehicle by stabilizing its lateral motion on the hypersonic leg of flight in the upper atmosphere.     

矿井搜救探测机器人驱动系统最小能耗建模与影响因素分析

研究了矿井搜救探测机器人最小能耗影响因素。针对直流伺服电机和减速齿轮驱动下的移动机器人驱动系统,利用Hamiltonian函数与最小值原理求解出驱动系统的最优速度函数和最优控制电流函数,建立了非线性摩擦条件下的最小能耗模型。通过MATLAB仿真,分析了减速器效率[η]、负载、库仑摩擦力和粘性摩擦系数等因素对最小能耗的影响,确定了各参数对驱动系统最小能耗的影响规律。为矿井搜救探测机器人驱动系统消耗能量最小进行参数配置提供依据,对实现机器人节能降耗具有现实的意义。     

关于最小分散镇定结构的研究

本文考察线性时不变多变量系统的分散镇定问题,揭示了局部控制站间的通信与消除固 定模间的内在联系,并由此把求最小(最经济)分散可镇定结构问题转化成一个显式的特殊0-1 规划问题,导出了一种求最小分散可镇定结构的有效算法.     

Time-optimal control of a self-propelled particle in a spatiotemporal flow field

We address a minimum-time problem that constitutes an extension of the classical Zermelo navigation problem in higher dimensions. In particular, we address the problem of steering a self-propelled particle to a prescribed terminal position with free terminal velocity in the presence of a spatiotemporal flow field. Furthermore, we assume that the norm of the rate of change of the particle's velocity relative to the flow is upper bounded by an explicit upper bound. To address the problem, we first employ Pontryagin's minimum principle to parameterise the set of candidate time-optimal control laws in terms of a parameter vector that belongs to a compact set. Subsequently, we develop a simple numerical algorithm for the computation of the minimum time-to-come function that is tailored to the particular parametrisation of the set of the candidate time-optimal control laws of our problem. The proposed approach bypasses the task of converting the optimal control problem to a parameter optimisation problem, which can be computationally intense, especially when one is interested in characterising the optimal synthesis of the minimum-time problem. Numerical simulations that illustrate the theoretical developments are presented.     

The Application of Connectionist Structures to Learning Impedance Control in Robotic Contact Tasks

The goal of this paper is to consider the synthesis of learning impedance control using recurrent connectionist structures for on-line learning of robot dynamic uncertainties in the case of robot contact tasks. The connectionist structures are integrated in non-learning impedance control laws that are intended to improve the transient dynamic response immediately after the contact. The recurrent neural network as a part of hybrid learning control algorithms uses fast learning rules and available sensor information in order to improve the robotic performance progressively for a minimum possible number of learning epochs. Some simulation results of deburring process with the MANUTEC r3 robot are presented here in order to verify the effectiveness of the proposed control learning algorithms.     

关于最小分散镇定结构的研究

本文考察线性时不变多变量系统的分散镇定问题,揭示了局部控制站间的通信与消除固定模间的内在联系,并由此把求最小(最经济)分散可镇定结构问题转化成一个显式的特殊0-1规划问题,导出了一种求最小分散可镇定结构的有效算法。     

Adaptive stabilization with exponential decay

Adaptive high gain stabilizers for classes of linear time-invariant state space systems are presented. The classes cover multi-input—multi-output systems where the state dimension is not known. Only standard assumptions such as minimum phase and known respectively unknown sign are required. The main result is, that the adaptive control laws of Byrnes and Willems can be modified to produce a gurantee of exponentially decaying states.     

Estimating the region of attraction for large-scale systems with uncertainties

In this paper, a new approach is proposed for estimating regions of attraction for large-scale dynamic systems. In designing control laws for such systems, it is essential to incorporate the underlying information structure constraints while keeping the number of optimization variables at a minimum. The proposed method successfully accomplishes both of these objectives. It is computationally efficient, and can produce decentralized control laws without imposing structural constraints on the Lyapunov function (a feature that can considerably improve the quality of the estimate). The design algorithm is based on linear matrix inequalities and can easily accommodate various types of uncertainties in the system model. An example with 300 states is provided to demonstrate the suitability of this approach for large, sparse systems.     

高频增益未知的受扰非线性系统输出反馈控制

针对一类具有未知定常参数,包括未知高频增益的受扰非线性最小相位系统,给出了一种鲁棒自适应输出反馈控制策略．系统所受的干扰假设有界,但其界值是未知的．通过采用自适应策略来对其界值进行在线估计,控制算法并不需要高频增益符号的先验知识．同时,系统中的非线性项并不要求满足增长性条件和匹配条件．算法使得估计参数量达到了最小,保证了闭环系统所有信号的有界性,同时使得跟踪误差渐近收敛于零．     

基于最小衰减率多面体不变集的鲁棒模型预测控制

针对一类具有输入输出约束的多胞体结构线性变参数系统,提出了一种基于最小衰减率多面体不变集的鲁棒模型预测控制算法,算法分为在线和离线两个部分.为增强系统控制效果,提高系统响应速度,离线算法首先采用寻求状态变量的最小衰减率的方法优化出一系列状态变量及相应的状态反馈控制律,然后构建出相应的多面体不变集序列;在线算法根据当前实测状态变量,在多面体不变集序列内确定状态变量所处的最小多面体不变集,通过在线优化得出系统的控制输入.给出了鲁棒模型预测控制算法的详细步骤和系统的闭环稳定性证明.仿真结果验证了本算法的有效性,表明本算法使系统的闭环响应更为快速和稳定.     

Stability and Control of Collective Systems

In this paper, decentralized, distributed feedback control laws are presented for cooperative robotic systems whose task is to localize unknown sources. The control laws follow from a second order representation of the source field. The stability of the proposed feedback control laws for the individual robots, and for the entire robot collective, is demonstrated using Lyapunov's direct method and a vector Lyapunov approach. Additional feedback control laws are proposed to achieve an additional level of coordination. In particular, control laws that achieve desired formations surrounding a localized source are developed.     

A method for synthesis of “soft” and “super-soft” control laws for final states of dynamic systems

The problem of synthesis of control laws for final states of dynamic systems in non-conventional state is considered, when, together with the boundary conditions for the parameters of the system state, boundary conditions for control actions are given. Control laws providing zero values of control actions at the terminal time instant are called laws of soft control. Control laws that provide zero values of not only control actions at the terminal time instant, but also of a number of their successive derivatives with respect to time are called laws of super-soft control. The practicability of application of these laws in control algorithms of flying vehicles and of objects of other types was shown in works cited in the references.     

一类不确定性时滞关联大系统的分散鲁棒稳定控制器设计

对一类满足匹配条件的不确定性关联大系统，设计了基于一组线性矩阵不等式（LMI）有解的系统状态反馈镇定的充分条件。本文采用了实对称矩阵集合的最小上界定理，并对设计参数进行优化处理，从而使得所设计的控制器具有较小反馈增益。同时系统的不确定性采用区间矩阵表示，因此该设计方法具有很好的普遍性。最后给出的仿真实例说明了该方法的有效性。