Dynamical properties of an adaptive control system with a nonlinear reference model. I

We show the results of solving the problem of constructing an adaptive control system with nonlinear reference model for a nonstationary dynamical object with a nonlinear actuator. To analyze the dynamical properties of a nonlinear nonstationary control system we use the method of pointwise transformations. We show the results of analysis and mathematical modeling of the considered control system with the adaptation circuit switched off.     

A polynomial approach for optimal control of switched nonlinear systems

Optimal control problems for switched nonlinear systems are investigated. We propose an alternative approach for solving the optimal control problem for a nonlinear switched system based on the theory of moments. The essence of this method is the transformation of a nonlinear, nonconvex optimal control problem, that is, the switched system, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high‐performance numerical computing. Consequently, we propose to convexify the control variables by means of the method of moments obtaining semidefinite programs. Copyright © 2013 John Wiley & Sons, Ltd.     

Learning control for a class of nonlinear differential-algebraic systems with application to constrained robots

In this article, a learning control design method for analyzing the convergence of nonlinear systems described by a class of differential-algebraic equations¹ is developed. This method generalizes the previous methods for the study of the learning control problem of constrained mechanical systems to include a wider class of systems. Furthermore, the proposed design method allows us to analyze the role of force learning in the learning control of such a control system. We derive a sufficient condition for the convergence of both the motion and contact force of the systems as the operations are repeated. An application of the proposed controller to robotic manipulators with holonomic constraints is discussed, and simulation results of a constrained cylindrical robot are presented. © 1996 John Wiley & Sons, Inc.     

双机电力系统周期轨道分析与滑模控制

针对一类双机电力系统,运用次谐波Melnikov函数分别对系统的振荡周期轨道和旋转周期轨道进行了分析计算。同时对系统的追踪控制问题,采用滑模控制方法,有效地实现了双机电力系统对给定参考信号的追踪控制。对双机电力系统滑模控制策略及周期轨道进行了数值仿真,验证了理论分析的正确性。     

嵌入式控制系统程序模式的自动分析方法

嵌入式控制系统通常都有模式,比如启动模式、正常工作模式以及紧急模式等。程序模式是由其输入变量值范围组合构成的输入变量约束表达式表示的。基于源程序,获取其模式,不仅能够验证实现的模式与设计是否一致,还能够更加精确地计算程序的WCET。在对源程序进行分析的基础上,提出了一种自动获取程序模式的新方法。该方法基于C语言源程序,针对程序控制流程图,通过调整循环中节点流向以及去除与输入变量无关的节点,获得输入变量相关控制流程图ICFG,通过对ICFG每条路径建立线性规划问题并求解,获得每一个潜在的程序模式及其输入变量约束表达式。对基准程序的实验结果,表明了该方法的可行性和有效性。     

Local control gradients criterion for selection of neuroemulators for model reference adaptive neurocontrol

We discuss neural identification and control of nonlinear dynamic plant. We analyze a problem of selection of a proper neuroemulator for training neurocontrollers, and we propose a new effective criterion based on the analysis of local gradients of neuroemulator??s input neurons. We present results of numerical simulations of neurocontroller training by a gradient descent method and by an Extended Kalman Filter method.     

real-time motion planning for multibody systems

The solution of constrained motion planning is an important task in a wide number of application fields. The real-time solution of such a problem, formulated in the framework of optimal control theory, is a challenging issue. We prove that a real-time solution of the constrained motion planning problem for multibody systems is possible for practical real-life applications on standard personal computers. The proposed method is based on an indirect approach that eliminates the inequalities via penalty formulation and solves the boundary value problem by a combination of finite differences and Newton–Broyden algorithm. Two application examples are presented to validate the method and for performance comparisons. Numerical results show that the approach is real-time capable if the correct penalty formulation and settings are chosen.     

最优控制问题的Legendre 伪谱法求解及其应用

伪谱法通过全局插值多项式参数化状态和控制变量,将最优控制问题(OCP)转化为非线性规划问题(NLP)进行求解,是一类具有更高求解效率的直接法。总结Legendre伪谱法转化Bolza型最优控制问题的基本框架,推导OCP伴随变量与NLP问题KKT乘子的映射关系,建立基于拟牛顿法的LGL配点数值计算方法,并针对非光滑系统,进一步研究分段伪谱逼近策略。基于上述理论开发通用OCP求解器,并对3个典型最优控制问题进行求解,结果表明了所提出方法和求解器的有效性。     

Notch filter feedback control in a class of chaotic systems

The paper focuses on the problem of the notch filter feedback control in the perturbed planar Hamiltonian systems. By Melnikov's method, a suitable range of parameters in the notch filter controller can be obtained to convert chaotic motions into desired low-period motions. The averaging method is introduced to analyze the stability of the low-period orbits, the subharmonic orbits inside the homoclinic loop, in the control systems. As a typical example, the design procedure for controlling Duffing's oscillator and its stability analysis are derived in detail, and the thorough simulation results are presented to demonstrate the effectiveness of the theoretical analysis. Finally, the further examples show the applicability of the notch filter controller in a wide range of chaotic systems and hyper-chaotic systems.     

混沌Lorenz系统延迟反馈控制的机理分析

利用广义Hamilton系统理论的Melnikov方法,严格分析了延迟反馈方法控制混沌Lorenz系统到周期解的机理,揭示了延迟时间与控制混沌的关系.延迟反馈项实际上是一个作用明显的扰动项,通过选择合适的参数,使得系统的稳定流形与不稳定流形不再横截相交,Smale意义下的混沌受到抑制,将Lorenz混沌系统引导到各种不同的周期轨道;可见,延迟时间关系到控制扰动量的大小,但不必是混沌吸引子内嵌不稳定周期轨道的周期整数倍.另外,通过数值仿真,其结果与理论分析相符,从而表明了该分析方法的有效性.     

Reduction of Cycle Unification of Type Cpg+r

In this paper a class of cycle unification problems called type Cpg r is studied,and an intuitive representation method,called the position supergraph,is presented to analyze or reduce the cycle problem.A typical case of Cpg r problems is considered and an algorithm is given,which tansforms the original problem of Cpg r that is of the exponential complexity into a new one that is of the linear complexity when using SLD resolution method.These results are of importance for practice and theory.     

Sufficient Relative Minimum Conditions in the Optimal Control Problem for Quasilinear Stochastic Systems

We consider the optimal control problem for quasilinear stochastic systems with continuous time whose coefficients have a generally non-linear dependence on the program control. We establish sufficient conditions for a strong and weak relative minimum. We give examples of using the resulting conditions for constructing optimal control in a nonlinear onedimensional problem and in a two-dimensional linear problem with information constraints and analyze the possible results.     

An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

This paper deals with the Ritz spectral method to solve a class of fractional optimal control problems (FOCPs). The developed numerical procedure is based on the function approximation by the Bernstein polynomials along with fractional operational matrix usage. The approximation method is computationally consistent and moreover, has a good flexibility in the sense of satisfying the initial and boundary conditions of the optimal control problems. We construct a new fractional operational matrix applicable in the Ritz method to estimate the fractional and integer order derivatives of the basis. As a result, we achieve an unconstrained optimization problem. Next, by applying the necessary conditions of optimality, a system of algebraic equations is obtained. The resultant problem is solved via Newton's iterative method. Finally, the convergence of the proposed method is investigated and several illustrative examples are added to demonstrate the effectiveness of the new methodology.     

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

This paper presents a new approach for the efficient solution of singular optimal control problems (SOCPs). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of solution. At first, the SOCP is converted into a binary optimal control problem. Then, by utilising the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the optimal control and to compute the approximating solution. The main advantages of the present method are that: (1) without a priori information, the structure of optimal control is detected; (2) it produces good results even using a small number of collocation points; (3) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.     

Division by zero,pseudo-division by zero,Zhang dynamics method and Zhang-gradient method about control singularity conquering

In this paper, the division-by-zero (DBO) problem in the field of nonlinear control, which is traditionally termed the control singularity problem (or specifically, controller singularity problem), is investigated by the Zhang dynamics (ZD) method and the Zhang-gradient (ZG) method. According to the impact of the DBO problem on the state variables of the controlled nonlinear system, the concepts of the pseudo-DBO problem and the true-DBO problem are proposed in this paper, which provide a new perspective for the researchers on the DBO problems as well as nonlinear control systems. Besides, the two classes of DBO problems are solved under the framework of the ZG method. Specific examples are shown and investigated in this paper to illustrate the two proposed concepts and the efficacy of the ZG method in conquering pseudo-DBO and true-DBO problems. The application of the ZG method to the tracking control of a two-wheeled mobile robot further substantiates the effectiveness of the ZG method. In addition, the ZG method is successfully applied to the tracking control of a pure-feedback nonlinear system.     

基于Radau伪谱法的非线性最优控制问题的收敛性

在过去的10年里,伪谱方法(如Legendre伪谱法、Gauss伪谱法、Radau伪谱法)逐步成为求解不同领域中非线性最优控制问题的一种高效、灵活的数值解法.本文从最优控制问题解的存在性、收敛性以及解的可行性3个方面对采用Radau伪谱法求解一般非线性最优控制问题解的收敛性进行研究.证明了原最优控制问题的离散解存在、存在收敛到原最优控制问题解上的离散解和离散形式的收敛解是原最优控制问题的最优解.在此基础上,证明了Radau伪谱法的收敛性.本文结论与现有文献相比,去掉了一些必要条件,更适合一般的非线性时不变系统.     

Accelerating Hair Rendering by Learning High-Order Scattered Radiance

Efficiently and accurately rendering hair accounting for multiple scattering is a challenging open problem. Path tracing in hair takes long to converge while other techniques are either too approximate while still being computationally expensive or make assumptions about the scene. We present a technique to infer the higher order scattering in hair in constant time within the path tracing framework, while achieving better computational efficiency. Our method makes no assumptions about the scene and provides control over the renderer's bias & speedup. We achieve this by training a small multilayer perceptron (MLP) to learn the higher-order radiance online, while rendering progresses. We describe how to robustly train this network and thoroughly analyze our resulting renderer's characteristics. We evaluate our method on various hairstyles and lighting conditions. We also compare our method against a recent learning based & a traditional real-time hair rendering method and demonstrate better quantitative & qualitative results. Our method achieves a significant improvement in speed with respect to path tracing, achieving a run-time reduction of 40%-70% while only introducing a small amount of bias.     

Multilayer Minimum Projection Method with Singular Point Assignment for Nonsmooth Control Lyapunov Function Design

Control Lyapunov function (CLF) design on a manifold is a difficult problem in control theory. To address this problem, we have proposed the multilayer minimum projection method. The method requires CLFs on different manifolds from the manifold where the control problem is defined. In this paper, we relax the requirement by desingularization of the functions on the manifolds. The paper focuses on the problem of desingularization in the multilayer minimum projection method. We show that the functions on other manifolds need not be CLFs by consideration of desingularization. Moreover, we propose a CLF design method by singular point assignment based on the advantage of desingularization. The method enables us to merge local CLFs into the global CLF. This paper proposes two CLF design methods: desingularization and singular point assignment. A CLF design example is provided for each method; the advantages of the proposed methods are confirmed by those two examples.     

A variational binary level-set method for elliptic shape optimization problems

We present a variational binary level-set method to solve a class of elliptic problems in shape optimization. By the ‘ersatz material’ approach, which amounts to fill the holes by a weak phase, the original shape optimization model is approximated by a two-phase optimization problem. Under the binary level-set framework, we need to optimize a smooth functional under a binary constraint. We propose an augmented Lagrangian method to solve the constrained optimization problem. Numerical results are presented and compared with those obtained by level-set methods, which demonstrate the robustness and efficiency of our method.     

Élie Cartan’s geometrical vision or how to avoid expression swell

The aim of the paper is to demonstrate the superiority of Cartan’s method over direct methods based on differential elimination for handling otherwise intractable equivalence problems. In this sense, using our implementation of Cartan’s method, we establish two new equivalence results. We establish when a system of second order ODEs is equivalent to flat system (second derivations are zero), and when a system of holomorphic PDEs with two independent variables and one dependent variable is flat. We consider the problem of finding transformation that brings a given equation to the target one. We shall see that this problem becomes algebraic when the symmetry pseudogroup of the target equation is zerodimensional. We avoid the swelling of the expressions, by using non-commutative derivations adapted to the problem.