On the solution of inverse problems of dynamics of linearly controlled systems by the negative discrepancy method

The paper is devoted to the substantiation of the negative discrepancy method for solving inverse problems of the dynamics of deterministic control systems that are nonlinear in state variables and linear in control. The problem statements include the known sampling history of trajectories measured inaccurately, with known error estimates. The investigation is based on the Pontryagin maximum principle. The results of simulation of an inverse problem for a macroeconomic model are presented.     

开环策略下多阶段均值-方差投资组合优化研究

首先研究开环策略下不同财富动态过程的多阶段均值-方差投资组合优化模型,讨论它们的实际意义和计算方法,其中投资比例财富动态过程模型为高度非线性非凸数学规划.进一步研究投资比例财富动态过程模型实际计算问题,并且通过构造辅助模型,给出投资比例两阶段模型的全局解求解方法并通过数值算例和仿真说明该方法的有效性和准确性.最后通过数值算例比较不同财富动态过程在开环策略下和闭环策略下前沿面的关系,结果表明在闭环策略下三种财富过程等价,但是在开环策略下资产财富模型的前沿面最高、资产调整模型的前沿面次之、投资比例多阶段模型的前沿面最低.     

结构动力学逆特征值问题的同伦解法

将结构动力学反问题视为拟乘法逆特征值问题,利用求解非线性方程组的同伦方法来解决结构动力学逆特征值问题,这种方法由于沿同伦路径求解,对初值的选取没有本质的要求,算例说明了这种方法是可行的．     

The Drazin inverse in multibody system dynamics

Summary The numerical analysis of multibody system dynamics is based on the equations of motion as differential-algebraic systems. A thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches. The main tool is the Drazin inverse, a generalized matrix inverse, which preserves the eigenvalues. The results are illustrated by a realistic truck model. Finally, the approach is extended to the nonlinear index 2 formulation.     

Computation of bounded feed-forward control for underactuated multibody systems using nonlinear optimization

Underactuation occurs, when only some generalized coordinates have a control input. For end-effector trajectory tracking a combined feed-forward and feedback control is often a suitable approach. Feed-forward control design based on an inverse model for underactuated multibody systems is presented. The starting point is the transformation of the multibody system into a nonlinear input-output normal-form. The inverse model follows from this and consists of chains of differentiators, driven internal dynamics and an algebraic part. Especially when using the end-effector as system output the internal dynamics is often unbounded. In order to obtain a viable feed-forward control, a bounded solution must be determined. For this task the internal dynamics is solved as a nonlinear optimization problem. Thereby, the coordinates of the internal dynamics define the objective function which is minimized. The equation of the internal dynamics must be fulfilled at each point of a discrete time grid. In addition continuity of the solution is achieved by adding as equality constraint an integration formula, e.g. trapezoidal rule. The optimization problem is then solved by a SQP-method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Terminal control of a nonlinear process under disturbances

We consider a nonlinear model of motion of a solid body with deficiency of control parameters. The model contains a disturbance parameter. We propose an open-loop control that takes the system from a given initial state to a given terminal state. Results of numerical calculations are presented for the dynamics of the components of the phase vector and of the controls.     

Neural network prediction of load from the morphology of trabecular bone

Bone adaptation models are often solved in the forward direction, meaning that the response of bone to a given set of loads is determined by running a bone tissue adaptation model. The model is generally solved using a numerical technique such as the finite element model. Conversely, one may be interested in the loads that have resulted in a given state of bone. This is the inverse of the former problem. Even though the inverse problem has several applications, it has not received as much attention as the forward problem, partly because solving the inverse problem is more difficult. A nonlinear system identification technique is needed for solving the inverse problem. In this study, we use artificial neural networks for prediction of tissue adaptation loads from a given density distribution of trabecular bone. It is shown that the proposed method can successfully identify the loading parameters from the density distribution of the tissue. Two important challenges for all load prediction algorithms are the non-uniqueness of the solution of the inverse problem and the inaccuracies in the measurement of the morphology of the tissue. Both challenges are studied, and it is shown that the load prediction technique proposed in this paper can overcome both.     

Constrained numerical optimization methods for blind deconvolution

This paper describes a nonlinear least squares framework to solve a separable nonlinear ill-posed inverse problem that arises in blind deconvolution. It is shown that with proper constraints and well chosen regularization parameters, it is possible to obtain an objective function that is fairly well behaved and the nonlinear minimization problem can be effectively solved by a Gauss–Newton method. Although uncertainties in the data and inaccuracies of linear solvers make it unlikely to obtain a smooth and convex objective function, it is shown that implicit filtering optimization methods can be used to avoid becoming trapped in local minima. Computational considerations, such as computing the Jacobian, are discussed, and numerical experiments are used to illustrate the behavior of the algorithms. Although the focus of the paper is on blind deconvolution, the general mathematical model addressed in this paper, and the approaches discussed to solve it, arise in many other applications.     

Robust intelligent sliding model control using recurrent cerebellar model articulation controller for uncertain nonlinear chaotic systems

In this paper, a robust intelligent sliding model control (RISMC) scheme using an adaptive recurrent cerebellar model articulation controller (RCMAC) is developed for a class of uncertain nonlinear chaotic systems. This RISMC system offers a design approach to drive the state trajectory to track a desired trajectory, and it is comprised of an adaptive RCMAC and a robust controller. The adaptive RCMAC is used to mimic an ideal sliding mode control (SMC) due to unknown system dynamics, and a robust controller is designed to recover the residual approximation error for guaranteeing the stable characteristic. Moreover, the Taylor linearization technique is employed to derive the linearized model of the RCMAC. The all adaptation laws of the RISMC system are derived based on the Lyapunov stability analysis and projection algorithm, so that the stability of the system can be guaranteed. Finally, the proposed RISMC system is applied to control a Van der Pol oscillator, a Genesio chaotic system and a Chua’s chaotic circuit. The effectiveness of the proposed control scheme is verified by some simulation results with unknown system dynamics and existence of external disturbance. In addition, the advantages of the proposed RISMC are indicated in comparison with a SMC system.     

A Pseudo-rigid model for the inverse dynamics of an Euler beam

Inversion technique has been very successful in the tracking control of nonlinear dynamical systems. However, when applied to manipulators constructed with elastic links, inverse dynamics through direct integration in temporal space causes unbounded controller command. It has been suggested that seeking an inverse dynamics solution for a given tip trajectory with given initial conditions is an ill-posed problem. It has also been suggested that increasing model accuracy by including more terms in a truncated beam model worsens the controller’s ability to stabilize the system dynamics. In this paper, we seek to understand the nature of the inverse dynamics instability and to find an alternative solution. We appeal to the notion of a pseudo-rigid model which describes the beam deflection by a homogeneous displacement field. Particularly, we derive the mode shape in order to yield a bounded inverse dynamics solution. Different from most of the existing solutions where solution stability was achieved through modifying the output function, we modified the inverse dynamics model. A bounded inverse solution and model simplicity provide much needed ease in the design and implementation of an inversion controller. Numerical simulations and experiments have both been conducted to prove the validity of the proposed method.     

Qualitative and quantitative combined nonlinear dynamics model and its application in analysis of price,supply–demand ratio and selling rate

The qualitative and quantitative combined nonlinear dynamics model proposed in this paper fill the gap in nonlinear dynamics model in terms of qualitative and quantitative combined methods, allowing the qualitative model and quantitative model to perfectly combine and overcome their weaknesses by learning from each other. These two types of models use their strengths to make up for the other’s deficiencies. The qualitative and quantitative combined models can surmount the weakness that the qualitative model cannot be applied and verified in a quantitative manner, and the high costs and long time of multiple construction as well as verification of the quantitative model. The combined model is more practical and efficient, which is of great significance for nonlinear dynamics. The qualitative and quantitative combined modeling and model analytical method raised in this paper is not only applied to nonlinear dynamics, but can be adopted and drawn on in the modeling and model analysis of other fields. Additionally, the analytical method of qualitative and quantitative combined nonlinear dynamics model proposed in this paper can satisfactorily resolve the problems with the price system’s existing nonlinear dynamics model analytical method. The three-dimensional dynamics model of price, supply–demand ratio and selling rate established in this paper make estimates about the best commodity prices using the model results, thereby providing a theoretical basis for the government’s macro-control of price. Meanwhile, this model also offer theoretical guidance to how to enhance people’s purchasing power and consumption levels through price regulation and hence to improve people’s living standards.     

Testing for Varying Dispersion in Exponential Family Nonlinear Models

A diagnostic model and several new diagnostic statistics are proposed for testing for varying dispersion in exponential family nonlinear models. A score statistic and an adjusted score statistic based on Cox and Reid (1987, J. Roy. Statist. Soc. Ser. B, 55, 467-471) are derived in normal, inverse Gaussian, and gamma nonlinear models. An adjusted likelihood ratio statistic is also given for normal and inverse Gaussian nonlinear models. The results of simulation studies are presented, which show that the adjusted tests keep their sizes better and are more powerful than the ordinary tests.     

Stable Model Inversion for Underactuated Multibody Systems

Inverse models for underactuated multibody systems are presented. The derivation of the inverse model is based on the transformation of the equation of motion into a nonlinear input–output normal–form. These inverse models often include a internal dynamics, for which different methods of solution are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical methods for simulation of large systems

The numerical solution of large initial value problems, including those that are derived as approximations to systems of partial differential equations, may encounter difficulties using conventional numerical methods because of stiffness (large range of eigenvalues of the associated linear system). In a nonlinear system, the eigenvalues may change greatly during the solution and a system that is initially well behaved may become stiff, yielding increased computer cost or inaccuracies. This paper contains a discussion of various definitions of stiffness, and several methods for overcoming it, including a new method for identifying and partitioning a two-time-scale system into fast and slow sub-systems. Also included are some experiences using the DARE continuous system simulation language for systems as large as 200 coupled nonlinear ordinary differential equations.     

Two Degree-Of-Freedom Online Compensation of a Piezoelectric Micro-Positioning Unit

Due to their almost unlimited resolution and fast dynamics, piezoelectric actuators are a common choice for mechatronic systems targeting positioning tasks with high demands on precision. However, these piezoelectric actuators inherently suffer from nonlinear characteristics (mainly hysteresis and creep effects) which need to be addressed by appropriate control strategies. The operator-based modified Prandtl-Ishlinksii (mPI) approach does not only model hysteresis effects with asymmetries and creep effects but also provides an analytical solution for its inverse model. Online feedforward compensation of the aforementioned nonlinear effects can be realized by using the inverse model and additional weight adaptation. In this paper, online compensation via the mPI model is applied to a commercial micro-positioning unit driven by piezoelectric actuators with more than one degree of freedom (DOF). For validation of the proposed approach, two coupled trajectories in the X-Y plane are utilized. Subsequent tracking error analysis validates the efficacy of the stated approach. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints

We consider a neoclassical (economic) growth model. A nonlinear Ramsey equation, modeling capital dynamics, in the case of Cobb-Douglas production function is reduced to the linear differential equation via a Bernoulli substitution. This considerably facilitates the search for a solution to the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. A proposed algorithm for constructing growth trajectories combines methods of open-loop control and closed-loop regulatory control. For some levels of constraints and initial conditions, a closed-form solution is obtained. We also demonstrate the impact of technological change on the economic equilibrium dynamics. Results are supported by computer calculations.     

大垂度柔索的动力学建模与仿真

用多刚体-球铰模型对大垂度柔索进行离散化建模,利用多刚体系统动力学理论建立了该模型的动力学方程.采用矩阵广义逆理论对其位移和速度进行修正,以消除该微分-代数方程在数值分析中的违约现象.数值仿真证明了该方法的正确性.     

非线性离散系统的一种实时建模方法

本文利用二维线性离散系统理论给出了非线性离散系统的一种实时建模方法,理 论及仿真实验显示这种实时模型能够任意逼近非线性离散动态.     

Standard passivity-based control for multi-hydro-turbine governing systems with surge tank

This paper addresses the problem of control design for hydro-turbine governing systems with surge tanks from the perspective of standard passivity-based control. The dynamic model of a synchronous machine is considered in conjunction with a model of the hydro-turbine to generate an eleventh-order nonlinear set of differential equations. An Euler–Lagrange representati of the system and its open-loop dynamics is developed. Then, the standard passivity-based control is applied to design a global and asymptotically stable controller in closed-loop operation. The proposed control is decentralized to avoid challenges of communication between the hydro-turbine governing systems. The proposed standard passivity-based control approach is compared with two control approaches. First, a classical standard cascade proportional-integral-derivative controller is applied for the governing system, the automatic voltage regulator, and the excitation system. Second, a sliding mode control is also implemented in the governing system. Two test systems were used to validate the performance of the proposed controller. The first test system is a single machine connected to an infinite bus, and the second test system is the well-known Western System Coordinating Council’s multimachine system. Overall, simulation results show that the proposed controller exhibits a better dynamic response with shorter stabilization times and lower peaks during the transient periods.