The solution for the generalized riccati algebraic equations of linear equality constraint system

Based on the dynamic equation, the performance functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper. Project supported by the National Natural Science Youth Foundation of China     

THE SOLUTION FOR THE GENERALIZED RICCATIALGEBRAIC EQUATIONS OF LINEAR EQUALITY CONSTRAINT SYSTEM

I.IntroductionTheestablishmentofanalogytheorybetweencomputationalstructuralmechanicsalldoptimalcontrolisbasedonthesubstructuralchaintheoryinstructuralmechanicsandthelillearquadratic(LQ)controlprobleminoptimalcontrolllJ.Whendealillgwiththecontrolproblenlbyapplyingtheabovethe.ry,wecanrealizetheadvantagesofmethodsinstrLlcturalInechanicsl2'3].Howevermostworkhasbeenlimitedtothenon-constraintsystemandlittleworkhasbeendonefortheconstraintsystemsofLlr.InRef.I4],basedonthegeneralizedvariationalprin…     

连续时间线性约束LQ控制问题的时程精细积分方法

本文基于连续时间线性约束LQ控制问题,给出时段的消元公式。由于消元过程与消元次序无关,故可在此过程中引入2N类高精度时程积分方法,求出Riccati方程后,对状态向量进一步采用时程精细积分法,可确定系统非常精确的状态向量,该方法不仅保证了系统的计算精度,而且有很好的数值稳定性,数值例题说明了本文方法的有效性。     

均衡实现降阶系统LQ控制器设计改进研究

土木工程结构振动控制由于模型的自由度数目很大,控制器阶次非常高,从而导致主动控制和半主动控制实施过程中时滞效应十分的突出。本文在采用均衡实现技术对结构模型进行降阶的基础上,进行LQ控制器的设计,考虑了对应较小奇异值的状态变量对低阶控制器的影响,对LQ控制器的设计进行改进,优化了控制效果。本文通过对Highway Bridge的Benchmark模型进行LQ控制器设计的改进研究,验证了本文方法的改进效果。     

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.     

Dynamic output-feedback guaranteed cost control for linear systems with uniform input quantization

There have existed the researches on guaranteed cost control for systems with the dynamic or logarithmic quantizer; however, intensive studies not only on continuous-time output-feedback linear systems with uniform input quantization but also on guaranteed cost control for such systems have not been carried out thus far. This paper first solves the problem of guaranteeing the linear quadratic (LQ) cost for continuous-time output-feedback linear systems with uniform input quantization. The proposed output-feedback controller comprises the main and the additional control parts. The former part is designed as a linear dynamic output-feedback controller for determining the fundamental characteristics of the system associated with the LQ cost, and the latter part is constructed as an integer multiple of the quantization level for eliminating the effect of uniform input quantization. The additional control part adopts the state estimate instead of the state itself, which causes the state estimation errors. By computing their quantity added to the upper bound of the LQ cost, this paper shows that the proposed controller still guarantees the LQ cost despite the state estimation errors.     

The construction of non-linear normal modes for systems with internal resonance

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.     

An algorithm for the global volume conservation constraint in lake circulation

We present an algorithm for introducing a global constraint of volume conservation in lake circulation problems. The algorithm is described for linear problems, and is then generalized to non-linear cases. Numerical examples are presented to show the influence of water-level conditions on convergence and to demonstrate the practical superiority of the global constraint algorithm for obtaining reliable convergent solutions.     

Optimal placement of piezoelectric active bars in vibration control by topological optimization

A continuous variable optimization method and a topological optimization method are proposed for the vibration control of piezoelectric truss structures by means of the optimal placements of active bars. In this optimization model, a zero-one discrete variable is defined in order to solve the optimal placement of piezoelectric active bars. At the same time, the feedback gains are also optimized as continuous design variables. A two-phase procedure is proposed to solve the optimization problem. The sequential linear programming algorithm is used to solve optimization problem and the sensitivity analysis is carried out for objective and constraint functions to make linear approximations. On the basis of the Newmark time integration of structural transient dynamic responses, a new sensitivity analysis method is developed in this paper for the vibration control problem of piezoelectric truss structures with respect to various kinds of design variables. Numerical examples are given in the paper to demonstrate the effectiveness of the methods.     

Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

In this paper, an analytical approximation of damped oscillations of some strongly non-linear, planar Hamiltonian systems is considered. To apply the Krylov–Bogoliubov–Mitropolsky method in this strongly non-linear case, we mainly provide the formal and exact solutions of the homogeneous part of the variational equations with periodic coefficients resulting from the Hamiltonian systems. It is shown that these are simply expressed in terms of the partial derivatives of the solutions, written in action-angle variables, of the Hamiltonian systems. Two examples, including a non-linear harmonic oscillator and the Morse oscillator, are presented to illustrate this extension of the method. The approximate first order solution obtained in each case is observed to be quite satisfactory.     

Optimal synthesis of function generator of four-bar linkages based on distribution of precision points

This article describes a method for optimizing the synthesis of a four-bar linkage mechanism to generate a definite mathematic function. In this research, the objective function was defined based on the least squares of error between the generated function and the desired function. Because of non-linearity of the objective function and the constraint, SQP method has been used to find the optimal mechanism. This method of optimization is a gradient-based method and its result is more trustable than heuristic methods. In this study, five precision points have been used to synthesize four-bar linkages; therefore, by using this amount of precision points, a set of non-linear equations was obtained for mechanism synthesis. Unlike the previous researches that dimensional parameters are used as optimization variables, precision points distribution was used in current research. The main innovation of this paper is presentation of some directions to have estimative prediction for distribution of precision points in optimal mechanism which maybe useful for designers. These directions were obtained based on the method that presented in current work and regard to shape of desired function and its first and second derivative.     

无限元与相同子结构串的一致性及其算法

讨论了无限元方法￣［１］与相同子结构串分析￣［２］的一致性，综合文［１］和［２］得到一种通用算法，依据文［３］尚可应用于ＬＱ控制问题，且提出一种直接求出转移矩阵的算法，具有简便和较高精度的优点．     

Generalized lie symmetries and the integrability of generalized Holmes-Rand non-linear oscillator

Generalized Lie symmetries and the integrability of generalized Holmes-Rand non-linear oscillator (GHRNO) are considered. The constraint which the variable-coefficient functions must satisfy for the GHRNO to have infinite-dimensional symmetry algebras is derived. The integral of motion for the GHRNO under this condition can be read off from the symmetry vector fields. The structure of the symmetry algebras is also presented.     

具有非线性homologous变形约束的桁架结构形态分析

研究了具有非线性homologous变形约束条件的桁架结构形态分析问题。在已有的线性homologous变形约束桁架形态分析的基础上,将结构的节点分成三类:homologous变形约束节点,形状可变节点和边界点。运用Moore-Penrose广义逆矩阵性质,将基础方程组解的存在条件表示为包含形状可变节点未知坐标的非线性方程组,为采用Newton-Raphson方法求解非线性方程组,对AA (A为任意矩阵,A 为A的Moore-Penrose广义逆矩阵)求偏导数,找到了满足保型要求的形态,给出的桁架算例说明了本文方法的有效性。     

Differential-algebraic equations of programmed motions of Lagrangian dynamical systems

We suggest a method for constructing the dynamic equations of manipulator systems in canonical variables. The system of differential dynamic equations has an integral manifold corresponding to the holonomic and nonholonomic constraint equations. The controls are determined so as to ensure the stability of this manifold. We state conditions for the exponential stability of the manifold and for constraint stabilization when solving the dynamic equations numerically by a simplest difference method. We also present the solution of the problem of control of a plane two-link manipulator.     

含时滞的LQ控制车辆悬架的研究

研究时滞对具有LQ最优控制的1／4汽车线性悬架模型的动特性影响。首先采用了广义的Sturm序列判断准则给出了不同控制增益下单自由度车模型全时滞稳定区,以及在给定时滞情况下绘出了控制增益稳定区;其次,给出了两自由度车模型的全时滞稳定区以及系统在给定的增益下临界时滞的值。此外,讨论了全状态反馈下的稳定性以及稳定性切换问题;最后研究了时滞对系统动力学行为的影响。     

频率约束板结构拓扑优化

本文利用ICM（独立、连续、映射）方法建立了频率约束下平板结构重量最轻的拓扑优化模型。采用指数函数作为单元重量、单元质量及单元刚度的过滤函数。通过瑞利商对刚度过滤函数倒变量的泰勒一阶展式，将频率约束近似显式化。利用对偶理论将含有大量设计变量的约束优化模型转化为易于求解的少设计变量拟无约束优化模型，通过序列二次规划将转化模型进行求解，提高了求解的效率。本文选择MSC.Patran&Nastran软件及PCL二次开发语言构架了平板结构频率约束拓扑优化问题的软件。数值算例表明：本文的方法具有迭代稳定性和收敛高效性。     

工程机械中的机构-结构一体化优化设计

工程机械中的运动机构同时往往也是承载机构,其结构优化问题须同时考虑机构运动和结构受力,较传统的结构优化问题需要考虑的因素更多,约束条件更为复杂。本文以广泛应用于工程机械中的四连杆式俯仰机构为对象,研究了工程机械中机构-结构一体化优化问题,探讨了此类优化问题的目标函数、设计变量和约束条件,并对实际工程机械进行了分析和优化,达到了理想的效果。     

Inverse Dynamics of Servo-Constraints Based on the Generalized Inverse

The acceleration form of constraint equations is utilized in this paper to solve for the inverse dynamics of servo-constraints. A condition for the existence of control forces that enforce servo-constraints is derived. For overactuated dynamical systems, the generalized Moore-Penrose inverse of the constraint matrix is used to parameterize the solutions for these control forces in terms of free parameters that can be chosen to satisfy certain requirements or optimize certain criterions. In particular, these free parameters can be chosen to minimize the Gibbsian (i.e., the acceleration energy of the dynamical system), resulting in ideal control forces (those satisfying the principle of virtual work when the virtual displacements satisfy the servo-constraint equations). To achieve this, the nonminimal nonholonomic form recently derived by the authors in the context of Kanes method is used to determine the accelerations of the system, and hence to determine the forces to be generated by the redundant manipulators. Finally, an extension to inverse dynamics of servo-constraints involving control variables is made. The procedures are illustrated by two examples.     

A non-equilibrium thermodynamic geometric structure for thermoviscoplasticity with maximum dissipation

A maximum dissipation principle induces a class of viscoplastic evolution equations within a non-linear, non-equilibrium thermodynamic structure which extends Gibbs thermostatics. The evolution of a relaxation process is determined through non-linear affinities, which generalize the linear Onsager construct, along with the maximum dissipation principle and the long term states. The relaxation modulus is the norm of the plastic strain rate. Forced processes, in which the control variables change with time, are assumed to be relaxation processes at each instant. This class of thermoviscoplastic models includes the three-dimensional Freed–Chaboche–Walker model in which the internal state variables are the back stress, the drag strength and the elastic yield strength for creep initiation.