Design of tensegrity structures using parametric analysis and stochastic search

Tensegrity structures are lightweight structures composed of cables in tension and struts in compression. Since tensegrity systems exhibit geometrically nonlinear behavior, finding optimal structural designs is difficult. This paper focuses on the use of stochastic search for the design of tensegrity systems. A pedestrian bridge made of square hollow-rope tensegrity ring modules is studied. Two design methods are compared in this paper. Both methods aim to find the minimal cost solution. The first method approximates current practice in design offices. More specifically, parametric analysis that is similar to a gradient-based optimization is used to identify good designs. Parametric studies are executed for each system parameter in order to identify its influence on response. The second method uses a stochastic search strategy called probabilistic global search Lausanne. Both methods provide feasible configurations that meet civil engineering criteria of safety and serviceability. Parametric studies also help in defining search parameters such as appropriate penalty costs to enforce constraints while optimizing using stochastic search. Traditional design methods are useful to gain an understanding of structural behavior. However, due to the many local minima in the solution space, stochastic search strategies find better solutions than parametric studies.     

On static analysis of tensile structures with sliding cables: the frictional sliding case

In some structural systems, such as cable structures, membranes and tensegrity structures, the use of sliding cables allows to reduce the number of elements required to be controlled during tensioning or activation. However, using sliding cables modifies the structural behavior of tensile structures since it alters the distribution of axial forces in structural members. This has been experienced in structures with continuous cables under the assumption of frictionless sliding. However, sliding-induced friction can further alter the behavior of the system. An enhancement of the static analysis of tensile structures with sliding-induced friction is investigated in this paper. In the proposed formulations, the finite-element analysis method and the dynamic relaxation method are combined with a linear complementary approach. Sliding-induced friction is integrated in the formulations through the consideration of the Euler–Eytelwein equation. The importance of considering sliding-induced friction in the static analysis of tensile structures is demonstrated through a series of examples, where it is shown that friction significantly affects the mechanical behavior of the structures. The examples also reveal that the proposed formulations do not affect the computational time of the static analyses.

     

Identification of nonlinear systems by Takagi–Sugeno fuzzy logic grey box modeling for real-time control

Identification of nonlinear systems by fuzzy models has been successfully applied in many applications. Fuzzy models are capable of approximating any real continuous function to a chosen accuracy. An algorithm for real-time identification of nonlinear systems using Takagi–Sugeno's fuzzy models is presented in this paper. A Takagi–Sugeno fuzzy system is trained incrementally each time step and is used to predict one-step ahead system output. Ability of the proposed identifier to capture the nonlinear behavior of a synchronous machine is illustrated. Effectiveness of the proposed identification technique is demonstrated by simulation and experimental studies on a power system.     

Form-finding of tensegrity structures using double singular value decomposition

A numerical form-finding procedure of tensegrity structures is developed. The only required information is the topology and the types of members. The singular value decompositions of the force density and equilibrium matrices are performed iteratively to find the feasible sets of nodal coordinates and force densities which satisfy the minimum required deficiencies of these two matrices, respectively. An approach of defining a unique configuration of tensegrity structure by specifying an independent set of nodal coordinates is provided. An explanation is given for the preservation in self-equilibrium status of the tensegrity structures under affine transformation. Two- and three-dimensional examples are illustrated to demonstrate the efficiency and robustness of the proposed method in searching stable self-equilibrium configurations of tensegrity structures.     

A gap metric based multiple model approach for nonlinear switched systems

Many applications in chemical engineering often exhibit a switching character due to the presence of discrete modes in the course of their operation. First principles models of such systems constructed using process simulators are far too complex for use in online applications, especially in model-based control. For such systems, numerous control-relevant modeling approaches have been reported in the literature such as mixed logic dynamical (MLD) models [1] and piece wise affine (PWA) [2] models among others. These models describe the evolution of states in each discrete mode using linear equations. Fewer control-relevant models have been reported that address the nonlinear behavior of switched systems. To model nonlinear hybrid systems, Nandola and Bhartiya [3] proposed a multiple linear model approach wherein multiple linear models are used to describe the dynamic behavior in each mode of the hybrid system. However, no guidelines were provided to select the number of models necessary in each mode and their region of validity. In this work, we address these lacunae by presenting a systematic multiple model approach to describe nonlinear switched systems. The method involves a trajectory based linearization and employs a model bank with a set of local linear models for each discrete operational mode. The model bank is generated by linearizing the first principles model across a carefully designed trajectory based on accuracy of multi-step ahead predictions. The numerous models thus obtained are clustered using the gap metric as the distance measure and representative models are selected. The selected linear models are aggregated using Bayesian or Fuzzy approaches to obtain the global model for the nonlinear switched system. A simulation case study of spherical two-tank system and an experimental case study of a benchmark problem consisting of three tanks are used to validate the proposed modeling strategy.     

Advanced automatic grouping for form-finding of tensegrity structures

An advanced automatic grouping method for form-finding of tensegrity structures is presented. In the proposed method, properties of self-equilibrium and stability in tensegrity structures can be obtained by using the force density method combined with a genetic algorithm. A constrained minimization problem is formulated using the standard deviation of the force density in the cables. As a result, the minimum number of member groups for tensegrity structures with automatic grouping can be obtained. This elicited regular tensegrity structures with uniform force density values. Moreover, the geometrical and mechanical parameters of tensegrity structures with multiple states of self-stress can be easily obtained by using the proposed method.     

Adaptive tracking control of nonlinear systems with dynamic uncertainties using neural network

In this paper, an adaptive neural tracking control approach is proposed for a class of nonlinear systems with dynamic uncertainties. The radial basis function neural networks (RBFNNs) are used to estimate the unknown nonlinear uncertainties, and then a novel adaptive neural scheme is developed, via backstepping technique. In the controller design, instead of using RBFNN to approximate each unknown function, we lump all unknown functions into a suitable unknown function that is approximated by only a RBFNN in each step of the backstepping. It is shown that the designed controller can guarantee that all signals in the closed-loop system are semi-globally bounded and the tracking error finally converges to a small domain around the origin. Two examples are given to demonstrate the effectiveness of the proposed control scheme.     

非线性不确定系统的非奇异快速终端滑模控制

针对存在非匹配干扰的非线性系统,设计了一种基于干扰观测器和反步法的非奇异快速终端滑模控制.引入非线性干扰观测器估计系统的不确定性,利用反步的思想处理高阶非线性系统,从而可以将非线性干扰观测器估计的干扰值引入反步法的虚拟控制量中,同时设计一种新颖的非奇异快速终端滑模控制律保证系统的收敛速度和精度.利用Lyapunov函数从理论上证明了所设计的控制器可以保证闭环系统的有限时间收敛.最后通过数值仿真验证了所设计的控制方法的有效性.     

一种带有色量测噪声的非线性系统辨识方法

利用最大似然判据, 本文提出了一种带有色量测噪声的非线性系统辨识方法. 首先, 利用量测差分方法将有色量测噪声白色化, 获得新的量测方程, 从而将带有色量测噪声的非线性系统辨识问题转化成带白色量测噪声和一步延迟状态的非线性系统辨识问题. 其次, 利用期望最大化(Expectation maximization, EM)算法提出了一种新的基于最大似然估计的非线性系统辨识方法, 该算法由期望步骤(Expectation step, E-step)和最大化步骤(Maximization step, M-step)两部分组成. 在期望步骤中, 基于当前估计的参数并利用带有色量测噪声的高斯近似滤波器和平滑器, 近似计算完整的对数似然函数的期望. 在最大化步骤中, 近似计算的似然函数期望值被最大化, 并且通过解析更新获得噪声参数估计, 通过Newton更新方法获得模型参数的估计. 最后, 数值仿真验证了本文提出算法的有效性.     

An affine-scaling derivative-free trust-region method for solving nonlinear systems subject to linear inequality constraints

In this paper, an affine-scaling derivative-free trust-region method with interior backtracking line search technique is considered for solving nonlinear systems subject to linear inequality constraints. The proposed algorithm is designed to take advantage of the problem structured by building polynomial interpolation models for each function in the nonlinear system function F. The proposed approach is developed by forming a quadratic model with an appropriate quadratic function and scaling matrix: there is no need to handle the constraints explicitly. By using both trust-region strategy and interior backing line search technique, each iteration switches to backtracking step generated by the trust-region subproblem and satisfies strict interior point feasibility by line search backtracking technique. Under reasonable conditions, the global convergence and fast local convergence rate of the proposed algorithm are established. The results of numerical experiments are reported to show the effectiveness of the proposed algorithms.     

Controller design based on similar skew‐symmetric structure for nonlinear plants

This paper proposes a novel controller design approach for nonlinear plants. A class of stable nonlinear systems with a similar skew‐symmetric structure is chosen as the objective closed loop system, and two design methods are proposed with backstepping and direct construction. Compared with the conventional backstepping method, the proposed backstepping method need not construct a Lyapunov function step by step, thus the design procedure is simplified. The direct construction method can be applied to some nonlinear plants for which the conventional backstepping is not feasible; and the design can be accomplished in only one step. Furthermore, for some nonlinear plants which have a lower triangular structure with two subsystems, simpler controllers can be derived by the proposed direct construction method than those derived by backstepping design. In addition, the proposed methods are both system structure oriented, therefore their designs are more intuitive than the conventional backstepping design. Two controllers are derived for satellite attitude control by employing the proposed methods; simulation results demonstrate their effectiveness. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

基于逆阵更新方法的局部非线性结构频响分析

针对含有非线性连接的大型局部非线性结构,采用描述函数表示其所连接的非线性内力,将非线性结构频响表示为拟线性动柔度矩阵(Quasilinear Receptance Matrix),提出一种逆阵更新(Inverse Matrix Updating Method,IMU)方法,将求解系统动柔度(频响特性)的高阶矩阵求逆转化为低阶矩阵的求逆,从而获得大型局部非线性结构主频响应的快速计算方法.仿真结果表明,本文的分析方法具有较好的稳定性,并能大幅提高大型局部非线性结构主频响应的计算效率.     

An adaptive full order sliding mode controller for mismatched uncertain systems

In this paper, an adaptive full order sliding mode (FOSM) controller is proposed for strict feedback nonlinear systems with mismatched uncertainties. The design objective of the controller is to track a specified trajectory in presence of significant mismatched uncertainties. In the first step the dynamic model for the first state is considered by the desired tracking signal. After the first step the desired dynamic model for each state is defined by the previous one. An adaptive tuning law is developed for the FOSM controller to deal with the bounded system uncertainty. The major advantages offered by this adaptive FOSM controller are that advanced knowledge about the upper bound of the system uncertainties is not a necessary requirement and the proposed method is an effective solution for the chattering elimination from the control signal. The controller is designed considering the full-order sliding surface. System robustness and the stability of the controller are proved by using the Lyapunov technique. A systematic adaptive step by step design method using the full order sliding surface for mismatched nonlinear systems is presented. Simulation results validate the effectiveness of the proposed control law.     

Control‐oriented modeling and deployment of tensegrity–membrane systems

This study addresses control‐oriented modeling and control design of tensegrity–membrane systems. Lagrange's method is used to develop a control‐oriented model for a generic system. The equations of motion are expressed as a set of differential‐algebraic equations (DAEs). For control design, the DAEs are converted into second‐order ordinary differential equations (ODEs) based on coordinate partitioning and coordinate mapping. Because the number of inputs is less than the number of state variables, the system belongs to the class of underactuated nonlinear systems. A nonlinear adaptive controller based on the collocated partial feedback linearization (PFL) technique is designed for system deployment. The stability of the closed‐loop system for the actuated coordinates is studied using the Lyapunov stability theory. Because of system complexity, numerical tests are used to conduct stability analysis for the dynamics of the underactuated coordinates, which represents the system's zero dynamics. For the tensegrity–membrane systems studied in this work, analytical proof of zero dynamics stability remains an open theoretical problem. An H_∞ controller is implemented for rapid stabilization of the system at the final deployed configuration. Simulations are conducted to test the performance of the two controllers. The simulation results are presented and discussed in detail. Copyright © 2016 John Wiley & Sons, Ltd.     

Tuning of a decoupling controller for a 2×2 system using iterative feedback tuning

Iterative feedback tuning is used to tune a decoupling controller for a 2×2 multivariable system. The tuning is carried out in two steps. In the first step the coefficients of two proportional decoupling controllers are tuned, and in the second step two PD-controllers are tuned. The approach is evaluated on a nonlinear simulation model of a two-link manipulator. The evaluation is carried out using two different reference trajectories and also in the presence of friction. The resulting closed loop systems show satisfactory behavior in all three cases.     

A new approach for multimodel identification of complex systems based on both neural and fuzzy clustering algorithms

The multimodel approach was recently developed to deal with the issues of complex systems modeling and control. Despite its success in different fields, it is still faced with several design problems, in particular the determination of the number and parameters of the different models representative of the system as well as the choice of the adequate method of validities computation used for multimodel output deduction.In this paper, a new approach for complex systems modeling based on both neural and fuzzy clustering algorithms is proposed, which aims to derive different models describing the system in the whole operating domain. The implementation of this approach requires two main steps. The first step consists in determining the structure of the model-base. For this, the number of models must be firstly worked out by using a neural network and a Rival Penalized Competitive Learning (RPCL). The different operating clusters are then selected referring to two different clustering algorithms (K-means and fuzzy K-means). The second step is a parametric identification of the different models in the base by using the clustering results for model orders and parameters estimation. This step is ended in a validation procedure which aims to confirm the efficiency of the proposed modeling by using the adequate method of validity computation. The proposed approach is implemented and tested with two nonlinear systems. The obtained results turn out to be satisfactory and show a good precision, which is strongly related to the dispersion of the data and the related clustering method.     

Modelling and control of non-minimal non-linear realisations of tensegrity systems

The dynamics of tensegrity systems have their simplest mathematical form when the realisation is non-minimal. In this article, the simplicity of this structure is exploited to design controls. A Lyapunov-based approach creates a non-linear feedback control law exploring the tools of linear algebra due to the linearity of the model on a new set of transformed control variables. Explicit solutions are given for different choices of norm minimisation of the instantaneous control. The theory is illustrated through numerical simulations of simple tensegrity structures.     

悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.     

Nonlinear system modeling via knot-optimizing B-spline networks

In using the B-spline network for nonlinear system modeling, owing to a lack of suitable theoretical results, it is quite difficult to choose an appropriate set of knot points to achieve a good network structure for minimizing, say, a minimum error criterion. In this paper, a novel knot-optimizing B-spline network is proposed to approximate the general nonlinear system behavior. The knot points are considered to be independent variables in the B-spline network and are optimized together with the B-spline expansion coefficients. The simulated annealing algorithm with an appropriate search strategy is used as an optimization algorithm for the training process in order to avoid any possible local minima. Examples involving dynamic systems up to six dimensions in the input space to the network are solved by the proposed method to illustrate the effectiveness of this approach.