State Variable Dimension Research Articles

Modified transfer matrix method for vibration analysis of beam structures including branches and rigid bodies

Beam frames can be found in many application scenarios, ranging from big-scale structure engineering of buildings, bridges to small systems of micromachines, sensors, robotics and metamaterials. However, several challenging issues, e.g. serial-parallel configurations, beam and rigid body coupling along with inconsistent dimension of state variables, often lead to a complicated dynamic modeling. In this paper, we show that the free vibration analysis of a class of beam structures with serial-parallel branches and rigid bodies can be significantly streamlined by modifying the transfer matrix method in a graphic manner. Treating general beam structures as a topological graph with standard building blocks enables a modular modeling process and hence offers a new way to divide the complex modeling issue into much easier steps. The transfer matrix of typical building blocks, such as a beam connected to rigid bodies, tree branches and parallel clamped sub-chains, are derived and standardized providing new perspectives and distinct forms in contrast to previous studies. The transfer matrix method exhibits several advantages of easy programming, accurate solution and small degrees-of-freedom for serial structures. The presented approach extends these characteristics to a range of multi-body systems consisting of rigid bodies and beams. Comparative validation with four application case studies confirms these advantages.

On the regularized risk of distributionally robust learning over deep neural networks

In this paper, we explore the relation between distributionally robust learning and different forms of regularization to enforce robustness of deep neural networks. In particular, starting from a concrete min-max distributionally robust problem, and using tools from optimal transport theory, we derive first-order and second-order approximations to the distributionally robust problem in terms of appropriate regularized risk minimization problems. In the context of deep ResNet models, we identify the structure of the resulting regularization problems as mean-field optimal control problems where the number and dimension of state variables are within a dimension-free factor of the dimension of the original unrobust problem. Using the Pontryagin maximum principles associated with these problems, we motivate a family of scalable algorithms for the training of robust neural networks. Our analysis recovers some results and algorithms known in the literature (in settings explained throughout the paper) and provides many other theoretical and algorithmic insights that to our knowledge are novel. In our analysis, we employ tools that we deem useful for a future analysis of more general adversarial learning problems.

An Improved System-Level Calibration Method of Strapdown Inertial Navigation System Based on Matrix Factorization

This paper develops an improved system-level calibration method for the strapdown inertial navigation system (SINS). It solves the problems of too many dimensions of state variables and installation error matrix coupling. To solve the two problems, the improved system-level calibration method derives the physical causes of the installation error matrix. The installation error matrix is decomposed into skew symmetric matrix and symmetric matrix. Then the appropriate system-level calibration scheme is designed based on the velocity error equations and attitude error equations. The designed system-level calibration scheme separates the coupled installation error matrix parameters by changing the value of strapdown matrix. Simulation results demonstrate that the proposed method is feasible. The relative errors of gyro calibration parameters are between 0.03% and 0.70%, and the relative errors of accelerometer calibration parameters are between 0.01% and 0.24%. For comparison, the navigation and positioning experiments are also performed to evaluate the proposed method. The results indicate that the improved system-level calibration scheme is better than the traditional system-level calibration scheme. The positioning error is reduced by more than 30%.

A neurodynamic approach to zero-one quadratic programming

This paper proposes a neurodynamic approach to zero-one quadratic programming with linear constraints. Compared with one existing neurodynamic approach to such problems, the proposed one has lower dimensions of state variables and less computational cost, which makes the implementation easier. Under some suitable conditions, the stability and convergence properties of the proposed approach are established. Numerical simulation results and related comparisons show the efficiency of this proposed method in practical computation.

Dynamic Modeling and Locomotion Control for Quadruped Robots Based on Center of Inertia on SE(3)

Quadruped robots have good mobility and agility in complex environments, but dynamic control of locomotion for quadruped robots has long been a big challenge for researchers. In this paper, we build the center of inertia (COI) dynamic model of a general quadruped robot and give the exponential coordinates of COI on the special Euclidean space SE(3). The COI model takes the whole quadruped robot as one body, so that the only concern is the movement of the COI rather than the body or legs when the robot walks. As a result, the COI model has fewer dimensions of state variables than the full dynamic model, which helps to reduce the computational load. A control method for quadruped robots is presented based on the dynamic model which is constituted of force loop and position loop. This method controls the movement of the COI directly, so it facilitates to guarantee the robot's stability. The virtual body of the quadruped robot is defined to describe the configuration of the quadruped robot. The proportional-derivative (PD) control method on SE(3) is applied to control the movement of the virtual body, which makes the movement more in line with the group theoretic viewpoint. Finally, some simulation experiments have been conducted to verify the validity of our method.

Structure of Spot Rates and Duration Hedging*

The present study proposes a three-factor model using spot rates as proxies for the state variables of the term structure of interest rates. Empirical analysis is carried out on the in-sample explanatory power and the out-of-sample prediction ability of spot-rate models, and comparison is made between the modified Macaulay duration and spot-rate duration hedging for bond portfolios. The results not only show that the optimal three-spot-rate model outperforms the optimal two-spot-rate model proposed by Elton et al. (Journal of Finance, 45, 1990, 629–642) with respect to explanation ability of unexpected changes in the term structure of interest rates, but also illustrate the importance of capturing the curvature characteristic of the term structure of interest rates for spot-rate duration hedging methods. Moreover, the impressive performance of three-spot-rate duration hedging implies that it is feasible to reduce the dimensions of state variables to three for the purposes of risk exposure prediction and risk management of bond portfolios.

INS/vSLAM system using distributed particle filter

In implementing an INS/SLAM integrated navigation system based on the vision sensor, a suboptimal nonlinear filter is used to figure out the nonlinear characteristics in measurement and noise model. When a conventional centralized filter is used, however, the entire state vectors need to be reconfigured in every necessary cycle as the number of feature points changes, which is hard to isolate potential faults. Furthermore, any change in the number of feature points and a subsequent increase in the dimension of state variables may result in an exponential growth in computation quantities. In order to address these issues, this paper presents a distributed particle filter approach for implementing a vision sensor based INS/SLAM system. The proposed system has several local filters which are subject to change flexibly by the number of feature points, and separates state vectors into sub-states for vehicle dynamics and feature points so that minimum state vectors can be estimated in the master filter. Simulation results show that the distributed particle filter performs competitively as with the centralized particle filter and is capable of improving computation quantities.

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.

Design of a Dynamic Positioning System for Self-Propulsive Barge under External Disturbances

A dynamic positioning system (DPS) is used in order to keep the specified position. By using the inverse linear quadratic (ILQ) optimal servo theory, it will be able to construct a control system of DPS which has no interaction among the controlled variables. In ILQ optimal servo theory, time constant Ti as the design parameters have much influence on the performance of the control system.In this paper, we applied the ILQ optimal servo theory to design the control system of DPS for a self-propulsive barge. We designed the system with Ti defined automatically in response to the dimensions of state variables. We examined the characteristics of the control system using Ti which is defined automatically.

Adaptive tuning of fuzzy logic identifier for unknown non‐linear systems

AbstractAn on‐line approximator using a linear fuzzy logic model with automatic tuning mechanism is presented in this paper. the proposed approximation method can be used to model a class of unknown non‐linear systems as long as they are Caratheodory ones. the structure of the fuzzy model is fixed but the parameters are tuned by the state errors. the fuzzy model is composed of several semi‐closed, continuous, totally ordered and well‐defined fuzzy numbers defined in each state variable dimension and a number of appropriately defined condition‐action rules. Either the min‐inference or product‐inference technique is utilized to generate the weighted average of the linear coefficients which make the whole unknown non‐linear system piecewise linearized. No off‐line preprocessing is needed. the initial values of the parameters of the fuzzy model can be arbitrarily assigned. Then they are tuned to their true values through adaptive update law and therefore it is guaranteed that the unknown non‐linear system is linearized and approximated to any degree of accuracy by the linear fuzzy logic model.

Parallel processing of large scale discrete-time unconstrained differential dynamic programming

Differential Dynamic Programming (DDP) is an optimal control algorithm that has been used in a variety of applications involving optimization of dynamic systems. This paper discusses modifications in the serial DDP algorithm required to develop an efficient parallel algorithm. The modifications change the order of calculations in order to minimize the number of required barrier synchronization points within a time step. Because DDP is a fine-to-medium grained algorithm, previous approaches to parallel processing of DDP and related algorithms have focused on parallel processing across time steps, which results in a substantial loss of efficiency. The approach described here is most efficient for problems with a high state variable dimension. Numerical results indicate that for problems with 1000 state variables, our method can obtain a speedup of 5.3 on six processors (95.6% of the Amdahl upper bound), which is substantially higher than published results from earlier studies investigating problems of smaller state dimension and larger time dimension.

Convergence in unconstrained discrete-time differential dynamic programming

The conditions under which the original differential dynamic programming (DDP) algorithm can be expected to converge are investigated and modifications in the algorithm to improve convergence properties are proposed. Quadratic convergence of DDP requires that the stagewise Hessian matrices computed in the algorithm be positive definite. Three procedures for modifying the stagewise Hessian matrices to guarantee convergence for situations in which the stagewise Hessian matrices are not positive definite are proposed. One of the procedures (an active shift) converges quadratically. The computational requirements for DDP with and without the shift procedures are presented, and the impact of the ratio of state variable dimension to control variable dimension on the relative efficiency of the different shift procedures is discussed. Numerical results for a series of problems, including one with 16 state variables and 200 time steps, are presented. The most robust shift procedure is an adaptive shift that utilizes a constant shift followed by an active shift. >

Eigenvector derivatives of repeated eigenvalues using singular value decomposition

An explicit formula is obtained for the first-order eigenvector derivative that corresponds to the eigenvector of a repeated eigenvalue, in the case of the nonself-adjoint eigenvalue problem. This method applies to the class of nondefective problems whose first eigenvalue derivatives of the repeated eigenvalues are distinct. A singular-value decomposition approach is used to compute four requisite bases for eigenspaces, as well as to keep track of the dimensions of state variables and the conditioning of the state equations.