Higher-order Dynamical Systems Research Articles

순응도 기반의 구동기 동적 성능 최적화

본 논문은 4차 동적 시스템인 구동기를 구성하는 부품들의 변량 및 공기외란의 변량으로 인해 구동기 성능에 변량이 존재할 때, 구동기의 성능을 순응도 기반으로 최적화한 설계 결과를 제시하였다. 구동기는 BLDC 모터와 평기어 및 웜기어 조립체, 카나드로 구성된다. 구동기의 성능은 시간에 따라 변화하는 카나드 각도 정보를 이용하여 평가하였다. 고차 시스템의 성능은 해석적 접근을 통해 시스템 변수들의 함수로 표현하기 어렵기 때문에 반응표면모델을 기반으로, F-검정을 수행하여 유효한 시스템 변수들을 선정한 후 성능을 유효한 시스템 변수들의 함수로 근사화하였다. 본 연구에서 변량을 가지는 구동기 성능의 규격에 대한 순응도를 확률값으로 정의하였다. 순응도를 최적화하여 구한 설계변수는 부품 및 공기외란에 변량이 존재하더라도 고려한 규격을 모두 만족시킬 수 구동기를 제공할 수 있다.

Research on the Identification for a Nonlinear System

The characteristic of the drift error of inertial platform is a high-order nonlinear dynamic system, using the neural networks’ abilities of universal approximation of differentiable trajectory and capturing system dynamic information, this paper presents the drift error identifying project of inertial platform based on Elman networks structure. First, the drift error model of inertial platform is established, after selecting the input and output for network, momentum and alterable speed algorithm is used to speed up the network convergence. On the basis of the algorithm, the extended nonlinear node function in the hidden network does not only improve the learning speed of network, but also satisfies the need of accuracy on system identification. Through the drift error data measured on inertial platform, the training result shows that the scheme achieves satisfied identification results.

Higher-Order Derivative Compensators for Controlling the Systems With Higher Dynamics

The PID controller is a popular choice for the control of industrial processes. The PDN controller introduced in this paper is a better alternative to PIDN. The paper presents the design and computer simulation of the PDN controller. The PDN controller is a simpler and more effective choice when dealing with higher-order dynamic systems. The gain components of the controller are introduced to improve transient characteristics of the higher-order industrial systems or processes by directly modifying the components of their characteristic equations. As a result the system under control becomes stable and able to respond faster to the inputs as compared to PID-based systems. The tunable command feed-forward steady-state gain (K S ) of the PDN controller helps to track the constant or time-varying input commands much faster and more precisely than the integral component of the PID controller. The paper discusses the steady-state performance, stability problems and the gain tuning strategies to improve the transient responses of the feedback system. The quality of the PDN performance is demonstrated by solving the set point input (step) and tracking a time-varying input (ramp) problems. The paper also presents the detailed higher-order derivative compensators tuning procedures for the general case of unknown order, stable/unstable plant and particular case of unknown order but stable plant.

Dynamical systems identification using Gaussian process models with incorporated local models

Gaussian process (GP) models form an emerging methodology for modelling nonlinear dynamic systems which tries to overcome certain limitations inherent to traditional methods such as e.g. neural networks (ANN) or local model networks (LMN). The GP model seems promising for three reasons. First, less training parameters are needed to parameterize the model. Second, the variance of the model's output depending on data positioning is obtained. Third, prior knowledge, e.g. in the form of linear local models can be included into the model. In this paper the focus is on GP with incorporated local models as the approach which could replace local models network. Much of the effort up to now has been spent on the development of the methodology of the GP model with included local models, while no application and practical validation has yet been carried out. The aim of this paper is therefore twofold. The first aim is to present the methodology of the GP model identification with emphasis on the inclusion of the prior knowledge in the form of linear local models. The second aim is to demonstrate practically the use of the method on two higher order dynamical systems, one based on simulation and one based on measurement data.

On consensus algorithms for double-integrator dynamics without velocity measurements and with input constraints

This note deals with consensus strategy design for double-integrator dynamics. Specifically, we consider the case where the control inputs are required to be a priori bounded and the velocity (second state) is not available for feedback. Two different design methods are proposed. First, based on the auxiliary system approach, we propose a consensus algorithm that extends some of the existing results in the literature to account for actuator saturations and the lack of velocity measurement. The proposed velocity-free control scheme, using local information exchange, achieves consensus among the team members with an a priori bounded control law, whose upper bound depends on the number of neighbors of the vehicle. Second, we propose another approach based on the use of a high order dynamic auxiliary system such that the upper bound of the control law is independent of the number of neighbors of the vehicle, and the performance of the closed loop system is improved in terms of the response damping. Finally, simulation results are provided to illustrate the effectiveness of the proposed algorithms.

Design and experimental validation of a model reduction algorithm for high-order port-Hamiltonian systems

This paper illustrates a model reduction procedure for port Hamiltonian systems able to preserve the frequency behavior of the original system in a neighborhood of a predefined set of frequencies of interest. This research is part of a wider activity carried out in collaboration with Tetra Pak concerning the modelling and simulation of the Ultrasonic Sealing System (USTS). Due to the presence of a Compact Transducer (CT) that can be modelled only by means of commercial finite element CAE software, which provide extremely high-order dynamical systems, it is not possible to perform a simulation of the complete system to test the validity of the controller and perform the diagnosis of the sealing process in detail. The proposed procedure drastically reduces the simulation time without loosing the essential dynamical information. The model reduction algorithm is experimentally validated.

Consensus seeking of high-order dynamic multi-agent systems with fixed and switching topologies

This article studies a consensus problem for a group of high-order dynamic agents with fixed and switching topologies. Two linear consensus protocols, which only depend on the agent's own information and its neighbours' partial information, are proposed, respectively, for the group with and without communication time-delays. For the case of fixed topology and zero communication time-delay, a necessary and sufficient condition for the consensus is established. The parameter design of the protocol is also discussed. Under switching topology, the consensus is investigated for two cases: the case of zero communication time-delay and the case of nonuniform communication time-delays. For each case, a sufficient condition for the consensus is provided. Moreover, for a group of double/triple-integrator agents with fixed topology and uniform constant time-delay, some necessary and sufficient conditions for the consensus are given. Finally, numerical simulations are worked out to illustrate the effectiveness of the theoretical results.

Consensus of high-order dynamic multi-agent systems with switching topology and time-varying delays

This paper studies the consensus problems for a group of agents with switching topology and time-varying communication delays, where the dynamics of agents is modeled as a high-order integrator. A linear distributed consensus protocol is proposed, which only depends on the agent’s own information and its neighbors’ partial information. By introducing a decomposition of the state vector and performing a state space transformation, the closed-loop dynamics of the multi-agent system is converted into two decoupled subsystems. Based on the decoupled subsystems, some sufficient conditions for the convergence to consensus are established, which provide the upper bounds on the admissible communication delays. Also, the explicit expression of the consensus state is derived. Moreover, the results on the consensus seeking of the group of high-order agents have been extended to a network of agents with dynamics modeled as a completely controllable linear time-invariant system. It is proved that the convergence to consensus of this network is equivalent to that of the group of high-order agents. Finally, some numerical examples are given to demonstrate the effectiveness of the main results.

Bond graph model of an induction machine with hysteresis nonlinearities

An induction machine is one of the most convenient devices for conversion of electrical energy to mechanical rotational energy. Induction machine is a typical member of a multi-domain, nonlinear, high-order dynamic system. To reduce its complexity, the mathematical models used for designing their control have several assumptions built into them. The most striking of these assumptions is that of linear magnetics. Bond graph is a convenient tool for modelling nonlinear elements. This paper proposes the use of the bond graph methodology to develop a model of an induction machine that includes the nonlinearities due to magnetics.

Bond graph model of doubly fed three phase induction motor using the Axis Rotator element for frame transformation

Induction motor is a typical member of a multi-domain, non-linear, high order dynamic system. For speed control a three phase induction motor is modelled as a d– q model where linearity is assumed and non-idealities are ignored. Approximation of the physical characteristic gives a simulated behaviour away from the natural behaviour. This paper proposes a bond graph model of an induction motor that can incorporate the non-linearities and non-idealities thereby resembling the physical system more closely. The model is validated by applying the linearity and idealities constraints which shows that the conventional ‘abc’ model is a special case of the proposed generalised model.

Krylov Subspaces Associated with Higher-Order Linear Dynamical Systems

A standard approach to model reduction of large-scale higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for model reduction of first-order systems. This paper presents some results about the structure of the block-Krylov subspaces induced by the matrices of such equivalent first-order formulations of higher-order systems. Two general classes of matrices, which exhibit the key structures of the matrices of first-order formulations of higher-order systems, are introduced. It is proved that for both classes, the block-Krylov subspaces induced by the matrices in these classes can be viewed as multiple copies of certain subspaces of the state space of the original higher-order system.

A note on the controllability of higher-order linear systems

In this paper, a new condition for the controllability of higher-order linear dynamical systems is obtained. The suggested test contains rank conditions of suitably defined matrices and is based on the notion of compound matrices and the Binet-Cauchy formula.

CONTROLLING CHAOS IN HIGHER-ORDER DYNAMICAL SYSTEMS

In this paper, we investigate a continuous extension of the Ott, Grebogi and Yorke control method [Ott et al., 1990]. This extension allows the control of chaotic behavior in higher-order dynamical systems. To begin with, studying the so-called Poincaré Sections (PS) enables determining fixed points and the effect of small changes in the control parameter on the system trajectory. Introduction of a stabilizer parameter in the control law keeps the dynamical properties and maintains them on a cyclic trajectory. The effectiveness of the proposed method is tested by numerical examples on the two typical Chen and Chua chaotic systems.

New Computational Framework for Trajectory Optimization of Higher-Order Dynamic Systems

Using tools from systems theory, it is now well known that classes of linear and nonlinear dynamic systems in firstorder form can be alternatively written in higher-order form, i.e., as sets of higher-order differential equations. Input-state linearization is one of the popular tools to achieve such a transformation. For mechanical systems, the equations of motion naturally have a second-order form. For real-time planning and control, a higher-order form offers a number of advantages compared to the first-order form. In this paper, we address the question of trajectory optimization of higher-order systems with general nonlinear constraints. First, we develop the optimality conditions directly using their higher-order form. These conditions are then used to develop computational approaches. A general purpose program has been developed to benchmark computations between problems posed in alternate higher-order and firstorder forms. The program implements both direct and indirect methods and uses collocation in conjunction with a nonlinear programming solver. infancy. However, a few studies in the literature address this issue applied to limited classes of problems. For example, a direct method was used for PVTOL systems to compute the optimal solution, where the inequality constraints were not considered [3]. In recent work, Agrawal and coworkers have used the indirect method to compute the optimal solution in the absence of inequality constraints. In these studies, the system equations were explicitly embedded into the cost functional. The results from higher-order variational theory were used to find the optimality conditions [4]. This approach was applied successfully to linear systems ([5], [6]) and feedback linearizable nonlinear systems [7]. The purpose of this paper is to extend the optimality theory applicable to higher-order systems, without converting the system equations to the first-order form. Using the resulting optimality conditions, direct and indirect computational algorithms are developed and implemented. The same direct or indirect algorithm is used to solve the firstorder and the higher-order optimization problems, thereby, allowing to benchmark the computation requirements in a systematic manner.

Numerical and Experimental Studies on Nonlinear Dynamic Behaviors of a Rotor-Fluid Film Bearing System With Squeeze Film Dampers

In this paper, the nonlinear oil film forces of bearings and dampers with free boundary conditions are determined by the finite element method (FEM) and the complementary solution for variational inequalities. The mode synthesis technique is used to reduce the linear degrees of the high order finite element model. The periodic solution of the system and its stability are determined by the Poincare´ mapping method and the Floquet theory, respectively. The results of experiment show that squeeze film dampers (SFDs) can effectively prevent subsynchronous and nonsynchronous vibrations and some structural parameters have significant effects on the dynamic behaviors of the system. Comparing the numerical results with those of experiment, it is shown that the above theories and schemes are feasible and efficient in analyzing nonlinear behaviors of the high-order dynamic system with local nonlinearities.

Some results for the comparative statics of steady states of higher-order discrete dynamic systems

Abstract Higher-order discrete dynamic systems arise naturally in many economic models in which the problem at hand requires an explicit treatment of dynamics involving lags of more than one period. In studying such models, one type of analysis that economists are often interested in is the assessment of the signs and the magnitudes of the effects of a change in the parameters of the dynamic system on the stable steady states of the system. This paper presents some results for such comparative statics analysis. Brief remarks explain the results and illustrate their potential economic usefulness.

An Efficient Analysis of High-Order Dynamical System with Local Nonlinearity

This paper deals with dynamical systems including spatially localized nonlinear substructures. In these cases, the differential equations of motion consist of the coupled linear and nonlinear subsets. According to the feature of such systems, a modal transformation is used, by means which the number of degrees of freedom of the linear subset is reduced significantly and the localized feature of the nonlinear subset still remains. In accordance with this reduced model, a modified Newmark method that is unconditionally stable is proposed to integrate the responses of the reduced system. The advantage of this method is that the nonlinear iterations only need to be executed on localized nonlinear parts of the system equations. The numerical schemes of this study are applied to a large-order flexible rotor with two elliptical bearing supports. The periodic solutions and long term behaviors of the system are investigated numerically, which reveals many interesting phenomena.

Lift order problems for ordinary differential equations on manifolds

Fork≥0, let ττk:T k+1(M)=T(T k(M))→T k(M) denote the (k+1)th iterated tangent bundle in relation to a base manifoldT 0(M)=M. LetV represent a possibly nonstationary vector field overT k(M), and letQ be a subset/submanifold inT k(M). Sufficient conditions (and, whenV is completely integrable inQ, necessary and sufficient conditions) are established to ensure that all solutionsg toy′=V(t, y) lying entirely inQ have the formG=f [k], wheref [k] is thekth-order differential lift of a curvef lying inM. The relevance of the issue for higher order dynamical systems (especially in mechanics) is discussed. Higher order involutions and complete vector field lifts are examined from the viewpoint of the differential equations they present. Collateral results on the general solvability of initial value problems are obtained and numerous examples are discussed in detail.

The representation of AC machine dynamics by complex signal flow graphs

Induction motors are modeled by nonlinear higher-order dynamic systems of considerable complexity. The dynamic analysis based on the complex notation exhibits a formal correspondence to the description using matrices of axes-oriented components; yet differences exist. The complex notation appears superior in that it allows the distinguishing between the system eigenfrequencies and the angular velocity of a reference frame which serves as the observation platform. The approach leads to the definition of single complex eigenvalues that do not have conjugate values associated with them. The use of complex state variables further permits the visualization of AC machine dynamics by complex signal flow graphs. These simple structures assist to form an understanding of the internal dynamic processes of a machine and their interactions with external controls. >

Control with a spherical classifier neural network

This paper presents the spherical classifier, an artificial neural network model for content addressable memory, pattern recognition, and control. The spherical classifier is a high-order dynamical system which stores as memories (attractors) any finite set of unit vectors in n-dimensional space. We illustrate applications of the spherical classifier in classification and control.