System Identification of Dynamic Systems With Cubic Nonlinearities Using Linear Time-Periodic Approximations

2009 ◽  
Author(s):  
Matthew S. Allen ◽  
Michael W. Sracic
Keyword(s):  
System Identification ◽  
Limit Cycle ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Linear Time ◽  
Stable Limit Cycle ◽  
Parametric Models ◽  
Stable Limit ◽  
Nonlinear Dynamic Model ◽  
Time Periodic

This work develops methods to identify parametric models of nonlinear dynamic systems from response measurements using tools for Linear Time Periodic (LTP) systems. The basic approach is to drive the system periodically in a stable limit cycle and then measure deviations of the response from that limit cycle. Under certain conditions, the resulting response can be well approximated as that of a linear-time periodic system. In the analytical realm it is common to linearize a system about a periodic trajectory and then use Floquet analysis to assess the stability of the limit cycle. This work is concerned with the inverse problem, using a measured time-periodic response to derive a nonlinear dynamic model for the system. Recently, a few new methods were developed that facilitate the experimental identification of linear time periodic systems, and those methods are exploited in this work. The proposed system identification methodology is evaluated by applying it to a Duffing oscillator, demonstrating that the nonlinear force-displacement relationship can be identified without a priori knowledge of its functional form. The proposed methods are also applied to simulated measurements from a cantilever beam with a cubic nonlinear spring on its tip, revealing that the model order of the system and the displacement dependent stiffness can be readily identified.

scholarly journals STUDY POINTS OF REST HEREDITARITY DYNAMIC SYSTEMS VAN DER POL-DUFFING

2019 ◽  
pp. 71-77
Author(s):  
Е.Р. Новикова ◽  
Р.И. Паровик
Keyword(s):  
Limit Cycle ◽  
Dynamic Systems ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Viscous Friction ◽  
Oscillatory System ◽  
Stable Limit ◽  
Van Der Pol ◽  
Nonlinear Oscillatory System ◽  
Stable Limit Cycles

Using numerical modeling, oscillograms and phase trajectories were constructed to study the limit cycles of a van der Pol Duffing nonlinear oscillatory system with a power memory. The simulation results showed that in the absence of a power memory (α = 2, β = 1) or the classical van der Pol Duffing dynamical system, there is a single stable limit cycle, i.e. Lienar theorem holds. In the case of viscous friction (α = 2, 0 < β < 1), there is a family of stable limit cycles of various shapes. In other cases, the limit cycle is destroyed in two scenarios: a Hopf bifurcation (limit cycle-limit point) or (limit cycle-aperiodic process). Further continuation of the research may be related to the construction of the spectrum of Lyapunov maximal exponents in order to identify chaotic oscillatory regimes for the considered hereditary dynamic system (HDS). В работе с помощью численного моделирования построены осциллограммы и фазовые траектории с целью исследования предельных циклов нелинейной колебательной системы Ван-дер-Поля Дуффинга со степенной памятью. Результаты моделирования показали, что в случае отсутствия степенной памяти (α = 2, β = 1) или классической динамической системы Ван-дер-Поля Дуффинга, существует единственный устойчивый предельный цикл, т.е. выполняется теорема Льенара. В случае вязкого трения (α = 2, 0 < β < 1), существует семейство устойчивых предельных циклов различной формы. В остальных случаях происходит разрушение предельного цикла по двум сценариям: бифуркация Хопфа (предельный цикл-предельная точка) или (предельный циклапериодический процесс). Дальнейшее продолжение исследований может быть связано с построением спектра максимальных показателей Ляпунова с целью идентификации хаотических колебательных режимов для рассматриваемой эредитарной динамической системы (ЭДС).

scholarly journals Linear Time-Periodic System Identification with Grouped Atomic Norm Regularization

2020 ◽  
Vol 53 (2) ◽  
pp. 1237-1242
Author(s):  
Mingzhou Yin ◽  
Andrea Iannelli ◽  
Mohammad Khosravi ◽  
Anilkumar Parsi ◽  
Roy S. Smith
Keyword(s):  
System Identification ◽  
Periodic System ◽  
Linear Time ◽  
Atomic Norm ◽  
Time Periodic

scholarly journals STUDYING THE STABILITY BY USING LOCAL LINEARIZATION METHOD

2020 ◽  
Vol 2 (1) ◽  
pp. 27-31
Author(s):  
Abdulghafoor Jasim Salim ◽  
Kais Ismail Ebrahem ◽  
Suhirman
Keyword(s):  
Singular Point ◽  
Limit Cycle ◽  
Autoregressive Model ◽  
Stable Limit Cycle ◽  
Linearization Method ◽  
Stable Limit ◽  
Local Linearization ◽  
Non Linear ◽  
The Stability ◽  
Local Linearization Method

Abstract: In this paper we study the stability of one of a non linear autoregressive model with trigonometric term  by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude  that the proposed model under certain conditions have a non-zero singular point which is  a asymptotically salable ( when  0 ) and have an  orbitaly stable limit cycle . Also we give some examples in order to explain the method. Key Words : Non-linear Autoregressive model; Limit cycle; singular point; Stability.

Bifurcation of Codimension 3 in a Predator–Prey System of Leslie Type with Simplified Holling Type IV Functional Response

2016 ◽  
Vol 26 (02) ◽  
pp. 1650034 ◽  
Author(s):  
Jicai Huang ◽  
Xiaojing Xia ◽  
Xinan Zhang ◽  
Shigui Ruan
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Positive Equilibrium ◽  
Stable Limit ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Stable Equilibria

It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

scholarly journals Height control for a two-mass hopping robot based on stable limit cycle

2016 ◽  
Vol 13 (6) ◽  
pp. 172988141665774
Author(s):  
Taihui Zhang ◽  
Honglei An ◽  
Qing Wei ◽  
Wenqi Hou ◽  
Hongxu Ma
Keyword(s):  
Limit Cycle ◽  
Control Algorithm ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Inverted Pendulum Model ◽  
Hopping Robot ◽  
Control Objective ◽  
Upper Level ◽  
Mass Spring ◽  
And Control

Differing from the commonly used spring loaded inverted pendulum model, this paper makes use of a two-mass spring model considering impact between the foot and ground which is closer to the real hopping robot. The height of upper mass which includes the upper leg and body is the main control objective. Then we develop a new kind of control algorithm acting on two levels: The upper level aims to achieve the desired velocity of the upper mass based on a stable limit cycle, where three different controllers are used to regulate the limit cycle; the target of the lower level is to drive the system to converge to the desired state and control the contact force between the foot and ground within an appropriate range based on the inner force control at the same time. Simulation results presented in this paper confirm the efficiency of this control algorithm.

Identification of Multivariable Nonlinear Dynamic System Based on PID Neural Network

2015 ◽  
Vol 719-720 ◽  
pp. 475-481
Author(s):  
Hua Shu ◽  
Huai Lin Shu
Keyword(s):  
Neural Network ◽  
System Identification ◽  
Dynamic System ◽  
Nonlinear Dynamic ◽  
Linear Time ◽  
Nonlinear Dynamic System ◽  
Identification Methods ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Pid Neural Network

System identification is the basis for control system design. For linear time-invariant systems have a variety of identification methods, identification methods for nonlinear dynamic system is still in the exploratory stage. Nonlinear identification method based on neural network is a simple and effective general method that does not require too much priori experience about the system to be identified. Through training and learning, the network weights are corrected to achieve the purpose of system identification. The paper is about the identification of multivariable nonlinear dynamic system based on PID neural network. The structure and algorithm of PID neural network are introduced and the properties and characteristics are analyzed. The system identification is completed and the results are fast convergence.

On the Poincaré–Hill cycle map of rotational random walk: locating the stochastic limit cycle in a reversible Schnakenberg model

2009 ◽  
Vol 466 (2115) ◽  
pp. 771-788 ◽  
Author(s):  
Melissa Vellela ◽  
Hong Qian
Keyword(s):  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Quantitative Measure ◽  
Power Spectral Analysis ◽  
Chemical System ◽  
Deterministic System ◽  
Stable Limit ◽  
Novel Device ◽  
Schnakenberg Model ◽  
Cycle Map

Recent studies on stochastic oscillations mostly focus on the power spectral analysis. However, the power spectrum yields information only on the frequency of oscillation and cannot differentiate between a stable limit cycle and a stable focus. The cycle flux, introduced by Hill (Hill 1989 Free energy transduction and biochemical cycle kinetics ), is a quantitative measure of the net movement over a closed path, but it is impractical to compute for all possible cycles in systems with a large state space. Through simple examples, we introduce concepts used to quantify stochastic oscillation, such as the cycle flux, the Hill–Qian stochastic circulation and rotation number. We introduce a novel device, the Poincaré–Hill cycle map (PHCM), which combines the concept of Hill’s cycle flux with the Poincaré map from nonlinear dynamics. Applying the PHCM to a reversible extension of an oscillatory chemical system, the Schnakenberg model, reveals stable oscillations outside the Hopf bifurcation region in which the deterministic system contains a limit cycle. Bistable behaviour is found on the small volume scale with high probabilities around both the fixed point and the limit cycle. Convergence to the deterministic system is found in the thermodynamic limit.

Floquet Experimental Modal Analysis for System Identification of Linear Time-Periodic Systems

2007 ◽  
Author(s):  
Matthew S. Allen
Keyword(s):  
System Identification ◽  
Linear Time ◽  
Time Varying ◽  
Response Model ◽  
State Matrix ◽  
Linear Time Invariant ◽  
Response Data ◽  
Time Varying Systems ◽  
Time Invariant ◽  
Time Periodic

A variety of systems can be faithfully modeled as linear with coefficients that vary periodically with time or Linear Time-Periodic (LTP). Examples include anisotropic rotorbearing systems, wind turbines, satellite systems, etc… A number of powerful techniques have been presented in the past few decades, so that one might expect to model or control an LTP system with relative ease compared to time varying systems in general. However, few, if any, methods exist for experimentally characterizing LTP systems. This work seeks to produce a set of tools that can be used to characterize LTP systems completely through experiment. While such an approach is commonplace for LTI systems, all current methods for time varying systems require either that the system parameters vary slowly with time or else simply identify a few parameters of a pre-defined model to response data. A previous work presented two methods by which system identification techniques for linear time invariant (LTI) systems could be used to identify a response model for an LTP system from free response data. One of these allows the system’s model order to be determined exactly as if the system were linear time-invariant. This work presents a means whereby the response model identified in the previous work can be used to generate the full state transition matrix and the underlying time varying state matrix from an identified LTP response model and illustrates the entire system-identification process using simulated response data for a Jeffcott rotor in anisotropic bearings.

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