scholarly journals Chaos and bifurcation in time delayed third order phase-locked loop

2021 ◽  
Vol 11 (2) ◽  
pp. 1431
Author(s):  
Bassam Harb ◽  
Mohammad Qudah ◽  
Ibrahim Ghareeb ◽  
Ahmad Harb
Keyword(s):  
Time Delay ◽  
Control Parameter ◽  
Gain Control ◽  
Phase Locked Loop ◽  
Stable Region ◽  
Space Representation ◽  
Third Order ◽  
State Space Representation ◽  
The One ◽  
Feedback Time

In this paper, the modern nonlinear theory is applied to a third order phase locked loop (PLL) with a feedback time delay. Due to this delay, different behaviors that are not accounted for in a conventional PLL model are identified, namely, oscillatory instability, periodic doubling and chaos. Firstly, a Pade approximation is used to model the time delay where it is utilized in deriving the state space representation of the PLL under investigation. The PLL under consideration is simulated with and without time delay. It is shown that for certain loop gain (control parameter) and time delay values, the system changes its stability and becomes chaotic. Simulations show that the PLL with time delay becomes chaotic for control parameter value less than the one without time delay, i.e, the stable region becomes narrower. Moreover, the chaotic region becomes wider as time delay increases.

Robust state derivative feedback LMI-based designs for discretized systems.

2020 ◽  
Author(s):  
Marco A. C. Leandro ◽  
Renan L. Pereira ◽  
Karl H. Kienitz
Keyword(s):  
Discrete Time ◽  
Cartesian Product ◽  
Stable Region ◽  
Space Representation ◽  
Linear Matrix ◽  
Feedback Controllers ◽  
Time Model ◽  
State Space Representation ◽  
Augmented System ◽  
Discretized Systems

This work addresses novel Linear Matrix Inequality (LMI)-based conditions for thedesign of discrete-time state derivative feedback controllers. The main contribution of this work consists of an augmented discretized model formulated in terms of the state derivative, such that uncertain sampling periods and parametric uncertainties in polytopic form can be propagated from the original continuous-time state space representation. The resulting discrete-time model is composed of homogeneous polynomial matrices with parameters lying in the Cartesian product of simplexes, plus an additive norm-bounded term representing the residual discretization error. Moreover, the referred condition allows for the closed-loop poles allocation of the augmented system in a D-stable region. Finally, numerical simulations illustrate the effectiveness of the proposed method.

Stability of Turning Process with a Continuous Delay Model

2012 ◽  
Vol 500 ◽  
pp. 20-25 ◽  
Author(s):  
Qing Hua Song ◽  
Xing Ai ◽  
Bing Guo
Keyword(s):  
Governing Equation ◽  
Distributed Delay ◽  
Space Representation ◽  
Force Model ◽  
Delay Model ◽  
Cutting Force Model ◽  
Third Order ◽  
State Space Representation ◽  
Continuous Delay ◽  
The Stability

An alternative physical explanation for process damping where a distributed cutting force model, along with a function distribution over the tool-chip interface, is assumed, is described. An exponential shape function is used to approximate the force distribution on the tool-chip interface. The distributed force model results in a more complicated governing equation, a second-order delayed integrodifferential equation, which involves both a discrete and distributed delay. An approach to transform and normalize the governing equation of motion into a third-order discrete system is described and the state-space representation of the new system is obtained. The semi-discretization method is then used to chart the stability boundaries for turning operation.

scholarly journals Delay-Dependent Stability Analysis of TS Fuzzy Switched Time-Delay Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Nawel Aoun ◽  
Marwen Kermani ◽  
Anis Sakly
Keyword(s):  
Time Delay ◽  
Stability Criterion ◽  
Delay Systems ◽  
Space Representation ◽  
Time Delay Systems ◽  
State Space Representation ◽  
Arbitrary Switching ◽  
The Stability ◽  
Delay Dependent ◽  
Arrow Form Matrix

This paper proposes a new approach to deal with the problem of stability under arbitrary switching of continuous-time switched time-delay systems represented by TS fuzzy models. The considered class of systems, initially described by delayed differential equations, is first put under a specific state space representation, called arrow form matrix. Then, by constructing a pseudo-overvaluing system, common to all fuzzy submodels and relative to a regular vector norm, we can obtain sufficient asymptotic stability conditions through the application of Borne and Gentina practical stability criterion. The stability criterion, hence obtained, is algebraic, is easy to use, and permits avoiding the problem of existence of a common Lyapunov-Krasovskii functional, considered as a difficult task even for some low-order linear switched systems. Finally, three numerical examples are given to show the effectiveness of the proposed method.

scholarly journals Random walks, random sequences, and nonlinear dynamics in human optokinetic nystagmus

2001 ◽  
Vol 91 (4) ◽  
pp. 1750-1759 ◽  
Author(s):  
P. Trillenberg ◽  
C. Gross ◽  
M. Shelhamer
Keyword(s):  
Nonlinear Dynamics ◽  
Time Delay ◽  
Random Walks ◽  
Random Process ◽  
Eye Movement ◽  
Optokinetic Nystagmus ◽  
Order Approximation ◽  
Space Representation ◽  
Nonlinear Interactions ◽  
State Space Representation

Optokinetic nystagmus (OKN) is a reflexive eye movement with target-following slow phases (SP) alternating with oppositely directed fast phases (FP). We measured the following from OKN in three humans: FP beginning and ending positions, amplitudes, and intervals and SP amplitudes and velocities. We sought to predict future values of each parameter on the basis of past values, using state-space representation of the sequence (time-delay embedding) and local second-order approximation of trajectories. Predictability is an indication of determinism: this approach allows us to investigate the relative contributions of random and deterministic dynamics in OKN. FP beginning and ending positions showed good predictability, but SP velocity was less predictable. FP and SP amplitudes and FP intervals had little or no predictability. FP beginnings and endings were as predictable as randomized versions that retain linear autocorrelation; this is typical of random walks. Predictability of FP intervals did not change under random rearrangement, which is characteristic of a random process. Only linear determinism was demonstrated; nonlinear interactions may exist that would not be detected by our present approach.

scholarly journals Avaliação da Curva de Juros Empregando Extensões do Modelo de Diebold & Li com Três Fatores

2015 ◽  
Vol 13 (2) ◽  
pp. 251
Author(s):  
Alberto Ronchi Neto ◽  
Osvaldo Candido
Keyword(s):  
Stochastic Volatility ◽  
Covariance Matrix ◽  
Latent Variables ◽  
Cholesky Decomposition ◽  
Space Representation ◽  
Interest Rate Parity ◽  
State Space Representation ◽  
Out Of Sample ◽  
Rate Parity ◽  
The One

This paper evaluates methods that employ Kalman Filter to estimate Diebold and Li (2006) extensions in a state-space representation, applying the Nelson and Siegel (1987) function as measure equation and different specifications for the transition equation that determines level, slope and curvature dynamics. The models that were analyzed have the following structures in transition equation: (1) AR(1) specification, employing a diagonal covariance matrix for the residuals; (2) VAR(1) specification, employing a covariance matrix calculated with Cholesky decomposition; (3) VAR(1) extension, inserting variables related to the Covered Interest Rate Parity (CIRP); (4) VAR(1) extension, including stochastic volatility components. The major findings of this paper were: (1) evaluating the latent variables dynamics, the curvature was the factor that fitted better to the stochastic volatility component; (2) in a broad sense, even though the simplest VAR(1) model was the one that provided the best out-of-sample performance in the most part of maturities and forecasting horizons, the extension inserting variables related to the CIRP was able to overcome the former specification in some of these simulations.

Robust D-stability via discrete controllers for continuous-time uncertain systems with multiple delays.

2020 ◽  
Author(s):  
Marco A. C. Leandro ◽  
Karl H. Kienitz
Keyword(s):  
Time Delay ◽  
Continuous Time ◽  
Closed Loop ◽  
Delay System ◽  
System Stability ◽  
Stable Region ◽  
Control Synthesis ◽  
Space Representation ◽  
Linear Matrix ◽  
Multiple Delays

This work addresses the allocation of closed-loop poles of a discretized system from a continuous-time one with multiple input delays, aiming at its control through a computer. In order to handle a practical challenge presented in Network Control System (NCS) approaches, uncertain sampling period, distinct input time delays and parametric uncertainties in polytopic form can be propagated from the original state space representation to the discretized state model. The resulting discrete-time time-delay system has a very specific feature, so that it can be converted into an augmented linear system without time-delay. In this context, the main contribution of the present paper consists of a Linear Matrix Inequality (LMI) based control synthesis condition composed of homogeneous polynomial matrices of arbitrary degree, which ensures the continuous-time system stability and simultaneously the allocation of the closed-loop poles of the augmented system in a D-stable region. Numerical simulations illustrate the exposed.

Modeling, Actuation, and Control of an Active Fluid Vibration Isolator

1997 ◽  
Vol 119 (1) ◽  
pp. 52-59 ◽  
Author(s):  
M. J. Panza ◽  
D. P. McGuire ◽  
P. J. Jones
Keyword(s):  
Electrical Power ◽  
Fluid Pressure ◽  
Space Representation ◽  
Vibration Isolator ◽  
Mass System ◽  
Integral Control ◽  
Active Fluid ◽  
State Space Representation ◽  
In Series ◽  
And Control

An integrated mathematical model for the dynamics, actuation, and control of an active fluid/elastomeric tuned vibration isolator in a two mass system is presented. The derivation is based on the application of physical principles for mechanics, fluid continuity, and electromagnetic circuits. Improvement of the passive isolator performance is obtained with a feedback scheme consisting of a frequency shaped notch compensator in series with integral control of output acceleration and combined with proportional control of the fluid pressure in the isolator. The control is applied via an electromagnetic actuator for excitation of the fluid in the track connecting the two pressure chambers of the isolator. Closed loop system equations are transformed to a nondimensional state space representation and a key dimensionless parameter for isolator-actuator interaction is defined. A numerical example is presented to show the effect of actuator parameter selection on system damping, the performance improvement of the active over the passive isolator, the robustness of the control scheme to parameter variation, and the electrical power requirements for the actuator.

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