A sampling-based approach for handling delays in continuous and hybrid systems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Erzana Berani Abdelwahab ◽  
Martin Fränzle
Keyword(s):  
Hybrid Systems ◽  
Formal Analysis ◽  
Distributed Delay ◽  
Continuous Systems ◽  
Digital Networks ◽  
Embedded Control ◽  
Physical Plant ◽  
Ongoing Work ◽  
Band Limited ◽  
Delay Differential

Abstract Delays in feedback dynamics of coupled dynamical systems arise regularly, especially in embedded control where the physical plant and the controller continuously interact through digital networks. Systems featuring delays are however notoriously difficult to analyze. Consequently, formal analysis often addresses simplified, delay-free substitute models, risking negligence of the adverse impact of delay on control performance. In this ongoing work, we demonstrate that for continuous systems such as delay differential equations, a major part of the delay-induced complexity can be reduced effectively when adding natural constraints to the model of the delayed feedback channel, namely that it transports a band-limited signal and implements a non-punctual, distributed delay. The reduction is based on a sampling approach which is applicable when the above conditions on the feedback are satisfied. We further discuss the possibilities of lifting this method to mixed discrete-continuous dynamics of delayed hybrid systems and the open issues thereof.

scholarly journals Sensitivity Analysis for Hybrid Systems and Systems With Memory

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
Radu Serban ◽  
Antonio Recuero
Keyword(s):  
Hybrid Systems ◽  
Exponential Model ◽  
Differential Algebraic Equations ◽  
Mode Switching ◽  
Continuous Systems ◽  
Algebraic Equations ◽  
Adjoint Sensitivity ◽  
Adjoint Systems ◽  
Systems With Memory ◽  
With Memory

We present an adjoint sensitivity method for hybrid discrete—continuous systems, extending previously published forward sensitivity methods (FSA). We treat ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) of index up to two (Hessenberg) and provide sufficient solvability conditions for consistent initialization and state transfer at mode switching points, for both the sensitivity and adjoint systems. Furthermore, we extend the analysis to so-called hybrid systems with memory where the dynamics of any given mode depend explicitly on the states at the last mode transition point. We present and discuss several numerical examples, including a computational mechanics problem based on the so-called exponential model (EM) constitutive material law for steel reinforcement under cyclic loading.

scholarly journals Oscillation of Fourth-Order Functional Differential Equations with Distributed Delay

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 61 ◽  
Author(s):  
Clemente Cesarano ◽  
Omar Bazighifan
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
All Solutions ◽  
Functional Differential ◽  
Delay Differential ◽  
Comparison Technique

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the fourth order delay differential equations. Some new oscillatory criteria are obtained by using the generalized Riccati transformations and comparison technique with first order delay differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. The effectiveness of the obtained criteria is illustrated via examples.

scholarly journals Temporal cavity solitons in a delayed model of a dispersive cavity ring laser

2020 ◽  
Vol 15 ◽  
pp. 47
Author(s):  
Alexander Pimenov ◽  
Shalva Amiranashvili ◽  
Andrei G. Vladimirov
Keyword(s):  
Continuous Wave ◽  
Modulational Instability ◽  
Distributed Delay ◽  
Delay System ◽  
Differential Equation Models ◽  
Dispersion Regime ◽  
Fiber Delay Line ◽  
Wave States ◽  
The Stability ◽  
Delay Differential

Nonlinear localised structures appear as solitary states in systems with multistability and hysteresis. In particular, localised structures of light known as temporal cavity solitons were observed recently experimentally in driven Kerr-cavities operating in the anomalous dispersion regime when one of the two bistable spatially homogeneous steady states exhibits a modulational instability. We use a distributed delay system to study theoretically the formation of temporal cavity solitons in an optically injected ring semiconductor-based fiber laser, and propose an approach to derive reduced delay-differential equation models taking into account the dispersion of the intracavity fiber delay line. Using these equations we perform the stability and bifurcation analysis of injection-locked continuous wave states and temporal cavity solitons.

Periodic Solution of a Stochastic Microorganism Flocculation Model with Distributed Delay

2021 ◽  
Author(s):  
Xiaojie Mu ◽  
Daqing Jiang
Keyword(s):  
Periodic Solution ◽  
Dynamic Change ◽  
Distributed Delay ◽  
Positive Periodic Solution ◽  
Point Of View ◽  
Ocean Current ◽  
Positive Equilibrium ◽  
Microorganism Population ◽  
External Conditions ◽  
Delay Differential

Abstract In this paper, a nonautonomous delay differential equation of microorganism flocculation is established by considering the influence of external conditions such as seasonal alternation and ocean current movement on the ecological function of microorganism population. At the same time, the dynamic change characteristics of microorganism population in oil spill environment were simulated, and on this basis, the effects of diurnal change and climate change on the parameters of microorganism system were analyzed. From a mathematical point of view, the stochastic microorganism flocculation model exists a T-positive periodic solution. The existence and uniqueness of globally positive equilibrium of the exploited model is studied. Finally, some numerical examples illustrate the results.

scholarly journals On Periodic Solutions of Delay Differential Equations with Impulses

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 523 ◽  
Author(s):  
Mostafa Bachar
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Distributed Delay ◽  
Impulsive Delay Differential Equations ◽  
Impulsive Delay ◽  
Delay Differential ◽  
Distributed Delay Differential Equations

The purpose of this paper is to study the nonlinear distributed delay differential equations with impulses effects in the vectorial regulated Banach spaces R ( [ − r , 0 ] , R n ) . The existence of the periodic solution of impulsive delay differential equations is obtained by using the Schäffer fixed point theorem in regulated space R ( [ − r , 0 ] , R n ) .

Response and Stability Analysis of Periodic Delayed Systems With Discontinuous Distributed Delay

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Oleg A. Bobrenkov ◽  
Morad Nazari ◽  
Eric A. Butcher
Keyword(s):  
Stability Analysis ◽  
Continuous Time ◽  
Delay Differential Equations ◽  
Monodromy Matrix ◽  
Distributed Delay ◽  
Mathieu Equation ◽  
Adaptive Meshing ◽  
Delayed Systems ◽  
Time Approximation ◽  
Delay Differential

In this paper, the analysis of delay differential equations with periodic coefficients and discontinuous distributed delay is carried out through discretization by the Chebyshev spectral continuous time approximation (ChSCTA). These features are introduced in the delayed Mathieu equation with discontinuous distributed delay which is used as an illustrative example. The efficiency of stability analysis is improved by using shifted Chebyshev polynomials for computing the monodromy matrix, as well as the adaptive meshing of the parameter plane. An idea for a method for numerical integration of periodic DDEs with discontinuous distributed delay based on existing MATLAB functions is proposed.

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