Time Delay Systems - Anand's Portfolio

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Hayes DDE : Hayes delay differential equation is a most basic delayed differential equation.
INPUT: History Function (Initial Condition, which is a function in DDEs case)
OUTPUT : Equation solution.
Solved Using dde23 on MATLAB.

Hayes Equation

Blue curve shows the history of the system which should be known and is treated as input.
Red curve is the predicted behavior when subjected to stable a, b values.

Stability Chart for Hayes through SemiDiscritization: Hayes Equation.
x'(t)=ax(t)+bx(t-tau)
Code to find the stability region in (a,b) plane for Hayes equation using semidistritization.

Grey area shows the stable region in the (a,b) plane

Equation of Delayed Oscillator: x”(t) + 2zw_nx'(t) + w_n^2(x(t) + (3/4)x(t-tau)) = 0
This is a 2nd order time delayed differential equation. Solved using dde23 in MATLAB.
P = exp(Adeltat) ; R = inv(A)*(P-I)*B : P and R will always have this formula. Just the dimension of A and B will change depending on the order of the differential equation.

Differential equation for delayed oscillator

Grey area shows the stable region in the omega_n and zeta plane

Human delayed chain following plant and string stability curves: Code for plant and string stability of a human driven delayed chain.
INPUT: tau, kappa, (A,B,C matrices) or (a_0,a_tau,beta_0,beta_tau C) matrices in case of time delay system.
OUTPUT: Plant and String stability curves.
Human car following model or Car following model with delay. Considering Optimal Velocity Model.

Human driven chain following plant and string stability curves

Two vehicles on the ring with human delay

Two vehicles on the ring with the human delay (Non-Linear)

Platoon_Following_Model : Code to study the platoon response of n vehicle to some given profile of the leader.
INPUT: Number of Vehicles, Leader’s Velocity, Initial Conditions and delay
OUTPUT: Follower vehicle’s response based on the an alpha,beta controller

Ring_Following_Model : Code to study the response of n vehicles in the ring to some given initial conditions.
INPUT: Number of Vehicles, Initial Conditions
OUTPUT: Follower vehicle’s response based on the an alpha,beta controller

code repository can be found at here.