Hi,
I am trying to solve the state space form of the dynamical equation of motion under earthquake excitation with time-delay in the control force as
dz/dt(t)=A*z(t)+B1*U(t-delta)+B2*ag(t)
where z={x; dx/dt}, with x as the displacement vector; U is the control force; delta is the time delay; ag is earthquake input vector with 0.02 s time step; A, B1, B2 are constant matrices.
I have difficulty in dealing with time-delay. I would appreciate any comments on numerical solutions.
Regards,
Keyvan
For linear system you can find the transfer function between z and u using Laplace. Then a time delay will simply be e^(-delta s). Example:
dz/dt = -z(t) + u(t-delta)
s z(s) = -z(s) + e^(-delta s) u(s)
---> Transfer function: z/u (s) = e^(-delta s) / (s+1)
You can design a control law u in the frequency domain (Bode digram) for this transfer function.
The function dde23 in MATLAB can solve delay differential equation numerically. For analytical solution, Laplace or Fourier transform of time delay equations are the best options. You can also read more about Delay differntial equation in general from here https://www.math.auckland.ac.nz/~hinke/meetings/NZMRI/materials/NZMRI16-Tony1.pdf
Dear Thor I. Fossen ,
Thank you for your comment. I am using sliding mode control (SMC) law which includes sign function as sgn(p*z(t)). Considering the system dynamics after applying the SMC method as dz/dt=p*z(t-delta)-eta*sign(p*z(t-delta))+d, I wonder if I could use Laplace transformation for this equation. (d is an external disturbance).
Dear D.E.C Belkhiat and Tanushree Roy ,
Thank you for the references.
The laplace transformations of sgn and z(t-delta) are pretty simple, vide any table of transform.
The laplace of composition of functions can be found here:
https://math.stackexchange.com/questions/171462/fourier-transform-of-function-composition/171473
Hope this helps!
Hi, For designing the suitable controller in your case I suggest you to adopt the Model Predictive Control approach.
Independently of the mathematical problems related to the appropriate representation of such dynamics, the delay systems are known to rise challenging control problems due to the instabilities introduced in the closed loop by the presence of delays. Discrete time control of continuous systems aected by delays has to face even more diculties due to the sampling which introduces an uncertainty in the discrete models. It is known that predictors can be used to overcome the efects of dead-time (with inherent problems linked to the sensitivity of predictions for unstable models). MPC-"Model Predictive Control" solves at each sampling time a nite-time optimal control problem over a receding prediction horizon and is no surprise that its use in connection with delay systems was proposed from the early approaches.
My Dear,
One more thing:
In my opinion you should adopt the Linear Matrix Inequality (LMI) strategy in order to design sliding surface s=c*x(t), if you are going to use SMC approach. Nevertheless, I still suggest you to choose the MPC upon your system.
Dear Hossein Nejatbakhsh Esfahani ,
Thank you for your input. I will try Model Predictive Control strategy.
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