The transfer function for a 2nd order continuous-time system with a time delay is[^1]
$$G(s)=\frac{K}{(1+2\zeta T_1+T_1^2s^2)}e^{-T_2s}$$
The state-space model of a second-order continuous system with time delay is[^2]
$$\begin{bmatrix}
\dot{x_1} \\ \dot{x_2}
\end{bmatrix}=
\begin{bmatrix}
0 & 1 \\
-\frac{1}{T_1^2} & -\frac{2\zeta}{T_1}
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}+
\begin{bmatrix}
0 \\ \frac{K_p}{T_1^2}
\end{bmatrix}u(t-\theta_p)$$
$$y=\begin{bmatrix}
0&0
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}+
[0]u
$$
And the differential equation form of the second-order system with time delay is
$$T_1^2\frac{d^2y}{dt^2}+2\zeta T_1\frac{dy}{dt}+y=K_pu(t-\theta_p)$$
LQR-PI Controller – sensitive to systems with time-delayed dynamics
[[Single-Axis Satellite Thruster Model]] – The non-ideal thruster model can be represented as a 2nd order system with a time delay
- IdentifyLowOrderTransfer
- DynamicSimulationPython
[[Differential Equations]]
[[Laplace Transform of a Time Delay]]
[[Second-Order Systems]]
[[State-Space Model]]
[[Transfer Functions]]