I am studying the application of Pade approximations on a system with a dead time. In most cases, we consider a transfer function with a dead time and apply the Pade approximation on it.
Now here is my question: can we introduce a delay in state space? If yes, how one can apply the Pade approximation on it? If no, why is it not possible?
Yes, it is possible to consider Pade approximants for delays in linear state space systems. The simplest approach is to construct the transfer function for the delay and convert this to a state space model. As an example, the second order Pade approximation for a time delay of $T$ is $$ H_{\text{delay}}(s) = \frac{s^2-6/T s+12/T^2}{s^2+6/Ts+12/T^2} $$ which corresponds to a state space system with matrices $$ A_{\text{delay}} = \left[ \begin{array}{cc} 0 & -12/T \\ 1 & -6/T \end{array} \right], \quad B_{\text{delay}} = \left[ \begin{array}{c} 1 \\ 0 \end{array} \right], \quad C_{\text{delay}} = \left[ \begin{array}{c} -12/T & 72/T^2 \end{array} \right] $$ where I have used the companion state space form.
Supposing we want to apply this delay at the input to another linear state space system with matrices $A_{\text{sys}}$, $B_{\text{sys}}$, $C_{\text{sys}}$, $D_{\text{sys}}$, we can achieve this by forming the larger system with matrices: $$ A = \left[ \begin{array}{cc} A_\text{delay} & 0 \\ B_\text{sys} & A_\text{sys} \end{array} \right], \quad B = \left[ \begin{array}{c} B_{\text{delay}} \\ 0 \end{array} \right], \quad C = \left[ \begin{array}{c} D_\text{sys} & C_\text{sys} \end{array} \right], \quad D = 0 $$ In practice, most of this can be carried out easily in Matlab by using the pade() and ss() functions.
I'll answer part of your question. Pade approximation is used to represent delays in rational form. By doing this, you transform exp(-sT) in something like a transfer function, which can be used naturally in state space representation.
