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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?

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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.

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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.

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