42. Padé approximation of time delay

42.1. Time delay

In time-domain

\[u_{\tau}(t) := u(t-\tau) \ , \quad \tau \ge 0 \ ,\]

with causal function \(u(t) = 0\), for \(t < 0\).

In Laplace domain,

\[\begin{split}\begin{aligned} \mathscr{L}\left\{ u(t-\tau) \right\} & := \int_{t=0}^{+\infty} u(t-\tau) e^{-st} \, dt = \\ & = \int_{t'=0}^{+\infty} u(t') e^{-s (t'+\tau)} \, dt' = \\ & = e^{-s\tau} \int_{t'=0}^{+\infty} u(t') e^{-s t'} \, dt' = \\ & = e^{-s \tau} U(s) \ . \end{aligned}\end{split}\]

42.2. Padé approximation

Padè approximation is a rational approximation of the exponential transfer function of the delay, \(e^{-s \tau}\).

• Padé approximation is a (reasonable) low-frequency approximation of time delay. This immediately follows as Padé approximation exploits Taylor expansion of the time-delay transfer function \(e^{-s \tau}\) for \(s = 0\).

• Padé is usually required in time-domain methods, in order to avoid delay differential equations

Examples.

• First-order system w/ time delay - referece tracking

• …

42.2.1. Examples

\((1,1)\)-Padé approximation.

\[e^{-s \tau} \simeq P_{1,1}(s) = \frac{1 - \frac{\tau}{2}s}{1 + \frac{\tau}{2}s}\]

\((1,1)\)-Padé approximation - details

As an example, a rational approximation with both numerator and denominator of first order

\[e^{-s \tau} \simeq \frac{1 + a s}{1 + b s} \ .\]

Using Taylor approximation \(e^{-s \tau}\)

\[e^{-s \tau} \simeq 1 - \tau s + \frac{\tau^2 s^2}{2} + o (s^2) \ ,\]

and equating

\[\left( 1 - \tau s + \frac{\tau^2 s^2}{2} + o(s^2) \right) ( 1 + b s ) = 1 + a s \ ,\]

the first terms

\[\begin{split}\begin{aligned} 1 : & \quad 1 = 1 \\ s : & \quad -\tau + b = a \\ s^2: & \quad \frac{\tau^2}{2} - \tau b = 0 \ , \end{aligned}\end{split}\]

it follows

\[\begin{split}\begin{aligned} b & = \frac{\tau}{2} \\ a & = -\frac{\tau}{2} \ , \end{aligned}\end{split}\]

\((2,2)\)-Padé approximation.

\[e^{-s \tau} \simeq P_{2,2}(s) = \frac{1 - \frac{\tau s}{2} + \frac{\tau^2 s^2}{12}}{1 + \frac{\tau s}{2} + \frac{\tau^2 s^2}{12}}\]

\((2,2)\)-Padé approximation - details

As an example, a rational approximation with both numerator and denominator of first order

\[e^{-s \tau} \simeq \frac{1 + a s + b s^2}{1 + c s + d s^2} \ .\]

Using Taylor approximation \(e^{-s \tau}\)

\[e^{-s \tau} \simeq 1 - \tau s + \frac{\tau^2 s^2}{2} - \frac{\tau^3 s^3}{3!} + \frac{\tau^4 s^4}{4!} + o (s^4) \ ,\]

and equating

\[\left( 1 - \tau s + \frac{\tau^2 s^2}{2} - \frac{\tau^3 s^3}{3!} + \frac{\tau^4 s^4}{4!} + o(s^2) \right) ( 1 + c s + d s^2 ) = 1 + a s + b s^2 \ ,\]

the first terms

\[\begin{split}\begin{aligned} 1 : & \quad 1 = 1 \\ s : & \quad -\tau + c = a \\ s^2: & \quad \frac{\tau^2}{2 } - \tau c + d = b \\ s^3: & \quad -\frac{\tau^3}{3!} + \frac{\tau^2}{2 } c - \tau d = 0 \\ s^4: & \quad \frac{\tau^4}{4!} - \frac{\tau^3}{3!} c + \frac{\tau^2}{2} d = 0 \ , \end{aligned}\end{split}\]

it follows

\[\begin{split}\begin{aligned} c & = \frac{1}{2}\tau \\ d & = \frac{1}{12}\tau^2 \\ a & = -\frac{1}{2}\tau \\ b & = \frac{1}{12}\tau^2 \\ \end{aligned}\end{split}\]

as

\[\begin{split}\begin{aligned} \frac{1}{6} c - \frac{1}{2 \cdot 3!} \tau = 0 \quad & \rightarrow \quad c = \frac{1}{2}\tau \\ d = \tau \frac{c}{2} - \frac{1}{6} \tau^2 \quad & \rightarrow \quad d = \frac{1}{12}\tau^2 \\ & \rightarrow \quad a = -\frac{1}{2}\tau \\ & \rightarrow \quad b = \frac{1}{12}\tau^2 \\ \end{aligned}\end{split}\]

42.2.2. Padé approximation in state-space representation

Let the LTI system

\[\begin{split}\begin{aligned} \dot{\mathbf{x}}(t) & = \mathbf{A} \mathbf{x}(t) + \mathbf{B} \mathbf{u}(t-\tau) \\ \mathbf{y} (t) & = \mathbf{C} \mathbf{x}(t) + \mathbf{D} \mathbf{u}(t-\tau) \ , \end{aligned}\end{split}\]

A Padé approximation results in a LTI whose output is the approximate delayed input, and whose input is the input itself. As an example, the \((1,1)\)-Padé approximation in Laplace domain reads

\[U_\tau(s) = e^{-s \tau} U(s) \simeq \frac{1 - \frac{\tau s}{2}}{1 + \frac{\tau s}{2}} U(s) = \left[ \frac{-s + \frac{2}{\tau}}{s + \frac{2}{\tau} } \right] U(s) = \left[ - 1 + \frac{\frac{4}{\tau}}{s + \frac{2}{\tau}} \right] U(s) ,\]

and it has a state-space realization using controllable canonical form,1

\[\begin{split}\left\{\begin{aligned} \dot{\mathbf{x}}_\tau & = \begin{bmatrix} -\frac{2}{\tau} \end{bmatrix} \mathbf{x}_\tau + \begin{bmatrix} 1 \end{bmatrix} u \\ u_\tau(t) & = \begin{bmatrix} \frac{4}{\tau} \end{bmatrix} \mathbf{x}_\tau + \begin{bmatrix} -1 \end{bmatrix} u \end{aligned}\right. \ .\end{split}\]

The augmented system reads

\[\begin{split}\left\{ \begin{aligned} \begin{bmatrix} \dot{\mathbf{x}} \\ \dot{\mathbf{x}}_{\tau} \end{bmatrix} & = \begin{bmatrix} \mathbf{A} & \mathbf{B} \mathbf{C}_\tau \\ \cdot & \mathbf{A}_\tau \end{bmatrix} \begin{bmatrix} \mathbf{x} \\ \mathbf{x}_{\tau} \end{bmatrix} + \begin{bmatrix} \mathbf{B} \mathbf{D}_\tau \\ \mathbf{B}_\tau \end{bmatrix} \mathbf{u} \\ \mathbf{y} & = \begin{bmatrix} \mathbf{C} & \mathbf{D} \mathbf{C}_\tau \end{bmatrix} \begin{bmatrix} \mathbf{x} \\ \mathbf{x}_{\tau} \end{bmatrix} + \begin{bmatrix} \mathbf{D} \mathbf{D}_\tau \end{bmatrix} \mathbf{u} \\ \end{aligned} \right.\end{split}\]

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1

Usually, controllable canonical form for delay on input; observable canonical form for delay on output. Here, in this example with a \((1,1)\)-Padé approximation, the state of the system is 1-dimensional, and the two canonical forms differ only by the “position” of the “\(b_{n-1} = \frac{4}{\tau}\) element. The not-strictly proper TF is first written as a sum of a constant and a strictly proper TF, whose realization follows the rules of canonical realizations.