TITLE: Wrong stability results when using Padé approximation
QUESTION [7 upvotes]: The Padé approximation of the exponential function,  $F(s) = e^{-\tau s}$, is used often in control theory. I wonder whether its use can lead to erroneous results regarding the stability properties of the closed-loop system.
Let's say we have a dynamical system with transfer function 
$$
G(s) = \frac{P(s)}{Q(s)}e^{-\tau s},
$$
with $\tau>0$, where $P$ and $Q$ are polynomials and the degree of $P$ is strictly smaller than the degree of $Q$. The corresponding closed-loop transfer function is defined as 
$$
G_{CL}(s) = \frac{G(s)}{1+G(s)}.
$$
If we approximate $e^{-\tau s}$ with the Padé approximation of order $[1,1]$ we obtain 
$$
G_{\mathrm{approx}}(s) = \frac{P(s)}{Q(s)}\frac{1-\frac{\tau}{2}s}{1+\frac{\tau}{2}s}.
$$
Then, $G_{CL, \mathrm{approx}} = \frac{G_{\mathrm{approx}}(s)}{1+G_{\mathrm{approx}}(s)}$ becomes a rational function and we can use an algebraic criterion to tell whether it is stable. However, I guess that this does not imply that $G_{CL}$ is stable, but I have not been able to find a counterexample.
I tried to take $G$ to be a first and a second order system and tried with low and high values of $\tau$, but I was not able to come up with a counterexample where Routh's criterion on $G_{CL, \mathrm{approx}}$ and the Bode or Nyquist stability criteria for the actual transfer function lead to different conclusions regarding the stability of the closed-loop system.

REPLY [5 votes]: You can find a counterexample by using a BIBO-stable open-loop system with poles close to the imaginary axis and an intermediate value of $\tau$ (not too low, not too high). Here is an example:
$$
G(s) = \frac{2}{(s+0.5)^2 + 1}e^{-0.6s}.
$$
The closed-loop function $G_c(s) = G(s)/(1+G(s))$ is approximated by 
$$
\tilde{G}_{c} = -\frac{80-24s}{12s^3 + 52s^2 + 31s + 130}.
$$
Its poles are $-0.0085 \pm j1.5842$ and $-4.3163$ (I determined these using MATLAB), so the approximate closed-loop system is BIBO stable (you can use an algebraic criterion to verify this), but the Nyquist plot of the actual open-loop transfer function is

Indeed, for $\tau > 0.5758$, the closed-loop system becomes BIBO unstable, but it is not until $\tau>0.6189$ that you will be able to detect this using a $[1,1]$-order Padé approximation.
Here is some MATLAB code to determine these values:
xt = 0;
for td = 0.5:0.0001:0.65,
   G = tf(2, [1 1 1.25]);
   H = G*tf(1,1,'InputDelay', td);
   Hpade =  G*tf([-0.5*td 1], [0.5*td 1]);
   T = feedback(Hpade, 1);

   Gm = margin(H);
   x = (abs(isstable(T)- (Gm > 1)));
   if x~=xt,
      disp(td);
   end
   xt = x;
end

Some notes:

The Padé approximation of order $[1,1]$ is not too good. Use a higher order approximation (but not too high as this will lead to numerical problems)
The the best of my knowledge, you cannot use the Padé approximation to guarantee that the actual system is stable (at least not without additional requirements)
Don't use Padé approximations if you want to study the stability properties of your system. Use the Bode criterion (if applicable) or the Nyquist criterion (recommended).