What is causing this long tail in the transient reponse?

Question

I have a fifth order transfer function for which I designed a controller using pole-zero cancellation technique on a root locus. I am targeting \$<5%\$ overshoot and \$<2s\$ settling time. Currently, the overshoot criteria are satisfied.

Note I know that exact p-z cancellation is nearly impossible in real life.

The controller and the original 5th order transfer function are shown in Simulink below

which give a response with a long tail in the transient response, and thus, a very long settling time.

As per Chu's comment here,

Placing zeros close to poles in an attempt to 'cancel' is not too clever. It's usually impossible to plonk a zero directly on top of a pole and expect both poles and zero to stay put. The result is a 'dipole' (a pole and zero in close proximity) that gives rise to a long-tail in the transient response.

and HermitianCrustacean's comment,

The 4th order controller you've chosen is difficult to numerically model ...

What would be the root cause of this unacceptably long settling time, inexact p-z cancellation (the controller which is difficult to numerically model), or both? Any suggestions on how to improve the response would also be greatly appreciated.

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Poles of 5th order system

   Poles =

   1.0e+02 *

  -9.9990 + 0.0000i
  -0.0004 + 0.0344i
  -0.0004 - 0.0344i
  -0.0002 + 0.0058i
  -0.0002 - 0.0058i

Zeros placed to cancel poles

4th order controller

I would be happy to provide further information if needed.

Answer 1

The slow oscillating behavior in the system results from a pole with real part close to zero and, by looking at your step response, with a frequency close to \$0.1 \; \text{Hz}\$ \$\; (0.62 \; \text{rad/s})\$. So the poles causing it are the ones at \$s_0 = -0.02+0.58i\$ , and \$s_1 = -0.02-0.58i\$.

You should check if they really have been cancelled, and if they have not, try using root locus and different gains to change the pole positions away from the complex axis (having real parts as negative as possible).

Answer 2

I think you need to check the residue corresponding to pole you want to cancel to check if pole-zero cancellation is valid. Residue is the constant multiplied to partial fraction term of this pole. For example, if $$F(s)=\frac{26.25(s+4)}{s(s+3.5)(s+6)}$$ the residue of the partial fraction term of \$s+3.5\$ pole is \$1\$ which cannot be neglected so \$s+3.5\$ and \$s+4\$ cannot cancel each other, and for $$F(s)=\frac{26.25(s+4)}{s(s+4.01)(s+6)}$$ the residue of the partial fraction term of \$s+4.01\$ pole is \$0.033\$ which can be neglected so \$s+4.04\$ and \$s+4\$ can cancel each other.

Reference: Example 4.10, page 195 in Control Systems Engineering, Norman S. Nise, 6th Edition, 2010, John Wiley.