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''Optimal PI and PID control of first-order plus delay processes and evaluation of the original and improved SIMC rules'' ,

Published in: J. Process Control, vol. 70 (2018), 36-46.

** **

The original SIMC-rules are presented in the following JPC-paper from 2003:

**Sigurd Skogestad.
''Simple analytic rules for model reduction and PID controller tuning'' (2003)
**Published in: J. Process Control, vol. 13 (2003), 291-309

The new paper (2018) shows that the original SIMC PI-rules are close to optimal, except for pure time delay processes. To rectify this, the 2018 JPC-paper proposes "improved" SIMC-rules, which basically boil down to adding taud=theta/3 (for the serial/cascade-form PID).

The following paper from the IFAC PID-conference in 2012 has some interesting formulas for the stability margins (GM, PM, DM, Ms) of the SIMC rules:

**Sigurd Skogestad and Chriss Grimholt.
''Optimal PI-Control and Verification of the SIMC Tuning Rule'' (2012)
**

**Sigurd Skogestad and Chriss Grimholt.
''The SIMC Method for Smooth PID Controller Tuning'' (2012)
**

Chapter 5 in: *
R. Vilanova, A. Visioli (eds.), PID Control in the Third Millennium,
Advances in Industrial Control, Springer, 2012 *

Maybe if you want to read a single paper on the SIMC PID-rules then this is the best.

Compared to the original JPC paper (2003), a method for closed-loop identification of the process (k, tau and theta) is presented ("Sham's method").
The choice of the tuning parameter tauc is discussed in more detail, and lower and upper limits are presented for tight and smooth tuning, respectively.
Finally, the optimality of the SIMC PI rules is studied by comparing the performance (IAE) versus robustness (Ms) trade-off with the Pareto-optimal curve.
The difference is small which leads to the conclusion that the SIMC rules are close to optimal for PI-control.

The following paper compares PI/PID with a Smithe Predictor controller. Should we forget the Smith Predictor?

** Chriss Grimholt and Sigurd Skogestad.
''Should we forget the Smith Predictor?'' (2018)
**

In 3rd IFAC conference on Advances in PID control, Ghent, Belgium,
9-11 May 2018. *
In IFAC papers Online (2018)*

The answer is "yes", we may as well forget the Smith Predictor, at least for first- or second-order plus delay processes. The main problem with the Smith Predictor is it's sensitivity to changes in the time delay. A tightly tuned Smith Predictor controller may go unstable both for postitive and negative time delay errors. This is well known, but in this paper we also demonstrate that the performance benefits compared to PID control are small or nonexistent even for the case with no time delay error.

The following paper considers the special case of a double integrating with time delay process:

** Chriss Grimholt and Sigurd Skogestad.
''Optimal PID control of double integrating processes'' (2016)
**

11th IFAC Symposium on Dynamics and Control of Process Systems,
including Biosystems June 6-8, 2016. NTNU, Trondheim, Norway. *
In IFAC papers Online (2016) *

In the Laplace domain the process can be written as g(s) = k'' exp(-theta*s)/s^2 where theta is the time delay and k'' is the high-frequency gain.

This is fundamentally a difficult process to control, so if we can measure some internal variable and use local feedback (cascade control) or state feedback then this is the best.
The paper consider the case when this is not possible, that is, we can only measure the output.
A PI-controller gives instability for a double integrating process, so we have to add D-action and use PID-control.

The paper derives IAE-optimal PID-controllers for a given robustness level (Ms) and the really surprising result is that the PID-tunings recommended in the original SIMC-paper from 2003 are essentially optimal (see Table 2).
For the series/cascade PID-controller the SIMC-rule is to select both the derivative and integral time equal to 4*(tauc+theta), where tauc is the tuning parameter.

The double integrating process may be considered as a special case of a second-order process with k''=k/(tau1*tau2).
Equation (7) in the paper generalizes the results from the 2003-paper into a single SIMC PID-rule that covers all second-order plus delay processes.