Electronic – Astrom-Hagglund Relay PID tuning – relay transfer function

It's not a complete answer but I hope that it could be of some help.

You could rewrite the first system as

$$ \begin{cases} P(n) = K_P E(n) \\ I(n) = I(n-1) + \frac{K_P}{T_I} E(n) \Delta t \\ D(n) = K_P T_D \frac{E(n) - E(n-1)}{\Delta t} \end{cases} $$

Where \$E(n) = G(n) - target(n)\$ and \$\Delta t\$ is your sampling interval. Note that \$T_D\$ and \$T_I\$ are not defined as gains. \$K_I = \frac{K_P}{T_I}\$ and \$K_D = K_P T_I\$ are respectively the integral gain and the derivative gain.

Now you can rewrite the system as a single function of the error.

$$ PID(n) = P(n) + I(n) + D(n) $$

$$ I(n-1) = PID(n-1) - P(n-1) - D(n-1) \\ = PID(n-1) - K_P E(n-1) - K_P T_D \frac{E(n-1) - E(n-2)}{\Delta t} $$

$$ PID(n) = K_P E(n) + PID(n-1) - K_P E(n-1) - K_P T_D \frac{E(n-1) - E(n-2)}{\Delta t} + \frac{K_P}{T_I} E(n) \Delta t + K_P T_D \frac{E(n) - E(n-1)}{\Delta t} \\ = PID(n-1) + K_P \left(\left(1 + \frac{\Delta t}{T_I} + \frac{T_D}{\Delta t} \right)E(n) - \left(1 + 2\frac{T_D}{\Delta t} \right)E(n-1) + \frac{T_D}{\Delta t} E(n-2) \right) $$

The second one is a bit more complex to rewrite as a single equation but you can do it in a similar way. The result should be

$$ R(n) = K_1 R(n-1) - (\gamma K_0 + K_2) R(n-2) + (1+\gamma) (PID(n) - K_1 PID(n-1) + K_2 PID(n-2)) $$

Now you only need to substitute the equation of the PID in order to obtain the equation of the regulator as function of the error.

I wouldn't go so far as to call PID outdated. But there certainly is room for improvement. One way in which I have auto-tuned PID control loops is to use the Nelder-Mead method which is a form of hill climbing simplex algorithm. It has the benefit of being able to converge and reconverge on a target parameter that moves over time.

From this paper:

For example in our case of PID parameters tuning {K_P, K_I, K_D} a simplex is tetrahedron. Nelder–Mead generates a new test position of simplex by extrapolating the behavior of the objective function measured at each test point arranged as the simplex. The algorithm then chooses to replace one of these test points with the new test point and so the technique progresses.

My particular application was for motor control. We had two loops, a PID current control loop and a PI velocity control loop. We set our vertices to P, I, and D respectively and ran statistics on the output of the loop. We then ran the reflection, expansion, contraction, and reduction over and over again until the current or velocity control targets generated were within a few standard deviations.

With our product, the VP was very concerned with how the motor "sounded". And as it turned out, it "sounded" better when the current target bounced a bit more than was mathematically optimal. So, our tuning was done "live" in that we let the algorithm seek while the motor was running so that user's perception of the motor sound was also taken into account. After we found parameters that we liked, they were hard-coded and not changed.

This probably would not be ideal for you since you state, "putting the system in oscillation even as a part of auto-tuning is not acceptable to the users". Our system would most certainly oscillate and do other horrible things while it was auto-tuning.

However, you could run two copies of the PID controller. One that was "live" and actually controlling the process. And a second that was constantly being auto-tuned while being fed the same inputs as the "live" controller. When the output of the auto-tuned controller became "better" or more stable, you could swap the coefficients into the "live" controller. The controller would then perform corrections to the process until the desired performance was achieved. This would prevent oscillations that can be perceived by the user during auto-tuning. But if the inputs change drastically and the PID controller is no longer optimal, the auto-tuning can swap in new coefficients as they become available.