Why does PID controller block output non-zero values even when the P, I, and D gains are set to 0, when using MATLAB R2023b?

I am trying to use a PID controller block. The P, I, and D gains are all set to 0, and the input to the block is finite (between -60 and 10 for the first ~60 seconds). The PID controller is set to discrete time with sample time of 10 seconds and the output of the controller is saturated between 5000 and 10000. The integrator and filter are initialized to 4000 each. Despite the gains being 0, the output is still non-zero and keeps on oscillating between 5000 and 10000. Why is the output non-zero despite zero gains?

Answer 1

[MathWorks Support Team]

When using the PID Controller block with all P, I, and D gains set to zero, the output can still be non-zero if the integrator and filter initial conditions are non-zero. This is because:

- With zero I gain, the integrator initial condition acts as a constant offset in the output.

- With zero D gain, the filter initial condition can cause the output to oscillate.

To ensure the output is zero when all gains are zero, set the integrator and filter initial conditions to zero.

For more information, see the block documentation for

PID controller

. 

[Sam Chak]

Hi @MathWorks Support Team

Thank you for this important observation and the explanation regarding the non-zero initial integrator state condition. However, I remain perplexed by the observation of non-zero control output despite the Integral gain I being zero, as the info shown in the PID controller documentation appears to be consistent with the mathematical axiom. Is it not an absolute mathematical truth that anything multiplied by zero will always equal zero?

If the Integral gain I is zero but the Integral action is non-zero, the only logical explanation is that the PID controller forms depicted in the documentation are incorrect! In technical terms, this implies that a Gain block is placed before the Integrator block, such that the non-zero error signal is pre-multiplied by the Integral gain, , before being integrated using a non-zero initial state value.

(A few moments later ...)

Upon reviewing the documentation further, I found that this is illustrated in the Main / Source section. However, it applies specifically to externally-supplied Integral gain. There is no mention of how the Integral gain is used in the internal case.

I stand up for the zero property of multiplication, which states that for any number (real, complex, etc.), when multiplied by zero, the result is always ZERO (). In other words, the integral control action should mathematically behave as follows:

, with the initial condition .

Hi @Paul, Do you have any opinion on this observation?

[Paul]

Hi Sam,

I think the documentation should be improved to clearly show how the initial condition on the integrator AND the derivative filter flow to the output. The documentation should also show how the derivative filter is realized. Having such information would not only clarify how the PID output behaves for the (pathological?) case where all the gains are zero, but, more importantly (IMO), would make it clear how to reset the integrator and the derivative filter for anti-windpup compensation.

[Sam Chak]

Hi @Paul.

Thank you for your inputs. YOu're right that the documentation should be improved to provide a detailed description of how to perform the reset for anti-windup. Under normal circumstances, we don't internally set the zeros for ​ and ​ if the integral and derivative actions are desired. However, this may occur when the PID gains are supplied externally and varied within the ranges specified by the gain schedule, causing some gains to be adjusted to zeros at certain operating conditions.

[Paul]

Hey Sam,

Some gains scheduled to zero at certain operating conditions? Maybe. I'd be curious to see an example of that. Maybe there's a case where the plant is transitioning from unstable to stable that causes one or two gains to change sign?

But all three gains scheduled to zero, as discussed in this thread, at any operating condition sounds problematic.

[Suyash]

Hi everyone, thank you for your comments and contributions to this post. As pointed out by Sam in the first comment, the integrator gain is applied on the input before the integration, for both externally supplied and internally supplied gains. Therefore, the initial condition is not subject to the integrator gain and therefore, the integration can result in a non-zero value even when the integrator gain is zero. Similarly, even when D gain is zero, the output can still be non-zero when the filter initial condition is a non-zero value.

The equation in the documentation here shows this, however, we will consider improving this to show how initial condition is incorporated.

Hope this helps! Let me know if you have any more questions.