In the context of a known disturbance $d(t)$ in a control loop, what is the $\Delta t$ at which the control loop has to be executed?

As an example, consider a P-T1-system with a PID-controller. First look only at the P-T1 system, set a $y_r$ and wait a long time - then we take a look on its output $x$ and see that it has still a disturbance $d$ which variates with time (see the plot, system output $= x$). In this model, the system output is, after you wait a long time, a constant plus $d(t)$.  The next step is to introduce a PID-controller: For this loop alone we could just use some experience-based technique like the Ziegler and Nichols procedure to adjust its parameters $K_p$, $K_i$ and $K_d$ optimally. If we switch to discrete control loop, because the controller is digital, we will have one additional parameter: The $\Delta t$ at which the controller operates.

What $\Delta t$ is required for the control loop to diminish the effects of $d$ on the system output? The trend will of course be the smaller $\Delta t$ the better, but is there a general rule for the maximum $\Delta t$?

• By 'in order to function' I guess you mean 'in order to eventually return to a steady state'. So you are asking what the time response (or responsiveness) of the controller should be, but the time response of the system must also be known to predict the overall dynamics of the system. My guess is that for the system in the chart, as long as the controller responds within 2000 units, it is likely to eventually reach a steady state. But I don't know of a general rule for estimating this responsiveness. Are you asking for such a general rule, and do you have a particular application in mind? – dcorking Jan 26 '15 at 14:56
• @dcorking Yes, if you mean that the system-ouput will stay, for this example, at 380 $\pm$ a tolerance. Im looking for a general rule. I thought it would be something like this: Calculate the highest rate of change in the uncontrolled system output. Use this highest rate of change to calculate a $\Delta t$. – John H. K. Jan 26 '15 at 16:49
• No I didn't mean within a tolerance of 380. If that is the case, then I think you have the hidden assumption that the disturbance goes away. If it does, then write that into your question. Hopefully someone with more knowledge of dynamic response will respond. (Perhaps that will be an expert in microfluidics, avionics, machine control or robotics.) – dcorking Jan 26 '15 at 17:29
• No, the a tolerance was a number which should be low compared to 380. The disturbance does not go away, it is always there. – John H. K. Jan 26 '15 at 17:32
• In general, the loop will not return to its setpoint in the presence of a disturbance. A P or PD controller, for example, will not. That is the purpose of the integrator in PID. So it may help to add something to the question that defines 'in order to function'. – dcorking Jan 26 '15 at 19:02

The choice of time step sets the bandwidth of the control loop. The highest unity gain frequency (UGF) you can hope to achieve in the closed loop is the Nyquist frequency $$f_N=\frac12 f_s=\frac{1}{2\,\Delta t}$$ where $\Delta t$ is the sample time. Practically, the UGF will be somewhat lower than this. This means that above this frequency your feedback will not be suppressing the disturbance fluctuations in your system.
The UGF also limits how much gain you can have at frequencies below but near the UGF. For frequencies within an order of magnitude of the UGF, $\text{UGF}/10$, you won't be able to have a gain much higher than $\sim10$. A gain of $10$ in the closed loop means that disturbance fluctuations at those frequencies are suppressed by a factor of 10.