The next step is to determine the constants *K _{p}, K_{i }*and

*K*

*. In order to do this we will use the first order plus dead time model (FOPDT) to determine the open loop response of the Piezo stage + Piezo amplifier and the latency of the Digital Accumeasure capacitance probe along with the latency of the CPU loop and the dead time in the Ethernet communications.*

_{d}## Determining PID Parameters

To derive the PID coefficients in (Fig 3 & 4), we first measure the FOPDT parameters θ,κ, and τ from an open-loop step response. The values obtained are then input to a set of correlation functions that return the proportional, integral and derivative values*K _{p}, K_{i }*and

*K*

*, respectively. The PID coefficients are further refined through several closed-loop tests aimed at moderate to aggressive control.*

_{d}### Open-Loop Test

The open-loop test consists of a single 30 micron step change that assumes a nominal 25 micron operating point. We perform this test for loop time and then again at . We do this for two reasons. The first reason is to determine how fast the loop time can be executed without introducing too much jitter. The second reason is to compare the time constant and dead time measured at the two loop rates. Higher loop rates correspond to higher sample rates, which in turn improve measurement accuracy, while lower loop rates allow more time to execute the loop statements, thus realizing lower jitter and more consistent response times.

The step input and corresponding system response for the two single step tests are shown in Figure 6 and Figure 7. The sample points are highlighted in the middle figures to emphasize the difference in resolution: approximately 12 points are available to compute the time constant from the higher rate data, while less than half that number are available for computing the time constant using the lower rate data.

Conversely, as expected, jitter performance is better at the lower loop rate, as shown by the bottom figures of **Figure 6** and **Figure 7.**

Table 1 compares the graphically computed FOPDT model parameters dead time , gain , and time constant obtained from Fig 6 and Fig 7. The results show strong agreement between the two loop rates for and . The difference of roughly for the time constant is on the order of the difference in loop times and is to be expected since this time is approximately the difference in -axis resolution between the two tests.

Using the FOPDT parameters calculated in Table 1, the PID values are determined from the following correlations:

The derived PI coefficients are shown in Table 2.

Figure 8 compares the closed-loop performance for the same test profile used in Figure 6 and Figure 7. All closed-loop tests were performed was *T* = 500. The top figure shows the result using moderate tuning values for *T* = 200 in Table 2. In this case, we have used a PI controller tuned for one loop rate (*T* = 200.) and applied it to another, slower rate (*T* = 500). The middle and bottom figures show the results for moderate and aggressive tuning values, respectively, for *T* = 500.

Comparing the results shown in Figure 8, the middle achieves a faster response time than the top, while the response time in the bottom is faster than the middle