# A Novel Method for Control Performance Assessment with Fractional Order Signal Processing and Its Application to Semiconductor Manufacturing

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## Abstract

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## 1. Introduction

- Use MFDFA to analyze semiconductor data, derive the multifractal spectrum and select the characteristic parameters sensitive to changes of the control system.
- Extract the characteristic parameters from the multifractal spectrum of reference data to form reference feature sets;
- Modify the standard single Hurst exponent estimation by the multiple Hurst exponent fitting method with crossover points;
- Select multifractal properties and modified Hurst exponents to distinguish different types of control actions (tunings).

## 2. Preliminaries

#### 2.1. Fractal and Fractional Gaussian Noise

#### 2.2. Hurst Parameter

#### 2.3. $\alpha $ Stable Distribution

## 3. MFDFA Algorithm

#### 3.1. Basic MFDFA Algorithm

- Define the “profile” E and transform original data into mean-reduced cumulative sums,$${E}_{j}={\displaystyle \sum _{i=1}^{j}}({e}_{i}-\overline{e}),j=1,\cdots ,N,$$
- Divide time series ${E}_{j}$ into ${N}_{s}=\mathrm{int}(\mathit{N}/\mathit{s})$ non-overlapping segments of equal length s, starting from the beginning. Since the length N of the series is often not a multiple of the considered time scale s, in order to not miss any piece of data, another set of segments starting from the end of data is made from the end coming to the beginning. As a result, $2{N}_{s}$ segments are obtained covering the whole dataset.
- Calculate the local trend ${p}_{k}$ for each of the segments $k=1,\cdots ,2{N}_{s}$ by a least-square fit of the series.
- Calculate the mean square error ${F}^{2}(k,s)$ for the estimate of each segment k of length s.$${F}^{2}(k,s)=\frac{1}{s}{\displaystyle \sum _{i=1}^{s}}{(E[(k-1)s+i]-{p}_{k}\left[i\right])}^{2},$$$${F}^{2}(k,s)=\frac{1}{s}{\displaystyle \sum _{i=1}^{s}}{(E[N-(k-{N}_{s})s+i]-{p}_{k}\left[i\right])}^{2},$$
- Average over all segments to obtain the qth order variance (or fluctuation) function ${F}_{q}\left(s\right)$ for each size s:$${F}_{q}\left(s\right)={\left(\frac{1}{2{N}_{s}}{\displaystyle \sum _{k=1}^{2{N}_{s}}}{\left[{F}^{2}(k,s)\right]}^{q/2}\right)}^{1/q}.$$For $q=0$ use$${F}_{0}\left(s\right)=\mathrm{exp}\left\{\frac{1}{4{N}_{s}}{\displaystyle \sum _{k=1}^{2{N}_{s}}}\mathrm{ln}\left({F}^{2}(k,s)\right)\right\},$$
- Repeat steps (2)–(5) for different s evaluating new sets of variances ${F}_{q}\left(s\right)$.
- Plot ${F}_{q}\left(s\right)$ for each q in log-log scale and estimate the linear fit with least squares. If slope $h\left(q\right)$ varies with q, multifractality is suspected. Single slope shows monofractal scaling.
- Calculate multifractal exponent $t\left(q\right)$ as$$t\left(q\right)=qh\left(q\right)-1.$$
- Use Legendre transform to evaluate Hölder exponent $\alpha \left(q\right)$ and multifractal spectrum $f\left(\alpha \right)$:$$\left\{\begin{array}{c}\alpha \left(q\right)=H\left(q\right)+q{H}^{\prime}\left(q\right)\\ f\left(\alpha \right)=q\left[\alpha \right(q)-H(q\left)\right]+1\end{array}\right.$$

#### 3.2. Defining the Source of the Multifractality

#### 3.3. Plot Fitting Hurst Exponents with Crossovers

## 4. Case Studies

`Var1`,

`Var2`,

`Var3`to represent the error signals (${r}_{1,2,3}-{y}_{1,2,3}$ ) from each loop depicted in Figure 3.

#### 4.1. Non-Gaussian Statistical Analysis

`Var1`,

`Var2`,

`Var3`with a probabilistic distribution function (PDF) fitting is frequently carried out to detect if the data are impacted by non-Gaussian noises or subject to human interventions. It is often used to measure the heavy-tailedness underlying process during assessment and frequently the a suitable moment to call for further insight. Histogram fitting is significant since some parts of the process can be more impacted by the disturbances or can be more exposed to the process nonstationarity, can be cross-correlated or might be subject to human interventions. In real situations it is a challenge to have similar properties for all the loops.

`Var1`is the major concern in the current MIMO system. As introduced in Section 2.3, the $\alpha $-stable distribution is performed to detect whether the distributions are Gaussian, Lévy or else. In Figure 4, time series

`Var1`of the MIMO system has significant fat tail properties. Therefore, it can be inferred that there are non-Gaussian noises coming to the system.

`Var2`is the variable of errors from flow loop. The higher flow rate can exchange heat efficiently whereas more turbulence could be introduced at the same time. Therefore, the flow loop is interacted with the temperature loop. From the Figure 4 and Figure 5, we could see the clear correlations between

`Var1`and

`Var2`.

`Var3`explicitly shows very poor or even not working control. The histogram is highly scattered, almost uniform. The loop is uncontrolled with possible oscillations and operating on the edge of the stability region or manually operated.

#### 4.2. Hurst Exponents Fitting with Crossovers

`Var1`and

`Var2`are observed in Figure 6 and Figure 7. Therefore, the variation of the memory scales with crossovers cannot be captured by the conventional Hurst R/S fitting method with the constant value. After processing data with MFDFA, the multiple Hurst exponents, along with crossover point position might carry on information of multifractality of the control performance.

`Var3`is the variable of errors from coolant level loop, which in reality to keep the coolant away from the alarm levels. It should be noted that the control priority is comparatively lower in the design of the MIMO system. Compared with

`Var1, Var2`, however, the monofractal behavior of

`Var3`can be validated through Figure 8, Figure 9 and Figure 10.

`Var1`and

`Var2`show multi-persistence behaviour with different Hurst exponents in different memory scales. In contrary

`Var3`is clearly mono-persistent. It means that this loop is rather not coupled with the other ones. Its Hurst exponent is approximately equal to 1. Its tuning is highly sluggish, almost not existent. It confirms initial observation done using the histogram. The other two loops are the most probably close coupled. Observation of the R/S plots and the shortest Hurst exponent also reveals dynamically sluggish control, however in the longer memory scale control improves and is back to the neutral one.

#### 4.3. Multifractal Analysis

`Var1, Var2, Var3`can remove correlations from data and any remaining scaling is caused by probability density function broadness. It is shown that the multifractal behaviors can be captured by the multifractal spectrum analysis with the shuffled datasets in Figure 11.

`Var3`, while the wider arcs of

`Var1, Var2`shows the multifractality in MIMO system. Moreover, the multifractality of the MIMO process have been detected and analyzed for the each loop with shuffled datasets

`Var1, Var2, Var3`.

## 5. Results and Discussion

`Var1, Var2, Var3`and

`Var1’, Var2’, Var3’`with different PID parameters are compared in the end of article, in which the former settings are conservative and latter ones are more aggressive. In Table 1, the proposed FOSP method has been conducted according to the previous sections. The aggressive tuning with small $\alpha $ depicts the broadness and fat tails of the distribution of the underlying process. Moreover, the absolute values of $\beta $, $\gamma $, $\delta $ indicate the worse behavior with the latter tuning.

`Var2`is slightly more sluggish. Loop

`Var3`is unchanged. However, the effect of tuning (whatever the set is) is evident in comparison with original MIMO system presented in R/S plots (Figure 6, Figure 7 and Figure 8).

`Var1, Var2`require disturbance decoupling compared with aggressive tuning while

`Var3`requires possible redesign of the control philosophy and fine tuning.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Fractional Gaussian noise and fractional Brown noise in monofractal analysis and multifractal analysis.

**Figure 3.**Output errors of three loops:

`Var1`,

`Var2`,

`Var3`with first tuning parameters in blue;

`Var1’`,

`Var2’`,

`Var3’`with second tuning parameters in red.

**Figure 10.**Multifractal spectrum for shuffled

`Var3`. The parallel behavior with different scales (

**left plot**) and the tiny arch (

**right plot**) indicate the monofractal behaviors of

`Var3`in the MIMO system.

**Figure 11.**Multifractal spectrum for shuffled

`Var1`. The non-parallel behavior with different scales (

**left plot**) and the arch with large width (

**right plot**) indicate the multifractal behaviors of

`Var1`in the MIMO system.

**Figure 12.**Multifractal spectrum for shuffled

`Var2`. The non-parallel behavior with different scales (

**left plot**) and the arch with large width (

**right plot**) indicate the multifractal behaviors of

`Var2`in the MIMO system.

Variables | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\delta}$ | ${\mathit{h}}_{\mathit{q}}(-5)$ | ${\mathit{h}}_{\mathit{q}}\left(5\right)$ | $\mathbf{\Delta}\left(\mathit{\alpha}\right)$ |
---|---|---|---|---|---|---|---|

Var1 | 1.241 | 0.074 | 0.411 | 0.101 | 1.07 | 0.21 | 0.881 |

Var2 | 1.634 | 0.056 | 0.025 | −0.030 | 0.82 | 0.06 | 0.757 |

Var3 | 2.000 | −1.000 | 0.456 | −0.235 | 0.58 | 0.46 | 0.127 |

Var1’ | 1.171 | −0.156 | 0.456 | −0.235 | 1.04 | 0.25 | 0.798 |

Var2’ | 1.695 | 0.287 | 0.123 | −0.246 | 0.71 | 0.41 | 0.307 |

Var3’ | 2.000 | 1.000 | 1.567 | −0.208 | 0.57 | 0.53 | 0.069 |

Variables | ${\mathit{H}}_{1}$ | ${\mathit{H}}_{1},{\mathit{H}}_{2}$ | ${\mathit{H}}_{1},{\mathit{H}}_{2},{\mathit{H}}_{3}$ |
---|---|---|---|

Var1 | 0.682 | 0.913, 0.450 | 0.991, 0.671, 0.392 |

Var2 | 0.714 | 0.926, 0.500 | 0.964, 0.728, 0.436 |

Var3 | 1.010 | 1.044, 0.974 | 1.041, 1.035, 0.930 |

Var1’ | 0.622 | 0.827, 0.416 | 0.905, 0.560, 0.415 |

Var2’ | 0.824 | 1.099, 0.546 | 1.062, 0.951, 0.350 |

Var3’ | 0.968 | 1.057, 0.878 | 1.006, 1.066, 0.751 |

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**MDPI and ACS Style**

Liu, K.; Chen, Y.; Domański, P.D.; Zhang, X.
A Novel Method for Control Performance Assessment with Fractional Order Signal Processing and Its Application to Semiconductor Manufacturing. *Algorithms* **2018**, *11*, 90.
https://doi.org/10.3390/a11070090

**AMA Style**

Liu K, Chen Y, Domański PD, Zhang X.
A Novel Method for Control Performance Assessment with Fractional Order Signal Processing and Its Application to Semiconductor Manufacturing. *Algorithms*. 2018; 11(7):90.
https://doi.org/10.3390/a11070090

**Chicago/Turabian Style**

Liu, Kai, YangQuan Chen, Paweł D. Domański, and Xi Zhang.
2018. "A Novel Method for Control Performance Assessment with Fractional Order Signal Processing and Its Application to Semiconductor Manufacturing" *Algorithms* 11, no. 7: 90.
https://doi.org/10.3390/a11070090