The equally spaced forward-difference formula with 2–7 sampling points.

## Abstract

Most of the existing differential methods focus on the differential effect and do not make full use of the differential link’s filtering effect of reducing order and smoothing. In Proportion Integral Differential (PID) control, the introduction of differential can improve the dynamic performance of the system. However, the actual differential (containing differential gain) will be subject to the impact of high-frequency noises. Therefore, this paper proposes a differential with filtering function, which has weak effect on noise amplification, and strong effect on reducing order and smoothing. Firstly, a discrete differentiator was constructed based on the Newton interpolation, and the concept of “algorithm bandwidth” was defined to ensure the differential effect. Then, the proposed algorithm was used to design a new PID controller with feedforward filtering function. In the experiments, the proposed PID controller is applied to a high-performance hot water supply system. The result shows that the system obtains better control quality. It verifies that the proposed PID controller has a feedforward filtering function and can effectively remove high-frequency noise.

### Keywords

- Newton interpolation
- algorithm bandwidth
- the discrete differentiator
- PID controller
- feedforward filtering

## 1. Introduction

As Proportion Integral Differential (PID) control is widely used in industrial control, engineering applications, and other fields [1, 2], a problem in Proportion Integral Differential (PID) control cannot be ignored: the introduction of differential signals can improve the dynamic characteristics of the system, but it is also easy to introduce high-frequency interference [3], the insufficiency of the differential term is especially obvious when the error perturbation is abruptly changed. The feedforward control can generate a compensation amount in advance according to the magnitude of the disturbance after the disturbance occurs and before the controlled variable has changed, thereby eliminating the influence of the disturbance on the controlled variable. Therefore, a filter or feedforward link is generally added to the PID control system to improve the system performance, but this makes the system structure more complicated.

The existing differentiation method focuses on the differential function of the differential link, but involves less on the filtering function of the differentiator. Although the differential link has the function of order reduction, harmonics will appear at the same time as the order reduction. If it is not smoothed, the control effect is not ideal when applied to the actual control system. Therefore, it is necessary to design a PID controller with filtering function that has weak effect on noise amplification and strong reduction and smoothing effect.

In the current methods of extracting input differential signals, wavelet [4] and neural network [5] rely on system models or may amplify noise. In recent years, sliding mode algorithms have been used to design differentiators or filters, but there is a problem of chattering elimination [6, 7, 8]. Literature [9] realized a dither-free sliding mode differentiator, which constructed discrete differential based on backward Euler, but the algorithm parameter adjustment is more complicated. Tracking differentiator (TD) can quickly track the input signal [3], but its more parameters increase the difficulty of application in practice [10, 11]. The structure of fractional differential is generally more complicated. For example, the fractional differential in literature [12] is implemented with a complex combination structure of low-pass, high-pass, and band-pass filter functions.

Numerical differentiation [13, 14, 15, 16] approximates the function value of the unknown objective function at certain points based on the information of the known finite discrete sampling points. According to the literature, numerical differentiation mainly includes: finite difference type [17], polynomial interpolation type [13, 15], regularization method [18, 19], and undetermined coefficients [20]. Among them, the commonly used method is finite difference, but its effect is not very ideal for the high-frequency noise existing in the measurement process [21].

In view of the above problems, due to the advantages of Newton interpolation that it is convenient to calculate a large number of interpolation points, this paper constructs a “discrete differentiator” based on the equidistant Newton interpolation polynomial [22]. The proposed differentiator can realize the effect of differential and filter, so it can replace the strategy of “controller and filter” in some application. Compared with the existing Gaussian filtering [23], Kalman filtering [23, 24, 25], and Wiener filtering [25] schemes, the proposed discrete differentiator has a simple structure, and it is easy to implement.

Then this article gives a simulation example, using the discrete differentiator as the differential link in PID control, applied to the hot water temperature supply system, and compared with the control effect of the feedforward and feedback compound control (using the traditional PID controller) strategy. The results show that the PID controller simplifies the structure of the system. Even if it does not have the feedforward link, it still has the feedforward filtering function. It overcomes the negative factors of the differential link in the traditional PID controller that amplify the noise and produce side effects. The system has obtained better control quality.

## 2. A numerical differential algorithm based on Newton’s interpolation

The distribution of sampling points for many problems in practical engineering is often equidistant. Eq. (1) shows that the algorithm based on the equally spaced forward-difference Newton’s interpolation polynomials only utilizes the previous sampling points. Therefore, the “on-line” control can be achieved. Namely, the differential of the point

Eq. (1) gives the Newton’s interpolation function based on equal interval, and we introduce variable

When

The numerical differential algorithm formulas involving two to seven sampling points can be obtained from Eq. (2). Table 1 lists the equally spaced forward-difference formula based on the first derivative of the Newton’s interpolation.

Points number | The forward-difference formula |
---|---|

2 | |

3 | |

4 | |

5 | |

6 | |

7 |

The closed-loop system’s bandwidth is determined by the cutoff frequency

The algorithm bandwidth is calculated by comparing the frequency characteristics of the proposed method and the ideal differential. Therefore, we can get two values of algorithm bandwidth, which are respectively in the sense of amplitude-frequency characteristics and phase-frequency characteristics.

For the convenience of description, the following contents of this paper will replace the numerical differential algorithm with the differentiator filter. For a differentiator filter when two sampling points are adopted, we will replace it with the two-point differentiator filter.

The standard for judging the effect of the proposed differentiator mainly considers the following three aspects:

Accuracy of differentiation in algorithm bandwidth;

Maximum transmission coefficient outside the range;

Easy implementation;

The most significant is the transmission coefficient of differentiator filter. When a differentiator filter is similar to an ideal differential in the high-frequency characteristic, it would amplify high-frequency noises. Nevertheless, this factor is undesirable when it occurs in practical applications.

In section III, we compared the proposed differentiator with the ideal differential and the actual differential by analyzing their frequency characteristics. Besides, the frequency characteristics of the above differentiators are calculated by Laplace Fourier transforms.

## 3. The discrete differentiator

### 3.1 The design of discrete differentiator

For the ideal differential, its transfer function and amplitude-frequency characteristics, and phase-frequency characteristics are in Eq. (4).

A differentiator filter can be constructed by Eq. (2) using a first derivative of the Newton’s interpolation polynomials. Eq. (5) gives the two-point differentiator:

Applying Laplace transform and Fourier transforms to Eq. (5), we can get transfer function of the two-point differentiator in Eq. (6).

Substituting the real part and the virtual part of Eq. (6) into Eq. (7) and the frequency characteristics of the two-point differentiator filter will be obtained:

The amplitude-frequency characteristics and the phase-frequency characteristics of the two-point differentiator filter are given in Eq. (8) and Eq. (9) by calculating Eq. (7). The calculation process is not difficult, so I will not explain it in detail here.

It should be noted that the amplitude-frequency characteristics of the proposed method and the ideal differential should be basically the same.

The algorithm bandwidth of the two-point differentiator filter is the frequency range when the frequency characteristics of the two-point differentiator and the ideal differential are consistent within a specific precision range. From now on, algorithm bandwidth also refers to this upper-frequency limit.

In terms of the amplitude-frequency characteristics, we can get the estimation of algorithm bandwidth in Eq. (10) through Eq. (8). That is the algorithm bandwidth of the two-point differentiator filter in the sense of amplitude-frequency characteristics.

The phase of ideal differential is

Similarly, the frequency characteristics and algorithm bandwidth of the

Points number | |
---|---|

2 | |

3 | |

4 | |

5 | |

6 | |

7 |

### 3.2 The frequency characteristics of discrete differentiator

Next, we will compare the proposed differentiator with ideal differential and the actual differential by analyzing their frequency characteristics. We give the simulation of the proposed method with various sampling points and the sample step

We give the simulation of the amplitude-frequency characteristics and phase-frequency characteristics of the differentiator filters and ideal differential in Figure 1. The differentiator filters respectively adopt 2, 3, 4 sample points and their sample step are all 0.1 second. Therefore, transfer functions of the two-discrete differentiator, the three-discrete differentiator and the four-discrete differentiator are

From the result, it can be seen that the cutoff frequency

We obtain

### 3.3 Influence of parameters the sampling step ∆ t and sampling points n on the filtering effect of the discrete differentiator

In order to study the Influence of parameter the sampling step

In Figure 2, the cutoff frequency

Figures 3 and 4 shows the Nyquist diagram of the differentiator filter with 2–7 points differentiator filter. We can infer that the high-frequency transmission coefficient increases as the number of sampling points in the algorithm increases. Furthermore, the two-point differentiator has the maximum error and minimum transmission coefficient in the high-frequency region. Oppositely, the seven-point differentiator has the minimum error in algorithm bandwidth and the maximum transmission coefficient in the high-frequency region.

Besides, in Figures 3 and 4, when the number of sampling points exceeds 4, it is observed that the root locus of the differentiator filter appears on the left side of the ** s** domain. The differentiator filter constructed at this time does not conform to the characteristics of the differential link. In addition, adopting too many sampling points will increase the algorithm complexity.

Therefore, the proposed differentiator takes up to four sampling points. It is not recommended to increase the number of sampling points for differentiator filters.

As the ideal differential could not be synthesized in practical scenes, the ideal differential does not exist. The actual differential generally has certain inertia so it can be realized in series with the ideal differential. Another essential feature of discrete differentiation is that the maximum transfer coefficient (gain) of the differentiator is subject to the bandwidth limit. Eq. (12) gives transfer function of an actual differential.

Eq. (13) gives the transfer function of two-point differentiator filters, three-point differentiator filters, and four-point differentiator filters:

While

For the two-point differentiator filter, its modulus can be got as the following relation when

Make Eq. (14) equal to Eq. (15), we can get Eq. (16):

Calculate the solution of Eq. (16) and get Eq. (17):

In the same way, we can find the relation between

In order to evaluate the effect of differentiator filters more effectively, the frequency characteristic of the differentiator filter is compared with the frequency characteristic of the actual differential link.

The differentiating time constant of the actual differential link is 0.25 s, which corresponds to the three-point differentiator filter when

It can be seen from the figure that the two-point discrete differentiator and the actual differential have the same maximum of 12 dB amplitude, so the proposed discrete differentiator can realize the effect of suppressing noise like the actual differentiator. The continuous simulation of the discrete algorithm in the high-frequency domain can be analyzed on the transfer coefficient of the differentiator filter so that the possible noise on the signal transmission path after the control device can be determined to a great extent.

With the improvement of microprocessor control equipment performance, the existing control equipment has high sampling frequency and high processing speed. Implementing discrete differential algorithms on these devices can achieve the same effect as continuous algorithms. When the differential link of the PID controller is realized by the differentiator filter and the continuous differentiator, the frequency characteristics of the two are close.

While constructing the differentiator filter

The standard deviation estimates of the input and output random signals of the actual and differentiator filters under different noise are given in Table 3. Where the sample step of the differentiator filter is 1 second (

Noise | Input | Actual differential | The proposed differentiator | ||
---|---|---|---|---|---|

2 | 3 | 4 | |||

Normal distribution signal | 0.98 | 1.99 | 1.249 | ||

3.79 | 2.29 | ||||

6.48 | 3.39 | ||||

White noise | 3.32 | 6.39 | 4.18 | ||

12.15 | 7.34 | ||||

19.06 | 10.72 | ||||

Uniform distribution signal | 1.81 | 3.51 | 2.286 | ||

6.52 | 4.47 | ||||

11.50 | 5.00 |

In Table 3, comparing with the actual differential, the output of the proposed method had smaller standard deviation under different noise. There are three common types of noise: normal distribution noise, white noise, and uniform distribution noise.

Normal distribution noise also calls Gaussian noise, which is a kind of noise whose probability density function obeys Gaussian distribution. Common Gaussian noise includes fluctuation noise, cosmic noise, thermal noise, shot noise, and so on. The three noises are very common, and it is convenient to quantify and evaluate the effect of the algorithm.

White noise is a random signal with constant power spectral density. In other words, the power spectral density of this signal is the same in each frequency band. Because white light is mixed with monochromatic light of various frequencies (colors), the property of this signal with flat power spectrum is called “white,” and this signal is also called white noise.

Uniformly distributed noise is a kind of noise whose probability density function obeys uniform distribution. Uniform distribution is one of the important distributions in probability and statistics.

We give the algorithm effect under these three noises in order to verify the adaptability and robustness of the algorithm. It shows that the algorithm can deal with them effectively for common noises and the differential accuracy of the differentiator filter is higher than that of the actual differential, and the differential effect is better.

In combination with the previous article, it is concluded that when taking the maximum allowable sample step

## 4. The experiment and result analysis

We apply the proposed method in the high-performance hot water supply systems. The system belongs to the closed-loop system, and the controlled variable is temperature. The system consists of the following parts: comparator, PID controller, three-position relay, actuator, heat exchanger, and temperature sensor. The transfer function and parameters of the high-performance hot water supply system are also given in Figure 6.

In Figure 6, the three-position relay is a relay with dead zone characteristics. The dead zone width is determined by the deviation of the actual water temperature from the water temperature setting value.

Then we should consider the method of adjusting the parameters of the PID controller. In the proposed algorithm, the differential link in the PID controller is realized by the differentiator filter and the actual differential. Next, the specific method of adjusting PID controller parameters is introduced.

To reduce the system’s sensitivity to high-frequency noise, the differentiator filter in the PID controller would adopt an increased sample step

For an actual differential whose transfer function is

After adopting the ideal differential in PID controller, a filter is added (

For the convenience of description, PID1 refers to the PID controller designed by the actual differentiator, and PID2 refers to the PID controller designed by the discrete differentiator with 2–4 points. Besides, the only difference between PID1 and PID2 is their differential link that PID1 adopted the actual differential link and PID2 adopted the proposed discrete differential link.

Table 4 shows the relative error of the modulus and phase of the system when the PID controller adopted the differentiator filter with 2–4 points and the actual differentiator at the cutoff frequency of the system, which are compared with the ideal differentiator.

The relative error | PID1 | PID2 | ||
---|---|---|---|---|

2 | 3 | 4 | ||

Modulus | 9.0% | 9.37% | ||

5.0% | 1.0% | |||

3.0% | 0.008% | |||

Phase | 7.0% | 7.0% | ||

4.0% | 1.0% | |||

2.0% | 0.001% |

Table 4 indicates that the phase-frequency characteristics of the proposed differentiators have the smaller relative error and slight distortion near the cutoff frequency, which illustrates that it can meet the system requirements for transient quality.

Then we give several simulations to analyze the effect of the proposed PID controller in the high-performance hot water supply system, where the system applied PID1 and PID2 respectively. Figure 7 shows the amplitude-frequency and phase-frequency characteristics of the PID controllers of the high-performance hot water supply system, where the differential link structure of the PID controllers is defined as follows:

PID1-PID controller with the actual differential link.

PID2-PID controller with the discrete differential with an increased sample step.

The PID controllers in above were applied in high-performance hot water supply system. The results of the system’s frequency characteristics were shown in Figure 8.

From Figures 7 and 8, it can be seen that the system with the differentiator filters has the more remarkable effect of high-frequency attenuation than the system with the actual differentiators near the frequency

Next, the PID control strategy with the proposed differentiator and the feedforward-feedback control strategy were adopted in the above high-performance hot water supply system. Figure 9 shows the simulation structure of the second strategy: feedforward-feedback control system. In which, the selected noise is a 2 KHz pulse signal with amplitude 5, and the PID controller is a traditional PID in the form of continuous time. The pulse signal was selected as system noise instead of the band-limited white noise because feedforward cannot overcome unmeasurable noise. Besides, the pulse signal can be regarded as sudden interference such as industrial interference caused by high-frequency electrical equipment and hot water consumption in the hot water supply system, which can represent the actual random system interference. In order to highlight the effectiveness of the proposed differentiator, the noise of the scheme in Figure 6 keeps up with feedforward-feedback control strategy.

As mentioned above, PID 1 refers to the PID controller with the actual differential and PID2 refers to the PID controller with the discrete differentiator. The comparison simulations of the PID control strategy with the proposed differentiator, the actual differentiator and the feedforward-feedback control strategies are given in Figure 10. PID1 adopted the three-point discrete differentiator with

It can be seen from the figure that the algorithm has less overshoot and oscillation compared with the feedforward-feedback control strategy and strategy 1, and the system finally reached a steady state of 50°C. The results proved that the PID realized by the discrete differential can achieve higher control quality than the feedforward-feedback compound control. This algorithm realizes the filtering effect of high-frequency noise in the high-performance hot water supply system even without the feedforward filtering link. Therefore, the structure of the high-performance hot water supply system could be simplified by introducing the proposed PID controller.

Figure 11 shows the effect diagram of the high-performance hot water supply system with a PID controller to adjust the hot water setting temperature step by step without external interference. Curve 1 corresponds to the system effect when the PID controller’s differential is an actual differential. Curve 2 corresponds to the system effect when the PID controller’s differential link adopts the three-point discrete differentiator with an increasing sample step with

In Figure 11, the temperature is between 56.99°C and 57°C, and the time is from 400 seconds to 1000 seconds. The amplitude error is within 5%, and the system is in a stable state that the water temperature can meet the demand of people’s life. Furthermore, the result indicates that the effect of the system with an actual differential in the PID controller is consistent with the system’s effect while introducing the two-point differentiator filter with an increased sample step in the PID controller. It shows that the proposed algorithm can ensure that the system’s transient response is not overshot.

Table 5 shows the standard deviation estimation of the input and output of the system under the different noise

Noise | Input | Actual differential | The proposed differentiator | ||
---|---|---|---|---|---|

2 | 3 | 4 | |||

Normal distribution Signal | 0.24 | 6.65 | 6.77 | ||

6.92 | 7.07 | ||||

7.11 | 6.94 | ||||

White noise | 0.77 | 7.2 | 7.16 | ||

8.31 | 8.14 | ||||

9.64 | 9.15 | ||||

Uniform distribution signal | 0.44 | 6.8 | 6.87 | ||

7.31 | 7.36 | ||||

7.86 | 7.55 |

The standard deviation estimation of the input and output of the continuous system (PID controller with actual differential) and the PID controller with the proposed differential is very similar. And when the four-point differentiator is adopted, the output standard deviation of the system is smaller than that of the system with actual differential. In addition, we have got the conclusion that the number of sampling points of the proposed method cannot exceed 4 in Figures 3 and 4. It shows that when the number of sampling points is 4, the differential error is the smallest and the control effect is the best.

The three-position relay with the dead zone characteristic can reduce the number of actuator actions and suppress the continuous small oscillations due to the quantization of the output of the PID controller and, at the same time, it simplifies the structure of the amplifier (heat exchanger) in the system.

However, the stability and control quality in such a nonlinear system depend on the relationship between

The description function (harmonic linearization coefficient) of the relay with dead zone characteristics is shown in Eq. (19):

In Eq. (19), _{.}

In the dead zone relay,

The dead zone width

Table 6 lists the number of actions of the control device under white noise interference. PID controllers in the system respectively adopt the proposed differentiator and the actual differential, the system with the former one has smaller operation times of equipment. And as the number of sampling points in the algorithm increases, the number of actions of the control equipment also increases.

Filter | |||||||||
---|---|---|---|---|---|---|---|---|---|

Number of the sampling points | 2 | 3 | 4 | 2 | 3 | 4 | 2 | 3 | 4 |

of control equipment 1 | 33 | 46 | 67 | 55 | 111 | 135 | 80 | 132 | 189 |

of control equipment 2 | 28 | 29 | 52 | 32 | 76 | 100 | 50 | 81 | 136 |

## 5. Conclusions

Aiming at the problem that differential link would introduce the high-frequency interference while improving the dynamic characteristics of the system, this paper proposes a numerical differential algorithm based on the equally-spaced Newton interpolation. Then, a discrete differentiator was constructed by and proposed the concept of “algorithm bandwidth” to ensure the accuracy of the effect of differential. Subsequently, the impact of the number of sampling points and the sampling step on the performance of the “discrete differentiator” was studied. It is concluded that the differentiator filter has the best differential accuracy and the strongest anti-noise ability for high-frequency noise when 2–4 sampling points and maximum allowable sample step are selected. Then, we designed a new PID controller based on a discrete differentiator. In order to verify the proposed PID controller’s effect, some numerical simulations were given to verify its filtering effect. The results demonstrated that the disturbance introduced by high-frequency noises could be suppressed without extra filtering operation. Finally, we given an actual case of the high-performance hot water supply system, which adopted the cascade control scheme, and the temperature controller of the system applied the proposed PID controller. The results show that the PID controller designed by this algorithm has a feedforward filtering function even if the control system did not have a complex feedforward link or add an additional filter to suppress the high frequency noise in the system. At the same time, the introduction of the three-position relay effectively decreases the number of actuator actions. Compared with the traditional PID controller, the proposed PID controller with the discrete differentiator can achieve the effect of signal differentiation faster with a smaller mean square error, that is, it reaches the effect of “differentiation and filtering.”

## Acknowledgments

We want to thank to Zheng Liu for kind suggestions in the early version of this manuscript. The authors gratefully acknowledge the financial support of the Key Fund of Natural Science Foundation of Shaanxi Province (2019JLZ-06) and Key Project of Ministry of Industry and Information Technology of the People’s Republic of China (2018-470).

## Conflict of interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. No potential conflict of interest was reported by the authors.

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