Frequency analysis of a discrete-time fast nonlinear tracking differentiator algorithm based on isochronic region method

1.   Introduction

Real-time differentiation estimation of measured signals is critical and challenging in signal processing, fault diagnosis, fault control, and many other fields ^[1,2,3,4]. In practice, engineers often use the numerical difference method to directly approximate the differential of a measurement signal. When the sampling period $T$ is reduced, the approximation accuracy of the differential signal tends to increase. However, differential operations inherently capture the rate of change of a signal, thus in the presence of noise, minor fluctuations in the noise can be greatly magnified through these operations. This magnification can lead to distortions in the differential results, potentially compromising the accuracy and reliability of the overall system ^[5]. Specifically when it comes to proportional-integral-derivative (PID) control, a method that has been extensively employed since the previous century, engineers frequently resort to directly utilizing the numerical difference method to acquire differential information. However, this approach is highly susceptible to signal noise, often leading to distortions in the differential results ^[6]. In the absence of a superior differentiator, PID control frequently degenerates into PI control in numerous instances, resulting in a significant decline in the feedback control effect. This underscores the need for fast and accurate differentiation techniques to ensure optimal performance in PID control systems ^[7].

To deal with this, researchers have proposed many different approaches for differentiator design such as the high-gain observer via differentiator ^[8,9], the robust exact differentiation based sliding mode ^[10,11], the robust differentiator based extended Kalman filter ^[12], adaptive tracking differentiator^[13,14], and so on ^[15,16,17]. Among them, the linear high-gain differentiator, designed by Kahlil, provides derivatives of signals from 1 to n-1, but each derivative may contain disturbances, leading to poor anti-interference ability ^[18]. Linear tracking differentiator (TD) converges faster when the system state is far from the equilibrium point, but slower as it approaches ^[19]. Levant introduced a second-order sliding mode TD, which exhibits strong robustness and high precision. However, the rapid switching of system variables with small amplitudes near the sliding mode surface can easily lead to chattering ^[20]. Wang and Zhang developed a hybrid TD, integrating linear and nonlinear elements, which exhibits rapid convergence and superior noise suppression, making it suitable for aircraft speed detection ^[21,22].

Furthermore, Han initially proposed a nonlinear TD that utilizes the method of time optimal control (TOC), commonly referred to as $fhan$-TD ^[1,23]. The TOC law of $fhan$-TD is not strictly a bang-bang control as it incorporates a linear region to facilitate a smoother transition to the extreme value ^[24]. Equipped with this unique characteristic, $fhan$-TD not only efficiently tracks the output signal with minimal delay but also effectively retrieves the differentiation from signals contaminated by noise and discontinuities. In practical applications, $fhan$-TD, along with an extended state observer and transient profile generator, forms an automatic disturbance rejection controller (ADRC). This controller has been successfully employed in various scenarios, including speed control of permanent-magnet synchronous motors ^[25,26] and induction motor drives ^[9,27], vehicle suspension displacement detection ^[28], and improved Tornambe controller for pneumatic muscle actuators ^[29].

In the TD algorithms, the control synthesis function (CSF) plays a crucial role in achieving fast speed, low delay, and high accuracy. Han ^[1,23] derived the general form of the control synthesis function of $fhan$-TD by the state feedback method. Xie et al. ^[30,31] constructed a simple linear function, referred to as $fast$-TD, by examining the boundary characteristic points of the linear area, where the control law takes non-extreme values. It should be noted that the $fast$-TD algorithm avoids the complex root operation. Zhang et al. ^[32] chose the appropriate control quantity based on the initial point position, determined the corresponding boundary and characteristic curves, and further derived a linearization criterion to introduce a TD featuring a straightforward structure.

An improved CSF can enable the differentiator to achieve better signal tracking and differential extraction results. From the perspective of the CSF, this paper introduces a novel fast nonlinear tracking differentiator (FNTD) with the hyperbolic tangent function. This TD employs a state feedback approach to determine the boundary curve of the linear region, the control characteristic curve, and the control laws governing both the reachable area and the linear area.

This paper is organized as follows: Section 2 introduces the background of the TOC problem, providing a foundation for the subsequent discussions. In Section 3, the derivation of the FNTD algorithm is presented, utilizing the isochronic region (IR) method. Section 4 offers a practical rule for adjusting the parameters of FNTD based on frequency-sweep testing. In Section 5, a comparative analysis is conducted to demonstrate the superiority of FNTD in terms of convergence speed, tracking performance, differentiation accuracy, and filtering capabilities when compared to other tracking differentiators. Finally, Section 6 summarizes the key findings and contributions of this paper.

2.   TOC problem description

In this section, we review some fundamental concepts related to the TOC problem. This is crucial for understanding the theoretical foundation of the tracking differentiator algorithms that will be discussed later in the paper.

The first-order high-pass filter, often employed as a classical differentiator in servo control systems, has a transfer function that characterizes its response to input signals. The output of the differentiator can be expressed as

where $T$ represents a relatively small time constant, $v$ denotes the input to the differential system, $y$ is the system output, and $w(s)$ represents the transfer function. Equation (2.1) can be rewritten as

Define $\frac{v}{Ts+1}$ as the differential signal $v'(t)$, then

According to the definition of derivative, we have

From (2.3) and (2.4), we have

and when the time constant $T$ is smaller, the output signal $y$ gets closer to the differential signal $v^{'}(t)$.

Assuming that the measured signal is only coupled with the random noise signal $r(t)$, from (2.5) we have

which satisfies the following equation:

The solution to the above differential equation is

From (2.8), the noise in the signal is amplified by $1/T$. In order to eliminate or reduce the noise amplification effect, the above equation can be rewritten as

The time constants of the two inertial links are denoted by $\tau_1$ and $\tau_2$. The transfer function corresponding to the above equation is

The equivalent state space realization of (2.3) is

where $x_1$ and $x_2$ are the state variables and $y$ is the output variable. Equation (2.11) can be rewritten as the following general form ^[1]:

When the above differential equation has a solution, it is guaranteed that $x_1$ converges to $v(t)$ and $x_2$ converges to $\dot{v}(t)$, then (2.12) constitutes a differentiator. It was the desire to find a differentiator that gave birth to the time-optimal control based tracking differentiator as presented in the following.

The double-integral system is defined as

where $u$ is the control signal and $r$ is a constant constraint of the control input.

The task of finding a control law that steers any initial state $[x_1(0), x_2(0)]$ of a system (2.13) to the origin in the shortest possible time is known as the TOC problem. Solving this problem involves determining the optimal control strategy that minimizes the time required for the system to reach its destination.

The solution to this problem is ^[33,34,35]

where the switching function $s$ is

Substitute $x_1(t)$ with $x_1(t)-v(t)$ and $x_2(t)$ with $x_2(t)-\dot{v}(t)$, respectively, in (2.15), which is easily extended to a tracking problem as follows:

where $v(t)$ and $\dot{v}(t)$ are the desired state trajectory and its derivative. Note that the conditions $x(t) = 0\mathrm{\; and\; }x_1(t)+\frac{x_2(t)|x_2(t)|}{2r} = 0$ are hard to achieve in the noisy environment.

Remark 2.1. From (2.8), the noise in the original signal could be amplified by 1/T. To reduce the noise amplification effect, it is proposed here that a tracking differentiator can be designed which guarantees that $x_1$ converges to $v(t)$ and $x_2$ converges to $\dot{v}(t)$. The issue of extracting the differential information of real signals is first transformed into designing a rapid TD problem, then it is reduced to solving the problem of the TOC issue in the shortest time.

3.   FNTD algorithm design

Given that the majority of control algorithms are nowadays implemented in the discrete-time domain, there is a desire to identify the TOC law for a discrete-time double-integrator plant, aiming to construct a TD. This problem, referred to as the discrete-time optimal control (DTOC) problem, was first proposed by Han ^[1]. However, directly discretizing the solution from continuous TOC can be problematic due to high-frequency chattering in the control signals.

In this section, the construction of a FNTD based on the hyperbolic tangent function is proposed by the IR method. The FNTD algorithm is a discrete time closed-form solution to the TOC problem.

For a discrete-time double-integral system

where $A = \left[\begin{matrix}1&h\\0 & 1\\\end{matrix}\right]$, $B = \left[\begin{matrix}0\\h\\\end{matrix}\right]$, and $h$ is the sampling period, $x(k) = [x_1(k), x_2(k)]^T$.

Definition 3.1. (DTOC Problem). For the system (3.1) and its initial state $x(0)$, there exists at least one control sequence $u(k^*)$, $\left|u(k)\right|\leq r$, $k \leq N$, such that $k^* = \min\left\{k \middle| x(k) = 0\right\}$, where $N$ is a finite number.

Definition 3.2. (IR $G(k)$). For any $x(0)\in G(k)$, $\exists u(0), u(1), \dots, u(k)$ makes the solution of (3.1) satisfy $x(k) = 0$.

Remark 3.1. The whole phase plane is divided into two regions $G(k)$ and $R^2-G(k)$.

Proof. For a DTOC problem, we need to find the control sequence $u(0), u(1), \dots, u(k)$ that makes $x(k) = 0$ for any $x(0)$. The proof is divided into two parts:

(ⅰ) Determine $G(k)$.

(ⅱ) $\forall x(0)\in G(k)$ and $\forall x(0)\notin G(k)$; find the function of $u(k)$ that satisfies $x(k) = 0$.

To begin, examine $G(k)$. For (3.1), we have the iteration formula of $x(k)$

For $x(k)$ = 0, we can solve $x(0)$

With $A^{-k} = \begin{bmatrix} 1 & -kh \\ 0 & 1 \end{bmatrix}$, (3.3) can be rewritten as

So, the IR $G(k)$ is

Case 1: When $x(0) \in G(1)$. From (3.5), we have

that is, $G(1)$ is a line that satisfies $x_1(0)+hx_2(0) = 0, \lvert x_2(0)\rvert \leq hr$. As shown in Figure 1, $G(1)$ is a line connecting $a_1$ and $a_{-1}$.

When $x(0) \in G(1)$, the time optimal control law is ^[24]

Case 2: When $x(0) \in \{ x \mid (x \notin G(1)) \cap (x_1+hx_2 = 0) \}$

In this region, x(0) is still on the same line of $G(1)$ but goes beyond points $a_1$, $a_{-1}$. The corresponding control signal is ^[24] $u(0) = -r\text{sign}(x_2(0)-hr)$ until $x(0)$ enters $G(1)$. Here, using the hyperbolic tangent function to transition to extreme value ^[36], we have

where $\alpha$ is a constant; when $\alpha$ is larger, the hyperbolic tangent function is closer to the sign function; and $u$ approaches the extreme value faster. Additionally, from (3.8), when x(0) is closer to the boundary curve, $u$ will decrease.

Case 3: When $x(0) \in G(2)$.

From (3.5), we have

as shown in Figure 1.

Let

Then, the boundaries of $G_2$ are the following four straight lines:

When $x(0) \in G(2)$, the TOC law is ^[24]

Case 4: When $x(0) \in \{ x \mid (x \notin G(2)) \cap (|y| \leq h^2r) \}$

In this region, x(0) is not $G(2)$, but it is still on the area surrounded by two parallel lines $a_{2}b_{-2}$, $a_{-2}b_{2}$. Here, $u$ uses the hyperbolic tangent function to transition to the extreme value, that is,

Case 5: When $x(0) \in \{ x \mid |y| \geq h^2r \}$

In this region, $x_0$ is located outside the parallel line $a_{2}b_{-2}$, $a_{-2}b_{2}$.

From (3.5), the broken line $a_k$, $a_{-k}$, $b_k$, $b_{-k}$ is defined as

Here, $\Gamma_0^+$ is defined as the boundary curve formed by connecting …, $a_k$, $a_{k-1}$, …, $a_2$, $b_2$, …, $b_{k-1}$, $b_k$, …, and $\Gamma_0^-$ is defined as the boundary curve formed by connecting …, $b_{-k}$, $b_{-(k-1)}$, …, $b_{-2}$, $a_{-2}$, …, $a_{-(k-1)}$, $a_{-k}$, …. Zhang^[28] rewrote the function of $\Gamma_0^+$ and $\Gamma_0^-$ together as $\Gamma^0$ as follows:

where $\Gamma^0 = -hr$ is the curve $\Gamma_0^-$, and $\Gamma^0 = hr$ is the curve $\Gamma_0^+$.

Using the hyperbolic tangent function to substitute the sign function, we have

The boundary curves $\Gamma_0$ and $\Gamma$ are shown in Figure 2.

Note that the region surrounded by the boundary curve $\Gamma$ is misaligned with the $G(2)$, but (3.19) is valid on the region $|y| \geq h^2r$. So, the phase plane $|y| \geq h^2r$ is divided into the following two parts:

According to ^[24], when $x(0) \in \{ x \mid |y| \geq h^2r \}$, the control signal is

Combining (3.12), (3.13), and (3.21), we redefine $\Gamma(x_1, x_2, r, h)$ as

The characteristic curves $G(2)$ and $\Gamma$ of the FNTD algorithm are shown in Figure 3. The complete FNTD algorithm based on the hyperbolic tangent function coded in a digital computer is as follows:

Changing two adjustable parameters $\alpha$ in (3.23) into independent new variables $\alpha_1, \alpha_2$, we rewrite the FNTD algorithm in a new form,

where $x_1$ is the desired trajectory, $x_2$ is its derivative, $h$ is the filtering factor, and $r$ is the speed factor ^[1,23].

Remark 3.2. The tunable parameter $\alpha_1$ primarily influences the shape of the boundary curve, while $\alpha_1, \alpha_2$ jointly modify the control law, thereby altering the tracking speed of the signal.

4.   Frequency-domain characteristics analysis

Given a reference signal sequence $v(k), k = 0, 1, 2, …$, the FNTD algorithm based on (3.24) can be constructed as

The objective of this section is to show how damping factors $\alpha_1$, $\alpha_2$ affect the outputs of TD in the frequency domain. For different $\alpha_1$ and $\alpha_2$, we use the method of the frequency-sweep test to get the Bode diagram and analyze frequency domain characteristics of FNTD.

Here, to test the frequency domain characteristics of FNTD under ideal conditions, the sinusoidal signal input does not contain any random noise. $T = 0.001, h = 0.001, r = 100$, and five groups of $\alpha_1, \alpha_2$ are set as follows: $[\alpha_1, \alpha_2] = [1, 1], [2, 1], [100, 1], [1, 10], [1,100]$. The input signal is $v(t) = \text{sin}(2\pi ft)$, where $f$ ranges from 0.1Hz to 10Hz with a time step of 0.01 Hz. For each frequency point, we run 20,000 samples and record the data in the sampling interval [10001, 15000]. The simulation test results of the amplitude-frequency and phase-frequency curves of the proposed TD are shown in Figures 4 and 5.

From Figure 4, the amplitude-frequency characteristic of the tracking signal approximates an ideal first-order low-pass filter. When $\alpha_2 = 1$ and $\alpha_1$ increases from 1 to 100, the cutoff frequency also increases. The FNTD can track the higher frequency of the input signal, and the corresponding amplitude-frequency characteristic moves right. However, after exceeding the cutoff frequency, the phase of the input signal quickly decreases to -180°. When $\alpha_1 = 1$, altering $\alpha_2$ has little effect on the amplitude-frequency characteristic curve.

In Figure 5, the phase-frequency characteristic of the differential signal approximates a band-pass filter. Before the turning frequency, the phase almost remains 90° ahead. However, the phase quickly decreases to -90° near the cutoff frequency. When $\alpha_2 = 1$ and $\alpha_1$ increases from 1 to 100, the filtering ability of differential signals against high-frequency noise is enhanced.

Remark 4.1. From Figures 4 and 5, a bigger $\alpha_1$ enables the improvement of the quality of amplitude-frequency characteristics, and we can see that the boundary curve has a great impact on the speed of the signal tracking combined (3.24). Considering the properties of tanh($\alpha y$), as $\alpha$ increases, tanh($\alpha y$) approaches 1 from 0 more quickly.

We suggest increasing the parameters $\alpha_1$ and $\alpha_2$ as much as possible to achieve a better control law as shown below:

This approach will yield improved simulation results when utilizing the proposed algorithm.

5.   Simulation analysis

To verify the effectiveness of the proposed FNTD, in this section, we present simulations comparing FNTD (denoted as $TD_0$) with the following three differentiators in terms of convergence and filtering characteristics.

$TD_1$: The classical differentiator ^[37]

where $\tau_1$ and $\tau_2$ are tuning parameters.

$TD_2$: The nonlinear differentiator ^[12]

where the parameters $R, a_0, a_1, b_0, b_1 > 0, m, n > 0$ are tunable odd numbers.

$TD_3$: Robust exact differentiator using the sliding-mode technique ^[20]

where $\alpha$ and $\beta$ are tunable parameters.

We use the Euler method to discretize the state space. In all simulations, we choose the initial state $[x_1(0), x_2(0)] = [-1, 1]$ and the sampling period $T = 0.01$. All parameters are determined by the trial and error method, which are presented in Table 1.

The state trajectories of four differentiators are shown in Figure 6. Additionally, we calculate the convergence performance indicators of the four differentiators, as shown in Tables 2 and 3. Compared with other differentiators, the system with FNTD has the shortest adjusting time and smallest overshoot, which optimally drives the initial condition to the origin.

In practical engineering, $TD_1$ and $TD_3$ are commonly used, which are compared with $TD_0$(FNTD) to verify the filtering performance of the proposed TD algorithm. We choose the input signal $v(t)$ = $awgn$($\text{sin}(t)$, SNR), where the function $awgn$ (additive white Gaussian noise) in Matlab adds white Gaussian noise to the signal. Suppose the initial state is $[x_1(0), x_2(0)] = [0, 0]$ and the sampling period is $T = 0.01$. All parameters of three kinds of TDs are shown in Table 4. To ensure that the differential signal has sufficient noise suppression performance, the simulation results with different signal-to-noise ratio (SNR) of Gaussian noise are shown in Figures 7 and 8.

When SNR = 100 dB, the transient and steady processes are shown in Figure 7. From Figure 7, it can be seen that all algorithms have a certain level of noise suppression ability, but compared to $TD_0$ and $TD_1$, $TD_3$ exhibits a more violent oscillation response. In Figure 7(a), FNTD exhibits the fastest response speed and the smallest overshoot among the three TD algorithms. $TD_1$ has superior speed, but with the greatest overshoot. On the other hand, while $TD_3$ displays smaller overshoot, its response speed is the slowest, as shown in Table 5. During the steady process in Figure 7(b), $TD_0$ has the shortest tracking phase delay compared with $TD_1$ and $D_3$.

From Figure 8(a), when SNR = $100t$ dB, during the transient process phase, $TD_3$ has the smallest overshoot and chattering; $TD_0$ is the next, and $TD_1$ has the largest. In Figure 8(b), $TD_0$ is most accurate in tracking and differentiation compared with $TD_1$ and $TD_3$ in steady process.

Remark 5.1. In Figure 8, a performance comparison simulation experiment of the three differentiators has been conducted by gradually increasing the SNR. (1) Under low SNR conditions. When SNR $\leq$ 35 dB, as shown in Figure 8(a), $TD_0$ and $TD_1$ exhibit overshoot and long transient time, using the same parameters as those for SNR = 100 dB during the transient process. (2) Under high SNR conditions. From Figure 8(b), it is evident that the three TD algorithms all demonstrate parameter adaptability in high SNR noise, as seen in Figure 7(b), but among three TD algorithms, $TD_0$ is the smoothest and most accurate in capturing differential signals, making it more suitable for capturing actual differential signals in higher SNR noise environments.

In summary, through the performance comparison in various scenarios such as the transition process and steady-state process under different noise conditions, the FNTD has achieved superior noise filtering and differential extraction effects compared to other differentiators.

6.   Conclusions

To quickly extract accurate differential signals from contaminated signals, a FNTD algorithm based on the hyperbolic tangent function is proposed in this paper. To begin, the differential extraction problem is transformed into a double integral system based on TOC law. Next, using the equal time zone (IR) method in the discrete time domain, a FNTD algorithm based on the hyperbolic tangent function is proposed. Then, the frequency domain characteristics of FNTD are analyzed, and the adjustment rules of parameters are given by the frequency scanning test method. Finally, the simulation results show that the proposed FNTD has fast and accurate tracking performance, and has excellent filtering and differential extraction abilities compared with other differentiators.

Future work includes the phase delay compensation algorithm and DTOC-based TD design suitable for engineering applications.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61304036)) and Key Laboratory of Cross-Domain Flight Interdisciplinary Technology (2024-KF01007).

Conflict of interest

The authors declare there is no conflicts of interest.