Adaptive Suppression Method for Elastic Vibrations in the Control System of Guided Rocket

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1. Introduction

Guided rocket is a kind of precision-guided weapon that can carry out precise strikes on different operational targets and is being widely studied and improved by countries all over the world. In order to reduce air resistance and increase the maximum flight speed and range, guided rockets tend to adopt a design scheme with a high slenderness ratio [1, 2]. Moreover, in order to increase the effective thrust of the engine and the weight of the payload, guided rockets often use lightweight composite materials in the overall design scheme, which will cause a decrease in the inherent elastic frequency of the projectile body, resulting in the coupling of elastic vibration with its rigid motion, posing great challenges to the design of the control system.

During the flight of the guided rocket, the inertial navigation component is sensitive to the elastic deformation and vibration of the projectile body, inevitably generating additional signals brought by them. These additional signals will cause the high-frequency oscillation of the rudder servosystem after entering the control loops and affect the stability of the control system [3]. At the same time, the response to the elastic vibration of the projectile body will also aggravate the wear of the rudder, which may lead to the response error of the rudder servosystem [4]. Therefore, it is necessary to adopt corresponding design strategies when designing the control system of guided rockets to suppress the impact of elastic vibration on the stability of the projectile body.

At present, there are two main methods for suppressing elastic vibration of projectile body. One method is to treat the elastic vibration as disturbance, focus on improving the robustness of control system, and design the controller with robustness [5]. Dash et al. proposed a filtering method based on disturbance observer, which considers the elastic vibration of missile as a disturbance and eliminates the interference signal from the feedback loop to achieve stable control of missile [6]. Tao et al. proposed a robust control method based on adaptive preset control for spacecraft elastic attitude control by using an elastic modal observer to estimate unmeasured elastic modal variables [7]. Zhang and Li treated the elastic vibration signal of projectile body as a disturbance signal and regarded the suppression of elastic vibration as a standard $H_{\infty }$ control problem. They designed a two degree of freedom controller based on $H_{\infty }$ to suppress the disturbance signal [8]. Devan and Chandar considered the elastic vibration of projectile body as an unmodeled dynamic part, designed a robust controller based on proportional linear quadratic regulator to address the uncertainty of the missile model, and verified the steady-state performance of the missile under the above uncertain conditions [9]. Xiaobing et al. regard the elastic vibration of projectile body as an uncertain disturbance and designs an inverse sliding mode controller with preset performance [10].

Another method is to add a notch filter before rudder system. By generating the reverse resonance peak at the elastic frequency of projectile body, the adverse effects caused by elastic vibration can be offset or reduced, and the instability of control system can be avoided. Eser et al. estimated the elastic vibration frequency of projectile body through Fourier transform module and then designed an adaptive notch filter according to the frequency change to solve the air-to-ground missiles’ wing elastic oscillation problem [11]. Lv et al. used amplitude and phase frequency stability analysis method to introduce a notch filter in angular velocity loop and a low-pass filter in acceleration loop of the elastic missile’s autopilot to suppress the elastic vibration of the missile body [12]. Wang et al. used the minimum mean square algorithm to iteratively calculate the measurement feedback signal and proposed a design method for an adaptive notch filter. By estimating and updating the elastic vibration frequency in real time through the feedback measurement signal, the frequency of notch filter was changed to avoid the occurrence of servoelasticity problems [13]. Heng et al. estimate the elastic vibration frequency of missiles by designing a cascaded second-order adaptive notch filter, which uses a recursive least square method with a forgetting factor to optimize the parameters of each notch filter in real time [14]. Wanle et al. present an adaptive notch filter based on wavelet transform and singular value decomposition to address the issue that traditional notch filters seriously affect system dynamic performance, as the elastic modal frequency is low and close to the control system band [15].

Notch filters with fixed parameters are widely used to suppress elastic vibration in engineering. However, in the working process of the engine, due to the constant consumption of propellant, the mass of the projectile body is constantly changing, which causes the elastic vibration frequency to change in a wide range. Therefore, this paper proposes a method for the autonomous identification of elastic frequency and the adaptive suppression of elastic vibration based on a priori information. This method is simple and feasible, with high-frequency identification accuracy and minimal impact on the rigid motion of the projectile body. Through theoretical calculations and simulation experiments, it has been proven that this method has a good effect on suppressing elastic vibration and is suitable for use in missile-borne computer, meeting practical engineering needs.

2. Mathematical Model of Guided Rocket

2.1. Basic Assumptions

The research object in this paper is a guided rocket with an axisymmetric aerodynamic shape and skid-to-turn control, which is also the design form adopted by most guided rockets in the world. The motion of the guided rocket in three-dimensional space can be divided into the displacement motion of the centroid and the attitude motion rotating around the centroid. In the analysis of the motion of the guided rocket, it is necessary to simplify and make assumptions about the actual problem and then establish the mathematical model.

1. Ignoring the aerodynamic force changes caused by elastic deformation of the guided rocket, divide the projectile body motion into the superposition of rigid motion and elastic vibration.

2. Decompose the elastic vibration of the guided rocket into bending vibration along the longitudinal axis of the projectile body and torsional vibration around the longitudinal axis of the projectile body and ignore the coupling between them.

3. Due to the high frequency of torsional vibration around the longitudinal axis of the projectile body, which does not affect the stability of the autopilot, it is ignored.

4. Consider the elastic body as a continuous medium, describe the vibrational motion using differential equations, and simplify it as a finite superposition of selected vibration modes.

Based on the above assumptions, the elastic vibration of the guided rocket is similar to the loaded motion of an elastic beam, and its attitude motion along the longitudinal axis of the projectile body is shown in Figure 1.

[figure(s) omitted; refer to PDF]

In Figure 1, $O_e{x}_e{y}_e{z}_e$ is the elastic reference coordinate system, $O_e$ is the head position of the guided rocket, $O_e{x}_e$ is the longitudinal axis of the projectile body, $O_e{y}_e$ is perpendicular to $O_e{x}_e$ in the pitch moment, and $O_e{z}_e$ is determined according to the right hand rule. $O$ is the centroid of projectile body. $x_A$ is the installation position of the inertial navigation component. $\alpha$ and $\vartheta$ are, respectively, the angle of attack and the pitch angle caused by rigid motion. $\Delta \dot{\vartheta }$ is the pitch angle velocity caused by elastic vibration. $V$ is the velocity of guided rocket.

2.2. Rigid Motion of Guided Rocket

The dynamic equations of the guided rocket are nonlinear, time-varying differential equations. In order to utilize various control theories in the design process of the control system, the assumption of small perturbations with fixed coefficients is often introduced.

Therefore, it is necessary to linearize the nonlinear dynamic equations with small perturbations to obtain a set of linearized time-varying differential equations. Then, assuming that the time-varying parameters do not change significantly during the transition time of the designed object and can be taken as constants, we can get a set of small perturbation linear time-invariant differential equations.

Since the guided rocket researched in this paper uses an axisymmetric layout, its pitch motion and yaw motion have similar characteristics, so they can be described by the same equation. Taking the pitch motion as an example, the state space equation of the small disturbance linearization model is as follows: $$\begin{array}{}\text{(1)}& \begin{array}{c}\dot{\alpha }\\ \ddot{\vartheta }\end{array}=\begin{array}{cc}-b_{\alpha }& 1\\ -a_{\alpha }& -a_{\omega }\end{array}\begin{array}{c}\alpha \\ \dot{\vartheta }\end{array}+\begin{array}{c}-b_{\delta }\\ -a_{\delta }\end{array}{\delta }_z,\text{(2)}& \begin{array}{c}{a}_y\\ \dot{\vartheta }\end{array}=\begin{array}{cc}{Vb}_{\alpha }& 0\\ 0& 1\end{array}\begin{array}{c}\alpha \\ \dot{\vartheta }\end{array}+\begin{array}{c}{Vb}_{\delta }\\ 0\end{array}{\delta }_z,\end{array}$$where $\dot{\vartheta }$ and $a_y$ are, respectively, the pitch angle velocity and the pitch acceleration caused by rigid motion. $a_{\alpha }$ is the pitching moment caused by the angle of attack. $a_{\omega }$ is the pitch damping moment caused by the angle velocity of pitch. $a_{\delta }$ is the pitch moment caused by the elevator rudder. $b_{\alpha }$ is the lift force generated by the angle of attack. $b_{\delta }$ is the lift force generated by the elevator rudder.

Assuming the initial condition is 0, perform the Laplace transform on Equations (1) and (2), and the system transfer function of rigid motion can be obtained: $$\begin{array}{}\text{(3)}& \frac{\dot{\vartheta }s}{{\delta }_{z}s}=\frac{-a_{\delta }s-a_{\delta }{b}_{\alpha }-a_{\alpha }{b}_{\delta }}{s^2+a_{\omega }+b_{\alpha }s+a_{\alpha }+a_{\omega }{b}_{\alpha }}.\text{(4)}& \frac{a_{y}s}{{\delta }_{z}s}=\frac{V{b}_{\delta }{s}^2+a_{\omega }{b}_{\delta }s-a_{\delta }{b}_{\alpha }-a_{\alpha }{b}_{\delta }}{s^2+a_{\omega }+b_{\alpha }s+a_{\alpha }+a_{\omega }{b}_{\alpha }}.\end{array}$$

2.3. Elastic Vibration of Guided Rocket

During the flight process, the guided rocket is free and unconstrained at both ends, with uneven mass distribution and varying stiffness at different parts. According to the continuous elastomer theory, the differential equation of elastic vibration of the guided rocket under external force is $$\begin{array}{}\text{(5)}& \frac{{\partial }^2}{\partial x_e^2}EJ{x}_e\frac{{\partial }^{2}y{x}_e,t}{\partial x_e^2}+mx\frac{{\partial }^2{y}_e{x}_e,t}{\partial x_e^2}=F{x}_e,t,\end{array}$$where $y_e{x}_e,t$ is the elastic deformation function, $m{x}_e$ is the mass distribution function, $J{x}_e$ is the bending moment of inertia function, $E$ is the elastic modulus, and $F{x}_e,t$ is the force function applied to projectile body.

According to the method of separating variables, the elastic deformation function can be expressed using this set: $$\begin{array}{}\text{(6)}& y_e{x}_e,t=\sum _{i=1}^n{q}_{i}t{\Phi }_i{x}_e.\end{array}$$

In Equation (6), $y_e{x}_e,t$ is the elastic deformation function, ${\Phi }_i{x}_e$ is the function of the inherent mode shape, and $q_{i}t$ is the generalized coordinate of the inherent mode shape ($i=1,2,\cdots ,n$, representing the order of the inherent mode shape).

With Equations (5) and (6), the generalized coordinate equation of elastic vibration can be obtained: $$\begin{array}{}\text{(7)}& {\ddot{q}}_{i}t+2{\xi }_i{\omega }_i{\dot{q}}_{i}t+{\omega }_i^2{q}_{i}t=\frac{Q_i}{m_i}\hspace{1em}i=1,2,\cdots ,n,\end{array}$$where ${\omega }_i$, ${\zeta }_i$, $Q_i$, and $M_i$, respectively, represent the frequency, damping ratio, generalized force, and generalized mass of the $i$-order inherent mode shape.

The elastic vibration of guided rocket is mainly caused by aerodynamic force, rudder control force, and damping force caused by the rocket’s rotational angular velocity. Therefore, Equation (7) can also be expressed as $$\begin{array}{}\text{(8)}& {\ddot{q}}_{i}t+2{\xi }_i{\omega }_i{\dot{q}}_{i}t+{\omega }_i^2{q}_{i}t=D_{{\omega }_i}\dot{\vartheta }+D_{{\alpha }_i}\alpha +D_{{\delta }_i}{\delta }_z+D_{{\ddot{\delta }}_i}{\ddot{\delta }}_z.\end{array}$$

In Equation (8), $D_{{\omega }_i}$, $D_{{\alpha }_i}$, $D_{{\delta }_i}$, and $D_{{\ddot{\delta }}_i}$ are the generalized force coefficients of the $i$-order inherent mode shape, which are referred to as elastic dynamic coefficients. By performing the Laplace transform on Equation (8), we can obtain $$\begin{array}{}\text{(9)}& q_{i}s=\frac{D_{{\omega }_i}\dot{\vartheta }s+D_{{\alpha }_i}\alpha s+D_{{\delta }_i}+D_{{\ddot{\delta }}_i}{s}^2{\delta }_{z}s}{s^2+2{\xi }_i{\omega }_{i}s+{\omega }_i^2}.\end{array}$$

The rudder control force dominates the elastic deformation of projectile body. Neglecting the damping force $D_{{\omega }_i}$ and the lifting force $D_{{\alpha }_i}$, we can obtain $$\begin{array}{}\text{(10)}& q_{i}s=\frac{D_{{\delta }_i}+D_{{\ddot{\delta }}_i}{s}^2}{s^2+2{\xi }_i{\omega }_{i}s+{\omega }_i^2}{\delta }_{z}s.\end{array}$$

3. Design of Control System Structure

3.1. Transfer Function of Projectile Body

In the structural design of the control system, how many orders of elastic mode shapes to be considered should be determined based on the stiffness and elastic characteristics of the guided rocket. The mode shape with a higher order also has a higher frequency and a smaller proportion of the whole elastic deformation.

Generally speaking, when the elastic frequency of a guided rocket is closer to the rigid motion frequency, the more orders need to be considered; conversely, the order that needs to be considered can be less, and even in the design of some small missile, there is no need to consider the impact of the elastic vibration of the projectile body.

Due to the high frequency of the second-order and higher order elastic mode shape of the guided rocket studied in this paper, as shown in Table 1, its influence on the rigid body motion is very small, and the adverse effect caused by it can be suppressed by a low-pass filter. So only the first-order vibration frequency of the projectile body is considered in this paper.

Table 1

Frequency of elastic mode shapes with different orders.

	First-order (Hz)	Second-order (Hz)	Third-order (Hz)
Fully load of propellant	14.16	36.05	66.87
Empty load of propellant	23.8	51.77	101.92

Therefore, the system transfer function of the guided rocket elastic vibration is $$\begin{array}{}\text{(11)}& \frac{\Delta \dot{\vartheta }s}{{\delta }_{z}s}=-\frac{D_{{\delta }_1}+D_{{\ddot{\delta }}_1}{s}^{2}s}{s^2+2{\xi }_1{\omega }_{1}s+{\omega }_1^2}\cdot {{\dot{\Phi }}_1{x}_e}_{x_e=x_A},\text{(12)}& \frac{\Delta a_{y}s}{{\delta }_{z}s}=\frac{D_{{\delta }_1}+D_{{\ddot{\delta }}_1}{s}^2{s}^2}{s^2+2{\xi }_1{\omega }_{1}s+{\omega }_1^2}\cdot {{\Phi }_1{x}_e}_{x_e=x_A},\end{array}$$where $\Delta \dot{\vartheta }$ and $\Delta a_y$ are the pitch angle velocity and the pitch acceleration caused by elastic vibration. $D_{{\delta }_1}$ and $D_{{\ddot{\delta }}_1}$ are the generalized force coefficients of the first-order inherent mode shape. ${\omega }_i$, ${\zeta }_i$, ${\Phi }_1{x}_e$, and ${\dot{\Phi }}_i{x}_e$, respectively, represent the frequency, damping ratio, function, and function slope of the first-order inherent mode shape. $x_A$ is the installation position of the inertial navigation component on the guided rocket.

3.2. Analysis of Aeroservoelasticity Stability

The pitch channel autopilot structure of the guided rocket consists of two loops: the damping loop composed of angular velocity feedback and the acceleration loop composed of acceleration feedback, as shown in Figure 2.

[figure(s) omitted; refer to PDF]

$K_{\omega }$, $K_{\omega i}$, $K_p$, and $K_i$ are the control parameters of autopilot.

The dynamic model of rudder is represented by a second-order model, whose undamped natural vibration frequency is 17 Hz and damping coefficient is 0.6. $a_y$ and $\dot{\vartheta }$ are measured by the inertial navigation component (including gyroscope and accelerometer). The frequency of inertial navigation component is 200 Hz, which has little effect on the autopilot, and its transfer function can be regarded as 1.

At the flight altitude of 14 km and flight velocity of 1327.8 m/s, the rigid dynamic coefficient and control parameters of the guided rocket in the pitch direction are shown in Tables 2 and 3.

Table 2

Rigid dynamic coefficients in pitch direction.

${\mathit{a}}_{\mathit{\alpha }}/{\mathit{s}}^{-\mathbf{2}}$	${\mathit{a}}_{\mathit{\delta }}/{\mathit{s}}^{-\mathbf{2}}$	${\mathit{a}}_{\mathit{\omega }}/{\mathit{s}}^{-\mathbf{1}}$	${\mathit{b}}_{\mathit{\alpha }}/{\mathit{s}}^{-\mathbf{1}}$	${\mathit{b}}_{\mathit{\delta }}/{\mathit{s}}^{-\mathbf{1}}$
78.893	54.205	0.231	0.389	0.047

Table 3

Control parameters for pitch direction.

${\mathit{K}}_{\mathit{\omega }}$	${\mathit{K}}_{\mathit{\omega }\mathit{i}}$	${\mathit{K}}_{\mathit{p}}$	${\mathit{K}}_{\mathit{i}}$
0.4649	10	0.0007	0.0015

The elastic parameters of the first-order elastic mode shape are shown in Table 4.

Table 4

Elastic parameters of the first-order mode.

${\mathit{D}}_{\mathit{\delta }\mathbf{1}}$ (m/s)	${\mathit{D}}_{\ddot{\mathit{\delta }}\mathbf{1}}$ (m)	${\mathit{\varphi }}_{\mathbf{1}}{\mathit{x}}_{\mathit{A}}$	${\mathit{\varphi }}_{\mathbf{1}}^{\prime }{\mathit{x}}_{\mathit{A}}$	${\mathit{\omega }}_{\mathbf{1}}$ (Hz)	${\mathit{\omega }}_{\mathit{n}}$ (Hz)	${\mathit{\xi }}_{\mathbf{1}}$
−762.32	−0.0033	−0.067	0.057	23.8	1.42	0.014

In Table 4, ${\omega }_n$ is the resonant frequency of rigid motion, and ${\omega }_1$ is the first-order frequency of elastic vibration.

From the above data, the Bode diagram of the damping loop and the acceleration loop are shown in Figure 3.

[figure(s) omitted; refer to PDF]

In Figure 3, there is a higher resonance peak at ${\omega }_1$ due to the low damping ratio of the first-order mode of the projectile body, and the magnitude at this frequence is 7.4 dB. This frequency characteristic indicates that the control system will experience unstable elastic vibration.

Given the above reasons, notch filters are often introduced before the rudder in control system design. By designing a notch filter for the elastic vibration frequency of the projectile body, a reverse resonance peak can be generated to cancel or reduce the resonance peak caused by elastic vibration, so as to ensure the stability of the control system.

Therefore, the main research content of this paper is how to accurately and quickly identify the elastic vibration frequency of the projectile body from the inertial measurement signal and design an adaptive notch filter with a suitable filtering width to suppress the elastic vibration. The structure diagram of the control system including elastic frequency identification and adaptive notch filter is shown in Figure 4.

[figure(s) omitted; refer to PDF]

4. Adaptive Suppression Method for Elastic Vibration

4.1. Identification of Elastic Vibration Frequency

The purpose of frequency identification is to identify the vibration frequency of the first-order elastic mode shape of the projectile body and then design an adaptive notch filter for this vibration frequency. The angular velocity data and acceleration data of the projectile body can be obtained by the sampling of the inertial navigation module, and the rigid motion frequency is not in the same frequency band as the elastic vibration frequency. Therefore, the inertial navigation sampling data can be directly analyzed in time frequency, and the higher frequency in the result is the elastic vibration frequency of the projectile body.

In order to improve the real-time performance of elastic vibration detection, rolling sampling is adopted for the inertial navigation output signal. Firstly, the sampling data is preliminarily screened, and the threshold value is set to eliminate outlier data. Then, the sampled data is cached in the storage area according to the “first in, first out” principle. Finally, the cached data is called each time when the elastic vibration frequency is calculated.

The formula for fast Fourier transform (FFT) is $$\begin{array}{}\text{(13)}& Xm=\sum _{n=0}^{N-1}xn{e}^{-jkn2\pi /N},\end{array}$$where $xn$ is the time domain signal to be identified,and $Xm$ is the frequency domain signal in the complex form after FFT. $N$ is the number of sampled data points. We can find the signal with the largest spectrum in $Xm$ and write its corresponding serial number $m$ as $m_p$, and the frequency corresponding to $m_p$ is the first-order frequency of the vibration of projectile body.

The higher the number of samples in the time domain sampling, the more accurate the frequency resolution. However, it also brings a larger sample delay, which may not be able to track rapid changes in elastic frequency. When the $N$ value takes different numbers, the frequency resolution and sample delay are also transformed. For example, if the sampling frequency is 200 Hz, the theoretical identification resolution and sample delay are shown in Table 5.

Table 5

Identification resolution and sample delay for different sampling values.

$$N$$	128	256	512	1024
Resolution (Hz)	0.7813	0.3906	0.1953	0.0977
Sample delay(s)	0.32	0.64	1.28	2.56

During the powered-flight phase, the mass of the projectile body gradually decreases with the consumption of engine propellant, and the elastic frequency changes greatly with time. In order to track the frequency change timely, the sampling value $N$ is set to 256. In the unpowered-flight phase, the elastic frequency changes little, and the identification error caused by sample delay is very small. To improve the accuracy of frequency identification, the sampling value $N$ is set to 512.

4.2. Correction Strategy for Frequency Identification

For the guided rocket, the vibration frequency of the projectile body in flight usually changes within a certain range, which can be measured by the modal test of the guided rocket with full load and empty load of propellant. Moreover, the vibration frequency of the projectile body changes with its mass, and the variation trend of mass can be obtained by the engine thrust data. Then, we can correlate the changes in mass and elastic frequency to obtain the prior value of elastic frequency.

During the actual flight, external interference, channel coupling, and other factors can affect the angular velocity information collected by inertial navigation components, thereby affecting the identification results of elastic vibrations. In order to improve the reliability of frequency identification, this paper proposes an iterative correction strategy based on the a priori information.

1. In the process of engine operation, with the propellant consumed continuously, the mass of guided rocket is gradually reduced, and the elastic frequency is gradually increased. The mass is set as $M_{\max }$ when fully loaded of propellant and $M_{\min }$ when empty loaded of propellant. Then, combined with the thrust test data of rocket engine and set the mass flow rate as $\dot{m}t$. The a priori value of the guided rocket mass variation with the flight time can be obtained as follows:

$$\begin{array}{}\text{(14)}& Mt=\begin{array}{cc}{M}_{\max }\hfill & t=0,\hfill \\ M_{\max }-{\int }_0^{t_1}\dot{m}tdt\hfill & 0<t\le t_1,\hfill \\ M_{\min }\hfill & t>t_1.\hfill \end{array}\end{array}$$

In Equation (14), $t$ is the current flight time, $Mt$ is the mass of guided rocket at the current flight time, and $t_1$ is the end time of powered-flight phase.

Then, according to the modal test of guided rocket, its elastic frequency range can be obtained. The elastic frequency of guided rocket is set as ${\omega }_{\min }$ when fully loaded of propellant and ${\omega }_{\max }$ when empty loaded of propellant. The trend of elastic frequency variation is related to the trend of mass variation, and the a priori value of elastic vibration frequency can be obtained as follows: $$\begin{array}{}\text{(15)}& {\omega }_a=\begin{array}{cc}{\omega }_{\min }\hfill & t=0,\hfill \\ \text{interp}1M_{\max },M_{\min },{\omega }_{\min },{\omega }_{\max },Mt\hfill & 0<t\le t_1,\hfill \\ {\omega }_{\max }\hfill & t>t_1.\hfill \end{array}\end{array}$$

In Equation (15), ${\omega }_a$ is the a priori value of the elastic vibration frequency.

2. Suppose the current identification value of elastic vibration frequency be ${\omega }_1$ and suppose the average value of the earlier 10 frequency identification results be ${\omega }_2$, the identification correction formula is as follows:

$$\begin{array}{}\text{(16)}& {\omega }_b=\begin{array}{cc}{\omega }_2+k\cdot {\Delta }_t\hfill & {\omega }_1>{\omega }_2+k\cdot {\Delta }_t,\hfill \\ {\omega }_1\hfill & {\omega }_2-k\cdot {\Delta }_t<{\omega }_1<{\omega }_2+k\cdot {\Delta }_t,\hfill \\ {\omega }_2-k\cdot {\Delta }_t\hfill & {\omega }_1<{\omega }_2-k\cdot {\Delta }_t,\hfill \end{array}\end{array}$$where ${\Delta }_t$ is the sampling step length of inertial navigation component and $k$ is the maximum change rate of elastic frequency. In this formula, ${\omega }_b$ is the identified value of elastic vibration frequency after correction processing.

3. Based on the a priori value and the identification value of elastic vibration frequency on the present moment, set the weight coefficient as $a$ ($0\le a\le 1$). The accepted value of elastic vibration frequency is

$$\begin{array}{}\text{(17)}& \omega =a\cdot {\omega }_a+1-a\cdot {\omega }_b.\end{array}$$

The weight coefficient value $a$ given in this paper is as follows: $$\begin{array}{}\text{(18)}& a=\begin{array}{cc}0\hfill & {\omega }_a-1<{\omega }_b\le {\omega }_a+1,\hfill \\ -{\omega }_b+{\omega }_a-1\hfill & {\omega }_a-2<{\omega }_b\le {\omega }_a-1,\hfill \\ {\omega }_b-{\omega }_a+1\hfill & {\omega }_a+1<{\omega }_b\le {\omega }_a+2,\hfill \\ 1\hfill & {\omega }_b>{\omega }_a+2.\hfill \end{array}\end{array}$$

4.3. Design of Adaptive Notch Filter

The transfer function of the adaptive notch filter model is $$\begin{array}{}\text{(19)}& G_{F}s=\frac{1/{\omega }_i^2{s}^2+1}{1/{\omega }_j^2{s}^2+2{\xi }_j/{\omega }_{j}s+1},\end{array}$$where ${\omega }_i$ is the frequency of elastic vibration and ${\omega }_j$ and ${\xi }_j$ are the center frequency and damping ratio of the adaptive notch filter, respectively.

Generally speaking, let ${\omega }_j$ be equal to ${\omega }_i$, that is, select the center frequency of adaptive notch filter as the identification value of elastic vibration frequency. The increase of ${\xi }_j$ not only increase the filter width and depth of the adaptive notch filter but also increase the phase lag and slow the response speed of the system.

As mentioned in Section 3.2, the system has a magnitude of 7.4 dB at the elastic frequency. To keep the control system stable and ensure a certain stability margin, the magnitude at this frequency should be reduced by at least 20 dB. Assuming the effective filtering width is the range over which the magnitude can be reduced by more than 20 dB around the elastic frequency as the center. The effective filter width, phase lag, and system delay caused by different ${\xi }_j$ are shown in Table 6.

Table 6

The effective filter width, phase lag, and system delay caused by different ${\xi }_j$.

${\mathit{\xi }}_{\mathit{j}}$	Effective filter width	Phase lag in ${\mathit{\omega }}_{\mathit{n}}$	System delay in ${\mathit{\omega }}_{\mathit{n}}$
0.1	0.5 Hz	0.68°	1.33 ms
0.3	1.4 Hz	2.06°	4.03 ms
0.6	2.71 Hz	4.11°	8.04 ms

Figure 5 shows the frequency domain characteristics of the adaptive notch filter with different values of ${\xi }_j$.

[figure(s) omitted; refer to PDF]

Therefore, the designer needs to make a comprehensive balance and choose the appropriate ${\xi }_j$. The key of adaptive notch filter design is adjusting the filter parameters ${\omega }_j$ and ${\xi }_j$ according to the identified elastic frequency, where ${\omega }_j$ varies with the identification frequency of elastic vibration and ${\xi }_j$ changes with flight time. Compared with the unpowered-flight phase, the elastic vibration frequency of the projectile body in the powered-flight phase is lower and closer to the rigid movement frequency band. Therefore, in the powered-flight phase, the value of ${\xi }_j$ could take lower to reduce the phase lag and system delay caused by the adaptive notch filter. With the consumption of propellant, the elastic vibration frequency of projectile body gradually moves away from the rigid motion band, and the damping ratio can be gradually increased to add the effective filter width. The value of ${\xi }_j$ is generally taken as 0.3~0.6.

Figure 6 shows the adaptive suppression method of elastic vibration designed in this paper.

[figure(s) omitted; refer to PDF]

5. Simulation Experiment and Analysis

5.1. Simulation Environment

The simulation object in this paper is based on a six-degree-of-freedom model of a guided rocket with rigid motion and first-order elastic vibration under consideration. The total simulation time is 380 s, and the simulation step size is 5 ms. The sampling frequency of the inertial navigation component is 200 Hz. Tables 7 and 8, respectively, provide a priori information and adaptive notch filter parameters for this guided rocket.

Table 7

A priori information of guided rocket.

Parameter	Value	Unit
$$t_1$$	22.5	s
$$M_{\max }$$	839.6	kg
$$M_{\min }$$	346.57	kg
$${\omega }_{\min }$$	14.16	Hz
$${\omega }_{\max }$$	23.8	Hz

Table 8

Adaptive notch filter parameters of guided rocket.

Parameter	Value	Unit
$$k$$	1	Hz/s
$N$ in powered-flight phase	256	\
$N$ in unpowered-flight phase	512	\
${\xi }_j$ in powered-flight phase	0.3	\
${\xi }_j$ in unpowered-flight phase	0.6	\

Combined with the data information from the thrust test of rocket engine and the modal test of guided rocket, the a priori information of elastic vibration frequency and mass is shown in Table 9.

Table 9

The a priori information of elastic vibration frequency and mass.

Time (s)	0	1.7	3.0	5.0	7.0	10.0	15.0	18.0	21.0	22.5
$Mt$ (kg)	839.6	782.25	739.42	682.32	644.37	586.05	467.83	389.49	353.81	346.57
${\omega }_a$ (Hz)	14.16	15.23	15.95	16.98	17.66	18.70	20.82	22.23	23.47	23.8

5.2. Simulation Result

In order to verify the accuracy of elastic frequency identification and the effect of elastic vibration suppression of the method designed in this paper, interference signals with amplitudes of 10°/s and 10 m/s² are applied to the pitch angular velocity signal and pitch acceleration signal, respectively. The frequency of these interference signals is based on the a priori frequency of the guided rocket, and sinusoidal and white noise biases are performed to verify the accuracy of frequency identification. The frequency identification curve is shown in Figure 7. The weighted coefficient curve is shown in Figure 8.

[figure(s) omitted; refer to PDF]

Figure 9 shows the frequency identification error curve between input value and accepted value. In the initial stage of simulation, the identification of elastic vibration frequency requires transition time due to insufficient valid data in the identification data cache. It can be seen that the elastic vibration frequency identification algorithm designed in this paper can reduce the identification error to within 0.4 Hz after 0.47 s of simulation, which is sufficient to meet the requirements of engineering applications.

[figure(s) omitted; refer to PDF]

Figure 10 is the damping ratio curve of the adaptive notch filter. Figures 11, 12, and 13, respectively, show the comparison of the pitch angular velocity curve, pitch acceleration curve, and pitch rudder angle curve before and after processing by the adaptive algorithm proposed in this paper. It can be clearly seen from the figures that the curves of angular velocity, acceleration, and rudder angle become smooth after being processed by the elastic vibration suppression algorithm.

[figure(s) omitted; refer to PDF]

And when the frequency of elastic vibration changes with time, the adaptive algorithm can accurately identify the vibration frequency and dynamically adjust the adaptive notch filter parameters to suppress the elastic vibration.

6. Conclusion

This paper takes the guided rocket with external disturbance and elastic frequency perturbation as the research object, analyzes the rigid motion and elastic motion of the projectile body, and establishes the elastic vibration model from the perspective of system dynamics. Aiming at the aerodynamic servoelasticity problem of this type of guided rocket, we design a method based on the a priori information to identify the elastic frequency of the projectile body and adaptive suppression of elastic vibration. Combined with the six-degree-of-freedom model of the guided rocket, the effectiveness and feasibility of the method were verified through simulation experiments. The results show that the method designed in this paper has high frequency identification accuracy, with a stable error of less than 0.4 Hz and minimal impact on elastic and rigid motion. The phase delay caused by the adaptive notch filter is less than 4.11°. The suppression effect of elastic vibration is satisfactory.

Disclosure

No persons or third-party services beyond the authors participated in research or manuscript preparation. The list of authors accurately reflects all individuals who have made substantial contributions to this research.

Author Contributions

Chen-ming Zheng: conceptualization, methodology, software, formal analysis, visualization, and writing—original draft. Cheng Zhang: project administration, supervision, and writing—review and editing. Jun Wang: investigation and resources. Zhang-yao Zheng: validation, formal analysis, and data curation.

Funding

No funds were received.

Acknowledgments

Throughout the process of conception, experimental design, data collection and analysis, interpretation of results, and manuscript writing and modification of this study, the authors did not use any artificial intelligence (AI) software or large language models (LLMs, such as ChatGPT).

References

[1] Y. Wei, M. Hao, G. Duan, J. Lei, "Control Law Design for Elastic Aircraft Based on Intelligent Variable Structure," Journal of Intelligent & Fuzzy Systems, vol. 35 no. 1, pp. 485-499, DOI: 10.3233/JIFS-169605, 2018.

[2] L. Peiyao, L. Wenguang, L. Li, Z. Yu, "Research on Influence of Pneumatic Servo Elasticity on Missile Flight Status," Journal of Projectiles, Rockets, Missiles and Guidance, vol. 40 no. 1, pp. 94-101, DOI: 10.15892/j.cnki.djzdxb.2020.01.019, 2020.

[3] J. Xingyu, X. Fengfeng, J. Fanghua, P. Shi, S. Gong, "An Adaptive Improved PD Controller for Rigid-Elastic Coupled Launch Vehicles," Journal of Beijing University of Aeronautics and Astronautics,DOI: 10.13700/j.bh.1001-5965.2024.0383, 2024.

[4] L. Wentao, B. Ruilih, Z. Yuahbo, "Optimization Method of Resonant Suppression Parameters of Notch Filters," Noise and Vibration Control, vol. 38 no. 3, pp. 20-25, DOI: 10.3969/j.issn.1006-1355.2018.03.004, 2018.

[5] S. Changpeng, W. Xuemei, X. Zhe, J. Li, X. Wang, "Research on Elastic Vibration Suppression of Missile Attitude Control System," Journal of Projectiles, Rockets, Missiles and Guidance, vol. 39 no. 5, pp. 154-162, DOI: 10.15892/j.cnki.djzdxb.2019.05.036, 2019.

[6] S. N. Dash, D. D. Dhadekar, K. K. Manglurkar, M. M. Gade, "Bending Mode Stabilization in Tactical Missiles Using Disturbance Observer vs Phase Stabilization Approach," Proceedings of the 2020 IEEE Students Conference on Engineering & Systems (SCES),DOI: 10.1109/SCES50439.2020.9236690, 2020.

[7] J. Tao, T. Zhang, Y. Nie, "Adaptive Prescribed Performance Control for Flexible Spacecraft With Input Saturation and Actuator Misalignment," Tsinghua Science and Technology, vol. 24 no. 6, pp. 694-705, DOI: 10.26599/TST.2018.9010094, 2019.

[8] Z. Zhang, S. Li, "Two-Degree Controller Design for Flexible Missile Based on H-inf Interference Suppression," Proceedings of the 2012 Second International Conference on Instrumentation, Measurement, Computer, Communication and Control, pp. 1653-1656, DOI: 10.1109/IMCCC.2012.384, 2012.

[9] V. Devan, T. S. Chandar, "Proportional Linear Quadratic Regulator Based Robust Missile Roll Autopilot," Proceedings of the TENCON 2019 - 2019 IEEE Region 10 Conference (TENCON), pp. 1421-1427, DOI: 10.1109/TENCON.2019.8929643, 2019.

[10] L. Xiaobing, H. Yangguang, Z. Siyuan, "Design of Vibration Controller for Elastic Missile With Prescribed Performance," Journal of Huazhong University of Science and Technology Natural Science Edition, vol. 47 no. 11, pp. 83-89, DOI: 10.13245/j.hust.191115, 2019.

[11] A. Ö. Eser, B. Kürkçü, S. Ayasun, "Adaptive Notch Filter Design for Flexible Modes of Guided Munitions," Proceedings of the 2022 IEEE 13th International Conference on Mechanical and Intelligent Manufacturing Technologies (ICMIMT), pp. 275-280, DOI: 10.1109/ICMIMT55556.2022.9845254, .

[12] D. Lv, Q. Long-jun, S. Rui-sheng, Y. Ting, "Autopilot Design for Elastic Missile Systems," Proceedings of the 2019 Chinese Control Conference (CCC), pp. 8229-8235, DOI: 10.23919/ChiCC.2019.8865914, 2019.

[13] Y. Wang, C. Mao, Z. Gongping, "Adaptive Notch Filter Design Approach for Aeroservoelastic System," Aeronautical Science & Technology, vol. 31 no. 3, pp. 73-78, DOI: 10.19452/j.issn1007-5453.2020.03.009, 2020.

[14] Z. Heng, R. Xiaoting, Y. Fufeng, "Adaptive Estimation and Simulation on the Vibration Frequency of Missile System," Journal of Vibration and Shock, vol. 36 no. 10, pp. 212-216, DOI: 10.13465/j.cnki.jvs.2017.10.034, 2017.

[15] Z. Wanle, W. Xiaohu, Z. Zhijian, "A WT and SVD Based Adaptive Notch Filter for Aircraft Elastic Constraint," Aerospace Control, vol. 36 no. 4,DOI: 10.16804/j.cnki.issn1006-3242.2018.04.011, 2018.