The tracing algorithm of re-entry vehicle based on SMO

The tracing problem of a re-entry vehicle includes the characters of lacking prior information on the target and disturbances. Thus, the classic nonlinear filtering algorithms are not suitable for high-accuracy tracing fields. In order to solve the problem, the tracing algorithm of a re-entry vehicle based on SMO is presented in this paper. SMO stems from the depth research of sliding mode control, which has nice robustness to disturbances and model error. The results of the simulation show that the tracing algorithm based on SMO has high accuracy and nice robustness.


Introduction
Real-time tracking of a re-entry vehicle is an extremely complex problem in the field of nonlinear filtering applications.The complexity of this problem comes not only from the excessive computational burden brought by the nonlinear motion formula of the aircraft, but also from the non-cooperation of the target [1] .Under the condition that the target does not maneuver or has prior knowledge of the target maneuver, the conventional filtering algorithm can track the algorithm effectively to a certain extent.However, when the target has unknown maneuvers and prior knowledge of the target is not mastered, the simple filtering algorithm cannot meet the requirement of accurately tracing the target.For maneuvering target tracing, polynomial filters [2] that do not consider aerodynamic effects and extended Kalman filters that only consider drag and do not consider maneuvering factors have proved unable to meet the requirements of accurate tracing.As a reasonable extension of the extended Kalman filter, Chang et al. proposed the extension of the Kalman filter for a maneuverable re-entry vehicle [3] , which can effectively track the position, velocity, and other parameters of the target to a certain extent.However, this filter is based on an orbit model.Only when the motion of the re-entry vehicle conforms to the set dynamic model can the ideal tracking effect be obtained.If the target changes the maneuver trajectory, especially when the target is partially predictable, the tracing accuracy will be critically decreased due to the mismatch of the model.Another method is to use a hybrid filter to switch the filter to trace the target when the target is maneuvering.This method requires a variety of models to realize the alternations of tracing algorithms, and there is a large transient error when switching the filter, so this algorithm is relatively outdated [4] .
Therefore, in order to achieve accurate tracking of re-entry vehicles, a target state tracing filter which can be applied to nonlinear motion and model uncertainty is needed.
With the continuous development of the Slide Mode Control (SMC) theory [5] , numerous scholars have developed the Slide Mode Observer (SMO) technology for observing the state of linear or nonlinear systems.The technology has been successfully applied to robots, AC motors, inverters, and other fields.
The basic principle of the SMO technology is that when the state observation error crosses the switching surface in the state space, the observer structure changes, so that the system observation error slides along the desired switching surface [6] .The biggest feature of this technology is that it can be applied to linear or nonlinear systems with unknown inputs, so that all observed states converge in a finite time, and when the invariance condition is met, it has strong robustness to external interference and model uncertainty [7] .Therefore, the SMO technology can effectively solve the tracing problem of a re-entry vehicle with a nonlinear motion and an uncertain model.
Based on the assumption that the dynamic model of a re-entry vehicle is a Singer model, in this paper, a tracing algorithm of a re-entry vehicle is designed based on the SMO technology.Simulation results show that the proposed method has strong robustness and high tracing accuracy.

SMO algorithm
It is assumed that the state variables of systems are divided into i groups, that is = [ ,⋯, ,⋯, ,⋯, ] , and for the i group of state variables, the corresponding subsystems that satisfy the following triangle form can be written as: In the formula, di is the system uncertainty, including external disturbances and model error.In addition, the system input that satisfies the following form is given by: For such systems, there exists , where i, j satisfies The following form of SMO is established [8] : where ̄ can be calculated by the following formula: In Formula (4), = 2,⋯, , and ̄ = = − .The convergence of the SMO Formula (3) is proved, and the dynamic formula of the observation error can be obtained according to the subsystem dynamic Formula (1).The observation error is taken as = − , = 1,⋯, , and Formula (5) can be written as: If the input is bounded, the system state will not diverge in finite time, so the observation error is bounded.Obviously, when meets the following conditions, will converge to zero in finite time t1 and step ( 6) is followed, When converges to zero, ̄ =[ ( ̄ )] = .Therefore, after t1, the second error formula can be given by: Obviously, when satisfies the following conditions, →0 > + ( , , )− ( , , ) By analogy, after , the condition under which ̇ converges to zero is: In addition, the equivalent control signal (v)eq of v in Formula (4) can be calculated by a low-pass filter.The equivalent control signal obtained by this method can prevent peaks from occurring.
In the formula, δ is a small positive number.

Re-entry vehicle dynamic model
The re-entry vehicle tracing problem can generally be described as the tracing of non-cooperative targets.
The most important characteristic of non-cooperative targets is that their maneuvering modes are unknown, that is, the established target tracking model is uncertain [9] .At present, the description of a non-cooperative target dynamic model can be divided into three categories: the Gaussian noise model, the Markov process, and the semi-Markov jump process.In this study, the more commonly used Singer model is used, which was proposed by R. A. Singer in 1969.The characteristic of this model is that the maneuvering of the target is considered as a time-correlated colored noise sequence rather than a white noise sequence.The Singer model assumes that the target acceleration is a stationary first-order Markov process with zero-expect.The process has autocorrelation given by: Its power spectral density is defined as: The maneuver can be described by the following differential formulae: In the formula, α is the target maneuvering frequency, and ω(t) is the zero-expect white noise process with power spectral density 2ασ 2 , in which σ 2 is the variance of maneuvering acceleration.The maneuvering frequency α determines the maneuvering characteristics of the target in the interval (t, t + τ).Usually, the empirical value is: turning maneuver, α = 1/60; escaping maneuver, α = 1/20; atmospheric disturbance, α = 1.
Therefore, the Singer model of the non-cooperative target in a rectangular coordinate system can be formatted as [10] : where, =[ , , , , The measurement of the target is generally provided following the natural sensor coordinate system.Like radar, its natural sensor coordinate system is a three-dimensional spherical coordinate system, so the measured values are generally distance γ, azimuth θ, and pitch angle η.In order to satisfy the designed SMO observer, the measured values are converted to the Cartesian coordinate system.
The measured value of the radar to the target is taken as rm, θm, and ηm, that is where ωr, ωθ, and ωη is an independent Gaussian white noise with zero-expect.
The measured value in the spherical coordinate system is converted to the Cartesian coordinate system and measured as: = Obviously, in the Cartesian coordinate system, the noise of each measurement value is no longer Gaussian white noise, so it will affect the estimation accuracy of traditional filtering methods, such as EKF.

Target tracing algorithm based on SMO
Aiming at the dynamic model and measurement model ( 12) and ( 14) of the re-entry vehicle, the SMObased tracing algorithm designed by Formula (3) is as follows:

Conclusion
In this paper, a tracing algorithm based on SMO is designed for the lack of prior information such as unknown maneuvers and external disturbances of a re-entry vehicle.The algorithm is simple and easy to implement, and the simulation results show that it has good robustness and accuracy.

Figure 3 z
Figure 3 z Distance tracing error.

Figure 4 x
Figure 4 x-direction velocity tracing error.

Figure 5 y
Figure 5 y-direction velocity tracing error.

Figure 6 z
Figure 6 z-direction velocity tracing error.