Neural Extended State Observer Based Command Filtered Back-stepping Hypersonic Vehicle Integrated Guidance and Autopilot

This paper focuses on the integrated guidance and autopilot design with control input saturation in the end-game phase of hypersonic flight. Firstly, uncertain nonlinear integrated guidance and autopilot model is developed with third actuator dynamics, where the control surface deflection has magnitude constraint. Secondly, neural network is implemented in extended state observer (ESO) design, which is used to estimate the complex model uncertainty, nonlinearity and state coupling. Thirdly, a command filtered back-stepping controller is designed with hybrid sliding surfaces to improve the terminal performance. In the process, different command filters are implemented to avoid the influences of disturbances and repetitive derivation, meanwhile solve the problem of unknown control direction caused by saturation. The stability of closed-loop system is proved by Lyapunov theory, and the principles abided by the controller parameters are concluded through the proof. Finally, series of 6-DOF numerical simulations are presented to show the feasibility and validity of the proposed controller.


Introduction
Since 1980s, there have been lots of studies on the area of integrated guidance and control (IGC).In the conventional design, the guidance and control system is treated as two separated processes based on different operation frequencies.The outer guidance system creates acceleration or angle of attack command, and the inner control system tracks it.When the two control loops are combined, the original performance objectives are lost and must be recovered.Thus the resulting iterative design may not produce an optimal overall system.Additionally, under the condition of high speed, imprecisely known aerodynamics, complicated uncertainties and external disturbances, the hypersonic flight dynamics is obviously characterized by nonlinearities.The approaches which involves linearization about a set of equilibrium conditions or trim points within the flight envelop suffer from several disadvantages.Comparatively, the IGC design has provided better solutions in low-cost, modular growth, design flexibility, simple logistics and attracted great interest of researchers.
In [1], the scheme of tactical missile IGC system is designed, where information produced by inertial sensors of guidance system are used for the attitude control system.Because the accuracy of sensors in guidance system is much greater than that of autopilot, the IGC can reduce cost and improve the entire guidance and control system performance.In [2], the discussion of IGC scheme goes a step further with three optimal control laws for a tactical missile.In [3,4], new optimal control methods are proposed to effectively design IGC system for missiles.However, the weighting matrices have too many elements, while the actuator dynamics is assumed sufficiently fast and are not modeled in the IGC development.In comparison, sliding mode control (SMC) is a robust method with not complicated structure for nonlinear control issues.It has been extensively used in IGC design.In [5], a brief review of SMC is made at first.A simple first order actuator dynamics is considered in model development.Then the missile IGC design is compared with two different separated designs.It's indicated that the inherent instability of decoupled guidance and control loops in terminal phase is postponed by the integrated design, so the interception accuracy is greatly improved.Besides, the control input chasing is depressed by using Zero-Error-Miss (ZEM) sliding surface.In [6], Predicted-Impact-Point (PIP) heading error is used to make sliding surface, and terminal second order sliding mode control is designed to achieve the convergence in a finite time without chattering.In [7,8], the IGC dynamics is built with two first order actuators for dual-control missiles, and the controller complexity becomes staggering.Nevertheless, the true missile dynamics has high order for second or third order actuator dynamics.For high order nonlinear system model in [9][10][11][12], back-stepping technique is a useful tool, which is always combined with SMC and disturbance observer.In [9], the nonlinear disturbance observer is implemented to estimate the nonlinearities, uncertainties, disturbances in the system dynamics, thus the decrease of undesired chattering in control is achieved and the robustness of closed-loop system is enhanced.In [10], the ESOs are used to estimate indirectly measured states and various parametric uncertainties.The nonlinear ESO in [11] has better performance under complex uncertainties and measurement noises than linear ESO in [12,13], however, it has more parameters to tune.In [14], velocity tracking error is used to design a composite-errors-based ESO in a feedback form, which makes estimation and tracking errors smaller without high gains.
Neural network are capable of providing arbitrarily good approximations to prescribed functionals of a finite number of real variables in [15].Application to dynamic system modeling, nonlinear complex valued signal processing associated with the RBF network are describe in [16].In [17], direct neural control is proposed to deal with the input nonlinearity in the model of hypersonic vehicle.An adaptive neural network is employed to estimate the structure uncertainties, then back-stepping controller is proposed to guarantee the uniform asymptotic stability of the uncertain system in [18].In [19], RBF NN approximation is combined with adaptive back-stepping technique to achieve boundedness of all closed-loop system states.In [20], radial basis function neural networks are applied to approximate the lumped unknown nonlinearities to satisfy robustness against system uncertainties for a constrained flexible air-breathing hypersonic vehicle.
In order to simplify the implementation and cancel out the system noises, command filter is introduced.In [21], command filtered back-stepping is proposed to offer a means to get the time derivatives of the pseudo control signals.In [22,23], low-pass filter is employed to construct the derivative of pseudo control input.It solves the problem of "explosion of complexity" caused by the repeated differentiations of the pseudo control signals in dynamic surface control (DSC).In [24], the second-order command filters instead of the first-order filters are applied in DSC.With the help of command filters, the performance of back-stepping control scheme significantly improves in stability and steady-state tracking accuracy, while the analysis is made in detail in [25].In [26], the stability of commander filtered back-stepping control is further improved by composite learning.In [27,28], a second order command filter is designed to impose magnitude and rate limitation on the control input.In [29], directly differentiating the pseudo control command respect to time is avoided and the global uniform ultimate boundlessness of the tracking errors is guaranteed in the presence of the input constraints.In [30], low-pass filters of the adaptive control scheme guarantee the fast adaptive performance and robustness of the missile integrated guidance and autopilot system.Deep discussion about control input constraint, finite time convergence in IGC design arises rapidly in [31][32][33].In [34], first order auxiliary dynamics is developed addition to the system model to deal with the input constraints.In [35,36], Nussbaum function is introduced to compensate for the nonlinear term arising from input saturation and solve the problem of unknown control direction.In [36], a novel Nussbaum gain is proposed for multiple unknown actuator directions and time-varying nonlinearities.
In summary, IGC study should be more focused on the model development with actuator dynamics, coefficients uncertainties and control input constraints.Then, the application is able to be studied and discussed more practically and effectively.Motivated by 1) the integrated guidance and control design, 2) neural network and ESO based disturbance estimate, and 3) command filtered back-stepping control in the presence of control input constraint, this work proposes a novel integrated guidance and autopilot scheme for hypersonic flight.
The paper is organized as follows: In Section 1, the integrated design and related control techniques are introduced.In Section 2, the hypersonic flight dynamics in end game phase with third order actuator model is developed, and the control surface deflection has constraint of magnitude saturation.In Section 3, neural network based ESO is designed to synthetically estimate the uncertainty and time-varying disturbance, and three ESOs are used in different channels of the integrated guidance and autopilot model.In Section 4, the hybrid command filtered back-stepping technique is implemented based on the estimations of ESOs.Hybrid differentiators are used to get the derivate of virtual inputs produced by the sliding mode controllers, and Nussbaum function based command filter is introduced to solve the problem of unknown control direction when differentiating the saturated control input.In Section 5, the stability proof of closed-loop system is given by Lyapunov theory, and the principles followed by the controller parameters are concluded.In Section 6, 6-DOF numerical simulations verify the proposed design with five scenarios, and Monte Carlo simulations are made to test the performance under initial flight states bias and measurement noises.Finally, section 7 concludes the paper.

IGA Model
In this section, the integrated guidance and autopilot model in the end-game phase is developed.In Figure 1, the coordinate systems are built without considering the earth rotation.The aerodynamics is built in the velocity coordinate system.V is the velocity of mass center, R is the distance between the mass center and the landing point, q is the angle between line of sight (LOS) and local horizon,  is the flight path angle,  is the angle of attack (AOA) and z  is the control surface deflection.The angle directions shown in Figure 1   The state vector is defined as , , , , , , , and the control surface deflection is saturated as . Then the state space form of integrated guidance and control system can be depicted as follows: where  , The lift force can be simplified as two parts: in Eq. (2a, 2b) are the unified parts of lift, m is the mass.The aerodynamics moment is also described with two parts: , both are unified by the moment of inertia with respect to body z-axis z I : in Eq. (2c).The high order nonlinearity and states coupling mainly caused by aerodynamics coefficients uncertainties are synthesized as .
The actuator dynamics is considered as a first order inertial element with time constant act  and a second order element with damping rate act  , natural frequency , n act  , constant gain act K .Then the transfer function is expressed as following: The state space model of actuator can be depicted as follows: , , , , , , , The nonlinear saturation characteristic of the actuator in Eq. (1c) can be modeled as: where ,max z  is the magnitude constraint of control surface deflection.The saturated signal can be approximated by a smooth function defined as:  ,  is the small approximation error.

Neural Extended State Observer
As the introduction section refers to, neural network has good approximation performance in dealing with complex system uncertainties and disturbances.In this section, in order to compensate the integrated guidance and autopilot model under complex aerodynamics uncertainties and unmatched disturbances, the technique of neural extended state observer (NESO) is exploited.Firstly, considering the following system: The uncertain nonlinearity and time-varying disturbance are considered as a whole where , , , , , , is the center vector of the Gaussian functions, d is the affect size, and O in Eq.( 8) is the construction error of neural network.By employing the  -modification type of adaptation law, the weight can be functioned as: where x as reference signal of p x , then the tracking error can be expressed as p p pr e x x   .
In the integrated guidance and autopilot design, three NESOs are used to compensate the unmatched uncertain nonlinearities and time-varying disturbances in Eq. ( 1), the estimate results are respectively  

Hybrid Command Filtered Controller
In this section, a hybrid command filtered back-stepping sliding mode controller is designed to achieve LOS rate convergence based on the estimate results of NESOs.
Step 1: The first sliding surface is chosen as x is designed as reference signal of 2 x according to a proportional reaching law with positive Step 2: Defining the second sliding surface 2 , and the virtual control input 3 x is obtained by the following control law: where 2r x  is the derivative of 2 x , if directly differentiating 2 x respect to time, the virtual control input 3 x is likely to peak.So a low-pass command filter is used to get the derivative: In Eq. ( 13), 1  is positive time constant.The error produced by the low-pass filter is defined  , and 1  can be sufficiently small when 1  is appropriately set.
Another low-pass filter in the same form of Eq. ( 13) is used to avoid directly differentiating of 3 x .
    Then the error produced by the low-pass filter can be expressed as 2 Step 3: In this step, there are two sub-steps to finish the tracking control.Firstly, the virtual control input 4 x is designed to achieve the convergence of 3 3 3r

s x x  
under the following control law: The third low-pass filter is used to get the first order derivative of 4 x : The error between 4 x and 4r x is given as  , then differentiating it respect to time: where 4 x is also given by a low-pass filter with time constant act  .
  Then the proportional reaching law with positive constant 4 k is chosen to achieve the convergence of 4 The Nussbaum function is defining with the following properties: According to the above properties, the following Nussbaum function is implemented: The parameter  is an adaptive parameter according to the following principle: In the end, a third order differentiator is also used as command filter to get first and second order derivatives of the virtual control input 4u x : where , the error between 3 CF x and 4u x can be enough small through choosing suitable  .
Step 4: The task of tracking 4u x is completed by a terminal sliding mode controller.
Considering the second order dynamics in Eq. (4b) and Eq.(4c), the uncertainty and the time-varying disturbance are bounded by positive constants , F D . , where T is the convergence time of the terminal sliding mode controller.
Finally, the control input is designed as follows: 5 Stability Analysis , , , e x e x x e x x e g x x        , and differentiating them with respect to time: Besides the following Lyapunov function is chosen: Differentiating the actuator dynamics related part of Eq. ( 33) with respect to time: Because the differentiator is set appropriately such that  is small positive constant, and , We can find a positive constant  to yield to: Through differentiating Eq. ( 33) and combining Eq. ( 34), we have: According to the Young's inequality, Eq. ( 36) can be rewritten as following: , then the following inequality is yielded to: Integrating Eq. ( 39) directly, then we have: According to the proof in [36],   is bounded, and   V t is bounded, which implied all the error is bounded.

Numerical Simulation
In this section, series of 6-DOF nonlinear numerical simulations are provided to illustrate the control schemes proposed in the previous sections.Firstly, the initial flight condition in end-game phase is set as follows: The IGA model is built in four scenarios with the following aerodynamics coefficients based on table in Appendix A. Scenario 1: Furthermore, the un-modeled measurement uncertainty of q  in Scenario 4 is considered as a first order transfer function: The third order actuator dynamics is given by the following transfer function: Dead-zone is also part of control input nonlinearity, which can be defined by Eq. ( 6) with the following function of ,max z  : The time-varying disturbances in different channels of IGA model are given as follows: , , act x x y , according to the bound of the states, the following center vector is employed: The second neural network input is chosen as   2 , act x y , according to the bound of the states, the following center vector is employed: The third neural network input is chosen as   2 3 , , act x x y , according to the bound of the states, the following center vector is employed: The other parameters of NESOs are set with   30, 30, 30   .
Thirdly, the first order command filters have time constants . The third order differentiator in Eq. ( 24) is set with 0.04   .The control surface deflection is constraint with ,max 10deg z   , and the Nussbaum function in Eq. ( 22) is set with . The proportional reaching law of back-stepping sliding mode controller is designed with constants . The terminal sliding surface in Eq. ( 26) is set with and the bound of uncertainty and time-varying disturbance in Eq. ( 28) is given as

A. Nominal Simulation
The nominal simulation results of Scenario 1 (S1) indicate the feasibility of the proposed method.As Figure 2(a) shows, q  finally converges to zero under the saturated control surface deflection.In Figure 2(d), the control surface deflection of the proposed method is well constraint under the saturation.The state estimate errors of NESO converge well in very small range, the NESO can estimate the system states with good accuracy, and all the states converge under saturated control surface deflection.The results indicate that the proposed control scheme performs very well.Figure 3 shows the comparison simulation results between S2 and S1.The proposed control scheme is able to cancel out influence of the nonlinear uncertainty in aerodynamics coefficients of lift and moment.Figure 4 shows the comparison simulation results between S3 and S1.The nonlinear parts of control surface deflection in aerodynamics coefficients have a limited impact on the convergence with small overshoot in Figure 4(a) and Figure 4(b).It's indicated that the performance improves under complex nonlinearity with the help of the great approximation ability of neural network.In Figure 6, simulation results in S1 and S5 are compared.Figure 5(d) shows that the control surface deflection of S5 has 1 second dead-zone.It's seen that the NESO approximation based IGA control scheme guarantees the closed-loop system stable under saturated control surface deflection with dead-zone nonlinearity.

C. Simulations with Velocity, Initial Distance Dispersion and Measurement Noises
With the help of Monte Carlo theory, large number of numerical simulations under a wide dispersion range of velocity, initial distance and measurement noises are made.The results are shown in Figure 7, 8, 9.In Figure 7, the simulation results under velocity dispersion between 1000 m/s and 1400 m/s are presented.The mean value is 1200 m/s.It is seen that the parameters setting of the proposed controller can satisfied large variations of flight velocity.Figure 8 presents the Monte Carlo simulation results under initial distance dispersion between 18km and 22km.The mean value is 20km.It can be seen that the corresponding flight states converge well, the main difference lies in the convergence time under initial distance dispersion.Briefly, the proposed scheme is able to achieve the control object under large dispersion of initial flight states.In Figure 9, the Monte Carlo simulation results under measurement noises with unknown gain and time constant in Eq. (42d) are presented.It can be concluded that the uncertain gain of sensor dynamics has a very limited effect on the proposed controller.

Conclusion
In this paper, a novel composite IGA scheme combined with third order actuator dynamics under control input saturation and extended state observer is developed to address hypersonic flight control with multiple uncertainties and control constraint.The complex nonlinearities and unmatched time-varying disturbances are well estimated by three neural ESOs.Four differentiators include a third order hybrid nonlinear differentiator is employed to calculate the derivatives of virtual control input.Thus, the peaking phenomenon and chasing of back-stepping sliding mode controller is greatly depressed.Besides, the noteworthy feature of the proposed IGC approach is that the control surface deflection is constraint under saturation with Nussbaum gain.It is very important in practical application.
are positive.

Figure 1 .
Figure 1.The planar geometry in end-game phase disturbance in actuator dynamics.The integrated guidance and autopilot model is developed by Eq. (1) and Eq.(4) with state vector

Figure 2 .
Figure 2. IGA and NESO states of Scenario 1: (a) comparison between q  and NESOI, (b) comparison between  and NESOI, (c) comparison between z  and NESOI, and (d) saturated

Figure 3 .
Figure 3.Comparison between Scenario 1 and Scenario 2: (a) curves of q  , (b) curves of  , (c) curves of z  , and (d) curves of saturated z  .

Figure 4 .Figure 5 .
Figure 4. Comparison between Scenario 1 and Scenario 3: (a) curves of q  , (b) curves of  , (c) curves of z  , and (d) curves of saturated z  .The comparison results of S3 and S4 are shown in Figure5, which indicate that the proposed method can cancel out the complex uncertainties and disturbances with unknown measurement dynamics.

Figure 6 .
Figure 6.Comparison between Scenario 1 and Scenario 5: (a) curves of q  , (b) curves of  , (c) curves of z  , and (d) curves of saturated z  .

Figure 7 .
Figure 7. Monte Carlo Simulation under V dispersion: (a) curves of q  , (b) curves of  , (c) curves of z  , and (d) curves of saturated z  .

Figure 9 .
Figure 9. Monte Carlo simulation under measurement noises: (a) Curves of q  , (b) Curves of  , (c) Curves of z  , (d) Curves of saturated z  .