Reentry Attitude Tracking Control for Hypersonic Vehicle with Reaction Control Systems Via Improved Model Predictive Control Approach

This paper studies the reentry attitude tracking control problem for hypersonic vehicles (HSV) equipped with reaction control systems (RCS) and aerodynamic surfaces. The attitude dynamical model of the hypersonic vehicles is established, and the simplified longitudinal and lateral dynamic models are obtained, respectively. Then, the compound control allocation strategy is provided and the model predictive controller is designed for the pitch channel. Furthermore, considering the complicated jet interaction effect of HSV during RCS is working, an improved model predictive control approach is presented by introducing the online parameter estimation of the jet interaction coefficient for dealing with the uncertainty and disturbance. Moreover, considering the strong coupling effect between the yaw channel and roll channel, a coupled model predictive controller is designed by introducing the feedback of sideslip angle into the roll control channel to eliminate the coupling effect. Finally, the comparison simulations using the classical control method, MPC and IMPC approach are given to demonstrate the effectiveness and efficiency of the presented IMPC scheme.


Introduction
Hypersonic vehicles (HSV) have huge value in both military and civil applications due to their high speed, strong reliability and good penetration capability [Feng and Zhou (2017); Greg (2006); Liu (2011); Weingarten (1993) ;Zhou, Liu and She (2015)], which have become a hot research topic in the aerospace area during the last few decades. The HSV will dramatically reduce the operational cost of space missions because of the fact that they can be used repeatedly by recovering and reusing after each launch mission. Due to the wide flight envelop, large uncertainties and unknown external disturbances, the attitude control design of HSV, particularly in the reentry phase [Cheng, Tang, Wang et al. (2015); He, Qi, Jiang et al. (2015); Joseph (1988); Li (2009) ;Michael (2009) ;Qian (2008); Van (2006); Wang (2018); Xiao and Jun (2009)], is a great challenging problem. Confronting with challenges and high requirement of the reentry attitude control, many researchers have made great effort, and various control schemes have been provided. The gain-scheduling control approach is one of the classical and effective approaches [Roenneke and Markl (1994)], but the limitation is that a large number of control gains to be designed and scheduled in the HSV attitude controller. The feedback linearization [Balas, Reiner and Garrard (1996)] and dynamic inversion [Bharadwaj, Rao and Mease (1998)] methods are alternate control approaches whereas the exact model of the HSV is needed for controller synthesis. Moreover, the sliding mode control (SMC) [Sagliano, Mooij and Theil (2017)] is a popular technique for handling the uncertainties and unknown external disturbances, which can derive the sliding variables into origin in finite time by discontinuous injection [Edwards and Spurgeon (1998)]. However, the uncertainties existing in the HSV attitude system may not satisfy the matching condition [Xu, Mirmirani and Ioannou (2004)]. The disturbance observer-based control method has been received much attention in the past few years [Chen, Ohnishi and Guo (2015); Tian, Liu, Lu et al. (2018)]. It has been extensively applied in the HSV control system  ;Yang, Li and Sun (2013)]. The idea of the method is to design a disturbance observer which is used to estimate external disturbances and plant uncertainties. Then, the disturbance observer-based compensation is added to improve the robustness and disturbance attenuation with a baseline controller. More recently, a new continuous output feedback multivariable uniform finite-time reentry attitude control scheme is developed for HSV with matched and mismatched disturbances in Chen et al. [Chen, Yang, Guo et al. (2016); Li, Zhou and Yu (2013); Mohammadi, Tavakoli, Marquez et al. (2013); ]. The uniform finite-time convergent controller and observer are designed using the bi-limit homogeneous and the sliding mode differentiator technique, respectively. The aeroservoelastic model of HSV is established by using hypothetical modal method and Galerkin method in Mao et al. [Mao, Dou, Tian et al. (2018)], and a novel reentry attitude control scheme via Type-2 AFSMC method is proposed that achieves stable and reliable reentry flight with aeroservoelastic effect. Furthermore, a novel actuator fault-tolerant control scheme for reentry hypersonic vehicle is equipped with aerodynamic surfaces and reaction control systems (RCS) in Zhai [Zhai (2019)], and a nonlinear disturbance observer based sliding mode controller is designed to calculate the required attitude control torque which can handle system uncertainties and external disturbances together. The model predicative control (MPC) approach is one of effective approaches to deal with model uncertainty and external unknown disturbances, and also provide the optimal tracking performance. The basic idea is to employ the system model to predict future states of the systems. State predictions are then used to determine future control moves while minimizing future errors and control effort [Cheng, Tang, Wang et al. (2015); Edwards and Spurgeon (1998) ;Sagliano, Mooij and Theil (2017); Xiao and Jun (2009);]. As we know, there is no existing research work on the model predicative control study on the reentry control problem considering the coupling of RCS and aerodynamic surfaces. Therefore, this paper focuses on the model predictive control problem of HSV with equipped with both RCS and aerodynamic surfaces. The rest of article is organized as follows: Section 1 introduces the attitude dynamical model of the hypersonic vehicles in reentry phase, and the dynamic characteristics in the wide flight envelope is analyzed. Section 2 presents the compound control allocation strategy and the model predictive controller design method for the pitch channel is proposed, and an improved model predictive control (IMPC) approach is presented by introducing the online parameter estimation of the jet interaction coefficient for dealing with the uncertainty and disturbance. In Section 3, considering the strong coupling effect between the yaw channel and roll channel, a coupled model predictive controller is designed by introducing the feedback of sideslip angle into the roll control channel to eliminate the coupling effect. Several representative simulations are conducted in Section 4. Finally, Section 5 concludes this paper briefly.

Nonlinear dynamic model
Considering the reentry phase from suborbit to atmosphere, the HSV is controlled by the RCS and aerodynamic surfaces together. The attitude dynamic model of the HSV in the reentry phase is as follows: where variables are represented as follows: , , V α β γ are angle of attack, sideslip angle, and bank angle; , , x y z ω ω ω are bank, pitch, and yaw angular rates; , , Y Z are lift force and lateral force;

Compound control strategy
In the reentry phase of HSV, when aerodynamic surfaces cannot provide the required attitude control torque due to low dynamic pressure, RCS are activated to assist aerodynamic surfaces to generate the residual torque. Therefore, a compound control strategy should be provided at first.

Figure 1: Illustration of compound control strategy
As shown in Fig. 1, the required control torque is firstly generated by the tracking controller to be designed, then a torque allocation strategy is given to distribute the required torque command to RCS and aerodynamic surfaces. Considering the optimization of fuel consumption, the aerodynamic surfaces work as the primary actuators and RCS serves as auxiliary actuators to complement the residual torque.

Model predictive control design of HSV
The MPC approach is one of effective approaches to deal with model uncertainties and unknown disturbances, and also provide the optimal tracking performance. The basic idea is to employ the system model to predict future states of the systems. State predictions are then used to determine future control moves while minimizing future errors and control effort. With the help of Taylor's expansion [Cheng, Tang, Wang et al. (2015); Van (2006); Xiao and Jun (2009)], the following longitudinal discrete prediction model can be derived from attitude dynamic model of the HSV.
The above predictive model can be reformulated as (3) The following quadratic performance index function is chosen as: represents the tracking error of the step k j + predicted in the step k , and ( 1) u k j + − represents the control sequence at step . Q denotes weight coefficient to reflect the importance level of tracking error and consumed control energy, and n denotes the maximum predicted length at one time.

Figure 2: Longitudinal MPC strategy
Considering of the optimization problem of the proposed performance index (4), the required MPC input can be obtained by the minimal principle [Balas, Reiner and Garrard (1996); Xiao and Jun (2009) Here, only the first value of the obtained optimal control input sequence is selected as the control torque required for the HSV at this moment:

IMPC via jet interference factor identification
When the RCS is activated, a complicated jet interaction flow field caused by the mixture of the jet and the freestream will bring complicated effect on the control properties of RCS and aerodynamic surfaces. In order to attenuate the effects of the above issues on the control performance of the reentry phase of HSV, an improved model predictive control (IMPC) approach is presented by introducing the online parameter identification of the jet interaction coefficient. First, the online estimation of the jet interference factor is performed, and it is assumed that the jet interference factor changes slowly [Balas, Reiner and Garrard (1996)]. The jet interference factor F K in this paper is defined as follows: where real u is the control torque that the HSV subjected under actual flight conditions, and theory u is the control torque calculated by the controller. After adding the effect of jet interference factor, the predictive model can be corrected as: Considering the slow variation property of F K , which can be treated as constant in one period. Therefore, the estimated value of F K can be obtained by the following relationship. By introducing the estimated jet interference factor into the online computation process, the model predictive controller is updated continually. The more accurate predictive dynamic result is, the better control performance will be achieved.

Pulse amplitude modulation
It can be found that the control torque is designed in the continuous torques, but RCS of HSV have the discrete working mechanism. Motivated on this observation, the following conversion method is proposed using the principle of impulse equivalence. For given sampling period, the discrete torques are modulated into pulse sequences of different widths and amplitudes. The pulse equivalent principle [Xiao and Jun (2009)] is shown in the figure:

Lateral coupled IMPC design
Considering the strong coupling effect between the yaw channel and roll channel, a coupled model predictive controller is designed by introducing the feedback of sideslip angle into the roll control channel to eliminate the coupling effect. Similar with the longitudinal case, the lateral discretization prediction model of HSV can be obtained via Taylor's expansion [Cheng, Tang, Wang et al. (2015); Xiao and Jun (2009)].
( ) *cos( ( ))* ( ) where z B and z C are as follows: ( 1) *cos( ( )) ( 2) *cos( ( )) 2 2 2 0 0 0 2 * cos( ( )) 0 0 2 2 2 * cos( ( )) 2 ( 1) *cos( ( )) ( 2) *cos( ( )) 2 2 2 The above predictive model can be reformulated as represent the control sequence at step in yaw and roll channel. 1 Q and 2 Q denote weight coefficient to reflect the importance level of tracking error and consumed control energy in yaw and roll channel, and n denotes the maximum predicted length at one time. are the jet interference factors of yaw and roll channel. Considering of the optimization problem of the proposed performance index (23), the required MPC input can be obtained by the minimal principle: Here, only the first value of the obtained optimal control input sequence is selected as the control torque required for the HSV at this moment:

Simulation input
To verify the effectiveness of the proposed method, the simulations are carried out. The dynamic parameters of HSV are given as following table.

Simulation results
In order to verify the robustness of the proposed control strategy, the aerodynamic coefficients are assumed to remain within the 20% uncertainty about the nominal value, the simulation results in this paper are as follows.   The results demonstrate the tracking accuracy and robustness of the MPC approach and control allocation algorithm. Particularly, from Fig. 15, it shows the effectiveness of RCS pulse amplitude modulation, where RCS activates only in parts of sample time.

Comparative analysis 5.3.1 Comparison of MPC and PID control approach
Employing the designed MPC and PID controller, the comparative simulations are carried out with the same initial condition, the results are as below.

Comparison of MPC and IMPC
The MPC and IMPC approaches are employed for the attitude control design respectively, and the comparative simulations are carried out with the same initial condition, the results are as below. Although it can be seen from Fig. 19 that both of MPC and IMPC have good tracking performance, the MPC approach require larger torque comparing with IMPC approach, which illustrates the advantage of IMPC by introducing the identification of jet interference factor.

Comparison of decoupled control and coupled control approach
The decoupled control and coupled control approaches are employed for the lateral attitude control design respectively, and the comparative simulations are carried out with the same initial condition, the results are as below. As shown above, the green dotted line is the sideslip angle and the bank angle tracking curve based on lateral decoupled control (LDC). The blue dashed line is the reference value and the red solid line is the sideslip angle and the bank angle tracking curve based on lateral coupled control (LCC). It can be seen that compared with the decoupled control; the coupled control approach has less overshot when reference bank angle reverses sign, which means the better control performance.

Conclusion
Aiming at the reentry attitude control problem of HSV with reaction control systems (RCS) and aerodynamic surfaces, this paper provides a novel model predictive control approach to cope with the dynamic uncertainty and coupling effect. On one hand, considering the jet interaction effect of HSV during RCS working, an improved model predictive (IMPC) control approach is proposed by introducing the online parameter estimation of the jet interaction coefficient for dealing with the uncertainty and disturbance; On the other hand, considering the strong coupling effect between the yaw channel and roll channel, a coupled model predictive controller is designed by introducing the feedback of sideslip angle into the roll control channel to eliminating the coupling effect. Especially, the comparison simulations results demonstrate the effectiveness and efficiency of the proposed approach.