0 Introduction

The research on control method and technology of hypersonic vehicles (HSVs) has attracted tremendous attention in recent years. One of the major causes is the challenge that HSVs have complex nonlinear aerodynamic characteristics in the flight envelope, always accompanied by model uncertainties, elastic deformation (airframe), and strong coupling of control channels^[1]. Besides, performing large angle attitude maneuvers in the near space may result in the actuator saturation which can violently damage the control performance of the HSVs and even cause the vehicle to enter an unstable state. Therefore, actuator amplitude, rate saturation and aerodynamic uncertainty need to be further studied for the control system of HSVs.

Many advanced control approaches, such as predictive control, sliding mode control, and backstepping control have been employed for tackling the above control problems of HSVs^[2]. The backstepping control method is widely used in the attitude control and uncertainty suppression of HSVs^[3]. However, the problem of "explosion of terms" still exists and is caused by repeated differentiations of the virtual control law. Many scholars have studied the problem and made gratifying progress^[4,5,6]. Dynamic surface control (DSC) technique by Swaroop^[7] is introduced to address this problem. In Ref.[5], a novel robust adaptive dynamic surface controller was adopted for a HSV in the presence of aerodynamics uncertainty and input saturation. A robust adaptive DSC scheme based radial basis function neural function was presented by Zong et al.^[6] for control problem with parametric uncertainty and input constraints. Waseem et al.^[8] introduced a novel integral filter to DSC design procedure of a HSV to ensure feasibility of tracking performance under actuator magnitude and rate constraints. In Ref.[9], a second⁃order sliding mode based integral filter possessed superior noise suppression performance compared to first⁃order filter. But the saturation or sign function consisting in the second⁃order⁃filter always improves damage on performance of control system. So a novel integral filter with smooth hyperbolic tangent function was investigated to ensure the outstanding tracking performance. There is no doubt that it is difficult to measure and estimate the aerodynamic uncertainty caused by the external space and the internal coupling of the vehicle. To enssure high performance and stabilization, the disturbance observer^[10] (DO) technique that has been widely used is recommended to cope with the uncertainty. In Refs. [11,12], a DO is developed to estimate the unknown compounded disturbance. It is defined to take account of the unknown nonsymmetric input saturation and the unknkown external disturbance, and shows a great estimation performance.

Additionally, it should be pointed out that the anti⁃windup auxiliary system is also the object concerned by this paper. The command filter consisting of a first⁃order filter is a nonlinear method to compensate the default between the control instruction and the deflection of the actual actuator. Investigation of previous studies indicates that there are few literatures^[13,14,15] considering amplitude and rate saturation at the same time, whereas it is obvious that the actuator rate has its limitation, and better maneuverability requires higher actuator rates. In Ref.[15], an anti⁃saturation control architecture incorporating feedback linearization and disturbance observer was designed to deal with input constraints. However, most studies^[15,16,17] did not involve rate constraints of actuator. In Ref.[13], an adaptive backstepping method based on sampling time was constructed to handle the rate and amplitude constraints of the elevator, but the actuator rate of this method can only oscillate between zero and maximum. Yuan et al.^[14] constructd a new adaptive backstepping controller for a class of uncertain multiple⁃input multiple⁃output nonlinearity systems with input magnitude and rate saturation, and this controller is not very effective in dealing with strong time⁃varying nonlinear system. In Ref. [18], a robust constrained autopilot control was developed to solve the angle of attack constraint and actuator input amplitude and rate constraints by utilizing dynamic surface control and intergral barrier Lyapunov functional technique, but the feasibility check^[19] of barrier Lyapunov function is always the difficulty of its design. Therefore, the vehicle attitude control problem with the amplitude and rate saturation of the actuator has not been solved effectively.

Motivated by the results of the above mentioned literatures, this paper proposes a disturbance observer⁃based adaptive backstepping control (DOABC) scheme to cope with the HSV precise and robust attitude control problems that include of actuator amplitude and rate saturation with aerodynamic uncertainty. A vehicle⁃actuator control model is presented by introducing actuator dynamics into the design procedure of attitude control system for HSVs. An advanced integer filter to handle the "explosion of terms" problem is adopted with the analysis of stability error. A new type of auxiliary compensation system is designed to deal with the problem of actuator amplitude and rate constraints.

1 The HSV Model

The HSV attitude control mathematical model with uncertainty can be written as

where Θ=[α,β,σ]Τ is the attitude angle vector whose three items represent the angle of attack, sideslip angle and bank angle, respectively; ω=[p,q,r]Τ the vector of angular rates consisting of roll, pitch and yaw rate; M0∈R3 the control moment vector; Δf1(Θ) and Δf2(Θ) are uncertainty terms induced by the uncertainty of aerodynamic parameters and modeling errors. The detailed expressions of f1(Θ) , f2(ω) , g1(Θ) and g2(ω) are shown as follows

where Jx , Jy and Jz are the main moments of inertia of the three axes, respectively; the detailed expression of fα , fβ and fσ and other details are given in Refs. [2,17].

Assumption The uncertainties, Δf1(Θ) and Δf2(Θ) , satisfy the following conditions

where η1 and η2 are all positive constants, and · stands for Euclidean norm of vectors.

The control moment M0 is described as

where the matrix gfδ is the sensitivity moment distribution matrix; δ control surface vector and its components are modeled by a group of second⁃order dynamics for control system, designed as follows

where κ1i and κ2i are the positive constants about actuator dynamic system and always far greater than 1 in reality; δic is the control deflection commanded. The actuator, meanwhile, has the following constraints because of physical limits

where um and vm are the lower and the upper bounds of the actuator amplitude and the rate⁃saturation, respectively.

Lemma 1 Define the continuous and positive function V(t) with ∀t∈R+ and V(0) bounded. If the following inequality holds

where c1 and c2 are positive constants, and ρ(t)∈L∞ is real‑valued function, then V(t) is bounded and the solution x(t) is uniformly bounded^[20].

Lemma 2 For the positive definite Lyapunov function given blow

where e(t)=x(t)-xd(t) and W˜(t)=Wˆ(t)-W(t) , and constants Q(t)=QΤ(t) and Φ(t)=ΦΤ(t) are dimensionally compatible positive matrices. If the following inequality is satisfied

and accompanied by any given initial compact set

then the following conclusions can be obtained^[20].

(1) The states and weight of closed⁃loop system will remain in the compact given by

(2) The states and weight of closed⁃loop system will eventually converge to the compact sets given by

where constants

with λQmin=minτ∈[0,t]λmin(Q(τ)) , λΦmin=minτ∈[0,t]λmin(Φ(τ)) .

2 Controller Design and Stability Analysis

In this section, a novel HSV attitude control scheme is proposed to deal with attitude tracking. It guarantee that the attitude angle tracks the reference signal Θc=[αc,βc,σc]Τ with aerodynamic uncertainty and input constraints.

The traditional hypersonic attitude control did not consider actuator dynamics or actuator input amplitude and rate constraints. But the actual control performance of the vehicle will decline sharply when the actuator is in the state of saturation for a long time. Therefore, actuator dynamic property should be considered in order to obtain the preferable control performance.

A novel reconfiguration vehicle⁃actuator model consists of vehicle dynamics and actuator dynamics is described as follows.

Define new state variables, x1=Θ , x2=ω , x3=δ , x4=δ˙ , and Eqs.(1), (2), (8) are converted to new strict feedback system

where g2f=g2gfδ , κ1=diag(κ11,κ12,κ13) , κ2=diag(κ21,κ22,κ23) .

2.1 DOABC scheme design

In this subsection, the controller for above vehicle⁃ actuator model is designed by combining adaptive backstepping control method with the DO.

For the design goal of tracking the desired command angle, a set of auxiliary variables is defined as

where xd is the reference command vector of attitude angle; λi1 , i=2,3,4 are the estimation of the designed virtual control variables and will be shown at a later stage; e is an auxiliary variable for solving actuator saturation problem and will be given in following Step 4.

Step 1 First, from Eqs.(9), (10), the time derivative of z1 is described by

Ref. [12] suggested a new idea for estimating the uncertainty Δf1 . It is given by

where ξ1 is the intermediate variable; Δˆf1 the estimation of Δf1 ;and kf1 a positive scalar to be designed.

The first Lyapunov function in Step 1 is employed as

where Δ˜f1=Δf1-Δˆf1 is the estimation error of Δf1 .

From Eqs.(7), (8), (12), the derivative of V˙11 satisfies

The controller is designed as virtual control law x2d to enforce z1→0 and expressed as the following form

where k1 is a designed positive constant.

As noted earlier, since the aerodynamic variables are not differentiable and associated with uncertainty, it is difficult to find the derivative of the virtual input Eq. (15). The problem of “explosion of terms” will occur with the increase of extension items.

For these problems, a novel first⁃order integral tanh filter is employed in this paper to eliminate the differential computation of x˙2d . It is described in Ref. [12] as

where τi1 is the time constant of the filter; ζi1 and li1 are scale positive constants to be designed.

Remark 1 It is noted that the above proposed second‑order filter degenerates into a classical integral filters when the parameter ζi1=0 . As known to all, the span of tanh function is [-1,1] , so the parameter ζi1 constrains the upper and the lower bounds of the tanh function. It is worth stressing that the hyperbolic tangent function guarantees the smoothness of state variables, and especially the smooth process required by the physical system.

Define the second Lyapunov function in

From Eqs.(10), (14), (15), Eq. (11) becomes

So the time derivative of Eq. (17) is

Step 2 From the definition of second error surface z2 , the time derivative of z˙2 is expressed as

The DO of Δf2 , same as Step 1, is designed as follows

where ξ2 is the intermediate variable; Δˆf2 the estimation of Δf2 ; and kf2 is a positive scalar to be designed.

The Lyapunov function for Eq.(18) is chosen as

where Δ˜f=Δf2-Δˆf2 is the estimation error of Δf2 .

Through the similar analysis process with Eq.(14), the time derivative of V˙21 is given by

The virtual control law of this step is chosen as

where k2 is a designed positive constant.

The approximate estimation variable λ31 can be achieved in a similar way by Eq.(16). The estimation error of the virtual control law x3d is defined as

Define a Lyapunov function candidate

From Eqs.(10), (24), (25), Eq.(20) becom⁃es

So the time derivative of Eq.(26) is described by

Step 3 The time derivative of the error z3 is expressed as

It is worth noting that the control law can not be completely implemented since the physical constraints of the actual system are unavoidable, such as the magnitude and rate constraints considered in this paper.

Inspired by the research in Refs.[6,13], a new auxiliary first⁃order system is constructed to reduce the damnification of the attitude control system due to the saturation of the actuator, given as follows

where k4 and ζe are parameters that should be designed according to the requirement of the compensation performance. Δu=u-uc is the error between the actual control input u that is control surface deflection and the desired input uc designed by the control law which will be given next.

Remark 2 Many auxiliary anti⁃windup saturation compensation schemes based on filtering idea were proposed^[10,21]. A noteworthy problem is that auxiliary system is likely to cause excessive deflection of the actuator, and it is also possible to cause strong bucket vibration of the actuator. So a novel tanh function unite is employed to solve this problem and this idea is enlightened by activation function in deep learning. From Eq.(30), the hyperbolic tangent function is adopted to accelerate the response of the compensation system compared with previous command filters. The parameter ζe is used to attenuate the magnitude of auxiliary output and avoid excessive deflection of the actuator. It is known from the response of the traditional first order system that we can get the desired the control performance by selecting the parameters k4 and ζe properly .

Remark 3 It is worth noting that traditional compensation system^[6,13,22] design is determined by the characteristics of the system itself. So the compensation signal of the compensation system has a highly relates to the system itself, which will cause uncontrollable size of the compensation signal. The design method of compensation system modified by this paper enhances the autonomy of compensation system design. This method not only ensures good attitude tracking performance, but also effectively solves the problem of actuator rate saturation, and the problem of actuator shake and reverse saturation(RS) (A detailed explanation about RS, is illustated by the example in section 3) caused by the previous compensation system.

So the virtual control law is chosen as

where k3 is a positive constant. The approximate estimation variable λ41 can be achieved in the similar way by Eq.(16). The estimation error of the virtual control law x4d is defined as

The Lyapunov function is defined as

From Eqs.(10), (31)， (32), Eq.(29) becom⁃es

So the time derivative of Eq.(33) can be given by

Step 4 In the final step, the derivative of z4 is given as

From Eqs.(9), (30), Eq.(36) becomes

When 1≪ζe<min(κ1i) , i<1,2,3 , using Eq. (10), we have

So we choose the virtual control law in Step 4.

Remark 4 The parameter κ1 is related to the performance of actuator. The parameter ζe is selected as ζe<min(κ1i) , which can attenuate the magnitude of compensation signals and accelerate the response of compensation system. The higher value of ζe corresponds to a worse smoothness and a faster response speed and vice versa. The actual value is determined by the actual actuator.

Define the Lyapunov function

Due to Eqs.(37)， (38), the derivative of Eq.(39) can be acquired

Remark 5 The analysis and explanation of the presented algorithm is described by two points.

First, the aim of reconfiguration vehicle⁃actuator model in Eq.(8) is to capture the amplitude and speed of the actuator. In addition, this cascade system is very suitable to adopt backstepping control method which can effectively unite disturbance observer and compensation system.

Second, employing tanh function to improve the corresponding rate of compensation system is the highlights of this paper. The traditional first⁃order system is described as e˙=-k4e+κ1(u-uc) . The contrast Eq.(30) can be found that the effect of tanh function is to accelerate the feedback of (u-uc) instead of merely depending on κ1(u-uc) whose κ1 belongs to the parameter of system feature itself and is uncontrollable.

2.2 Stability analysis

To facilitate the stability analysis, we define the Lyapunov function

Using Eqs.(14), (19), (23), (28), (35)， Eq.(40) yields

The time derivative equation of Eq.(42) is

From Eq.(16) and the filter error yi1 , if ζi1>x˙idmax ，on the basis of Polycarpou and Ioannou^[3], we get

From the research in Ref.[10] , the ultimate form of Eq.(43) is derived as

Eq.(45) can be rewritten as

where c1=mink1-12,k2-12,k3,k4,kf1-1,kf2-1,1τ21,1τ31,1τ41 , c2=∑i=12ηi2+(ρi1) .

By the differential inequality in Eq.(46), the solution can be obtained as

According to Lemmas (1,2), the estimation error of the DO can converge to the following compact set

The convergence domain of zi is given in a similar way

From Eqs.(48，49), the command angle errors can converge to arbitrarily small neighborhoods around zero by properly choosing parameters c1 and c2 which can guarantee the stability of vehicle⁃actuator control system.

3 Simulation

To verify the effectiveness of the proposed control scheme, several simulations are carried out in this section. The winged⁃cone simulation model, a widely used simulation model of verification actuator constraints, is used to verify the effectiveness of the attitude control system proposed in this paper. The desired attitude command is expressed as Θc=[αc,βc,σc]Τ . The initial attitude and attitude angular rates are chosen as Θo=[0,0,0]Τ and ωo=[0,0,0]Τ , respectively. The parameters of actuator dynamics^[23] are chosen as κ1i=900 and k2i=35 . The deflection angles and angular rate of control surfaces are limited within ±30 °, ±100 °/s, respectively. Other parameters of control scheme proposed in this paper are as follows: k1=9.1 , k2=0.9 , k3=15 , k4=5.7 , kf1=2.5 , kf2=4 , ζi1=1 , ζe=77 , li1=0.5 , τi1=0.06 . Other initial states about this vehicle are H0=40 km, τ=118.5 °, δ=32 °, V0= 3 km/s, χ0=0 ° and γ0=0 °.

In simulations tests, a first⁃order filter is adopted to smooth the reference signals

where xr , xc and Tr are output state of the filter, input command and time parameter, respectively. The selection of the parameters Tr is related to the dynamic performance of the control system. In consideration of the control system design of this paper, it is appropriate to set Tr=2 .

In order to compare the control effect, a conventional adaptive dynamic surface control scheme(ADSC) with parameter estimation and actuator compensation in Ref.[21] is used in the simulation tests. Ref.[21] proposed the control scheme ADSC focusing on the flight longitudinal control problem of an HSV with aerodynamic uncertainties and input constraints. The main difference between the two schemes is that the DOABC uses tanh function to design auxiliary compensation system. So the two have the same simulation parameters except ζe . Next, the control architectures DOABC and ADSC are tested in two cases, the normal and the uncertainty case. There are 16 design parameters of the controller and the observer in the simulation system and the calculation during one step time is non⁃iterative. The computer we used is Intel I5 processor, 1.8 GHz. The fourth order method of Runge‑Kutta based on MATLAB is used. The simulation step size is 0.02 s. The simulation time is set as 40 s. The actual average running time of 2 000 steps is about 18.57 s. Therefore, it shows that the control method presented is suitable for real⁃time computation of flight control computer.

3.1 Case 1: Normal case

First, the normal cases of the ADSC and DOABC are implemented to test the dynamic performance when the actuator is within the range of constraints.

The corresponding results with attitude tracking and the amplitude and rate of actuator deflection are shown in Figs.2—4. Fig.1shows the three⁃dimensional trajectory of the maneuvering flight of a vehicle. As depicted in Fig2, the attitude angles stably follow the desired commands Figs.3—4 show the difference between the amplitude and the rate of actuator deflection of the two control architectures.

Fig.1 Three⁃dimensional trajectory

Fig.4 Rate of actuator deflection of Case 1

Fig.2demonstrates the deflection of actuator amplitude. The third subgraph of Fig.2shows actuator deflection of control scheme ADSC and DOABC. It is obvious that the actuator is accompanied by obvious bucket vibration for ADSC. However, it should be noted that the rapid deflection may cause the excessive wear of the actuator and other adverse damage for the vehicle. From the comparison above, we can see that the deflection of actuator of the control scheme DOABC shows better smoothness. A similar situation prevails in Fig.3. Severe rate changes are most likely to cause actuator failure. The architecture and new algorithm DOABC proposed in this paper effectively avoid this situation. The actuator’s rate change is smoother. It also shows the effectiveness of the architecture and algorithm proposed in this paper when dealing with the amplitude and rate saturation of the actuator.

Fig.2 Attitude angle tracking of Case 1

Fig.3 Amplitude of actuator deflection of Case 1

3.2 Case 2: Uncertainty case

In this section, the dynamic performance of ADSC and DOABC is tested to validate the aggregation effect of aerodynamic uncertainty and actuator constraints. Therefore, the aerodynamic uncertainty in simulation is selected as ±20%fi randomly. With the same input command, Figs.5—7 show the attitude angle tracking effect and the deflection of the amplitude and rate of the actuator. Fig.5shows that the tracking performance of two control architectures is great and stable. It can be seen that the two architectures have good results in dealing with uncertainty suppression. The tremendous difference in actuator deflection is shown in Figs.6—7.

Fig.5 Attitude angle tracking of Case 2

The actuator amplitude of control scheme ADSC exhibits a violent shake in a small range, as shown in Fig.6. The strenuous bucket vibration of the actuator not only brings wear to the actuator, but also leads to the instability and even damage of the vehicle. Instead, the actuator deflection of the control architecture DOABC shows absolute smoothness and almost no shake. This is more evident in Fig.7. We can observe the rate variation of the actuator and its violent behavior that rapid shifts from one side to the other is what we call reverse RS. However, in the control architecture and algorithm proposed in this paper, RS phenomenon which usually occurs at the moment when the angle of attack begins to change occurs only in a few points. In summary, as shown in the Figs.5—7, the control architecture and algorithm presented herein demonstrate superior performance in processing actuator saturation.

Fig.6 Amplitude of actuator deflection Case 2

Fig.7 Rate of actuator deflection of Case 2

Remark 6 We must realize that this shake and the occurrence of reverse RS saturation of the actuator are due to the inadequate design of the previous anti⁃saturation auxiliary system and the overall control architecture. The new control architecture and algorithm proposed in this paper takes a step forward in solving this problem. Furthermore, we know that only by improving the algorithm and control architecture can we solve this kind of constraint problem further.

4 Conclusions

This paper investigates the actuator amplitude and rate constraint problem occurring at the maneuvering flight of a HSV with aerodynamic uncertainties. An adaptive backstepping based disturbance observer with anti⁃saturation auxiliary compensation system control strategy is proposed to deal with such problems. Simulation results show that the proposed control architecture has a satisfactory performance in handling actuator amplitude and rate constraints compared with previous algorithms. For future work, as shown in this paper, the tanh function plays a key role, and other applications about it still attract research interests. Besides, the adaptive adjustment of the parameter ζe is worth exploring and will be able to improve the control effect of the presented method.

References