Velocity Control Based on Active Disturbance Rejection for Air-Breathing Supersonic Vehicles

This paper investigates a velocity tracking control approach for air-breathing supersonic vehicles with uncertainties and external disturbances. Considering angle of attack is difficult to be precisely measured in practice, extended state observer technique is introduced into the state reconstruction design. In order to avoid possible oscillations in the design of the traditional extended state observer (TESO), a modified extended state observer (MESO) is developed, where a new smooth function is proposed to replace nonsmooth function of TESO. On the basis of it, an active disturbance rejection controller (ADRC) is designed for velocity control systems. Simultaneously, the closed-loop stability is rigorously proved by using Lyapunov theory. Finally, numeric simulations are conducted to validate the effectiveness of the proposed method.


Introduction
Air-breathing supersonic vehicles (ASV) are becoming crucial in recent decades because they may provide a feasible and cost-efficient access to space for both civilian and military applications [1].In comparison to traditional rocket propulsion weapon systems, ASV represents a series of advantages of higher payload capacity, lower flight cost, and rapid global precision strike capability.However, most ASV adopt the design of airframe integrated with scramjet engine configuration [2], which leads to strong couplings between the flight attitude and propulsion.For instance, the compression of the flow through the scramjet engine inlet depends on the characteristic of the bow shock wave under the vehicle fore-body, which is mainly up to the angle of attack (AOA).In addition, the dynamic model built by using aerodynamic experiments is usually imprecise due to the unmodelled dynamics and external disturbances in practice.Thus, the control design of such flight control systems is a challenging issue due to its high nonlinearities, strong couplings, and significant uncertainties.
In recent decades, plenty of control strategies have also been explored for air-breathing hypersonic vehicles (AHV), for example, feedback linearization [3], backstepping [4], adaptive control [5,6], and sliding mode control (see [7,8] and references therein).Generally speaking, feedback linearization is an effective means to analyse stability of the nonlinear system, and it was used to design AHV control system in [3].However, the feedback linearization excessively depends on the accurate information of the dynamic model, and it cannot deal with the unknown changes of the dynamic system.Additionally, a backstepping control scheme is proposed by combining the dynamic surface control technique in [4], but there still exist the fatal shortcomings of "explosion of term" in the backstepping design, because of the repetitive computation of differentiation for virtual control laws.From [5,6], adaptive control technique has provided a new way to design the control system.However, adaptive controllers always have overly complicated computation and the unknown parameters of system need to be recognized online, which cannot be implemented in engineering.Among previous control approaches, sliding model control (SMC) attracts extensive attention due to its simplicity and robustness to parameter uncertainties and external disturbances [7,8].Unfortunately, there are also some disadvantages in SMC, and the well-known one is chattering phenomenon, which limits application of SMC in the AHV control system.

Complexity
It is worthwhile to mention that the active disturbance rejection controller (ADRC) has been well developed as an effective robust control strategy to achieve satisfactory performance for nonlinear systems with uncertainties and external disturbances [9,10].Compared with the methods mentioned above, ADRC does not depend on the accurate information of unknown dynamic model.It can retain better static and dynamic performances and stronger robustness and adaptability, where a nonlinear control strategy is designed by estimating and compensating the internal and external disturbances in real time.Thus, the ADRC is widely applied in industrial control, for example, motion control [11,12], microelectromechanical gyroscopes [13,14], wind energy system [15,16], and robot control [17,18].In recent research, the concept of ADRC has also been applied in the field of AHV [19][20][21].Reference [19,20] presented ADRC control schemes for AHV attitude tracking control system.Based on this work, a novel ADRC approach was designed for hypersonic vehicle attitude tracking system by introducing trajectory linearization control technique in [21].In all these attitude system control designs for AHV, there are two aspects of issues.One is that AOA is assumed to be precisely measurable.In practice, AOA is unfortunately difficult to be measured.The other is that in the framework of ADRC the extended state observer (ESO) is used to estimate the lumped uncertainties without any parametric condition [22,23], and estimation error can converge to zero in the specified condition.Nevertheless, the traditional ESO (TESO) is not able to obtain high precision and enough smooth response because the general nonlinear function used in TESO is nonderivable at the dividing point.Consequently, there is a potential unsatisfactory chattering phenomenon in the estimated dynamics if the width of fraction is fairly small [24].
In terms of these two issues, the objective of this paper is to further study the ADRC design for nonlinear ASV velocity systems subject to parameter uncertainties, external disturbances, and immeasurable AOA.Firstly, we investigate a MESO by introducing a new smooth function to obtain more precise estimation of lumped unknown dynamics.Then considering the AOA cannot be measured directly, the MESObased reconstruction design of AOA is conducted in the ADRC velocity systems, and the closed-loop system stability is also proved based on Lyapunov theory.Finally, extensive simulations are conducted to verify the performance of the proposed control method.
The remainder of this paper is organized as follows.Section 2 describes the longitudinal dynamic model of ASV and states the problem formulation and some preliminaries.In Section 3, a MESO is developed by using a new smooth function and the MESO convergence is analysed.In Section 4, the ADRC-based controller is designed in detail and the closed-loop system stability is also proved.Comparative simulations are conducted in Section 5 and some conclusions are involved in Section 6.

Problem Formulation
This paper will investigate the velocity control design for air-breathing supersonic vehicle systems.A typical velocity control process of nonlinear air-breathing supersonic vehicle systems can be given in Figure 1.The task of the velocity controller is to calculate the control   (i.e., fuel mass flow) in terms of the desired velocity command   (i.e., control input) and current velocity .The thrust force  is produced by the scramjet engine system, which can be decided by the velocity, fuel mass flow, and angle of attack .The current velocity is measured by inertial navigation system while angle of attack is not precisely measurable in practice.Thus it is essential to reconstruct the angle of attack by using the measurable attitude, which will be discussed in the next section.
With design simplification and without loss of generality, we only consider the longitudinal dynamics of ASV here.The nonlinear dynamics can be described by a set of differential equations in terms of velocity , flight-path angle , altitude ℎ, angle of attack , pitch rate   , and vehicle mass , respectively.Define the state vector [ 1 ,  2 ,  3 ,  4 ] = [, , ,   ], and the nonlinear dynamics can be written in a state-space form [25] as follows: Here where ,   represent the acceleration of gravity and the moment of inertia, respectively; fuel mass flow   is the control;   means the total disturbance of velocity system.The lift , drag , thrust force , and pitching moment   can be formulated as where dynamic pressure  = 0.5 Owing to the couplings between the aerodynamic force and the propulsion system, a precise curve-fitted approximation of thrust coefficient is expressed as where the coefficients and  0 depend on the altitude ℎ, Mach number Ma, and .
Meanwhile, the parameter uncertainties and external disturbance are considered in this study, and the parametric uncertainties are depicted as a perturbation Δ to its nominal values; that is, where subscript 0 denotes the nominal value.The total disturbance   is unknown and bounded with the parameter uncertainties and external disturbances  in the velocity system, which is expressed as

Modified ESO Design
In this section, MESO is developed based on a new smooth function to prevent undesirable chattering phenomenon in TESO, and the MESO will be employed to address the unknown lumped dynamics and the unmeasured AOA.The design of MESO can be processed as follows.
To begin, the concept of the traditional ESO is presented, and an th order nonlinear dynamic system is considered as where (⋅) is the function of the system states  and unknown disturbance .
By introducing an extended state  +1 , system (7) can be rewritten in a state-space form; that is, where the uncertain item ẇ is unknown and bounded.Consequently, we can estimate the uncertain item by using state observer technique, which is defined as follows: . . .
where  1 =  − ẑ1 is observer output error, and ẑ+1 is estimation value of the unknown disturbance. 0 is the th observer gain and   ( 1 ) represents a set of suitably constructed nonlinear functions satisfying  1   ( 1 ) > 0, ∀ 1 ̸ = 0 and   (0) = 0.If the nonlinear functions   (⋅) and the related parameters are chosen properly, the estimated state variables ẑ are expected to converge to the respective states of the system   ; that is, ẑ →   ,  = 1, 2, . . .,  + 1.
Obviously, the nonlinear function is crucial for the design of ESO [26], and it can be expressed as where  > 0 and 0 <  < 1.
It is easily seen that fal( 1 , , ) is not derivable in the dividing point ±.Furthermore, if  is too small, the derivative of function fal( 1 , , ) will become nonsmooth, which will degrade the performance of the system and even seriously trigger divergent phenomenon.
Consequently, it is imperative to find a smooth and derivable function fal 2 ( 1 , , ) to replace the traditional fal( 1 , , ), and the modified function fal 2 ( 1 , , ) is depicted as follows.
When | 1 | > , fal 2 ( 1 , , ) is same as (10) such that When Complexity subject to where (13) can guarantee that ( 12) is smoothly convergent to zero; exp(⋅) is the exponential function based on the natural constant .Substituting ( 12) into ( 13), then we can obtain that Then, we analyse the convergence of the general secondorder MESO based on the self-stable region method [27] by introducing the following lemmas.8)-( 9), the MESO errors are modelled as

Lemma 1. In terms of the defined observer estimation errors
where  01 ,  02 are the observer gain.
(2) For ( 1 (),  2 ()), which is out of the self-stable region , for example, Introduce the function  2 as follows: The time derivative of  2 is conducted as (30) According to ( 24) and (30), we can obtain that Thus, the error of the observer will converge to zero, that is,  1 () → 0,  2 () → 0 ( → ∞).This completes the proof.In terms of ( 11)-( 12), the functions   ( 1 ) are According to Lemma 2, the self-stable condition of ( 15) is Therefore, the MESO can be employed to estimate the total disturbances and reconstruct the unmeasured AOA.

Control System Design
In this section, a robust control method is proposed based on the ADRC technique to handle parameter uncertainties and external disturbances.It is assumed that the attitude had been adjusted to the command value when velocity control design is proceeding; that is, the velocity is mainly related to fuel mass flow   since the thrust force  is affected by .Additionally, AOA reconstruction method is developed by using the available input/output information of attitude system via MESO.

AOA Reconstruction.
Practically, the deflection of the control surface   and the pitch rate   are measurable, but the AOA  is difficult to be precisely measured.To address this problem, the modified extended state observer technique is introduced into the design of the reconstruction for AOA.From the AOA reconstruction depicted in Figure 2, it can be seen that this strategy only needs the information of the measured   and   .
The AOA reconstruction based on the MESO is formulated as where  0 ( = 1, 2, 3) is the observer gain.If  0 is chosen appropriately, the state variables of MESO will converge as follows: Obviously,  is reconstructed by  2 without  2 ; that is, α =  2 .Thus, the reconstruction  2 can be used in the velocity system instead of the measured AOA.

Velocity Controller Design.
In this subsection, the ADRC is designed for the velocity control of nonlinear ASV system in the presence of uncertainties and disturbances.From the structure of the ADRC for ASV velocity control shown in Figure 3, it can be seen that the ADRC system mainly includes three parts.In part 1, MESO is designed to estimate the total disturbances.Then, a nonlinear state error feedback (NLSEF) law is conducted to obtain the initial control law  0 according to the error between state values and command values in part 2, and the velocity control law  is implemented by augmenting the NLSEF law  0 with the disturbance compensation  V2 which is completed by MESO in part 3.
In this way, the velocity can be controlled in set value, and the ADRC design procedure of velocity control is presented in detail as follows.

Total Disturbance Estimation.
The key of this design is to interpret the total disturbance including the uncertainties and disturbances as an additional state to be estimated.Different from the traditional ADRC, the MESO with the modified smooth function fal 2 ( 1 , , ) is adopted to estimate the total disturbances   .For this purpose, we add an extended state  5 as the total disturbance   which is calculated by (6), and then the velocity system is rewritten as where  V =  1   and the function ℎ() is the derivative of   , which is unknown and bounded.Then a second-order MESO is proposed as where the observer gains are  01 > 0,  02 > 0, and corresponding to (24) the stability condition of MESO is satisfied.That is,  V1 →  1 ,  V2 →   can be guaranteed in finite time.In this case,  V2 is the estimated value of the total disturbances; that is, the estimated errors can converge to zero in finite time.This indicates that an improved tracking performance can be achieved by using the estimated state  V2 in the controller design to compensate for the total disturbances   .

Nonlinear State Error Feedback Law.
A nonlinear feedback control law is the nonlinear combination of the errors between the command and the real state.Here, by introducing fal 2 ( 1 , , ), the feedback control law is selected as where  * 1 =   is the velocity command and  is the ADRC parameter.

Disturbance Dynamic Compensation.
In order to reduce the influence of the disturbances, the disturbance dynamics need to be substituted into the control law based on the estimated state  V2 which is obtained by the MESO.According to the structure of ADRC, the final control law of system (36) can be expressed as Then the control 4.3.Stability Analysis.This subsection will study the stability and the convergence of the closed-loop system.Define the estimation errors of total disturbance Substituting (39) into (36), then the velocity control system is rewritten as Choose the following Lyapunov function: Then calculate the time derivate of  3 ; that is, that is, the control system is stable.

Simulation Results and Discussions
In this section, a numerical simulation is conducted to illustrate the effectiveness of the control method and the state  reconstruction strategy.In order to show the performance improvement of the MESO, the total disturbances of the system are estimated by comparing with TESO method.Three conditions with uncertainties and external disturbance are listed in Table 1.
As shown in Figure 4, the system state  can accurately and quickly track the given command.Moreover, the tracking error remains remarkably small and converges to zero asymptotically in all cases; that is, a fairly satisfactory tracking error response is achieved with no steady-state error, which indicates the effectiveness of the suggested control method.Figures 5 and 6 show that the AOA reconstruction error rapidly converges to zero with a maximal error about 0.1 ∘ , which indicates that the developed state reconstruction method is efficient by using the MESO.From the altitude and flight-path angle histories presented in Figures 7 and 8, it is apparent that there are no significant differences among the listed cases.Notably the control   is fairly smooth without chattering phenomenon (see Figure 9).From Figures  10-13, it is shown that the suggested MESO can estimate the total disturbances.Meanwhile, a fairly satisfactory estimation error response is achieved, while both the transient and steady-state performance are retained with the MESO.Also it is found that TESO leads to significant estimation steady-state error, and the convergence rate is slower than that of MESO.Furthermore, the influence of uncertainties and disturbances in the flight control system is inhibited by compensating the estimated value into the control system, which is owing to the precise and rapid estimation of the total disturbances via MESO.Consequently, the simulation results demonstrate that the proposed control scheme achieves good velocity tracking performance and enhanced robustness against parameter uncertainties and external disturbances.

Conclusions
In this paper, a novel robust controller is addressed for the velocity system of a generic ASV.A MESO is proposed by developing a new smooth function to replace the traditional nonderivable function.Then an ADRC technique is introduced to track the desired velocity by using the MESO to estimate and compensate the total disturbance so as to improve the robustness performance against the parameters uncertainties and external disturbances.Additionally, a state reconstruction method by utilizing the available input/output information is presented based on the MESO to reconstruct the unmeasured AOA.Simulation results demonstrate the effectiveness of the suggested control approach.The extension of presented control method to a flexible ASV and the influence of structural flexibility on control system will be potential areas of further research.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Figure 1 :
Figure 1: Velocity control process of ASV systems.

Figure 2 :
Figure 2: The structure of AOA reconstruction.

Figure 5 :
Figure 5: Reconstruction of angle of attack.

Figure 6 :
Figure 6: The reconstruction error of angle of attack.
2, and  and  denote reference area and length, respectively. is density of air;   ,   , and   define the lift, drag, and thrust force coefficient, separately.   ,   represent the moment coefficient due to the angle of attack, pitch rate, and deflection of the control surface   , respectively.Here,   and   are usually obtained by wind tunnel test.

Table 1 :
Parameter uncertainties of different cases.