#### Abstract

This paper investigates the design problem of the attitude controller for air-breathing supersonic vehicle subject to uncertainties and disturbances. Firstly, the longitudinal model is established for the attitude controller design which is devised as a strict feedback formulation, and a transformed tracking error is derived with the prescribed performance control technique such that it can limit the tracking error to a predefined region. Then, a novel linear active disturbance rejection control scheme is proposed for the attitude system to enhance the steady-state and transient-state performances by incorporating the transformed tracking error. On the basis of the Lyapunov stability theorem, the convergence and stability characteristics are both rigorously proved for the closed-loop system. Finally, extensive contrast simulations are conducted to demonstrate the effectiveness, robustness, and advantage of the proposed control strategy.

#### 1. Introduction

Hypersonic technique is a wide development prospect in both military and civil fields [1, 2]. As a derivative of hypersonic technique, air-breathing supersonic vehicle (ASV) is a new type of aircraft with large airspace, super flight velocity, long range, and high precision compared with the traditional aircraft [3], and flight guidance-control design is a key technology in the ASV system. However, as the special structure and changeable flight conditions, the vehicle dynamics are peculiar including the fast time varying, nonlinearity, uncertainty, and multiple disturbances [4], which lead to more difficulties and complexities in the control design and analysis. In addition, the ASV usually attacks the target at a supersonic velocity such that the flight states are changed rapidly which imposes a higher requirement for the transient characteristic of the control system [5]. Therefore, the control performance index such as the overshoot, steady-state error, convergence rate, and robustness must be considered the major designed indexes in the control system design of such flight vehicle.

In the past few years, the air-breathing vehicle technique has attracted the wide attention of domestic and foreign scientific research institutions and scholars due to the aforementioned superiorities, and various control methods have been explored for such vehicles, such as linear parameter varying (LPV) method [6], dynamic inversion [7, 8], trajectory linearization control [9, 10], fuzzy control method [11], dynamic surface control [12, 13], back-stepping control method [14, 15], neural network [16, 17], and sliding mode control [18–21].

It is worth noting that the above listed control method achieved an outstanding control effect for air-breathing vehicle, but most mainly fasten the attention on the stability, robustness, and accuracy of the control system. However, the situation of autodisturbance rejection is seldom considered in control design. As we all know that the active disturbance rejection control (ADRC) method can achieve a satisfactory performance for nonlinear systems, where the parameter perturbations and external disturbances can be estimated and rejected actively, it is successfully applied in industrial control [22, 23]. However, there is an inevitable problem that many parameters are needed to be tuned which limits the application of ADRC in practice. In this connection, a linear active disturbance rejection control (LADRC) was firstly designed by Prof. Gao [24] which is extremely simple and easily implementable and it has been widely extended to various industrial control fields, such as electric erection system [25], hovercraft system [26], servo systems [27], wind power systems [28], and photobioreactor [29]. Meanwhile, [30, 31] adopt the LADRC approach to design the attitude control for a supersonic missile. As far as we know, there are few literatures which concern with the attitude control system design by adopting the LADRC method. Furthermore, in the design process of the above LADRC scheme, both did not consider the transient performance design. Therefore, it is vitally essential for the ASV attitude control system design that can guarantee the system for both the steady-state performance and transient performance with LADRC method.

Recently, a newly emerging control method called the prescribed performance control (PPC) for the nonlinear system was firstly proposed by Na et al. [32], where the tracking error can converge to an arbitrarily predefined small set and the convergence rate and maximum overshoot can be delimited less than a prespecified constant. Owing to the significant advantages in improving the control performance, the PPC is introduced to the vehicle suspension system [33], manipulator system [34], servo mechanisms [35, 36], unmanned aerial vehicle [37], spacecraft [38–40], etc., and several new ideas of the PPC design are emerged; [41] proposes a new error transformation method and the performance function so that the limitation of PPC on the known initial error can be relaxed. [42] investigates a new PPC methodology for the longitudinal dynamic model of an air-breathing hypersonic vehicle via neural approximation that the satisfactory transient performance with small overshoot can be achieved. [43] presents a new error transformation to reduce the complexity of the control system which is caused by conventional error constraint approaches. Berger [44] proposes a novel prescribed performance controller which can guarantee the output to stay within a prescribed performance funnel bound by incorporating the funnel control with the PPC technique. [45] designs a new prescribed performance controller without an approximation structure which can avoid the large amount of calculation and some specific problems of the fuzzy or neural network method in the approximation process. Although extensive achievements have been yielded in theory and application of the PPC technique, the previous controller design methods are mainly neural network control, backstepping control, dynamic surface control, and sliding model control. To our best knowledge, no results on the LADRC design with the PPC technique have been reported for nonlinear systems.

Motivated by the aforementioned discussions, this study investigates a novel LADRC scheme for the longitudinal attitude mode of the air-breathing supersonic vehicle in the presence of uncertain dynamics and external disturbances with a prescribed performance constraint. The main contributions of this paper are summarized as follows: (1) A novel LADRC approach design method with a prescribed performance constraint is firstly proposed for the ASV attitude system with multiple disturbances. (2) The proposed control scheme does not need the knowledge of the flight dynamic model, and the uncertainty and disturbance can be actively estimated and compensated into the control signal. (3) The proposed controller can improve the steady-state and transient performances of the ASV attitude system compared with the traditional LADRC method. (4) The system stability and convergence characteristic are both proved strictly.

The rest of this paper is outlined as follows. The considered ASV attitude control model is established and the prior knowledge and problem formulation is given in Section 2. Section 3 presents the LADRC-based prescribed performance attitude controller design procedures. Comparative simulation results are provided in Section 4. Finally, some conclusions of this work are presented in Section 5.

#### 2. Problem Statement and Preliminaries

In the section, the considered ASV attitude dynamic model is presented; then, some basic definitions of the prescribed performance control method are provided for the subsequent analysis, and finally, the objective of controller design is outlined.

##### 2.1. Vehicle Attitude Model

For the subsequent design, the nonlinear longitudinal model of the ASV attitude system in this paper is derived on the basis of the model which includes the altitude, velocity, and attitude motion of the hypersonic vehicle modeled in [46, 47]. In addition, the main purpose is to investigate a novel attitude control method of ASV, and the attitude dynamics belongs to a fast motion compared with the change of velocity and altitude. With design simplification and no loss of generality, the changes of altitude and velocity can be ignored due to the limited influence on attitude dynamics. Hence, the ASV attitude system can be simplified as follows: where the pitch moment is defined as where elevator deflection is the control input, and , , , and are the aerodynamic coefficients.

For simplicity, defining the state and combining with (1) and (2), the ASV attitude control system can be described with the strict feedback state-space form: where

Here, the term is described as “total disturbance,” which consists of the unknown dynamic uncertainty and external disturbance.

*Remark 1. *The uncertainty term is mainly the result of parameter uncertainties and external disturbances, e.g., the aerodynamic parameters perturbations and wind interference.

*Assumption 1. *The total disturbance is differentiable, and its derivative is described as . The disturbance in system (3) is unknown, but the disturbance and its derivative are all bounded.

##### 2.2. Prescribed Performance Theory

The prescribed performance denotes that the prescribed transient and steady-state performances can strictly limit the tracking error to a predesigned small residual set. Based on the prescribed performance concepts [31], the prescribed performance can be obtained in the condition that the tracking error evolves in the predefined bounds with a decreasing smooth function as follows:

The prescribed performance function (PPF) is selected as [38] where and . The maximum overshoot, steady-state error, and convergence rate will be limited by selecting the appropriate parameters , , and , respectively. For a more intuitive insight of the concept, Figure 1 illustrates the schematic diagram of the aforementioned prescribed performance theory.

It is worth to mention that the complexity of the controller design will be significantly increased by directly adopting (5) and (6) which correspond to an additional constraint of the controlled system. In order to evade this problem, the system with constrains can be converted to an equivalent disengaged one by introducing an error transformation function , i.e., where is the transformed tracking error, and is smooth and strictly increasing which satisfies the conditions and .

Here, we design the error transformation function as

According to the property of the function , we can inversely derive the error as follows: where is the normalized tracking error.

*Remark 2. *In the subsequent parts, the transformed error will be used into the controller in place of the tracking error to deal with the problem of control system design with prescribed performance constraint.

##### 2.3. Control Objective

The control objective is the proposed novel controller for the ASV attitude system (3) that the state track the desired command accurately, and the tracking error can be limited within a predefined bound with a satisfactory prescribed performance in spite of multiple disturbances including unmodeled dynamics, uncertainties, and external disturbances, i.e., the objective can be expressed as follows: (1)The state can accurately track the desired command with the unknown multiple disturbances(2)The output tracking error is stabilized at the origin with a prescribed maximum overshoot, the steady-state error, and convergence rate(3)The closed-loop system states are both stable and robust to the uncertainties and disturbances

#### 3. Control System Design and Stability Analysis

In this part, a novel ASV attitude controller is proposed based on the LADRC method by introducing the PPC technique, which can improve the steady-state and transient performances of the ASV attitude control system. Then, the convergence property and stability analysis for the control system are provided on the basic of the Lyapunov method.

##### 3.1. Attitude Control System Design

The block diagram of the presented control system is depicted in Figure 2, which mainly consists of three parts: prescribed performance control (PPC), linear extended state observer (LESO), and linear state error feedback (LSEF).

In view of the previous description, the design process of the ASV attitude control system is illustrated as follows:

Step 1.

The LESO is the core part of the LADRC; it can generate the estimation of the states and the disturbances in real time; the estimated value can be used to compensate the disturbances to the controller which can enhance the robustness of the system. According to system (3), a three-order LESO is constructed as follows: where , , and are the observer states, and , , and are the designed gains which can be chosen with the pole-placement method [24] as follows: where is the observer bandwidth.

With the properly selected gains, , will converge to the system states and , respectively, and will accurately track the total disturbance term , i.e., .

Note that the LESO can estimate the states and total disturbance exactly without the mathematical precise model, it is only dependent of the system input and output information.

Step 2.

The PPC can transform the tracking error into an equivalent unconstrained one by incorporating the transformation function and the prescribed performance function such that the tracking error can be limited in the envelope of the prescribed performance bounds (PPB).

Here, we define the error as where is the desired command and is the estimation value of obtained by LESO.

Subsequently, the normalized error is given by where is a similar PPF defined in (6), and it is expressed as where,, and are all positive constants.

Then, the transformed error is obtained by

Step 3.

Based on the accurate estimation and the designed transformed error, the LSEF can approximately simplify the system to a disturbance-free form meanwhile reconciling the performance of the system. The final control law is design as follows: where term denotes a linear state error feedback control term that guarantees the system is asymptotically stable, and term represents the dynamic compensation control term to suppress the unfavourable consequence of the total disturbance such that it can enhance the robustness of the ASV attitude control system.

Here, the control subitem adopted the linear proportional and derivative (PD) control framework by introducing the transformed error of PPC which is designed as follows: where , are the controller gains and is the estimation value of obtained by LESO, and expression of is given in (3).

Meanwhile, the control subitem compensates the disturbances with the estimated value , which is given by

Thus, the controller is obtained in terms of (16), (17), and (18) as follows: Note that the controller introduces the transformed error instead of which can enhance the transient and steady-state performances with the PPC technique.

##### 3.2. Stability Analysis

In this subsection, the observer convergence and the closed-loop system stability will be analysed with the listed theorem. Prior to investigating, the following lemmas are introduced.

Lemma 1. *For system , where is a matrix, and , , if and is Hurwitz, then holds.*

This lemma has been proofed in detail which can be seen in [48].

Lemma 2. *If the transformed error can be controlled to be bounded, i.e., the condition holds where is constant; then, the tracking error can be controlled within a prescribed boundary, i.e., holds.*

This lemma has been proofed in detail which can be seen in [38].

Theorem 1. *For system (3) and LESO (10), if satisfies the Lipschitz condition in the definition domain, there exist a constant such that the estimated states , , and can converge to the state , , and , respectively, i.e., the observer errors satisfy that , where the observer errors are defined as , , and .*

*Proof. *Define the observer errors as

From (3) and (10), the observer error dynamic can be given by

Let and , then (21) can be rewritten as follows: where , .

As satisfies the Hurwitz stability, the condition holds, where the matrix is 3-order identity matrix and is a positive definite Hermitian matrix and is expressed as

Design the following Lyapunov function candidate

Substituting (23) into (24) yields

The time derivative of along (25) is

The term satisfies [49]. where is constant.

Meanwhile, as the equation holds, we can obtain that

In addition, the following inequality holds Thus, we obtain where .

Based on (26), (28), and (30), we can obtain that

That is, if such that . Therefore, can be achieved, i.e., the estimation errors of the LESO (10) are asymptotically stable. The proof of Theorem 1 is completed.

Theorem 2. *For system (3), LESO (10), and the proposed controller (19) with the prescribed performance function (14), if satisfies the Lipschitz condition in definition domain, there exists constants and that can guarantee the system (3) to be asymptotically stable. Furthermore, the tracking error can be maintained in a predefined set, i.e., the inequality holds.*

*Proof. *Define the tracking error of system (3) as

As the control command is assumed as a constant, we have . The time derivative of and along (3) is

Based on the expression of the transformed error , it yields

Then, the function and can be derived using Taylor’s expansion as follows: where is the 2-order Taylor remainder. As the error is relatively small near zero, thus the can be neglected.

Substituting (36) into (35), the transformed error can be approximately expressed as

Substituting (38) into (19), the proposed controller (18) can be rewritten as

Substituting (39) into (34) yields

Combining the expressions of the observer error and system error, we can obtain that

Substituting (41) into (40) yields

From (33) and (44), the tracking error model is given by where and are the system tracking error vector and the observer error vector, respectively. The matrix and are described as

Meanwhile, the prescribed performance function satisfies

Thus, under the condition , the matrix and can be given as

According to Theorem 1, we can obtain that . Furthermore, if the characteristic polynomial satisfies the Routh criterion, here, we select , where is the controller bandwidth, i.e., such that the matrix is Hurwitz. Based on Lemma 1, we can obtain that the holds, i.e., the system (3) is asymptotically stable.

Besides that, we can obtain the following equation by conducting the inverse operation based on the transformed error (15) as

Based on Lemma 2, if the transformed errors can be controlled to be bounded, i.e., holds for positive constants . This further implies

From the fact and , the error can be maintained within a predefined set, holds. This completes the proof.

#### 4. Simulation Results and Analysis

In order to evaluate the performance of the proposed ASV attitude controller, three numerical simulation cases are conducted in different configurations and scenarios.

##### 4.1. Contrast Scheme

In order to demonstrate effectiveness and superiority of the proposed controller with the prescribed performance control-based LADRC (LADRC-PPC) method, the LADRC method [30] is introduced to the simulation scenarios for comparison study, and the controller is devised as follows: where , , and is obtained by LESO.

##### 4.2. Flight Conditions

The vehicle is assumed to maintain altitude km cruise at constant velocity m/s such that the flight path angle is , i.e., the variation of the angle of attack is the same as that of the pitch angle based on the system (1). The aerodynamic coefficients are , , _{,} and . The initial conditions for the ASV are set as , .

##### 4.3. Controller Parameters

The relevant parameters of PPF is designed as , , and , i.e., the PPF is selected as . The observer bandwidth is set as , and controller bandwidth is designed as . Then, the observer gains , , and and controller gains , can be obtained according to the correspondence with the bandwidth and .

##### 4.4. Evaluation Index

For quantitatively contrasting the performance, the following performance indexes are introduced to evaluate the above control schemes. (1)Convergence time (CT) of the tracking error (TE) The convergence time is assumed as the corresponding time that the tracking error is and continues at least 10 sampling periods.(2)Average value (AV) of the TE where is the sample point number(3)Standard deviation (SD) of the TE (4)Amount of control consumption (ACC) where and are the start time and end time, respectively, is the control input.

*Case 1. *This simulation is conducted in a standard condition. The aerodynamic coefficients are set to be the nominal values without external disturbance, i.e., the term in system (3) is .

The comparative results are summarized in Table 1 and Figures 3–10.

As depicted in Figures 3 and 4, the ASV can track the desired attitude command with the listed control approaches accurately. It is worth noting that the tracking error is limited in a prescribed set with a fairly satisfactory tracking error response by utilizing the presented LADRC-PPC controller. As listed in Table 1, the tracking error remains remarkably small with a faster rate and a much smaller average error which the average error is 0.03 deg, and converges to the neighbourhood of zero in approximately 0.99 s with the LADRC-PPC controller. However, the corresponding values with the LADRC scheme are 0.07 deg and 2.55 s, respectively. That is, the steady-state and transient performances of the ASV attitude system under the designed LADRC-PPC controller is obviously superior than the contrast LADRC which is owing to the help of introducing the PPC technology. We can see that initial value with the LADRC-PPC controller is slightly larger than the one with LADRC which is due to the faster convergence in the initial phase with the LADRC-PPC approach as shown in Figures 5 and 6, and the amount of control consumption under the two schemes is virtually identical, i.e., the novel proposed LADRC-PPC scheme can achieve an excellent control performance with approximately the same control consumption in comparison with the LADRC method. Figures 7 and 8 illustrate the estimations of disturbances; it can be seen that LESO can precisely and rapidly estimate the total disturbances of the system, and the convergence rate of LESO under the LADRC-PPC is faster than that of LADRC as show in Figures 9 and 10.

*Case 2. *This case is performed in a perturbed condition to verify the robustness of the proposed method. The uncertainty term is mainly considered two parts: (1) the model uncertainties , and the aerodynamic coefficients are presumed to decrease by 10% on the basis of standard values, the density of air is perturbed to be +5% and (2) external disturbance , and it is considered an abrupt one , i.e., the uncertainty term is set as follows

The comparative results are summarized in Table 2 and Figures 11–18.

From Figure 11, the ASV attitude system can track the desired command successfully by using the mentioned two control approaches in the presence of unknown dynamics, uncertainties, and disturbances. The tracking error response of the system with the proposed LADRC-PPC scheme is also better than the classic LADRC method in Figure 12. Especially, when the abrupt external disturbance occurs, the tracking error under the LADRC-PPC controller can converge to a compact set less than 0.5 s, and the corresponding convergence time under the LADRC approach is nearly 2.0 s. That is, the transient performance under the LADRC-PPC controller is significantly improved than the contrast LADRC in spite of unknown uncertainties and disturbances. From Figures 13 and 14, it can be seen that the variations of the states under the perturbed condition are generally accordant with the standard condition. Meanwhile, the amount of control consumption under the two controller is similar. In addition, the two controllers can achieve an excellent tracking performance owing to the total disturbances can be precisely estimated with the LESO as shown in Figures 15–18.

*Case 3. *For further evaluating the robust performance of the proposed control scheme, a Monte Carlo analysis consisting of sample runs is developed, and the parameter perturbations are considered
(1)There is a random variation between -20% and +20% obeying normal distribution for the aerodynamics coefficients(2)The perturbation of parameter and is -10% ~ +10% of the standard value, which follows the random variation with normal distribution

Figures 19–24 depicts the 1000-run Monte Carlo simulation results, and Table 3 illustrates the corresponding statistical results including the expectation value and average variance of the above indexes CT , AV , and ACC .

The Monte Carlo simulation results are depicted in Figures 19–24, and Table 3 lists the statistical results of 1000 Monte Carlo simulations which included the expectation and average variance of the concerned indexes CT , AV , and ACC .

Figures 19 and 20 depict the convergence time of the tracking error history for 1000 times with the mentioned two controllers, we can obtain that the expectation of CT with the proposed LADRC-PPC 1.07 s that is shorter than the one of the classic LADRC approach, and it can achieve a smaller dispersion with LADRC-PPC. The statistical results exhibit the similar regularity as shown in Figures 21 and 22. The distributions of the amount of control consumption with the above controllers are illustrated in Figures 23 and 24, and the expectation of under LADRC-PPC is 1.228, which is very close to the one of the LADRC method. The above results further verify that the proposed LADRC-PPC approach exhibits enhanced robustness despite multiple disturbances, and it can achieve an obviously superior steady-state and transient performances than the classic LADRC-PPC method.

#### 5. Conclusion

In this paper, a novel LADRC scheme is proposed for thr ASV attitude control system with multiple disturbances and prescribed performance constraint. The chief feature is that it introduces the PPC technique into the LADRC design process so that the tracking error can be strictly confined to an adjustable residual set with the prescribed steady-state and transient performances. Meanwhile, the LESO is employed to estimate the total disturbances such that it can effectively solve the unknown model dynamic problem. Theoretical analysis and contrast simulation results validate that a superior robust tracking performance of the ASV attitude system with the proposed control strategy design. Furthermore, both the transient and steady-state performances of the classical LADRC can be significantly improved by introducing the PPC method. Future work will consider the completely unknown dynamic control system design with actuator saturation via the presented control approach.

#### Nomenclature

: | Pitch angle |

: | Pitch angular rate |

: | Flight path angle |

: | Angle of attack |

: | Pitch moment of inertia |

: | Pitch moment |

: | Reference area |

: | Reference length |

: | Dynamic pressure |

: | Density of air |

: | Velocity |

: | Elevator deflection |

: | Exponential function with natural constant. |

#### Data Availability

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

#### Conflicts of Interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

#### Acknowledgments

The authors are grateful for the projects supported by the National Natural Science Foundation of China (Grant No. 11176012).