Research Article  Open Access
Active Disturbance Rejection Control for AirBreathing Hypersonic Vehicles Based on Prescribed Performance Function
Abstract
Aiming at the longitudinal motion model of the airbreathing hypersonic vehicles (AHVs) with parameter uncertainties, a new prescribed performancebased active disturbance rejection control (PPADRC) method was proposed. First, the AHV model was divided into a velocity subsystem and altitude system. To guarantee the reliability of the control law, the design process was based on the nonaffine form of the AHV model. Unlike the traditional prescribed performance control (PPC), which requires accurate initial tracking errors, by designing a new performance function that does not depend on the initial tracking error and can ensure the small overshoot convergence of the tracking error, the error convergence process can meet the desired dynamic and steadystate performance. Moreover, the designed controller combined with an active disturbance rejection control (ADRC) and extended state observer (ESO) further enhanced the disturbance rejection capability and robustness of the method. To avoid the differential expansion problem and effectively filter out the effects of input noise in the differential signals, a new tracking differentiator was proposed. Finally, the effectiveness of the proposed method was verified by comparative simulations.
1. Introduction
Airbreathing hypersonic vehicles (AHVs) are a new type of aircraft, which fly at speeds greater than Mach 5 at near space altitudes. AHVs exhibit fast flying speeds, strong penetration abilities, and long combat distances. It is difficult to detect and intercept AHVs. AHVs have strong survivability, and they have outstanding advantages in strategy, tactics, and costeffectiveness compared to traditional aerospace vehicles [1–3]. AHVs have become a priority development direction for all of the aerospace powers competing for air and space rights. However, AHVs are multivariable and strongly coupled nonlinear systems. AHVs have large flight airspaces, and the flight environments are complex and variable, which results in large and fast timevarying characteristics and altitude uncertainty of the AHV model. Thus, the design of the AHV control system involves unprecedented difficulties and challenges [4, 5].
Most of the previous research on the modelling and control of AHVs has mainly focused on the AHV’s longitudinal motion plane. On the one hand, the longitudinal motion model is complex enough to require flight control. On the other hand, due to the scramjet engine’s extreme sensitivity to the flight attitude, AHVs should avoid horizontal manoeuvres during actual flight [6]. In a previous study [7], a robust gain control method was designed for AHVs, which used the TakagiSugeno (TS) fuzzy system to approximate the unknown state of the model. To guarantee the robustness of the control scheme, a new fuzzy disturbance observer was designed to estimate the disturbance. A similar longitudinal elastic model of the AHV was expressed as a TS fuzzy system [8], and a tracking control law was designed to achieve the robust tracking of the speed and altitude reference inputs. To improve the tracking effect, a nonlinear adaptive backstepping controller was designed for the AHV based on the backstepping design [9]. Based on the traditional backstepping control method, a nonsingular fast terminal sliding model control was applied to control the pitch angle and pitch rate, which optimized the control structure of the backstepping method and achieved finite time convergence of the system [10]. In one study, an integral sliding model control method was proposed [11]. When there are model uncertainties and external disturbances, the control scheme can still guarantee finite time convergence of the velocity and altitude tracking errors. Aiming at the attitude control problem of AHVs, a robust fuzzy control method based on a nonlinear switching system was proposed [12], which used a fuzzy system to approximate the unknown function of the model and guaranteed the robustness of the control scheme. Neural control methods for scenarios where the control input of the AHV contained a dead zone [13] and the actuator of the AHV contained failures [14] were studied. Two new neural backstepping control methods were put forward [15, 16]. The altitude subsystem was rewritten in strict and pure feedback forms. Based on the improved backstepping strategy, the control scheme was designed, and a minimal learning parameter (MLP) algorithm was applied to reduce the online learning parameters. The tracking simulation results of the velocity and altitude reference input showed that the proposed method exhibited better robustness and control effects. To solve the control problem of a strongly nonlinear model, such as an AHV, in addition to the fuzzy system or neural network used in the above literature to estimate the uncertainty of the model, active disturbance rejection control (ADRC) can also be considered. The ADRC term was first used in [17] where Professor J.Q. Han’s unique ideas were first systematically introduced into the English literature. In recent years, many scholars have conducted meaningful work on ADRC. In a previous study [18], ADRC was offered as the basis of a paradigm shift, providing the framework, the objectives, and constraints for future control theory development. For a class of multipleinput multipleoutput (MIMO) lowertriangular systems which have uncertain dynamics and disturbance, an ADRC method was designed to solve the control problem [19]. An overview of the concept, principles, and practice of the ADRC was presented in [20]. The key idea of ADRC is to use an extended state observer (ESO) to estimate the total disturbance, which is then compensated in the feedback loop. For a class of nonlinear systems with large uncertainties that come from both internal unknown dynamics and external stochastic disturbance, a novel ESO was designed to estimate both state and total disturbance which included the internal uncertain nonlinear part and the external uncertain stochastic disturbance [21]. To improve the performance of ADRC, a falbased, singleparametertuning ESO was proposed, and the convergence of the falbased ESO and the output tracking were established [22]. To improve the performance of the tracking differentiator (TD), an ADRC method based on a radial basis function neural network (RBFNN) was proposed [23], which enhanced the robustness and disturbance rejection ability of the method. To assist in the tuning of the parameters of the linear ADRC controller, a tuning rule was proposed to minimize the load disturbance attenuation performance [24]. Based on the existing swarm intelligence algorithms, a parameter tuning optimization design of ADRC was achieved [25]. An indepth study on the scheme design of an ADRC was conducted and applied to a threedegreeoffreedom pneumatic motion system subject to actuator saturation [26], which achieve good control.
Although the methods in the above literature achieved certain control effects, the research focus was on the robustness and steadystate performance of the AHV closedloop control system, neglecting the dynamic performance of the control system. However, AHV’s supermanoeuvrable, largeenvelope, and hypersonic flight demands better dynamic performance of the control system than any other existing aircraft. In most cases, a small control delay will cause significant errors in the hypersonic flight. Therefore, to guarantee the robustness and steadystate accuracy of the control system, more attention should be paid to the dynamic performance and realtime performance of the control system. To consider both the steadystate and dynamic performances, the concept of prescribed performance control was proposed by Charalampos and George [27]. The prescribed performance ensured that the tracking error converged to a prescribed arbitrary small area. Meanwhile, the convergence rate and overshoot met the prescribed conditions. Based on the prescribed performance control, a longitudinal innerloop controller of the hypersonic vehicle was designed [28]. However, the vehicle was considered to be a pure rigid body and the elastic problem was not considered, which resulted in the limitations of the proposed method. Aiming at the elastic body of the AHV model, a prescribed performance fuzzy backstepping control method and performance neural backstepping control method were proposed [29, 30], which guaranteed the steadystate performance and dynamic performance of the control system. However, the above two methods considered the AHV model to be an affine model, which made the reliability of the design method not guaranteed. An AHV adaptive neural control method was designed based on the prescribed performance function, which avoided the cumbersome virtual control law design process [31]. Meanwhile, the accuracy and rapidity of the control system were achieved. However, the proposed method relied too much on the initial value of the tracking error, which resulted in the poor practicality and operability.
In view of the deficiencies of the research in the above studies, an elastic hypersonic vehicle was taken as the research object. In this study, a new prescribed performance function was designed based on the hyperbolic cosine function, which avoided the singular control problem caused by the improper initial value setting. Thus, the steadystate performance and dynamic performance of the control system could be guaranteed. Meanwhile, active disturbance rejection control was introduced and an ESO was designed for each unknown nonaffine function in the AHV system [32], which further guaranteed the control accuracy and the robustness of the method. To address the complexity of the derivative of the virtual control law, a track differentiator was applied to estimate the related signals and signal derivatives. The effectiveness and superiority of the proposed method was verified through the simulation and comparison.
2. AHV Model and Preliminaries
2.1. Model Description
To better describe the longitudinal motion of the AHV, American scholar Parker used the research conclusions of Bolender and Doman [33, 34] combined with Hooke’s law and the Lagrange equation to establish the following controloriented AHV longitudinal motion parameter fitting model:
Equations (1)–(5) describe the rigid body part of the AHV. The five rigid states are the velocity , altitude , flightpath angle , pitch angle , and the pitch rate . is the mass of the AHV, is the gravitational acceleration constant, and is the moment of inertia of the AHV. The force condition of the AHV is shown in Figure 1. The parameter fitting form of the thrust , drag , lift , pitching moment , and generalized force can be expressed as follows [35]: with where the attack angle , the fuel equivalence ratio , and the elevator angular deflection are control inputs, and are the reference area and aerodynamic chord of the AHV, respectively, is the thrust moment arm, is the coefficient of in , and are the nominal altitude and corresponding air density, respectively, is the air density decay rate, and are dynamic pressure and air density at , respectively, is the th order coefficient of in , is the constant coefficient in , is the contribution of the coefficient of in , is the th order coefficient of in , is the constant coefficient in , is the th order contribution of to , is the constant term in , is the contribution of to , and is the th thrust fit parameter.
Equations (6) and (7) describe the elastic body part of the AHV. The elastic states are and . and are the damping ratio and vibrational frequency of the AHV elastic state. and can be expressed as follows: where and are the front and rear beam length of the AHV, respectively, and are the mass distributions of the front and the rear beams, respectively, and and are the vibration mode functions of the front and rear beam, respectively. The specific values of the model parameters and aerodynamic parameters of the abovementioned AHV are shown in [35].
2.2. Prescribed Performance
The AHV prescribed performance control method based on the nonaffine model is studied in this paper. To make the tracking error convergence processes meet the desired dynamic and steadystate performance, the following new performance function is designed to limit the tracking error. where are parameters to be designed.
has following properties: (1) is a positive monotonically decreasing function(2)(3)(4)
The prescribed performance is defined as follows: where is the tracking error. By choosing a small enough , it is guaranteed that and based on the property (3). Therefore, for any unknown but bounded , the following inequality always holds:
The prescribed performance defined by Equation (12) is shown in Figure 2. is the upper bound of the steadystate value, which means . Therefore, the desired steadystate accuracy of can be guaranteed by choosing an appropriate . is the maximum overshoot allowed by . directly influences the decreasing rate of . With the increase in , decreases more rapidly.
Remark 1. For any arbitrary bounded whether it is known or not, as long as the is chosen to be small enough, must be included in the prescribed area defined by Equation (12), which avoids the singular control problem caused by the improper initial value setting of the traditional performance function [36].
Remark 2. When the is chosen to be small, the value of will be very large, which can lead to too large of an overshoot of . However, because the response speed of is limited, the desired dynamic performance, including the overshoot and setting time, can be guaranteed by choosing a larger . This suggests that if the tracking error satisfies Equation (12), the overshoot and the adjusting time can be constrained within a certain range, as shown in Figure 2. Thus, by designing the controller which can make the tracking error meet the constraint of Equation (12), the desired dynamic performance of the control system can be guaranteed.
The transformed error is defined as follows: where . The following theorem can be obtained.
Theorem 1. If is bounded, then .
Proof. Because is bounded, there must be a bounded constant that makes . Furthermore, the inverse transformation of Equation (14) is as follows: Based on Equation (15), Substituting into Equation (16) yields
Therefore, Theorem 1 is established.
Remark 3. The control law below will be designed based on the transformed error . Theorem 1 shows that as long as is bounded, can be limited to the prescribed area defined by Equation (12). By choosing appropriate design parameters for , the desired dynamic performance and steadystate accuracy of can be guaranteed.
2.3. Model Conversion and Control Objective
On the one hand, since the thrust is directly influenced by the fuel equivalence ratio , the velocity of the AHV is mainly controlled by . On the other hand, since the elevator angular deflection directly influences the pitch rate , it further changes the pitch rate and the flightpath angle and finally influences the change of the altitude. The altitude change of the AHV is mainly controlled by the elevator angular deflection [37, 38]. Therefore, most previous studies on AHV control issues first divide the AHV model into a velocity subsystem controlled by (Equation (1)) and an altitude subsystem controlled by (Equations (2)–(5)) and then derived control laws [39, 40].
Based on the previous studies [37, 38], the velocity subsystem of the AHV was considered to be a nonaffine form with the control input: where is the output of the system, and is a completely unknown continuous differentiable function. For the velocity subsystem, the control goal is to design an appropriate control law based on the nonaffine model (Equation (18)). Thus, the robustness tracking of the velocity to the reference input can be achieved, and the velocity tracking error can be limited to the prescribed area.
Similarly, the altitude subsystem of the AHV can be expressed as the following nonaffine pure feedback system: where is the output of the system, and , , and are completely unknown continuous differentiable function. For the altitude subsystem, the control goal is to design an appropriate control law . Thus, the stable tracking of the altitude to the reference input can be achieved and the altitude tracking error can be limited to the prescribed area. Also, the desired dynamic performance and steadystate accuracy can be guaranteed.
Remark 4. Most previous studies on AHV control issues designed the control law based on the affine model. However, the AHV motion model is nonaffine. If the nonaffine model of the AHV is forcibly simplified to an affine model, the loss of certain key dynamics is inevitably. The designed control law has the risk of partial or complete failure. The proposed control law in this paper will be designed based on the nonaffine model (Equations (18) and (19)), which guarantees the reliability of the control law.
2.4. Extend States Observer and Tracking Differentiator
To design the active disturbance rejection control law, the extended state observer is applied to estimate the uncertainty of the AHV model and the external disturbance. The following system is considered: where is the unknown term, and is the input of the subsystem. The state of the system is measurable. Therefore, the state of the system can be expanded to the following system: where is the unknown derivative of the unknown term . Therefore, the ESO can be established as follows: where is the estimation error of the ESO, and are the outputs of the ESO, and are the observer gains, and parametric function is an appropriately constructed nonlinear function, which satisfies , , and . The parametric function in this paper is chosen to have the following form: where .
Theorem 2. Aiming at the system specified by Equation (20), if the ESO (Equation (22)) is applied, there will be gain parameters and , which makes the output of the ESO and converge to the actual state and a compact set of the unknown term , respectively. Bounded constants exist, which make:
Proof. The proof process is shown in the appendix.
Some signals in the control law design process are often difficult to obtain by the model construction. Many scholars have proposed using the tracking differentiator to estimate the signal [41]. The design principle is to achieve the highest precision extraction of the differential signal and to ensure certain robustness to the input noise of the signal. A new TD is proposed in this paper to estimate the differential signal. The specific form of the new TD is as follows: where is the input signal to be estimated, is the estimated value of , and are the estimated values of the th derivative of , respectively; is the parameter to be designed.
Theorem 3. If the new TD (Equation (25)) is applied to estimate the input signal and the derivatives, there exists an that makes converge to a compact set of , respectively. Bounded constants exist, which make:
Proof. The certification process is detailed elsewhere [42].
Remark 5. Compared with traditional tracking differentiators [41], the new TD proposed in this paper has two advantages. First, the structure is simple and can estimate the arbitrary derivative of the input signal. Second, the new TD has only one parameter to be designed, and the parameter adjustment process is easier.
3. Controller Design
3.1. Velocity Controller Design
The velocity subsystem (Equation (18)) is assumed to be affected by external disturbances. According to the idea of selfinterference, it can be expressed as follows: where is the external disturbance, is the unknown term, and is the parameter to be designed.
The velocity tracking error can be defined as follows:
The first derivative of Equation (28) with respect to time is obtained, and Equation (27) is substituted into the result, yielding the following:
According to Equation (14), the velocity transformed error can be defined as follows: where , and are parameters to be designed.
The first derivative of Equation (30) with respect to time is obtained, and Equation (29) is substituted into the result, yielding the following: where
To estimate the unknown term , the following ESO is designed for Equation (31): where , is the parameter to be designed, is the estimated value of , and is the estimated value of the unknown term .
The active disturbance rejection control law can be designed as follows: where is the parameter to be designed.
Theorem 4. Considering the velocity subsystem (Equation (18)), if the active disturbance rejection control law (Equation (33)) and the ESO (Equation (32)) are applied, the closedloop control system is semiglobally uniformly asymptotically stable, and the speed tracking error is limited to the prescribed area. The following inequality holds .
Proof. Substituting Equation (33) into Equation (31) yields:
The following Lyapunov function was selected:
The first derivative of with respect to time is obtained, and Equation (34) is substituted into the result, yielding the following:
Combined with Equation (24), a constant exists such that:
Considering that , Equation (37) is substituted into Equation (36), yielding:
With , the following compact set is defined:
Combining Equations (38) and (39), if , then . Therefore, the closedloop control system is semiglobally uniformly asymptotically stable, and the velocity transformed error will finally converge to the compact set . If the estimated accuracy of the ESO is high enough ( is small enough) and the is sufficiently large, the radius of the compact and the can be sufficiently small.
The above proves that the transformed error is bounded. According to Theorem 1, . The velocity transformed error is limited to the prescribed area.
3.2. Altitude Controller Design
Considering the altitude subsystem (Equation (19)), the altitude tracking error can be defined as . The altitude transformed error can be defined as follows: where , and are parameters to be designed.
Furthermore, the first derivative of with respect to time yields the following: where
The flightpath angle command is defined as follows: where are parameters to be designed.
If , in Equation (41) is replaced with and a Laplace transform is applied, yielding:
The two characteristics roots of Equation (43), and , are negative real numbers. Therefore, the altitude transformed error is convergent and bounded. For the altitude subsystem (Equation (19)), the control goal can be achieved through the control law design which makes [43].
For the rest of the altitude subsystem (Equation (19)), the active disturbance rejection control law of is designed in three steps.
Step 1. For , the pitch angle command is designed.
Combined with the concept of active disturbance rejection, can be expressed as follows: where is the external disturbance, , and is a parameter to be designed.
The flightpath angle tracking error is defined as follows:
The first derivative of Equation (45) is obtained, and Equation (44) is substituted into the result, yielding:
The mathematical expression of is complicated. Thus, to avoid the differential expansion problem and effectively filter out the effects of input noise in the differential signals, the new TD proposed in this paper is applied to estimate . where is the parameter to be designed, and and are the estimated values of and , respectively.
To estimate the unknown term , the following ESO is designed for Equation (46): where , is the parameter to be designed, and and are the estimated values of and , respectively.
The pitch angle command can be designed as follows: where is the parameter to be designed.
Step 2. For , the pitch rate command is designed.
The pitch angle tracking error is defined as follows:
If the AHV is affected by an external disturbance, can be expressed as follows: where is the unknown external disturbance.
The first derivative of with respect to time is obtained, and Equation (51) is substituted into the result, yielding:
To avoid the differential expansion problem and effectively filter out the effects of input noise in the differential signals, the new TD proposed in this paper is applied to estimate : where is the parameter to be designed, and and are the estimated values of and , respectively.
To estimate the unknown term , the following ESO is designed for Equation (52): where , is the parameter to be designed, is the estimated value of , and is the estimated value of the unknown term .
The pitch rate command can be designed as follows: where is the parameter to be designed.
Step 3. For , the control law is designed.
The pitch rate tracking error is defined as follows:
If the AHV is affected by an external disturbance, combined with the concept of active disturbance rejection, the equation can be expressed as follows: where is the external disturbance, is the unknown term, and is the parameter to be designed.
The first derivative of with respect to time is obtained, and Equation (57) is substituted into the result, yielding:
The following new TD is applied to estimate : where is the parameter to be designed, and and are the estimated values of and , respectively.
To estimate the unknown term , the following ESO is designed for Equation (58): where , is the parameter to be designed, is the estimated value of , and is the estimated value of the unknown term .
The control law can be designed as follows: where is the parameter to be designed.
Theorem 5. Considering the altitude subsystem (Equation (19)), if the active disturbance rejection control law (Equation (61)), the ESO (Equation (60)), and the new TD are applied, the closedloop control system is semiglobally uniformly asymptotically stable. The tracking errors , , , and are bounded, and the altitude tracking error is limited to the prescribed area. The following inequality holds .
Proof. The following Lyapunov function is selected: The first derivative of with respect to time is obtained, and Equations (46), (49), (50), (52), (55), (56), (58), and (61) are substituted into the result, yielding: According to Equations (24) and (26), bounded constants , , , , , and exist, which make: where Combining Equations (63), (64), and (65) yields the following: With , , and , the following compact sets are defined:
According to Equation (67), if , , or , . decreases until , , and converge to the compact sets , , and , respectively. Therefore, the closedloop control system is semiglobally uniformly asymptotically stable. , , and are ultimately bounded. However, if the estimated error of the ESO and new TD are sufficiently small ( is sufficiently small), the radii of the compact sets , , and can be sufficiently small, and , , and can be sufficiently small. If the is sufficiently small, then . According to Equations (40)–(43), the altitude transformed error is bounded. Thus, by Theorem 1, , which means the altitude transformed error is limited to the prescribed area.
4. Simulation Results
Taking the longitudinal motion model of the AHV (Equations (1)–(10)) as the controlled object, the tracking simulations of speed and altitude reference input were carried out. The simulations were solved using the fourthorder RungeKutta method, and the simulation step was taken to be 0.01 s. The initial values of the AHV state variables are shown in Table 1.

The velocity reference command and the altitude reference command are specified by the following secondorder system: where the damping ratio , the natural frequency , and and are the input signals of the secondorder system.
When the control algorithm proposed in this paper was used for the simulation, the prescribed performance parameters were selected as follows: , , , , , and . The ADRC controller parameters were selected as follows: , , , , , , , , and . The designed parameters of the new TD were selected as follows: , , and . The designed parameters of the ESO were selected as follows: , , , , , , , , , , , and . Simulations were carried out in the following two scenarios.
Scenario 1. The prescribed performancebased active disturbance rejection control (PPADRC) method proposed in this paper with the robust backstepping control method (RBC) from a previous study [44] was compared. The simulation time was set to 100 s. and were selected as a step signal with an amplitude of 200 m/s. To verify the robustness of the proposed method, it was assumed that a 40% perturbation of the aerodynamic parameters existed in the AHV model, which is expressed as . represents the nominal value of the aerodynamic parameter and represents the value of the aerodynamic parameter in the simulation. In addition, an external disturbance was added after 50 s of simulation:
The simulation results for Scenario 1 are shown in Figure 3. Figures 3(a)–3(d) show that the velocity and altitude tracking errors were limited to the prescribed area when using the PPADRC. Compared with the RBC, the PPADRC can guarantee better dynamic performances of the velocity and altitude tracking errors. When there was parameter perturbation and external disturbance, the PPADRC exhibited a higher control accuracy and stronger robustness. Figure 3(e) shows that the flightpath angle response of the PPADRC was smoother than that of the RBC. Moreover, the PPADRC proposed in this paper could estimate the unknown term of the model through the ESO (Figures 3(g) and 3(h)). The virtual reference command and the differential signals could be effectively estimated through the new TD (Figure 3(f)). Thus, the control accuracy of the method is further guaranteed.
Scenario 2. The PPADRC was compared with the neural backstepping control method (NBC) from a previous study [45]. The simulation time was set to 300 s. To better reflect the actual manoeuvre of the AHV, was assumed to be a “step” type signal with a step of 150 m/s per 100 s. was assumed to be a square wave signal with a amplitude of 200 m and a period of 200 s. Also, it was assumed that the 40% perturbation of the aerodynamic parameters existed in the AHV model. The following definitions were made: where represents the nominal value of the aerodynamic parameter, and represents the value of the aerodynamic parameter in the simulation. In addition, external disturbances were added after 150 s of simulation: .
(a) Velocity tracking performance
(b) Velocity tracking error
(c) Altitude tracking performance
(d) Altitude tracking error
(e) Flightpath angle
(f) Estimation of and
(g)
(h)
(i) Elastic state
(j) Elastic state
(k) Fuel equivalence ratio
(l) Elevator angular deflection
The simulation results of the Scenario 2 are shown in Figure 4. Figures 4(a)–4(d) show that the dynamic performances and control accuracy of the velocity and altitude tracking errors of the PPADRC were better than those of the NBC. Figure 4(e) shows that the flightpath angle response of the PPADRC was smoother than that of the NBC. Meanwhile, the PPADRC proposed in this paper could estimate the unknown term of the model through the ESO (Figures 4(g) and 4(h)). The virtual reference command and the differential signals could be effectively estimated through the new TD (Figure 4(f)). Thus, the control accuracy of the method was further guaranteed. Figures 4(i)–4(l) show that both of the control methods could achieve the effective suppression of the elastic vibrations.
(a) Velocity tracking performance
(b) Velocity tracking error
(c) Altitude tracking performance
(d) Altitude tracking error
(e) Flightpath angle
(f) Estimation of and
(g)
(h)
(i) Elastic state
(j) Elastic state
(k) Fuel equivalence ratio
(l) Elevator angular deflection
5. Conclusions
(1)An active disturbance rejection control for an AHV based on the prescribed performance function is proposed in this paper. The proposed method guarantees the stability of the AHV closedloop control system. The desired dynamic and steadystate performances of the convergence process of the tracking error were ensured(2)In the controller design process, the adopted active disturbance rejection method and extended state observer further enhanced the capacity to resist the disturbances, which guaranteed the robustness of the method(3)The simulation results in the paper proved the effectiveness of the proposed method. The comparison with the related publications showed that the dynamic and steadystate performances of the proposed method were superior
Appendix
A.Proof of Theorem 2
Based on Equations (20)–(22), we set . The first derivatives of and with respect to time can be expressed as follows:
The following Lyapunov function was selected: because
is positive semidefinite. Below, is studied. The partial derivative of with respect to and can be expressed as follows:
The first derivative of with respect to time was obtained, and with Equations (70)–(A.1), this yields the following: